Fractal and Fractional Latest open access articles published in Fractal Fract. at https://www.mdpi.com/journal/fractalfract https://www.mdpi.com/journal/fractalfract MDPI en Creative Commons Attribution (CC-BY) MDPI support@mdpi.com Fractal Fract, Vol. 5, Pages 132: Finite Element Formulation of Fractional Constitutive Laws Using the Reformulated Infinite State Representation https://www.mdpi.com/2504-3110/5/3/132 In this paper, we introduce a formulation of fractional constitutive equations for finite element analysis using the reformulated infinite state representation of fractional derivatives. Thereby, the fractional constitutive law is approximated by a high-dimensional set of ordinary differential and algebraic equations describing the relation of internal and external system states. The method is deduced for a three-dimensional linear viscoelastic continuum, for which the hydrostatic and deviatoric stress-strain relations are represented by a fractional Zener model. One- and two-dimensional finite elements are considered as benchmark problems with known closed form solutions in order to evaluate the performance of the scheme. 2021-09-21 Fractal Fract, Vol. 5, Pages 132: Finite Element Formulation of Fractional Constitutive Laws Using the Reformulated Infinite State Representation

Fractal and Fractional doi: 10.3390/fractalfract5030132

Authors: Matthias Hinze André Schmidt Remco I. Leine

In this paper, we introduce a formulation of fractional constitutive equations for finite element analysis using the reformulated infinite state representation of fractional derivatives. Thereby, the fractional constitutive law is approximated by a high-dimensional set of ordinary differential and algebraic equations describing the relation of internal and external system states. The method is deduced for a three-dimensional linear viscoelastic continuum, for which the hydrostatic and deviatoric stress-strain relations are represented by a fractional Zener model. One- and two-dimensional finite elements are considered as benchmark problems with known closed form solutions in order to evaluate the performance of the scheme.

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Finite Element Formulation of Fractional Constitutive Laws Using the Reformulated Infinite State Representation Matthias Hinze André Schmidt Remco I. Leine doi: 10.3390/fractalfract5030132 Fractal and Fractional 2021-09-21 Fractal and Fractional 2021-09-21 5 3 Article 132 10.3390/fractalfract5030132 https://www.mdpi.com/2504-3110/5/3/132
Fractal Fract, Vol. 5, Pages 131: Some Dynamical Models Involving Fractional-Order Derivatives with the Mittag-Leffler Type Kernels and Their Applications Based upon the Legendre Spectral Collocation Method https://www.mdpi.com/2504-3110/5/3/131 Fractional derivative models involving generalized Mittag-Leffler kernels and opposing models are investigated. We first replace the classical derivative with the GMLK in order to obtain the new fractional-order models (GMLK) with the three parameters that are investigated. We utilize a spectral collocation method based on Legendre’s polynomials for evaluating the numerical solutions of the pr. We then construct a scheme for the fractional-order models by using the spectral method involving the Legendre polynomials. In the first model, we directly obtain a set of nonlinear algebraic equations, which can be approximated by the Newton-Raphson method. For the second model, we also need to use the finite differences method to obtain the set of nonlinear algebraic equations, which are also approximated as in the first model. The accuracy of the results is verified in the first model by comparing it with our analytical solution. In the second and third models, the residual error functions are calculated. In all cases, the results are found to be in agreement. The method is a powerful hybrid technique of numerical and analytical approach that is applicable for partial differential equations with multi-order of fractional derivatives involving GMLK with three parameters. 2021-09-20 Fractal Fract, Vol. 5, Pages 131: Some Dynamical Models Involving Fractional-Order Derivatives with the Mittag-Leffler Type Kernels and Their Applications Based upon the Legendre Spectral Collocation Method

Fractal and Fractional doi: 10.3390/fractalfract5030131

Authors: Hari Srivastava Abedel-Karrem Alomari Khaled Saad Waleed Hamanah

Fractional derivative models involving generalized Mittag-Leffler kernels and opposing models are investigated. We first replace the classical derivative with the GMLK in order to obtain the new fractional-order models (GMLK) with the three parameters that are investigated. We utilize a spectral collocation method based on Legendre’s polynomials for evaluating the numerical solutions of the pr. We then construct a scheme for the fractional-order models by using the spectral method involving the Legendre polynomials. In the first model, we directly obtain a set of nonlinear algebraic equations, which can be approximated by the Newton-Raphson method. For the second model, we also need to use the finite differences method to obtain the set of nonlinear algebraic equations, which are also approximated as in the first model. The accuracy of the results is verified in the first model by comparing it with our analytical solution. In the second and third models, the residual error functions are calculated. In all cases, the results are found to be in agreement. The method is a powerful hybrid technique of numerical and analytical approach that is applicable for partial differential equations with multi-order of fractional derivatives involving GMLK with three parameters.

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Some Dynamical Models Involving Fractional-Order Derivatives with the Mittag-Leffler Type Kernels and Their Applications Based upon the Legendre Spectral Collocation Method Hari Srivastava Abedel-Karrem Alomari Khaled Saad Waleed Hamanah doi: 10.3390/fractalfract5030131 Fractal and Fractional 2021-09-20 Fractal and Fractional 2021-09-20 5 3 Article 131 10.3390/fractalfract5030131 https://www.mdpi.com/2504-3110/5/3/131
Fractal Fract, Vol. 5, Pages 130: Jafari Transformation for Solving a System of Ordinary Differential Equations with Medical Application https://www.mdpi.com/2504-3110/5/3/130 Integral transformations are essential for solving complex problems in business, engineering, natural sciences, computers, optical science, and modern mathematics. In this paper, we apply a general integral transform, called the Jafari transform, for solving a system of ordinary differential equations. After applying the Jafari transform, ordinary differential equations are converted to a simple system of algebraic equations that can be solved easily. Then, by using the inverse operator of the Jafari transform, we can solve the main system of ordinary differential equations. Jafari transform belongs to the class of Laplace transform and is considered a generalization to integral transforms such as Laplace, Elzaki, Sumudu, G\_transforms, Aboodh, Pourreza, etc. Jafari transform does not need a large computational work as the previous integral transforms. For the Jafari transform, we have studied some valuable properties and theories that have not been studied before. Such as the linearity property, scaling property, first and second shift properties, the transformation of periodic functions, Heaviside function, and the transformation of Dirac’s delta function, and so on. There is a mathematical model that describes the cell population dynamics in the colonic crypt and colorectal cancer. We have applied the Jafari transform for solving this model. 2021-09-20 Fractal Fract, Vol. 5, Pages 130: Jafari Transformation for Solving a System of Ordinary Differential Equations with Medical Application

Fractal and Fractional doi: 10.3390/fractalfract5030130

Authors: Ahmed Ibrahim El-Mesady Yaser Salah Hamed Abdullah M. Alsharif

Integral transformations are essential for solving complex problems in business, engineering, natural sciences, computers, optical science, and modern mathematics. In this paper, we apply a general integral transform, called the Jafari transform, for solving a system of ordinary differential equations. After applying the Jafari transform, ordinary differential equations are converted to a simple system of algebraic equations that can be solved easily. Then, by using the inverse operator of the Jafari transform, we can solve the main system of ordinary differential equations. Jafari transform belongs to the class of Laplace transform and is considered a generalization to integral transforms such as Laplace, Elzaki, Sumudu, G\_transforms, Aboodh, Pourreza, etc. Jafari transform does not need a large computational work as the previous integral transforms. For the Jafari transform, we have studied some valuable properties and theories that have not been studied before. Such as the linearity property, scaling property, first and second shift properties, the transformation of periodic functions, Heaviside function, and the transformation of Dirac’s delta function, and so on. There is a mathematical model that describes the cell population dynamics in the colonic crypt and colorectal cancer. We have applied the Jafari transform for solving this model.

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Jafari Transformation for Solving a System of Ordinary Differential Equations with Medical Application Ahmed Ibrahim El-Mesady Yaser Salah Hamed Abdullah M. Alsharif doi: 10.3390/fractalfract5030130 Fractal and Fractional 2021-09-20 Fractal and Fractional 2021-09-20 5 3 Article 130 10.3390/fractalfract5030130 https://www.mdpi.com/2504-3110/5/3/130
Fractal Fract, Vol. 5, Pages 129: Pseudo-Likelihood Estimation for Parameters of Stochastic Time-Fractional Diffusion Equations https://www.mdpi.com/2504-3110/5/3/129 Although stochastic fractional partial differential equations have received increasing attention in the last decade, the parameter estimation of these equations has been seldom reported in literature. In this paper, we propose a pseudo-likelihood approach to estimating the parameters of stochastic time-fractional diffusion equations, whose forward solver has been investigated very recently by Gunzburger, Li, and Wang (2019). Our approach can accurately recover the fractional order, diffusion coefficient, as well as noise magnitude given the discrete observation data corresponding to only one realization of driving noise. When only partial data is available, our approach can also attain acceptable results for intermediate sparsity of observation. 2021-09-18 Fractal Fract, Vol. 5, Pages 129: Pseudo-Likelihood Estimation for Parameters of Stochastic Time-Fractional Diffusion Equations

Fractal and Fractional doi: 10.3390/fractalfract5030129

Authors: Guofei Pang Wanrong Cao

Although stochastic fractional partial differential equations have received increasing attention in the last decade, the parameter estimation of these equations has been seldom reported in literature. In this paper, we propose a pseudo-likelihood approach to estimating the parameters of stochastic time-fractional diffusion equations, whose forward solver has been investigated very recently by Gunzburger, Li, and Wang (2019). Our approach can accurately recover the fractional order, diffusion coefficient, as well as noise magnitude given the discrete observation data corresponding to only one realization of driving noise. When only partial data is available, our approach can also attain acceptable results for intermediate sparsity of observation.

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Pseudo-Likelihood Estimation for Parameters of Stochastic Time-Fractional Diffusion Equations Guofei Pang Wanrong Cao doi: 10.3390/fractalfract5030129 Fractal and Fractional 2021-09-18 Fractal and Fractional 2021-09-18 5 3 Article 129 10.3390/fractalfract5030129 https://www.mdpi.com/2504-3110/5/3/129
Fractal Fract, Vol. 5, Pages 128: Dynamics of Fractional-Order Digital Manufacturing Supply Chain System and Its Control and Synchronization https://www.mdpi.com/2504-3110/5/3/128 Digital manufacturing is widely used in the production of automobiles and aircrafts, and plays a profound role in the whole supply chain. Due to the long memory property of demand, production, and stocks, a fractional-order digital manufacturing supply chain system can describe their dynamics more precisely. In addition, their control and synchronization may have potential applications in the management of real-word supply chain systems to control uncertainties that occur within it. In this paper, a fractional-order digital manufacturing supply chain system is proposed and solved by the Adomian decomposition method (ADM). Dynamical characteristics of this system are studied by using a phase portrait, bifurcation diagram, and a maximum Lyapunov exponent diagram. The complexity of the system is also investigated by means of SE complexity and C0 complexity. It is shown that the complexity results are consistent with the bifurcation diagrams, indicating that the complexity can reflect the dynamical properties of the system. Meanwhile, the importance of the fractional-order derivative in the modeling of the system is shown. Moreover, to further investigate the dynamics of the fractional-order supply chain system, we design the feedback controllers to control the chaotic supply chain system and synchronize two supply chain systems, respectively. Numerical simulations illustrate the effectiveness and applicability of the proposed methods. 2021-09-17 Fractal Fract, Vol. 5, Pages 128: Dynamics of Fractional-Order Digital Manufacturing Supply Chain System and Its Control and Synchronization

Fractal and Fractional doi: 10.3390/fractalfract5030128

Authors: He Zheng Yuan

Digital manufacturing is widely used in the production of automobiles and aircrafts, and plays a profound role in the whole supply chain. Due to the long memory property of demand, production, and stocks, a fractional-order digital manufacturing supply chain system can describe their dynamics more precisely. In addition, their control and synchronization may have potential applications in the management of real-word supply chain systems to control uncertainties that occur within it. In this paper, a fractional-order digital manufacturing supply chain system is proposed and solved by the Adomian decomposition method (ADM). Dynamical characteristics of this system are studied by using a phase portrait, bifurcation diagram, and a maximum Lyapunov exponent diagram. The complexity of the system is also investigated by means of SE complexity and C0 complexity. It is shown that the complexity results are consistent with the bifurcation diagrams, indicating that the complexity can reflect the dynamical properties of the system. Meanwhile, the importance of the fractional-order derivative in the modeling of the system is shown. Moreover, to further investigate the dynamics of the fractional-order supply chain system, we design the feedback controllers to control the chaotic supply chain system and synchronize two supply chain systems, respectively. Numerical simulations illustrate the effectiveness and applicability of the proposed methods.

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Dynamics of Fractional-Order Digital Manufacturing Supply Chain System and Its Control and Synchronization He Zheng Yuan doi: 10.3390/fractalfract5030128 Fractal and Fractional 2021-09-17 Fractal and Fractional 2021-09-17 5 3 Article 128 10.3390/fractalfract5030128 https://www.mdpi.com/2504-3110/5/3/128
Fractal Fract, Vol. 5, Pages 127: Unified Scale Theorem: A Mathematical Formulation of Scale in the Frame of Earth Observation Image Classification https://www.mdpi.com/2504-3110/5/3/127 In this research, the geographic, observational, functional, and cartographic scale is unified into a single mathematical formulation for the purposes of earth observation image classification. Fractal analysis is used to define functional scales, which then are linked to the other concepts of scale using common equations and conditions. The proposed formulation is called Unified Scale Theorem (UST), and was assessed with Sentinel-2 image covering a variety of land uses from the broad area of Thessaloniki, Greece. Provided as an interactive excel spreadsheet, UST promotes objectivity, rapidity, and accuracy, thus facilitating optimal scale selection for image classification purposes. 2021-09-17 Fractal Fract, Vol. 5, Pages 127: Unified Scale Theorem: A Mathematical Formulation of Scale in the Frame of Earth Observation Image Classification

Fractal and Fractional doi: 10.3390/fractalfract5030127

Authors: Christos G. Karydas

In this research, the geographic, observational, functional, and cartographic scale is unified into a single mathematical formulation for the purposes of earth observation image classification. Fractal analysis is used to define functional scales, which then are linked to the other concepts of scale using common equations and conditions. The proposed formulation is called Unified Scale Theorem (UST), and was assessed with Sentinel-2 image covering a variety of land uses from the broad area of Thessaloniki, Greece. Provided as an interactive excel spreadsheet, UST promotes objectivity, rapidity, and accuracy, thus facilitating optimal scale selection for image classification purposes.

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Unified Scale Theorem: A Mathematical Formulation of Scale in the Frame of Earth Observation Image Classification Christos G. Karydas doi: 10.3390/fractalfract5030127 Fractal and Fractional 2021-09-17 Fractal and Fractional 2021-09-17 5 3 Article 127 10.3390/fractalfract5030127 https://www.mdpi.com/2504-3110/5/3/127
Fractal Fract, Vol. 5, Pages 126: A Note on Existence of Mild Solutions for Second-Order Neutral Integro-Differential Evolution Equations with State-Dependent Delay https://www.mdpi.com/2504-3110/5/3/126 This article is mainly devoted to the study of the existence of solutions for second-order abstract non-autonomous integro-differential evolution equations with infinite state-dependent delay. In the first part, we are concerned with second-order abstract non-autonomous integro-differential retarded functional differential equations with infinite state-dependent delay. In the second part, we extend our results to study the second-order abstract neutral integro-differential evolution equations with state-dependent delay. Our results are established using properties of the resolvent operator corresponding to the second-order abstract non-autonomous integro-differential equation and fixed point theorems. Finally, an application is presented to illustrate the theory obtained. 2021-09-17 Fractal Fract, Vol. 5, Pages 126: A Note on Existence of Mild Solutions for Second-Order Neutral Integro-Differential Evolution Equations with State-Dependent Delay

Fractal and Fractional doi: 10.3390/fractalfract5030126

Authors: Shahram Rezapour Hernán R. Henríquez Velusamy Vijayakumar Kottakkaran Sooppy Nisar Anurag Shukla

This article is mainly devoted to the study of the existence of solutions for second-order abstract non-autonomous integro-differential evolution equations with infinite state-dependent delay. In the first part, we are concerned with second-order abstract non-autonomous integro-differential retarded functional differential equations with infinite state-dependent delay. In the second part, we extend our results to study the second-order abstract neutral integro-differential evolution equations with state-dependent delay. Our results are established using properties of the resolvent operator corresponding to the second-order abstract non-autonomous integro-differential equation and fixed point theorems. Finally, an application is presented to illustrate the theory obtained.

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A Note on Existence of Mild Solutions for Second-Order Neutral Integro-Differential Evolution Equations with State-Dependent Delay Shahram Rezapour Hernán R. Henríquez Velusamy Vijayakumar Kottakkaran Sooppy Nisar Anurag Shukla doi: 10.3390/fractalfract5030126 Fractal and Fractional 2021-09-17 Fractal and Fractional 2021-09-17 5 3 Article 126 10.3390/fractalfract5030126 https://www.mdpi.com/2504-3110/5/3/126
Fractal Fract, Vol. 5, Pages 125: Design, Convergence and Stability of a Fourth-Order Class of Iterative Methods for Solving Nonlinear Vectorial Problems https://www.mdpi.com/2504-3110/5/3/125 A new parametric family of iterative schemes for solving nonlinear systems is presented. Fourth-order convergence is demonstrated and its stability is analyzed as a function of the parameter values. This study allows us to detect the most stable elements of the class, to find the fractals in the boundary of the basins of attraction and to reject those with chaotic behavior. Some numerical tests show the performance of the new methods, confirm the theoretical results and allow to compare the proposed schemes with other known ones. 2021-09-17 Fractal Fract, Vol. 5, Pages 125: Design, Convergence and Stability of a Fourth-Order Class of Iterative Methods for Solving Nonlinear Vectorial Problems

Fractal and Fractional doi: 10.3390/fractalfract5030125

Authors: Alicia Cordero Cristina Jordán Esther Sanabria-Codesal Juan R. Torregrosa

A new parametric family of iterative schemes for solving nonlinear systems is presented. Fourth-order convergence is demonstrated and its stability is analyzed as a function of the parameter values. This study allows us to detect the most stable elements of the class, to find the fractals in the boundary of the basins of attraction and to reject those with chaotic behavior. Some numerical tests show the performance of the new methods, confirm the theoretical results and allow to compare the proposed schemes with other known ones.

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Design, Convergence and Stability of a Fourth-Order Class of Iterative Methods for Solving Nonlinear Vectorial Problems Alicia Cordero Cristina Jordán Esther Sanabria-Codesal Juan R. Torregrosa doi: 10.3390/fractalfract5030125 Fractal and Fractional 2021-09-17 Fractal and Fractional 2021-09-17 5 3 Article 125 10.3390/fractalfract5030125 https://www.mdpi.com/2504-3110/5/3/125
Fractal Fract, Vol. 5, Pages 124: Thermophysical Investigation of Oldroyd-B Fluid with Functional Effects of Permeability: Memory Effect Study Using Non-Singular Kernel Derivative Approach https://www.mdpi.com/2504-3110/5/3/124 It is well established fact that the functional effects, such as relaxation and retardation of materials, can be measured for magnetized permeability based on relative increase or decrease during magnetization. In this context, a mathematical model is formulated based on slippage and non-slippage assumptions for Oldroyd-B fluid with magnetized permeability. An innovative definition of Caputo-Fabrizio time fractional derivative is implemented to hypothesize the constitutive energy and momentum equations. The exact solutions of presented problem, are determined by using mathematical techniques, namely Laplace transform with slipping boundary conditions have been invoked to tackle governing equations of velocity and temperature. The Nusselt number and limiting solutions have also been persuaded to estimate the heat emission rate through physical interpretation. In order to provide the validation of the problem, the absence of retardation time parameter led the investigated solutions with good agreement in literature. Additionally, comprehensively scrutinize the dynamics of the considered problem with parametric analysis is accomplished, the graphical illustration is depicted for slipping and non-slipping solutions for temperature and velocity. A comparative studies between fractional and non-fractional models describes that the fractional model elucidate the memory effects more efficiently. 2021-09-15 Fractal Fract, Vol. 5, Pages 124: Thermophysical Investigation of Oldroyd-B Fluid with Functional Effects of Permeability: Memory Effect Study Using Non-Singular Kernel Derivative Approach

Fractal and Fractional doi: 10.3390/fractalfract5030124

Authors: Muhammad Bilal Riaz Jan Awrejcewicz Aziz-Ur Rehman Ali Akgül

It is well established fact that the functional effects, such as relaxation and retardation of materials, can be measured for magnetized permeability based on relative increase or decrease during magnetization. In this context, a mathematical model is formulated based on slippage and non-slippage assumptions for Oldroyd-B fluid with magnetized permeability. An innovative definition of Caputo-Fabrizio time fractional derivative is implemented to hypothesize the constitutive energy and momentum equations. The exact solutions of presented problem, are determined by using mathematical techniques, namely Laplace transform with slipping boundary conditions have been invoked to tackle governing equations of velocity and temperature. The Nusselt number and limiting solutions have also been persuaded to estimate the heat emission rate through physical interpretation. In order to provide the validation of the problem, the absence of retardation time parameter led the investigated solutions with good agreement in literature. Additionally, comprehensively scrutinize the dynamics of the considered problem with parametric analysis is accomplished, the graphical illustration is depicted for slipping and non-slipping solutions for temperature and velocity. A comparative studies between fractional and non-fractional models describes that the fractional model elucidate the memory effects more efficiently.

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Thermophysical Investigation of Oldroyd-B Fluid with Functional Effects of Permeability: Memory Effect Study Using Non-Singular Kernel Derivative Approach Muhammad Bilal Riaz Jan Awrejcewicz Aziz-Ur Rehman Ali Akgül doi: 10.3390/fractalfract5030124 Fractal and Fractional 2021-09-15 Fractal and Fractional 2021-09-15 5 3 Article 124 10.3390/fractalfract5030124 https://www.mdpi.com/2504-3110/5/3/124
Fractal Fract, Vol. 5, Pages 123: Well Posedness of New Optimization Problems with Variational Inequality Constraints https://www.mdpi.com/2504-3110/5/3/123 In this paper, we studied the well posedness for a new class of optimization problems with variational inequality constraints involving second-order partial derivatives. More precisely, by using the notions of lower semicontinuity, pseudomonotonicity, hemicontinuity and monotonicity for a multiple integral functional, and by introducing the set of approximating solutions for the considered class of constrained optimization problems, we established some characterization results on well posedness. Furthermore, to illustrate the theoretical developments included in this paper, we present some examples. 2021-09-15 Fractal Fract, Vol. 5, Pages 123: Well Posedness of New Optimization Problems with Variational Inequality Constraints

Fractal and Fractional doi: 10.3390/fractalfract5030123

Authors: Savin Treanţă

In this paper, we studied the well posedness for a new class of optimization problems with variational inequality constraints involving second-order partial derivatives. More precisely, by using the notions of lower semicontinuity, pseudomonotonicity, hemicontinuity and monotonicity for a multiple integral functional, and by introducing the set of approximating solutions for the considered class of constrained optimization problems, we established some characterization results on well posedness. Furthermore, to illustrate the theoretical developments included in this paper, we present some examples.

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Well Posedness of New Optimization Problems with Variational Inequality Constraints Savin Treanţă doi: 10.3390/fractalfract5030123 Fractal and Fractional 2021-09-15 Fractal and Fractional 2021-09-15 5 3 Article 123 10.3390/fractalfract5030123 https://www.mdpi.com/2504-3110/5/3/123
Fractal Fract, Vol. 5, Pages 122: CMOS OTA-Based Filters for Designing Fractional-Order Chaotic Oscillators https://www.mdpi.com/2504-3110/5/3/122 Fractional-order chaotic oscillators (FOCOs) have shown more complexity than integer-order chaotic ones. However, the majority of electronic implementations were performed using embedded systems; compared to analog implementations, they require huge hardware resources to approximate the solution of the fractional-order derivatives. In this manner, we propose the design of FOCOs using fractional-order integrators based on operational transconductance amplifiers (OTAs). The case study shows the implementation of FOCOs by cascading first-order OTA-based filters designed with complementary metal-oxide-semiconductor (CMOS) technology. The OTAs have programmable transconductance, and the robustness of the fractional-order integrator is verified by performing process, voltage and temperature variations as well as Monte Carlo analyses for a CMOS technology of 180 nm from the United Microelectronics Corporation. Finally, it is highlighted that post-layout simulations are in good agreement with the simulations of the mathematical model of the FOCO. 2021-09-14 Fractal Fract, Vol. 5, Pages 122: CMOS OTA-Based Filters for Designing Fractional-Order Chaotic Oscillators

Fractal and Fractional doi: 10.3390/fractalfract5030122

Authors: Martín Alejandro Valencia-Ponce Perla Rubí Castañeda-Aviña Esteban Tlelo-Cuautle Victor Hugo Carbajal-Gómez Victor Rodolfo González-Díaz Yuma Sandoval-Ibarra Jose-Cruz Nuñez-Perez

Fractional-order chaotic oscillators (FOCOs) have shown more complexity than integer-order chaotic ones. However, the majority of electronic implementations were performed using embedded systems; compared to analog implementations, they require huge hardware resources to approximate the solution of the fractional-order derivatives. In this manner, we propose the design of FOCOs using fractional-order integrators based on operational transconductance amplifiers (OTAs). The case study shows the implementation of FOCOs by cascading first-order OTA-based filters designed with complementary metal-oxide-semiconductor (CMOS) technology. The OTAs have programmable transconductance, and the robustness of the fractional-order integrator is verified by performing process, voltage and temperature variations as well as Monte Carlo analyses for a CMOS technology of 180 nm from the United Microelectronics Corporation. Finally, it is highlighted that post-layout simulations are in good agreement with the simulations of the mathematical model of the FOCO.

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CMOS OTA-Based Filters for Designing Fractional-Order Chaotic Oscillators Martín Alejandro Valencia-Ponce Perla Rubí Castañeda-Aviña Esteban Tlelo-Cuautle Victor Hugo Carbajal-Gómez Victor Rodolfo González-Díaz Yuma Sandoval-Ibarra Jose-Cruz Nuñez-Perez doi: 10.3390/fractalfract5030122 Fractal and Fractional 2021-09-14 Fractal and Fractional 2021-09-14 5 3 Article 122 10.3390/fractalfract5030122 https://www.mdpi.com/2504-3110/5/3/122
Fractal Fract, Vol. 5, Pages 121: On a New Modification of the Erdélyi–Kober Fractional Derivative https://www.mdpi.com/2504-3110/5/3/121 In this paper, we introduce a new Caputo-type modification of the Erdélyi–Kober fractional derivative. We pay attention to how to formulate representations of Erdélyi–Kober fractional integral and derivatives operators. Then, some properties of the new modification and relationships with other Erdélyi–Kober fractional derivatives are derived. In addition, a numerical method is presented to deal with fractional differential equations involving the proposed Caputo-type Erdélyi–Kober fractional derivative. We hope the presented method will be widely applied to simulate such fractional models. 2021-09-13 Fractal Fract, Vol. 5, Pages 121: On a New Modification of the Erdélyi–Kober Fractional Derivative

Fractal and Fractional doi: 10.3390/fractalfract5030121

Authors: Zaid Odibat Dumitru Baleanu

In this paper, we introduce a new Caputo-type modification of the Erdélyi–Kober fractional derivative. We pay attention to how to formulate representations of Erdélyi–Kober fractional integral and derivatives operators. Then, some properties of the new modification and relationships with other Erdélyi–Kober fractional derivatives are derived. In addition, a numerical method is presented to deal with fractional differential equations involving the proposed Caputo-type Erdélyi–Kober fractional derivative. We hope the presented method will be widely applied to simulate such fractional models.

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On a New Modification of the Erdélyi–Kober Fractional Derivative Zaid Odibat Dumitru Baleanu doi: 10.3390/fractalfract5030121 Fractal and Fractional 2021-09-13 Fractal and Fractional 2021-09-13 5 3 Article 121 10.3390/fractalfract5030121 https://www.mdpi.com/2504-3110/5/3/121
Fractal Fract, Vol. 5, Pages 120: Multi-Model Selection and Analysis for COVID-19 https://www.mdpi.com/2504-3110/5/3/120 In the face of an increasing number of COVID-19 infections, one of the most crucial and challenging problems is to pick out the most reasonable and reliable models. Based on the COVID-19 data of four typical cities/provinces in China, integer-order and fractional SIR, SEIR, SEIR-Q, SEIR-QD, and SEIR-AHQ models are systematically analyzed by the AICc, BIC, RMSE, and R means. Through extensive simulation and comprehensive comparison, we show that the fractional models perform much better than the corresponding integer-order models in representing the epidemiological information contained in the real data. It is further revealed that the inflection point plays a vital role in the prediction. Moreover, the basic reproduction numbers R0 of all models are highly dependent on the contact rate. 2021-09-13 Fractal Fract, Vol. 5, Pages 120: Multi-Model Selection and Analysis for COVID-19

Fractal and Fractional doi: 10.3390/fractalfract5030120

Authors: Nuri Ma Weiyuan Ma Zhiming Li

In the face of an increasing number of COVID-19 infections, one of the most crucial and challenging problems is to pick out the most reasonable and reliable models. Based on the COVID-19 data of four typical cities/provinces in China, integer-order and fractional SIR, SEIR, SEIR-Q, SEIR-QD, and SEIR-AHQ models are systematically analyzed by the AICc, BIC, RMSE, and R means. Through extensive simulation and comprehensive comparison, we show that the fractional models perform much better than the corresponding integer-order models in representing the epidemiological information contained in the real data. It is further revealed that the inflection point plays a vital role in the prediction. Moreover, the basic reproduction numbers R0 of all models are highly dependent on the contact rate.

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Multi-Model Selection and Analysis for COVID-19 Nuri Ma Weiyuan Ma Zhiming Li doi: 10.3390/fractalfract5030120 Fractal and Fractional 2021-09-13 Fractal and Fractional 2021-09-13 5 3 Article 120 10.3390/fractalfract5030120 https://www.mdpi.com/2504-3110/5/3/120
Fractal Fract, Vol. 5, Pages 119: Enhancement in Thermal Energy and Solute Particles Using Hybrid Nanoparticles by Engaging Activation Energy and Chemical Reaction over a Parabolic Surface via Finite Element Approach https://www.mdpi.com/2504-3110/5/3/119 Several mechanisms in industrial use have significant applications in thermal transportation. The inclusion of hybrid nanoparticles in different mixtures has been studied extensively by researchers due to their wide applications. This report discusses the flow of Powell–Eyring fluid mixed with hybrid nanoparticles over a melting parabolic stretched surface. Flow rheology expressions have been derived under boundary layer theory. Afterwards, similarity transformation has been applied to convert PDEs into associated ODEs. These transformed ODEs have been solved the using finite element procedure (FEP) in the symbolic computational package MAPLE 18.0. The applicability and effectiveness of FEM are presented by addressing grid independent analysis. The reliability of FEM is presented by computing the surface drag force and heat transportation coefficient. The used methodology is highly effective and it can be easily implemented in MAPLE 18.0 for other highly nonlinear problems. It is observed that the thermal profile varies directly with the magnetic parameter, and the opposite trend is recorded for the Prandtl number. 2021-09-13 Fractal Fract, Vol. 5, Pages 119: Enhancement in Thermal Energy and Solute Particles Using Hybrid Nanoparticles by Engaging Activation Energy and Chemical Reaction over a Parabolic Surface via Finite Element Approach

Fractal and Fractional doi: 10.3390/fractalfract5030119

Authors: Yu-Ming Chu Umar Nazir Muhammad Sohail Mahmoud M. Selim Jung-Rye Lee

Several mechanisms in industrial use have significant applications in thermal transportation. The inclusion of hybrid nanoparticles in different mixtures has been studied extensively by researchers due to their wide applications. This report discusses the flow of Powell–Eyring fluid mixed with hybrid nanoparticles over a melting parabolic stretched surface. Flow rheology expressions have been derived under boundary layer theory. Afterwards, similarity transformation has been applied to convert PDEs into associated ODEs. These transformed ODEs have been solved the using finite element procedure (FEP) in the symbolic computational package MAPLE 18.0. The applicability and effectiveness of FEM are presented by addressing grid independent analysis. The reliability of FEM is presented by computing the surface drag force and heat transportation coefficient. The used methodology is highly effective and it can be easily implemented in MAPLE 18.0 for other highly nonlinear problems. It is observed that the thermal profile varies directly with the magnetic parameter, and the opposite trend is recorded for the Prandtl number.

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Enhancement in Thermal Energy and Solute Particles Using Hybrid Nanoparticles by Engaging Activation Energy and Chemical Reaction over a Parabolic Surface via Finite Element Approach Yu-Ming Chu Umar Nazir Muhammad Sohail Mahmoud M. Selim Jung-Rye Lee doi: 10.3390/fractalfract5030119 Fractal and Fractional 2021-09-13 Fractal and Fractional 2021-09-13 5 3 Article 119 10.3390/fractalfract5030119 https://www.mdpi.com/2504-3110/5/3/119
Fractal Fract, Vol. 5, Pages 118: On Weighted (k, s)-Riemann-Liouville Fractional Operators and Solution of Fractional Kinetic Equation https://www.mdpi.com/2504-3110/5/3/118 In this article, we establish the weighted (k,s)-Riemann-Liouville fractional integral and differential operators. Some certain properties of the operators and the weighted generalized Laplace transform of the new operators are part of the paper. The article consists of Chebyshev-type inequalities involving a weighted fractional integral. We propose an integro-differential kinetic equation using the novel fractional operators and find its solution by applying weighted generalized Laplace transforms. 2021-09-13 Fractal Fract, Vol. 5, Pages 118: On Weighted (k, s)-Riemann-Liouville Fractional Operators and Solution of Fractional Kinetic Equation

Fractal and Fractional doi: 10.3390/fractalfract5030118

In this article, we establish the weighted (k,s)-Riemann-Liouville fractional integral and differential operators. Some certain properties of the operators and the weighted generalized Laplace transform of the new operators are part of the paper. The article consists of Chebyshev-type inequalities involving a weighted fractional integral. We propose an integro-differential kinetic equation using the novel fractional operators and find its solution by applying weighted generalized Laplace transforms.

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On Weighted (k, s)-Riemann-Liouville Fractional Operators and Solution of Fractional Kinetic Equation Muhammad Samraiz Muhammad Umer Artion Kashuri Thabet Abdeljawad Sajid Iqbal Nabil Mlaiki doi: 10.3390/fractalfract5030118 Fractal and Fractional 2021-09-13 Fractal and Fractional 2021-09-13 5 3 Article 118 10.3390/fractalfract5030118 https://www.mdpi.com/2504-3110/5/3/118
Fractal Fract, Vol. 5, Pages 117: On the General Solutions of Some Non-Homogeneous Div-Curl Systems with Riemann–Liouville and Caputo Fractional Derivatives https://www.mdpi.com/2504-3110/5/3/117 In this work, we investigate analytically the solutions of a nonlinear div-curl system with fractional derivatives of the Riemann–Liouville or Caputo types. To this end, the fractional-order vector operators of divergence, curl and gradient are identified as components of the fractional Dirac operator in quaternionic form. As one of the most important results of this manuscript, we derive general solutions of some non-homogeneous div-curl systems that consider the presence of fractional-order derivatives of the Riemann–Liouville or Caputo types. A fractional analogous to the Teodorescu transform is presented in this work, and we employ some properties of its component operators, developed in this work to establish a generalization of the Helmholtz decomposition theorem in fractional space. Additionally, right inverses of the fractional-order curl, divergence and gradient vector operators are obtained using Riemann–Liouville and Caputo fractional operators. Finally, some consequences of these results are provided as applications at the end of this work. 2021-09-10 Fractal Fract, Vol. 5, Pages 117: On the General Solutions of Some Non-Homogeneous Div-Curl Systems with Riemann–Liouville and Caputo Fractional Derivatives

Fractal and Fractional doi: 10.3390/fractalfract5030117

Authors: Briceyda B. Delgado Jorge E. Macías-Díaz

In this work, we investigate analytically the solutions of a nonlinear div-curl system with fractional derivatives of the Riemann–Liouville or Caputo types. To this end, the fractional-order vector operators of divergence, curl and gradient are identified as components of the fractional Dirac operator in quaternionic form. As one of the most important results of this manuscript, we derive general solutions of some non-homogeneous div-curl systems that consider the presence of fractional-order derivatives of the Riemann–Liouville or Caputo types. A fractional analogous to the Teodorescu transform is presented in this work, and we employ some properties of its component operators, developed in this work to establish a generalization of the Helmholtz decomposition theorem in fractional space. Additionally, right inverses of the fractional-order curl, divergence and gradient vector operators are obtained using Riemann–Liouville and Caputo fractional operators. Finally, some consequences of these results are provided as applications at the end of this work.

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On the General Solutions of Some Non-Homogeneous Div-Curl Systems with Riemann–Liouville and Caputo Fractional Derivatives Briceyda B. Delgado Jorge E. Macías-Díaz doi: 10.3390/fractalfract5030117 Fractal and Fractional 2021-09-10 Fractal and Fractional 2021-09-10 5 3 Article 117 10.3390/fractalfract5030117 https://www.mdpi.com/2504-3110/5/3/117
Fractal Fract, Vol. 5, Pages 116: On Discrete Delta Caputo–Fabrizio Fractional Operators and Monotonicity Analysis https://www.mdpi.com/2504-3110/5/3/116 The discrete delta Caputo-Fabrizio fractional differences and sums are proposed to distinguish their monotonicity analysis from the sense of Riemann and Caputo operators on the time scale Z. Moreover, the action of Q− operator and discrete delta Laplace transform method are also reported. Furthermore, a relationship between the discrete delta Caputo-Fabrizio-Caputo and Caputo-Fabrizio-Riemann fractional differences is also studied in detail. To better understand the dynamic behavior of the obtained monotonicity results, the fractional difference mean value theorem is derived. The idea used in this article is readily applicable to obtain monotonicity analysis of other discrete fractional operators in discrete fractional calculus. 2021-09-09 Fractal Fract, Vol. 5, Pages 116: On Discrete Delta Caputo–Fabrizio Fractional Operators and Monotonicity Analysis

Fractal and Fractional doi: 10.3390/fractalfract5030116

The discrete delta Caputo-Fabrizio fractional differences and sums are proposed to distinguish their monotonicity analysis from the sense of Riemann and Caputo operators on the time scale Z. Moreover, the action of Q− operator and discrete delta Laplace transform method are also reported. Furthermore, a relationship between the discrete delta Caputo-Fabrizio-Caputo and Caputo-Fabrizio-Riemann fractional differences is also studied in detail. To better understand the dynamic behavior of the obtained monotonicity results, the fractional difference mean value theorem is derived. The idea used in this article is readily applicable to obtain monotonicity analysis of other discrete fractional operators in discrete fractional calculus.

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On Discrete Delta Caputo–Fabrizio Fractional Operators and Monotonicity Analysis Pshtiwan Othman Mohammed Thabet Abdeljawad Faraidun Kadir Hamasalh doi: 10.3390/fractalfract5030116 Fractal and Fractional 2021-09-09 Fractal and Fractional 2021-09-09 5 3 Article 116 10.3390/fractalfract5030116 https://www.mdpi.com/2504-3110/5/3/116
Fractal Fract, Vol. 5, Pages 115: Jacobi Spectral Collocation Technique for Time-Fractional Inverse Heat Equations https://www.mdpi.com/2504-3110/5/3/115 In this paper, we introduce a numerical solution for the time-fractional inverse heat equations. We focus on obtaining the unknown source term along with the unknown temperature function based on an additional condition given in an integral form. The proposed scheme is based on a spectral collocation approach to obtain the two independent variables. Our approach is accurate, efficient, and feasible for the model problem under consideration. The proposed Jacobi spectral collocation method yields an exponential rate of convergence with a relatively small number of degrees of freedom. Finally, a series of numerical examples are provided to demonstrate the efficiency and flexibility of the numerical scheme. 2021-09-09 Fractal Fract, Vol. 5, Pages 115: Jacobi Spectral Collocation Technique for Time-Fractional Inverse Heat Equations

Fractal and Fractional doi: 10.3390/fractalfract5030115

Authors: Mohamed A. Abdelkawy Ahmed Z. M. Amin Mohammed M. Babatin Abeer S. Alnahdi Mahmoud A. Zaky Ramy M. Hafez

In this paper, we introduce a numerical solution for the time-fractional inverse heat equations. We focus on obtaining the unknown source term along with the unknown temperature function based on an additional condition given in an integral form. The proposed scheme is based on a spectral collocation approach to obtain the two independent variables. Our approach is accurate, efficient, and feasible for the model problem under consideration. The proposed Jacobi spectral collocation method yields an exponential rate of convergence with a relatively small number of degrees of freedom. Finally, a series of numerical examples are provided to demonstrate the efficiency and flexibility of the numerical scheme.

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Jacobi Spectral Collocation Technique for Time-Fractional Inverse Heat Equations Mohamed A. Abdelkawy Ahmed Z. M. Amin Mohammed M. Babatin Abeer S. Alnahdi Mahmoud A. Zaky Ramy M. Hafez doi: 10.3390/fractalfract5030115 Fractal and Fractional 2021-09-09 Fractal and Fractional 2021-09-09 5 3 Article 115 10.3390/fractalfract5030115 https://www.mdpi.com/2504-3110/5/3/115
Fractal Fract, Vol. 5, Pages 114: Estimating Conditional Power for Sequential Monitoring of Covariate Adaptive Randomized Designs: The Fractional Brownian Motion Approach https://www.mdpi.com/2504-3110/5/3/114 Conditional power based on classical Brownian motion (BM) has been widely used in sequential monitoring of clinical trials, including those with the covariate adaptive randomization design (CAR). Due to some uncontrollable factors, the sequential test statistics under CAR procedures may not satisfy the independent increment property of BM. We confirm the invalidation of BM when the error terms in the linear model with CAR design are not independent and identically distributed. To incorporate the possible correlation structure of the increment of the test statistic, we utilize the fractional Brownian motion (FBM). We conducted a comparative study of the conditional power under BM and FBM. It was found that the conditional power under FBM assumption was mostly higher than that under BM assumption when the Hurst exponent was greater than 0.5. 2021-09-08 Fractal Fract, Vol. 5, Pages 114: Estimating Conditional Power for Sequential Monitoring of Covariate Adaptive Randomized Designs: The Fractional Brownian Motion Approach

Fractal and Fractional doi: 10.3390/fractalfract5030114

Authors: Yiping Yang Hongjian Zhu Dejian Lai

Conditional power based on classical Brownian motion (BM) has been widely used in sequential monitoring of clinical trials, including those with the covariate adaptive randomization design (CAR). Due to some uncontrollable factors, the sequential test statistics under CAR procedures may not satisfy the independent increment property of BM. We confirm the invalidation of BM when the error terms in the linear model with CAR design are not independent and identically distributed. To incorporate the possible correlation structure of the increment of the test statistic, we utilize the fractional Brownian motion (FBM). We conducted a comparative study of the conditional power under BM and FBM. It was found that the conditional power under FBM assumption was mostly higher than that under BM assumption when the Hurst exponent was greater than 0.5.

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Estimating Conditional Power for Sequential Monitoring of Covariate Adaptive Randomized Designs: The Fractional Brownian Motion Approach Yiping Yang Hongjian Zhu Dejian Lai doi: 10.3390/fractalfract5030114 Fractal and Fractional 2021-09-08 Fractal and Fractional 2021-09-08 5 3 Article 114 10.3390/fractalfract5030114 https://www.mdpi.com/2504-3110/5/3/114
Fractal Fract, Vol. 5, Pages 113: Analytic Fuzzy Formulation of a Time-Fractional Fornberg–Whitham Model with Power and Mittag–Leffler Kernels https://www.mdpi.com/2504-3110/5/3/113 This manuscript assesses a semi-analytical method in connection with a new hybrid fuzzy integral transform and the Adomian decomposition method via the notion of fuzziness known as the Elzaki Adomian decomposition method (briefly, EADM). Moreover, we use the aforesaid strategy to address the time-fractional Fornberg–Whitham equation (FWE) under gH-differentiability by employing different initial conditions (IC). Several algebraic aspects of the fuzzy Caputo fractional derivative (CFD) and fuzzy Atangana–Baleanu (AB) fractional derivative operator in the Caputo sense, with respect to the Elzaki transform, are presented to validate their utilities. Apart from that, a general algorithm for fuzzy Caputo and AB fractional derivatives in the Caputo sense is proposed. Some illustrative cases are demonstrated to understand the algorithmic approach of FWE. Taking into consideration the uncertainty parameter ζ∈[0,1] and various fractional orders, the convergence and error analysis are reported by graphical representations of FWE that have close harmony with the closed form solutions. It is worth mentioning that the projected approach to fuzziness is to verify the supremacy and reliability of configuring numerical solutions to nonlinear fuzzy fractional partial differential equations arising in physical and complex structures. 2021-09-08 Fractal Fract, Vol. 5, Pages 113: Analytic Fuzzy Formulation of a Time-Fractional Fornberg–Whitham Model with Power and Mittag–Leffler Kernels

Fractal and Fractional doi: 10.3390/fractalfract5030113

Authors: Saima Rashid Rehana Ashraf Ahmet Ocak Akdemir Manar A. Alqudah Thabet Abdeljawad Mohamed S. Mohamed

This manuscript assesses a semi-analytical method in connection with a new hybrid fuzzy integral transform and the Adomian decomposition method via the notion of fuzziness known as the Elzaki Adomian decomposition method (briefly, EADM). Moreover, we use the aforesaid strategy to address the time-fractional Fornberg–Whitham equation (FWE) under gH-differentiability by employing different initial conditions (IC). Several algebraic aspects of the fuzzy Caputo fractional derivative (CFD) and fuzzy Atangana–Baleanu (AB) fractional derivative operator in the Caputo sense, with respect to the Elzaki transform, are presented to validate their utilities. Apart from that, a general algorithm for fuzzy Caputo and AB fractional derivatives in the Caputo sense is proposed. Some illustrative cases are demonstrated to understand the algorithmic approach of FWE. Taking into consideration the uncertainty parameter ζ∈[0,1] and various fractional orders, the convergence and error analysis are reported by graphical representations of FWE that have close harmony with the closed form solutions. It is worth mentioning that the projected approach to fuzziness is to verify the supremacy and reliability of configuring numerical solutions to nonlinear fuzzy fractional partial differential equations arising in physical and complex structures.

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Analytic Fuzzy Formulation of a Time-Fractional Fornberg–Whitham Model with Power and Mittag–Leffler Kernels Saima Rashid Rehana Ashraf Ahmet Ocak Akdemir Manar A. Alqudah Thabet Abdeljawad Mohamed S. Mohamed doi: 10.3390/fractalfract5030113 Fractal and Fractional 2021-09-08 Fractal and Fractional 2021-09-08 5 3 Article 113 10.3390/fractalfract5030113 https://www.mdpi.com/2504-3110/5/3/113
Fractal Fract, Vol. 5, Pages 112: Controllability for Fuzzy Fractional Evolution Equations in Credibility Space https://www.mdpi.com/2504-3110/5/3/112 This article addresses exact controllability for Caputo fuzzy fractional evolution equations in the credibility space from the perspective of the Liu process. The class or problems considered here are Caputo fuzzy differential equations with Caputo derivatives of order β∈(1,2), 0CDtβu(t,ζ)=Au(t,ζ)+f(t,u(t,ζ))dCt+Bx(t)Cx(t)dt with initial conditions u(0)=u0,u′(0)=u1, where u(t,ζ) takes values from U(⊂EN),V(⊂EN) is the other bounded space, and EN represents the set of all upper semi-continuously convex fuzzy numbers on R. In addition, several numerical solutions have been provided to verify the correctness and effectiveness of the main result. Finally, an example is given, which expresses the fuzzy fractional differential equations. 2021-09-08 Fractal Fract, Vol. 5, Pages 112: Controllability for Fuzzy Fractional Evolution Equations in Credibility Space

Fractal and Fractional doi: 10.3390/fractalfract5030112

Authors: Azmat Ullah Khan Niazi Naveed Iqbal Rasool Shah Fongchan Wannalookkhee Kamsing Nonlaopon

This article addresses exact controllability for Caputo fuzzy fractional evolution equations in the credibility space from the perspective of the Liu process. The class or problems considered here are Caputo fuzzy differential equations with Caputo derivatives of order β∈(1,2), 0CDtβu(t,ζ)=Au(t,ζ)+f(t,u(t,ζ))dCt+Bx(t)Cx(t)dt with initial conditions u(0)=u0,u′(0)=u1, where u(t,ζ) takes values from U(⊂EN),V(⊂EN) is the other bounded space, and EN represents the set of all upper semi-continuously convex fuzzy numbers on R. In addition, several numerical solutions have been provided to verify the correctness and effectiveness of the main result. Finally, an example is given, which expresses the fuzzy fractional differential equations.

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Controllability for Fuzzy Fractional Evolution Equations in Credibility Space Azmat Ullah Khan Niazi Naveed Iqbal Rasool Shah Fongchan Wannalookkhee Kamsing Nonlaopon doi: 10.3390/fractalfract5030112 Fractal and Fractional 2021-09-08 Fractal and Fractional 2021-09-08 5 3 Article 112 10.3390/fractalfract5030112 https://www.mdpi.com/2504-3110/5/3/112
Fractal Fract, Vol. 5, Pages 111: Numerical Solutions of Fractional Differential Equations by Using Laplace Transformation Method and Quadrature Rule https://www.mdpi.com/2504-3110/5/3/111 This paper introduces an efficient numerical scheme for solving a significant class of fractional differential equations. The major contributions made in this paper apply a direct approach based on a combination of time discretization and the Laplace transform method to transcribe the fractional differential problem under study into a dynamic linear equations system. The resulting problem is then solved by employing the numerical method of the quadrature rule, which is also a well-developed numerical method. The present numerical scheme, which is based on the numerical inversion of Laplace transform and equal-width quadrature rule is robust and efficient. Some numerical experiments are carried out to evaluate the performance and effectiveness of the suggested framework. 2021-09-07 Fractal Fract, Vol. 5, Pages 111: Numerical Solutions of Fractional Differential Equations by Using Laplace Transformation Method and Quadrature Rule

Fractal and Fractional doi: 10.3390/fractalfract5030111

This paper introduces an efficient numerical scheme for solving a significant class of fractional differential equations. The major contributions made in this paper apply a direct approach based on a combination of time discretization and the Laplace transform method to transcribe the fractional differential problem under study into a dynamic linear equations system. The resulting problem is then solved by employing the numerical method of the quadrature rule, which is also a well-developed numerical method. The present numerical scheme, which is based on the numerical inversion of Laplace transform and equal-width quadrature rule is robust and efficient. Some numerical experiments are carried out to evaluate the performance and effectiveness of the suggested framework.

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Numerical Solutions of Fractional Differential Equations by Using Laplace Transformation Method and Quadrature Rule Samaneh Soradi-Zeid Mehdi Mesrizadeh Carlo Cattani doi: 10.3390/fractalfract5030111 Fractal and Fractional 2021-09-07 Fractal and Fractional 2021-09-07 5 3 Article 111 10.3390/fractalfract5030111 https://www.mdpi.com/2504-3110/5/3/111
Fractal Fract, Vol. 5, Pages 110: A q-Gradient Descent Algorithm with Quasi-Fejér Convergence for Unconstrained Optimization Problems https://www.mdpi.com/2504-3110/5/3/110 We present an algorithm for solving unconstrained optimization problems based on the q-gradient vector. The main idea used in the algorithm construction is the approximation of the classical gradient by a q-gradient vector. For a convex objective function, the quasi-Fejér convergence of the algorithm is proved. The proposed method does not require the boundedness assumption on any level set. Further, numerical experiments are reported to show the performance of the proposed method. 2021-09-03 Fractal Fract, Vol. 5, Pages 110: A q-Gradient Descent Algorithm with Quasi-Fejér Convergence for Unconstrained Optimization Problems

Fractal and Fractional doi: 10.3390/fractalfract5030110

Authors: Shashi Kant Mishra Predrag Rajković Mohammad Esmael Samei Suvra Kanti Chakraborty Bhagwat Ram Mohammed K. A. Kaabar

We present an algorithm for solving unconstrained optimization problems based on the q-gradient vector. The main idea used in the algorithm construction is the approximation of the classical gradient by a q-gradient vector. For a convex objective function, the quasi-Fejér convergence of the algorithm is proved. The proposed method does not require the boundedness assumption on any level set. Further, numerical experiments are reported to show the performance of the proposed method.

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A q-Gradient Descent Algorithm with Quasi-Fejér Convergence for Unconstrained Optimization Problems Shashi Kant Mishra Predrag Rajković Mohammad Esmael Samei Suvra Kanti Chakraborty Bhagwat Ram Mohammed K. A. Kaabar doi: 10.3390/fractalfract5030110 Fractal and Fractional 2021-09-03 Fractal and Fractional 2021-09-03 5 3 Article 110 10.3390/fractalfract5030110 https://www.mdpi.com/2504-3110/5/3/110
Fractal Fract, Vol. 5, Pages 109: On the Operator Method for Solving Linear Integro-Differential Equations with Fractional Conformable Derivatives https://www.mdpi.com/2504-3110/5/3/109 The methods for constructing solutions to integro-differential equations of the Volterra type are considered. The equations are related to fractional conformable derivatives. Explicit solutions of homogeneous and inhomogeneous equations are constructed, and a Cauchy-type problem is studied. It should be noted that the considered method is based on the construction of normalized systems of functions with respect to a differential operator of fractional order. 2021-09-02 Fractal Fract, Vol. 5, Pages 109: On the Operator Method for Solving Linear Integro-Differential Equations with Fractional Conformable Derivatives

Fractal and Fractional doi: 10.3390/fractalfract5030109

Authors: Batirkhan Kh. Turmetov Kairat I. Usmanov Kulzina Zh. Nazarova

The methods for constructing solutions to integro-differential equations of the Volterra type are considered. The equations are related to fractional conformable derivatives. Explicit solutions of homogeneous and inhomogeneous equations are constructed, and a Cauchy-type problem is studied. It should be noted that the considered method is based on the construction of normalized systems of functions with respect to a differential operator of fractional order.

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On the Operator Method for Solving Linear Integro-Differential Equations with Fractional Conformable Derivatives Batirkhan Kh. Turmetov Kairat I. Usmanov Kulzina Zh. Nazarova doi: 10.3390/fractalfract5030109 Fractal and Fractional 2021-09-02 Fractal and Fractional 2021-09-02 5 3 Article 109 10.3390/fractalfract5030109 https://www.mdpi.com/2504-3110/5/3/109
Fractal Fract, Vol. 5, Pages 108: Qualitative Study on Solutions of a Hadamard Variable Order Boundary Problem via the Ulam–Hyers–Rassias Stability https://www.mdpi.com/2504-3110/5/3/108 In this paper, the existence, uniqueness and stability of solutions to a boundary value problem of nonlinear FDEs of variable order are established. To do this, we first investigate some aspects of variable order operators of Hadamard type. Then, with the help of the generalized intervals and piecewise constant functions, we convert the variable order Hadamard FBVP to an equivalent standard Hadamard BVP of the fractional constant order. Further, two fixed point theorems due to Schauder and Banach are used and, finally, the Ulam–Hyers–Rassias stability of the given variable order Hadamard FBVP is examined. These results are supported with the aid of a comprehensive example. 2021-09-02 Fractal Fract, Vol. 5, Pages 108: Qualitative Study on Solutions of a Hadamard Variable Order Boundary Problem via the Ulam–Hyers–Rassias Stability

Fractal and Fractional doi: 10.3390/fractalfract5030108

Authors: Amar Benkerrouche Mohammed Said Souid Sina Etemad Ali Hakem Praveen Agarwal Shahram Rezapour Sotiris K. Ntouyas Jessada Tariboon

In this paper, the existence, uniqueness and stability of solutions to a boundary value problem of nonlinear FDEs of variable order are established. To do this, we first investigate some aspects of variable order operators of Hadamard type. Then, with the help of the generalized intervals and piecewise constant functions, we convert the variable order Hadamard FBVP to an equivalent standard Hadamard BVP of the fractional constant order. Further, two fixed point theorems due to Schauder and Banach are used and, finally, the Ulam–Hyers–Rassias stability of the given variable order Hadamard FBVP is examined. These results are supported with the aid of a comprehensive example.

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Qualitative Study on Solutions of a Hadamard Variable Order Boundary Problem via the Ulam–Hyers–Rassias Stability Amar Benkerrouche Mohammed Said Souid Sina Etemad Ali Hakem Praveen Agarwal Shahram Rezapour Sotiris K. Ntouyas Jessada Tariboon doi: 10.3390/fractalfract5030108 Fractal and Fractional 2021-09-02 Fractal and Fractional 2021-09-02 5 3 Article 108 10.3390/fractalfract5030108 https://www.mdpi.com/2504-3110/5/3/108
Fractal Fract, Vol. 5, Pages 107: Influence of Fin Length on Magneto-Combined Convection Heat Transfer Performance in a Lid-Driven Wavy Cavity https://www.mdpi.com/2504-3110/5/3/107 In the existent study, combined magneto-convection heat exchange in a driven enclosure having vertical fin was analyzed numerically. The finite element system-based GWR procedure was utilized to determine the flow model’s governing equations. A parametric inquiry was executed to review the influence of Richardson and Hartmann numbers on flow shape and heat removal features inside a frame. The problem’s resulting numerical outcomes were demonstrated graphically in terms of isotherms, streamlines, velocity sketches, local Nusselt number, global Nusselt number, and global fluid temperature. It was found that the varying lengths of the fin surface have a substantial impact on flow building and heat line sketch. Further, it was also noticed that a relatively fin length is needed to increase the heat exchange rate on the right cool wall at a high Richardson number. The fin can significantly enhance heat removal performance rate from an enclosure to adjacent fluid. 2021-08-31 Fractal Fract, Vol. 5, Pages 107: Influence of Fin Length on Magneto-Combined Convection Heat Transfer Performance in a Lid-Driven Wavy Cavity

Fractal and Fractional doi: 10.3390/fractalfract5030107

Authors: Md. Fayz-Al-Asad Mehmet Yavuz Md. Nur Alam Md. Manirul Alam Sarker Omar Bazighifan

In the existent study, combined magneto-convection heat exchange in a driven enclosure having vertical fin was analyzed numerically. The finite element system-based GWR procedure was utilized to determine the flow model’s governing equations. A parametric inquiry was executed to review the influence of Richardson and Hartmann numbers on flow shape and heat removal features inside a frame. The problem’s resulting numerical outcomes were demonstrated graphically in terms of isotherms, streamlines, velocity sketches, local Nusselt number, global Nusselt number, and global fluid temperature. It was found that the varying lengths of the fin surface have a substantial impact on flow building and heat line sketch. Further, it was also noticed that a relatively fin length is needed to increase the heat exchange rate on the right cool wall at a high Richardson number. The fin can significantly enhance heat removal performance rate from an enclosure to adjacent fluid.

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Influence of Fin Length on Magneto-Combined Convection Heat Transfer Performance in a Lid-Driven Wavy Cavity Md. Fayz-Al-Asad Mehmet Yavuz Md. Nur Alam Md. Manirul Alam Sarker Omar Bazighifan doi: 10.3390/fractalfract5030107 Fractal and Fractional 2021-08-31 Fractal and Fractional 2021-08-31 5 3 Article 107 10.3390/fractalfract5030107 https://www.mdpi.com/2504-3110/5/3/107
Fractal Fract, Vol. 5, Pages 106: Approximate Solutions of Nonlinear Partial Differential Equations Using B-Polynomial Bases https://www.mdpi.com/2504-3110/5/3/106 A multivariable technique has been incorporated for guesstimating solutions of Nonlinear Partial Differential Equations (NPDE) using bases set of B-Polynomials (B-polys). To approximate the anticipated solution of the NPD equation, a linear product of variable coefficients ai(t) and Bi(x) B-polys has been employed. Additionally, the variable quantities in the anticipated solution are determined using the Galerkin method for minimizing errors. Before the minimization process is to take place, the NPDE is converted into an operational matrix equation which, when inverted, yields values of the undefined coefficients in the expected solution. The nonlinear terms of the NPDE are combined in the operational matrix equation using the initial guess and iterated until converged values of coefficients are obtained. A valid converged solution of NPDE is established when an appropriate degree of B-poly basis is employed, and the initial conditions are imposed on the operational matrix before the inverse is invoked. However, the accuracy of the solution depends on the number of B-polys of a certain degree expressed in multidimensional variables. Four examples of NPDE have been worked out to show the efficacy and accuracy of the two-dimensional B-poly technique. The estimated solutions of the examples are compared with the known exact solutions and an excellent agreement is found between them. In calculating the solutions of the NPD equations, the currently employed technique provides a higher-order precision compared to the finite difference method. The present technique could be readily extended to solving complex partial differential equations in multivariable problems. 2021-08-31 Fractal Fract, Vol. 5, Pages 106: Approximate Solutions of Nonlinear Partial Differential Equations Using B-Polynomial Bases

Fractal and Fractional doi: 10.3390/fractalfract5030106

Authors: Muhammad I. Bhatti Md. Habibur Rahman Nicholas Dimakis

A multivariable technique has been incorporated for guesstimating solutions of Nonlinear Partial Differential Equations (NPDE) using bases set of B-Polynomials (B-polys). To approximate the anticipated solution of the NPD equation, a linear product of variable coefficients ai(t) and Bi(x) B-polys has been employed. Additionally, the variable quantities in the anticipated solution are determined using the Galerkin method for minimizing errors. Before the minimization process is to take place, the NPDE is converted into an operational matrix equation which, when inverted, yields values of the undefined coefficients in the expected solution. The nonlinear terms of the NPDE are combined in the operational matrix equation using the initial guess and iterated until converged values of coefficients are obtained. A valid converged solution of NPDE is established when an appropriate degree of B-poly basis is employed, and the initial conditions are imposed on the operational matrix before the inverse is invoked. However, the accuracy of the solution depends on the number of B-polys of a certain degree expressed in multidimensional variables. Four examples of NPDE have been worked out to show the efficacy and accuracy of the two-dimensional B-poly technique. The estimated solutions of the examples are compared with the known exact solutions and an excellent agreement is found between them. In calculating the solutions of the NPD equations, the currently employed technique provides a higher-order precision compared to the finite difference method. The present technique could be readily extended to solving complex partial differential equations in multivariable problems.

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Approximate Solutions of Nonlinear Partial Differential Equations Using B-Polynomial Bases Muhammad I. Bhatti Md. Habibur Rahman Nicholas Dimakis doi: 10.3390/fractalfract5030106 Fractal and Fractional 2021-08-31 Fractal and Fractional 2021-08-31 5 3 Article 106 10.3390/fractalfract5030106 https://www.mdpi.com/2504-3110/5/3/106
Fractal Fract, Vol. 5, Pages 105: Uniqueness of Solutions of the Generalized Abel Integral Equations in Banach Spaces https://www.mdpi.com/2504-3110/5/3/105 This paper studies the uniqueness of solutions for several generalized Abel’s integral equations and a related coupled system in Banach spaces. The results derived are new and based on Babenko’s approach, Banach’s contraction principle and the multivariate Mittag–Leffler function. We also present some examples for the illustration of our main theorems. 2021-08-31 Fractal Fract, Vol. 5, Pages 105: Uniqueness of Solutions of the Generalized Abel Integral Equations in Banach Spaces

Fractal and Fractional doi: 10.3390/fractalfract5030105

Authors: Chenkuan Li Hari M. Srivastava

This paper studies the uniqueness of solutions for several generalized Abel’s integral equations and a related coupled system in Banach spaces. The results derived are new and based on Babenko’s approach, Banach’s contraction principle and the multivariate Mittag–Leffler function. We also present some examples for the illustration of our main theorems.

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Uniqueness of Solutions of the Generalized Abel Integral Equations in Banach Spaces Chenkuan Li Hari M. Srivastava doi: 10.3390/fractalfract5030105 Fractal and Fractional 2021-08-31 Fractal and Fractional 2021-08-31 5 3 Article 105 10.3390/fractalfract5030105 https://www.mdpi.com/2504-3110/5/3/105
Fractal Fract, Vol. 5, Pages 104: An Experimental Approach towards Motion Modeling and Control of a Vehicle Transiting a Non-Newtonian Environment https://www.mdpi.com/2504-3110/5/3/104 The present work tackles the modeling of the motion dynamics of an object submerged in a non-Newtonian environment. The mathematical model is developed starting from already known Newtonian interactions between the submersible and the fluid. The obtained model is therefore altered through optimization techniques to describe non-Newtonian interactions on the motion of the vehicle by using real-life data regarding non-Newtonian influences on submerged thrusting. For the obtained non-Newtonian fractional order process model, a fractional order control approach is employed to sway the submerged object’s position inside the viscoelastic environment. The presented modeling and control methodologies are solidified by real-life experimental data used to validate the veracity of the presented concepts. The robustness of the control strategy is experimentally validated on both Newtonian and non-Newtonian environments. 2021-08-25 Fractal Fract, Vol. 5, Pages 104: An Experimental Approach towards Motion Modeling and Control of a Vehicle Transiting a Non-Newtonian Environment

Fractal and Fractional doi: 10.3390/fractalfract5030104

Authors: Isabela Birs Cristina Muresan Ovidiu Prodan Silviu Folea Clara Ionescu

The present work tackles the modeling of the motion dynamics of an object submerged in a non-Newtonian environment. The mathematical model is developed starting from already known Newtonian interactions between the submersible and the fluid. The obtained model is therefore altered through optimization techniques to describe non-Newtonian interactions on the motion of the vehicle by using real-life data regarding non-Newtonian influences on submerged thrusting. For the obtained non-Newtonian fractional order process model, a fractional order control approach is employed to sway the submerged object’s position inside the viscoelastic environment. The presented modeling and control methodologies are solidified by real-life experimental data used to validate the veracity of the presented concepts. The robustness of the control strategy is experimentally validated on both Newtonian and non-Newtonian environments.

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An Experimental Approach towards Motion Modeling and Control of a Vehicle Transiting a Non-Newtonian Environment Isabela Birs Cristina Muresan Ovidiu Prodan Silviu Folea Clara Ionescu doi: 10.3390/fractalfract5030104 Fractal and Fractional 2021-08-25 Fractal and Fractional 2021-08-25 5 3 Article 104 10.3390/fractalfract5030104 https://www.mdpi.com/2504-3110/5/3/104
Fractal Fract, Vol. 5, Pages 103: Numerical Solutions for Systems of Fractional and Classical Integro-Differential Equations via Finite Integration Method Based on Shifted Chebyshev Polynomials https://www.mdpi.com/2504-3110/5/3/103 In this paper, the finite integration method and the operational matrix of fractional integration are implemented based on the shifted Chebyshev polynomial. They are utilized to devise two numerical procedures for solving the systems of fractional and classical integro-differential equations. The fractional derivatives are described in the Caputo sense. The devised procedure can be successfully applied to solve the stiff system of ODEs. To demonstrate the efficiency, accuracy and numerical convergence order of these procedures, several experimental examples are given. As a consequence, the numerical computations illustrate that our presented procedures achieve significant improvement in terms of accuracy with less computational cost. 2021-08-25 Fractal Fract, Vol. 5, Pages 103: Numerical Solutions for Systems of Fractional and Classical Integro-Differential Equations via Finite Integration Method Based on Shifted Chebyshev Polynomials

Fractal and Fractional doi: 10.3390/fractalfract5030103

Authors: Ampol Duangpan Ratinan Boonklurb Matinee Juytai

In this paper, the finite integration method and the operational matrix of fractional integration are implemented based on the shifted Chebyshev polynomial. They are utilized to devise two numerical procedures for solving the systems of fractional and classical integro-differential equations. The fractional derivatives are described in the Caputo sense. The devised procedure can be successfully applied to solve the stiff system of ODEs. To demonstrate the efficiency, accuracy and numerical convergence order of these procedures, several experimental examples are given. As a consequence, the numerical computations illustrate that our presented procedures achieve significant improvement in terms of accuracy with less computational cost.

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Numerical Solutions for Systems of Fractional and Classical Integro-Differential Equations via Finite Integration Method Based on Shifted Chebyshev Polynomials Ampol Duangpan Ratinan Boonklurb Matinee Juytai doi: 10.3390/fractalfract5030103 Fractal and Fractional 2021-08-25 Fractal and Fractional 2021-08-25 5 3 Article 103 10.3390/fractalfract5030103 https://www.mdpi.com/2504-3110/5/3/103
Fractal Fract, Vol. 5, Pages 102: Spectral Galerkin Approximation of Space Fractional Optimal Control Problem with Integral State Constraint https://www.mdpi.com/2504-3110/5/3/102 In this paper spectral Galerkin approximation of optimal control problem governed by fractional advection diffusion reaction equation with integral state constraint is investigated. First order optimal condition of the control problem is discussed. Weighted Jacobi polynomials are used to approximate the state and adjoint state. A priori error estimates for control, state, adjoint state and Lagrangian multiplier are derived. Numerical experiment is carried out to illustrate the theoretical findings. 2021-08-24 Fractal Fract, Vol. 5, Pages 102: Spectral Galerkin Approximation of Space Fractional Optimal Control Problem with Integral State Constraint

Fractal and Fractional doi: 10.3390/fractalfract5030102

Authors: Fangyuan Wang Xiaodi Li Zhaojie Zhou

In this paper spectral Galerkin approximation of optimal control problem governed by fractional advection diffusion reaction equation with integral state constraint is investigated. First order optimal condition of the control problem is discussed. Weighted Jacobi polynomials are used to approximate the state and adjoint state. A priori error estimates for control, state, adjoint state and Lagrangian multiplier are derived. Numerical experiment is carried out to illustrate the theoretical findings.

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Spectral Galerkin Approximation of Space Fractional Optimal Control Problem with Integral State Constraint Fangyuan Wang Xiaodi Li Zhaojie Zhou doi: 10.3390/fractalfract5030102 Fractal and Fractional 2021-08-24 Fractal and Fractional 2021-08-24 5 3 Article 102 10.3390/fractalfract5030102 https://www.mdpi.com/2504-3110/5/3/102
Fractal Fract, Vol. 5, Pages 101: Control and Robust Stabilization at Unstable Equilibrium by Fractional Controller for Magnetic Levitation Systems https://www.mdpi.com/2504-3110/5/3/101 The problem of control and stabilizing inherently non-linear and unstable magnetic levitation (Maglev) systems with uncertain equilibrium states has been studied. Accordingly, some significant works related to different control approaches have been highlighted to provide robust control and enhance the performance of the Maglev system. This work examines a method to control and stabilize the levitation system in the presence of disturbance and parameter variations to minimize the magnet gap deviation from the equilibrium position. To fulfill the stabilization and disturbance rejection for this non-linear dynamic system, the fractional order PID, fractional order sliding mode, and fractional order Fuzzy control approaches are conducted. In order to design the suitable control outlines based on fractional order controllers, a tuning hybrid method of GWO–PSO algorithms is applied by using the different performance criteria as Integrated Absolute Error (IAE), Integrated Time Weighted Absolute Error (ITAE), Integrated Squared Error (ISE), and Integrated Time Weighted Squared Error (ITSE). In general, these objectives are used by targeting the best tuning of specified control parameters. Finally, the simulation results are presented to determine which fractional controllers demonstrate better control performance, achieve fast and robust stability of the closed-loop system, and provide excellent disturbance suppression effect under nonlinear and uncertainty existing in the processing system. 2021-08-20 Fractal Fract, Vol. 5, Pages 101: Control and Robust Stabilization at Unstable Equilibrium by Fractional Controller for Magnetic Levitation Systems

Fractal and Fractional doi: 10.3390/fractalfract5030101

Authors: Banu Ataşlar-Ayyıldız Oğuzhan Karahan Serhat Yılmaz

The problem of control and stabilizing inherently non-linear and unstable magnetic levitation (Maglev) systems with uncertain equilibrium states has been studied. Accordingly, some significant works related to different control approaches have been highlighted to provide robust control and enhance the performance of the Maglev system. This work examines a method to control and stabilize the levitation system in the presence of disturbance and parameter variations to minimize the magnet gap deviation from the equilibrium position. To fulfill the stabilization and disturbance rejection for this non-linear dynamic system, the fractional order PID, fractional order sliding mode, and fractional order Fuzzy control approaches are conducted. In order to design the suitable control outlines based on fractional order controllers, a tuning hybrid method of GWO–PSO algorithms is applied by using the different performance criteria as Integrated Absolute Error (IAE), Integrated Time Weighted Absolute Error (ITAE), Integrated Squared Error (ISE), and Integrated Time Weighted Squared Error (ITSE). In general, these objectives are used by targeting the best tuning of specified control parameters. Finally, the simulation results are presented to determine which fractional controllers demonstrate better control performance, achieve fast and robust stability of the closed-loop system, and provide excellent disturbance suppression effect under nonlinear and uncertainty existing in the processing system.

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Control and Robust Stabilization at Unstable Equilibrium by Fractional Controller for Magnetic Levitation Systems Banu Ataşlar-Ayyıldız Oğuzhan Karahan Serhat Yılmaz doi: 10.3390/fractalfract5030101 Fractal and Fractional 2021-08-20 Fractal and Fractional 2021-08-20 5 3 Article 101 10.3390/fractalfract5030101 https://www.mdpi.com/2504-3110/5/3/101
Fractal Fract, Vol. 5, Pages 100: Orthonormal Ultraspherical Operational Matrix Algorithm for Fractal–Fractional Riccati Equation with Generalized Caputo Derivative https://www.mdpi.com/2504-3110/5/3/100 Herein, we developed and analyzed a new fractal–fractional (FF) operational matrix for orthonormal normalized ultraspherical polynomials. We used this matrix to handle the FF Riccati differential equation with the new generalized Caputo FF derivative. Based on the developed operational matrix and the spectral Tau method, the nonlinear differential problem was reduced to a system of algebraic equations in the unknown expansion coefficients. Accordingly, the resulting system was solved by Newton’s solver with a small initial guess. The efficiency, accuracy, and applicability of the developed numerical method were checked by exhibiting various test problems. The obtained results were also compared with other recent methods, based on the available literature. 2021-08-17 Fractal Fract, Vol. 5, Pages 100: Orthonormal Ultraspherical Operational Matrix Algorithm for Fractal–Fractional Riccati Equation with Generalized Caputo Derivative

Fractal and Fractional doi: 10.3390/fractalfract5030100

Authors: Youssri Hassan Youssri

Herein, we developed and analyzed a new fractal–fractional (FF) operational matrix for orthonormal normalized ultraspherical polynomials. We used this matrix to handle the FF Riccati differential equation with the new generalized Caputo FF derivative. Based on the developed operational matrix and the spectral Tau method, the nonlinear differential problem was reduced to a system of algebraic equations in the unknown expansion coefficients. Accordingly, the resulting system was solved by Newton’s solver with a small initial guess. The efficiency, accuracy, and applicability of the developed numerical method were checked by exhibiting various test problems. The obtained results were also compared with other recent methods, based on the available literature.

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Orthonormal Ultraspherical Operational Matrix Algorithm for Fractal–Fractional Riccati Equation with Generalized Caputo Derivative Youssri Hassan Youssri doi: 10.3390/fractalfract5030100 Fractal and Fractional 2021-08-17 Fractal and Fractional 2021-08-17 5 3 Article 100 10.3390/fractalfract5030100 https://www.mdpi.com/2504-3110/5/3/100
Fractal Fract, Vol. 5, Pages 99: Advancement of Non-Newtonian Fluid with Hybrid Nanoparticles in a Convective Channel and Prabhakar’s Fractional Derivative—Analytical Solution https://www.mdpi.com/2504-3110/5/3/99 The present paper deals with the advancement of non-Newtonian fluid containing some nanoparticles between two parallel plates. A novel fractional operator is used to model memory effects, and analytical solutions are obtained for temperature and velocity fields by the method of Laplace transform. Moreover, a parametric study is elaborated to show the impact of flow parameters and presented in graphical form. As a result, dual solutions are predicted for increasing values of fractional parameters for short and long times. Furthermore, by increasing nanoparticle concentration, the temperature can be raised along with decreasing velocity. A fractional approach can provide new insight for the analytical solutions which makes the interpretation of the results easier and enable the way of testing possible approximate solutions. 2021-08-17 Fractal Fract, Vol. 5, Pages 99: Advancement of Non-Newtonian Fluid with Hybrid Nanoparticles in a Convective Channel and Prabhakar’s Fractional Derivative—Analytical Solution

Fractal and Fractional doi: 10.3390/fractalfract5030099

The present paper deals with the advancement of non-Newtonian fluid containing some nanoparticles between two parallel plates. A novel fractional operator is used to model memory effects, and analytical solutions are obtained for temperature and velocity fields by the method of Laplace transform. Moreover, a parametric study is elaborated to show the impact of flow parameters and presented in graphical form. As a result, dual solutions are predicted for increasing values of fractional parameters for short and long times. Furthermore, by increasing nanoparticle concentration, the temperature can be raised along with decreasing velocity. A fractional approach can provide new insight for the analytical solutions which makes the interpretation of the results easier and enable the way of testing possible approximate solutions.

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Advancement of Non-Newtonian Fluid with Hybrid Nanoparticles in a Convective Channel and Prabhakar’s Fractional Derivative—Analytical Solution Muhammad Imran Asjad Noman Sarwar Muhammad Bilal Hafeez Wojciech Sumelka Taseer Muhammad doi: 10.3390/fractalfract5030099 Fractal and Fractional 2021-08-17 Fractal and Fractional 2021-08-17 5 3 Article 99 10.3390/fractalfract5030099 https://www.mdpi.com/2504-3110/5/3/99
Fractal Fract, Vol. 5, Pages 98: Quarter-Sweep Preconditioned Relaxation Method, Algorithm and Efficiency Analysis for Fractional Mathematical Equation https://www.mdpi.com/2504-3110/5/3/98 Research into the recent developments for solving fractional mathematical equations requires accurate and efficient numerical methods. Although many numerical methods based on Caputo’s fractional derivative have been proposed to solve fractional mathematical equations, the efficiency of obtaining solutions using these methods when dealing with a large matrix requires further study. The matrix size influences the accuracy of the solution. Therefore, this paper proposes a quarter-sweep finite difference scheme with a preconditioned relaxation-based approximation to efficiently solve a large matrix, which is based on the establishment of a linear system for a fractional mathematical equation. The paper presents the formulation of the quarter-sweep finite difference scheme that is used to approximate the selected fractional mathematical equation. Then, the derivation of a preconditioned relaxation method based on a quarter-sweep scheme is discussed. The design of a C++ algorithm of the proposed quarter-sweep preconditioned relaxation method is shown and, finally, efficiency analysis comparing the proposed method with several tested methods is presented. The contributions of this paper are the presentation of a new preconditioned matrix to restructure the developed linear system, and the derivation of an efficient preconditioned relaxation iterative method for solving a fractional mathematical equation. By simulating the solutions of time-fractional diffusion problems with the proposed numerical method, the study found that computing solutions using the quarter-sweep preconditioned relaxation method is more efficient than using the tested methods. The proposed numerical method is able to solve the selected problems with fewer iterations and a faster execution time than the tested existing methods. The efficiency of the methods was evaluated using different matrix sizes. Thus, the combination of a quarter-sweep finite difference method, Caputo’s time-fractional derivative, and the preconditioned successive over-relaxation method showed good potential for solving different types of fractional mathematical equations, and provides a future direction for this field of research. 2021-08-14 Fractal Fract, Vol. 5, Pages 98: Quarter-Sweep Preconditioned Relaxation Method, Algorithm and Efficiency Analysis for Fractional Mathematical Equation

Fractal and Fractional doi: 10.3390/fractalfract5030098

Authors: Andang Sunarto Praveen Agarwal Jumat Sulaiman Jackel Vui Lung Chew Shaher Momani

Research into the recent developments for solving fractional mathematical equations requires accurate and efficient numerical methods. Although many numerical methods based on Caputo’s fractional derivative have been proposed to solve fractional mathematical equations, the efficiency of obtaining solutions using these methods when dealing with a large matrix requires further study. The matrix size influences the accuracy of the solution. Therefore, this paper proposes a quarter-sweep finite difference scheme with a preconditioned relaxation-based approximation to efficiently solve a large matrix, which is based on the establishment of a linear system for a fractional mathematical equation. The paper presents the formulation of the quarter-sweep finite difference scheme that is used to approximate the selected fractional mathematical equation. Then, the derivation of a preconditioned relaxation method based on a quarter-sweep scheme is discussed. The design of a C++ algorithm of the proposed quarter-sweep preconditioned relaxation method is shown and, finally, efficiency analysis comparing the proposed method with several tested methods is presented. The contributions of this paper are the presentation of a new preconditioned matrix to restructure the developed linear system, and the derivation of an efficient preconditioned relaxation iterative method for solving a fractional mathematical equation. By simulating the solutions of time-fractional diffusion problems with the proposed numerical method, the study found that computing solutions using the quarter-sweep preconditioned relaxation method is more efficient than using the tested methods. The proposed numerical method is able to solve the selected problems with fewer iterations and a faster execution time than the tested existing methods. The efficiency of the methods was evaluated using different matrix sizes. Thus, the combination of a quarter-sweep finite difference method, Caputo’s time-fractional derivative, and the preconditioned successive over-relaxation method showed good potential for solving different types of fractional mathematical equations, and provides a future direction for this field of research.

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Quarter-Sweep Preconditioned Relaxation Method, Algorithm and Efficiency Analysis for Fractional Mathematical Equation Andang Sunarto Praveen Agarwal Jumat Sulaiman Jackel Vui Lung Chew Shaher Momani doi: 10.3390/fractalfract5030098 Fractal and Fractional 2021-08-14 Fractal and Fractional 2021-08-14 5 3 Article 98 10.3390/fractalfract5030098 https://www.mdpi.com/2504-3110/5/3/98
Fractal Fract, Vol. 5, Pages 97: Synchronization Analysis of Multiple Integral Inequalities Driven by Steklov Operator https://www.mdpi.com/2504-3110/5/3/97 We construct a subclass of Copson’s integral inequality in this article. In order to achieve this goal, we attempt to use the Steklov operator for generalizing different inequalities of the Copson type relevant to the situations ρ&amp;gt;1 as well as ρ&amp;lt;1. We demonstrate the inequalities with the guidance of basic comparison, Holder’s inequality, and the integration by parts approach. Moreover, some new variations of Hardy’s integral inequality are also presented with the utilization of Steklov operator. We also formulate many remarks and two examples to show the novelty and authenticity of our results. 2021-08-14 Fractal Fract, Vol. 5, Pages 97: Synchronization Analysis of Multiple Integral Inequalities Driven by Steklov Operator

Fractal and Fractional doi: 10.3390/fractalfract5030097

Authors: Wedad Albalawi Zareen A. Khan

We construct a subclass of Copson’s integral inequality in this article. In order to achieve this goal, we attempt to use the Steklov operator for generalizing different inequalities of the Copson type relevant to the situations ρ&amp;gt;1 as well as ρ&amp;lt;1. We demonstrate the inequalities with the guidance of basic comparison, Holder’s inequality, and the integration by parts approach. Moreover, some new variations of Hardy’s integral inequality are also presented with the utilization of Steklov operator. We also formulate many remarks and two examples to show the novelty and authenticity of our results.

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Synchronization Analysis of Multiple Integral Inequalities Driven by Steklov Operator Wedad Albalawi Zareen A. Khan doi: 10.3390/fractalfract5030097 Fractal and Fractional 2021-08-14 Fractal and Fractional 2021-08-14 5 3 Article 97 10.3390/fractalfract5030097 https://www.mdpi.com/2504-3110/5/3/97
Fractal Fract, Vol. 5, Pages 96: Particularities of Forest Dynamics Using Higuchi Dimension. Parâng Mountains as a Case Study https://www.mdpi.com/2504-3110/5/3/96 The legal or illegal losses and the natural disturbance regime of forest areas in Romania generate major imbalances in territorial systems. The main purpose of the current research was to examine the dynamics of the complexity of forests under the influence of forest loss but also to compare the applicability of Higuchi dimension. In this study, two fractal algorithms, Higuchi 1D (H1D) and Higuchi 2D (H2D), were used to determine qualitative and quantitative aspects based on images obtained from a Geographic Information System (GIS) database. The H1D analysis showed that the impact of forest loss has led to increased fragmentation of the forests, generating a continuous increase in the complexity of forest areas. The H2D analysis identified the complexity of forest morphology by the relationship between each pixel and the neighboring pixels from analyzed images, which allowed us to highlight the local characteristics of the forest loss. The H1D and H2D methods showed that they have the speed and simplicity required for forest loss analysis. Using this methodology complementary to GIS analyses, a relevant status of how forest loss occurred and their impact on tree-cover dynamics was obtained. 2021-08-13 Fractal Fract, Vol. 5, Pages 96: Particularities of Forest Dynamics Using Higuchi Dimension. Parâng Mountains as a Case Study

Fractal and Fractional doi: 10.3390/fractalfract5030096

Authors: Adrian Gabriel Simion Ion Andronache Helmut Ahammer Marian Marin Vlad Loghin Iulia Daniela Nedelcu Cristian Mihnea Popa Daniel Peptenatu Herbert Franz Jelinek

The legal or illegal losses and the natural disturbance regime of forest areas in Romania generate major imbalances in territorial systems. The main purpose of the current research was to examine the dynamics of the complexity of forests under the influence of forest loss but also to compare the applicability of Higuchi dimension. In this study, two fractal algorithms, Higuchi 1D (H1D) and Higuchi 2D (H2D), were used to determine qualitative and quantitative aspects based on images obtained from a Geographic Information System (GIS) database. The H1D analysis showed that the impact of forest loss has led to increased fragmentation of the forests, generating a continuous increase in the complexity of forest areas. The H2D analysis identified the complexity of forest morphology by the relationship between each pixel and the neighboring pixels from analyzed images, which allowed us to highlight the local characteristics of the forest loss. The H1D and H2D methods showed that they have the speed and simplicity required for forest loss analysis. Using this methodology complementary to GIS analyses, a relevant status of how forest loss occurred and their impact on tree-cover dynamics was obtained.

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Particularities of Forest Dynamics Using Higuchi Dimension. Parâng Mountains as a Case Study Adrian Gabriel Simion Ion Andronache Helmut Ahammer Marian Marin Vlad Loghin Iulia Daniela Nedelcu Cristian Mihnea Popa Daniel Peptenatu Herbert Franz Jelinek doi: 10.3390/fractalfract5030096 Fractal and Fractional 2021-08-13 Fractal and Fractional 2021-08-13 5 3 Article 96 10.3390/fractalfract5030096 https://www.mdpi.com/2504-3110/5/3/96
Fractal Fract, Vol. 5, Pages 95: Qualitative Behavior of Unbounded Solutions of Neutral Differential Equations of Third-Order https://www.mdpi.com/2504-3110/5/3/95 New oscillatory properties for the oscillation of unbounded solutions to a class of third-order neutral differential equations with several deviating arguments are established. Several oscillation results are established by using generalized Riccati transformation and a integral average technique under the case of unbounded neutral coefficients. Examples are given to prove the significance of new theorems. 2021-08-12 Fractal Fract, Vol. 5, Pages 95: Qualitative Behavior of Unbounded Solutions of Neutral Differential Equations of Third-Order

Fractal and Fractional doi: 10.3390/fractalfract5030095

Authors: M. Sathish Kumar R. Elayaraja V. Ganesan Omar Bazighifan Khalifa Al-Shaqsi Kamsing Nonlaopon

New oscillatory properties for the oscillation of unbounded solutions to a class of third-order neutral differential equations with several deviating arguments are established. Several oscillation results are established by using generalized Riccati transformation and a integral average technique under the case of unbounded neutral coefficients. Examples are given to prove the significance of new theorems.

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Qualitative Behavior of Unbounded Solutions of Neutral Differential Equations of Third-Order M. Sathish Kumar R. Elayaraja V. Ganesan Omar Bazighifan Khalifa Al-Shaqsi Kamsing Nonlaopon doi: 10.3390/fractalfract5030095 Fractal and Fractional 2021-08-12 Fractal and Fractional 2021-08-12 5 3 Article 95 10.3390/fractalfract5030095 https://www.mdpi.com/2504-3110/5/3/95
Fractal Fract, Vol. 5, Pages 94: Novel Computations of the Time-Fractional Fisher’s Model via Generalized Fractional Integral Operators by Means of the Elzaki Transform https://www.mdpi.com/2504-3110/5/3/94 The present investigation dealing with a hybrid technique coupled with a new iterative transform method, namely the iterative Elzaki transform method (IETM), is employed to solve the nonlinear fractional Fisher’s model. Fisher’s equation is a precise mathematical result that arose in population dynamics and genetics, specifically in chemistry. The Caputo and Antagana-Baleanu fractional derivatives in the Caputo sense are used to test the intricacies of this mechanism numerically. In order to examine the approximate findings of fractional-order Fisher’s type equations, the IETM solutions are obtained in series representation. Moreover, the stability of the approach was demonstrated using fixed point theory. Several illustrative cases are described that strongly agree with the precise solutions. Moreover, tables and graphs are included in order to conceptualize the influence of the fractional order and on the previous findings. The projected technique illustrates that only a few terms are sufficient for finding an approximate outcome, which is computationally appealing and accurate to analyze. Additionally, the offered procedure is highly robust, explicit, and viable for nonlinear fractional PDEs, but it could be generalized to other complex physical phenomena. 2021-08-12 Fractal Fract, Vol. 5, Pages 94: Novel Computations of the Time-Fractional Fisher’s Model via Generalized Fractional Integral Operators by Means of the Elzaki Transform

Fractal and Fractional doi: 10.3390/fractalfract5030094

Authors: Saima Rashid Zakia Hammouch Hassen Aydi Abdulaziz Garba Ahmad Abdullah M. Alsharif

The present investigation dealing with a hybrid technique coupled with a new iterative transform method, namely the iterative Elzaki transform method (IETM), is employed to solve the nonlinear fractional Fisher’s model. Fisher’s equation is a precise mathematical result that arose in population dynamics and genetics, specifically in chemistry. The Caputo and Antagana-Baleanu fractional derivatives in the Caputo sense are used to test the intricacies of this mechanism numerically. In order to examine the approximate findings of fractional-order Fisher’s type equations, the IETM solutions are obtained in series representation. Moreover, the stability of the approach was demonstrated using fixed point theory. Several illustrative cases are described that strongly agree with the precise solutions. Moreover, tables and graphs are included in order to conceptualize the influence of the fractional order and on the previous findings. The projected technique illustrates that only a few terms are sufficient for finding an approximate outcome, which is computationally appealing and accurate to analyze. Additionally, the offered procedure is highly robust, explicit, and viable for nonlinear fractional PDEs, but it could be generalized to other complex physical phenomena.

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Novel Computations of the Time-Fractional Fisher’s Model via Generalized Fractional Integral Operators by Means of the Elzaki Transform Saima Rashid Zakia Hammouch Hassen Aydi Abdulaziz Garba Ahmad Abdullah M. Alsharif doi: 10.3390/fractalfract5030094 Fractal and Fractional 2021-08-12 Fractal and Fractional 2021-08-12 5 3 Article 94 10.3390/fractalfract5030094 https://www.mdpi.com/2504-3110/5/3/94
Fractal Fract, Vol. 5, Pages 93: Homotopy Perturbation Method for the Fractal Toda Oscillator https://www.mdpi.com/2504-3110/5/3/93 The fractal Toda oscillator with an exponentially nonlinear term is extremely difficult to solve; Elias-Zuniga et al. (2020) suggested the equivalent power-form method. In this paper, first, the fractal variational theory is used to show the basic property of the fractal oscillator, and a new form of the Toda oscillator is obtained free of the exponential nonlinear term, which is similar to the form of the Jerk oscillator. The homotopy perturbation method is used to solve the fractal Toda oscillator, and the analytical solution is examined using the numerical solution which shows excellent agreement. Furthermore, the effect of the order of the fractal derivative on the vibration property is elucidated graphically. 2021-08-11 Fractal Fract, Vol. 5, Pages 93: Homotopy Perturbation Method for the Fractal Toda Oscillator

Fractal and Fractional doi: 10.3390/fractalfract5030093

Authors: Ji-Huan He Yusry O. El-Dib Amal A. Mady

The fractal Toda oscillator with an exponentially nonlinear term is extremely difficult to solve; Elias-Zuniga et al. (2020) suggested the equivalent power-form method. In this paper, first, the fractal variational theory is used to show the basic property of the fractal oscillator, and a new form of the Toda oscillator is obtained free of the exponential nonlinear term, which is similar to the form of the Jerk oscillator. The homotopy perturbation method is used to solve the fractal Toda oscillator, and the analytical solution is examined using the numerical solution which shows excellent agreement. Furthermore, the effect of the order of the fractal derivative on the vibration property is elucidated graphically.

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Homotopy Perturbation Method for the Fractal Toda Oscillator Ji-Huan He Yusry O. El-Dib Amal A. Mady doi: 10.3390/fractalfract5030093 Fractal and Fractional 2021-08-11 Fractal and Fractional 2021-08-11 5 3 Article 93 10.3390/fractalfract5030093 https://www.mdpi.com/2504-3110/5/3/93
Fractal Fract, Vol. 5, Pages 92: The Proof of a Conjecture Relating Catalan Numbers to an Averaged Mandelbrot-Möbius Iterated Function https://www.mdpi.com/2504-3110/5/3/92 In 2021, Mork and Ulness studied the Mandelbrot and Julia sets for a generalization of the well-explored function ηλ(z)=z2+λ. Their generalization was based on the composition of ηλ with the Möbius transformation μ(z)=1z at each iteration step. Furthermore, they posed a conjecture providing a relation between the coefficients of (each order) iterated series of μ(ηλ(z)) (at z=0) and the Catalan numbers. In this paper, in particular, we prove this conjecture in a more precise (quantitative) formulation. 2021-08-11 Fractal Fract, Vol. 5, Pages 92: The Proof of a Conjecture Relating Catalan Numbers to an Averaged Mandelbrot-Möbius Iterated Function

Fractal and Fractional doi: 10.3390/fractalfract5030092

Authors: Pavel Trojovský K Venkatachalam

In 2021, Mork and Ulness studied the Mandelbrot and Julia sets for a generalization of the well-explored function ηλ(z)=z2+λ. Their generalization was based on the composition of ηλ with the Möbius transformation μ(z)=1z at each iteration step. Furthermore, they posed a conjecture providing a relation between the coefficients of (each order) iterated series of μ(ηλ(z)) (at z=0) and the Catalan numbers. In this paper, in particular, we prove this conjecture in a more precise (quantitative) formulation.

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The Proof of a Conjecture Relating Catalan Numbers to an Averaged Mandelbrot-Möbius Iterated Function Pavel Trojovský K Venkatachalam doi: 10.3390/fractalfract5030092 Fractal and Fractional 2021-08-11 Fractal and Fractional 2021-08-11 5 3 Article 92 10.3390/fractalfract5030092 https://www.mdpi.com/2504-3110/5/3/92
Fractal Fract, Vol. 5, Pages 91: State of Charge Estimation of Lithium-Ion Batteries Based on Fuzzy Fractional-Order Unscented Kalman Filter https://www.mdpi.com/2504-3110/5/3/91 The covariance matrix of measurement noise is fixed in the Kalman filter algorithm. However, in the process of battery operation, the measurement noise is affected by different charging and discharging conditions and the external environment. Consequently, obtaining the noise statistical characteristics is difficult, which affects the accuracy of the Kalman filter algorithm. In order to improve the estimation accuracy of the state of charge (SOC) of lithium-ion batteries under actual working conditions, a fuzzy fractional-order unscented Kalman filter (FFUKF) is proposed. The algorithm combines fuzzy inference with fractional-order unscented Kalman filter (FUKF) to infer the measurement noise in real time and take advantage of fractional calculus in describing the dynamic behavior of the lithium batteries. The accuracy of the SOC estimation under different working conditions at three different temperatures is verified. The results show that the accuracy of the proposed algorithm is superior to those of the FUKF and extended Kalman filter (EKF) algorithms. 2021-08-08 Fractal Fract, Vol. 5, Pages 91: State of Charge Estimation of Lithium-Ion Batteries Based on Fuzzy Fractional-Order Unscented Kalman Filter

Fractal and Fractional doi: 10.3390/fractalfract5030091

Authors: Liping Chen Yu Chen António M. Lopes Huifang Kong Ranchao Wu

The covariance matrix of measurement noise is fixed in the Kalman filter algorithm. However, in the process of battery operation, the measurement noise is affected by different charging and discharging conditions and the external environment. Consequently, obtaining the noise statistical characteristics is difficult, which affects the accuracy of the Kalman filter algorithm. In order to improve the estimation accuracy of the state of charge (SOC) of lithium-ion batteries under actual working conditions, a fuzzy fractional-order unscented Kalman filter (FFUKF) is proposed. The algorithm combines fuzzy inference with fractional-order unscented Kalman filter (FUKF) to infer the measurement noise in real time and take advantage of fractional calculus in describing the dynamic behavior of the lithium batteries. The accuracy of the SOC estimation under different working conditions at three different temperatures is verified. The results show that the accuracy of the proposed algorithm is superior to those of the FUKF and extended Kalman filter (EKF) algorithms.

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State of Charge Estimation of Lithium-Ion Batteries Based on Fuzzy Fractional-Order Unscented Kalman Filter Liping Chen Yu Chen António M. Lopes Huifang Kong Ranchao Wu doi: 10.3390/fractalfract5030091 Fractal and Fractional 2021-08-08 Fractal and Fractional 2021-08-08 5 3 Article 91 10.3390/fractalfract5030091 https://www.mdpi.com/2504-3110/5/3/91
Fractal Fract, Vol. 5, Pages 90: Spline Collocation for Multi-Term Fractional Integro-Differential Equations with Weakly Singular Kernels https://www.mdpi.com/2504-3110/5/3/90 We consider general linear multi-term Caputo fractional integro-differential equations with weakly singular kernels subject to local or non-local boundary conditions. Using an integral equation reformulation of the proposed problem, we first study the existence, uniqueness and regularity of the exact solution. Based on the obtained regularity properties and spline collocation techniques, the numerical solution of the problem is discussed. Optimal global convergence estimates are derived and a superconvergence result for a special choice of grid and collocation parameters is given. A numerical illustration is also presented. 2021-08-07 Fractal Fract, Vol. 5, Pages 90: Spline Collocation for Multi-Term Fractional Integro-Differential Equations with Weakly Singular Kernels

Fractal and Fractional doi: 10.3390/fractalfract5030090

Authors: Arvet Pedas Mikk Vikerpuur

We consider general linear multi-term Caputo fractional integro-differential equations with weakly singular kernels subject to local or non-local boundary conditions. Using an integral equation reformulation of the proposed problem, we first study the existence, uniqueness and regularity of the exact solution. Based on the obtained regularity properties and spline collocation techniques, the numerical solution of the problem is discussed. Optimal global convergence estimates are derived and a superconvergence result for a special choice of grid and collocation parameters is given. A numerical illustration is also presented.

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Spline Collocation for Multi-Term Fractional Integro-Differential Equations with Weakly Singular Kernels Arvet Pedas Mikk Vikerpuur doi: 10.3390/fractalfract5030090 Fractal and Fractional 2021-08-07 Fractal and Fractional 2021-08-07 5 3 Article 90 10.3390/fractalfract5030090 https://www.mdpi.com/2504-3110/5/3/90
Fractal Fract, Vol. 5, Pages 89: New Results on Controllability for a Class of Fractional Integrodifferential Dynamical Systems with Delay in Banach Spaces https://www.mdpi.com/2504-3110/5/3/89 The present work addresses some new controllability results for a class of fractional integrodifferential dynamical systems with a delay in Banach spaces. Under the new definition of controllability , first introduced by us, we obtain some sufficient conditions of controllability for the considered dynamic systems. To conquer the difficulties arising from time delay, we also introduce a suitable delay item in a special complete space. In this work, a nonlinear item is not assumed to have Lipschitz continuity or other growth hypotheses compared with most existing articles. Our main tools are resolvent operator theory and fixed point theory. At last, an example is presented to explain our abstract conclusions. 2021-08-04 Fractal Fract, Vol. 5, Pages 89: New Results on Controllability for a Class of Fractional Integrodifferential Dynamical Systems with Delay in Banach Spaces

Fractal and Fractional doi: 10.3390/fractalfract5030089

Authors: Daliang Zhao

The present work addresses some new controllability results for a class of fractional integrodifferential dynamical systems with a delay in Banach spaces. Under the new definition of controllability , first introduced by us, we obtain some sufficient conditions of controllability for the considered dynamic systems. To conquer the difficulties arising from time delay, we also introduce a suitable delay item in a special complete space. In this work, a nonlinear item is not assumed to have Lipschitz continuity or other growth hypotheses compared with most existing articles. Our main tools are resolvent operator theory and fixed point theory. At last, an example is presented to explain our abstract conclusions.

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New Results on Controllability for a Class of Fractional Integrodifferential Dynamical Systems with Delay in Banach Spaces Daliang Zhao doi: 10.3390/fractalfract5030089 Fractal and Fractional 2021-08-04 Fractal and Fractional 2021-08-04 5 3 Article 89 10.3390/fractalfract5030089 https://www.mdpi.com/2504-3110/5/3/89
Fractal Fract, Vol. 5, Pages 88: Applications of the (G′/G2)-Expansion Method for Solving Certain Nonlinear Conformable Evolution Equations https://www.mdpi.com/2504-3110/5/3/88 The core objective of this article is to generate novel exact traveling wave solutions of two nonlinear conformable evolution equations, namely, the (2+1)-dimensional conformable time integro-differential Sawada–Kotera (SK) equation and the (3+1)-dimensional conformable time modified KdV–Zakharov–Kuznetsov (mKdV–ZK) equation using the (G′/G2)-expansion method. These two equations associate with conformable partial derivatives with respect to time which the former equation is firstly proposed in the form of the conformable integro-differential equation. To the best of the authors’ knowledge, the two equations have not been solved by means of the (G′/G2)-expansion method for their exact solutions. As a result, some exact solutions of the equations expressed in terms of trigonometric, exponential, and rational function solutions are reported here for the first time. Furthermore, graphical representations of some selected solutions, plotted using some specific sets of the parameter values and the fractional orders, reveal certain physical features such as a singular single-soliton solution and a doubly periodic wave solution. These kinds of the solutions are usually discovered in natural phenomena. In particular, the soliton solution, which is a solitary wave whose amplitude, velocity, and shape are conserved after a collision with another soliton for a nondissipative system, arises ubiquitously in fluid mechanics, fiber optics, atomic physics, water waves, and plasmas. The method, along with the help of symbolic software packages, can be efficiently and simply used to solve the proposed problems for trustworthy and accurate exact solutions. Consequently, the method could be employed to determine some new exact solutions for other nonlinear conformable evolution equations. 2021-08-04 Fractal Fract, Vol. 5, Pages 88: Applications of the (G′/G2)-Expansion Method for Solving Certain Nonlinear Conformable Evolution Equations

Fractal and Fractional doi: 10.3390/fractalfract5030088

Authors: Supaporn Kaewta Sekson Sirisubtawee Sanoe Koonprasert Surattana Sungnul

The core objective of this article is to generate novel exact traveling wave solutions of two nonlinear conformable evolution equations, namely, the (2+1)-dimensional conformable time integro-differential Sawada–Kotera (SK) equation and the (3+1)-dimensional conformable time modified KdV–Zakharov–Kuznetsov (mKdV–ZK) equation using the (G′/G2)-expansion method. These two equations associate with conformable partial derivatives with respect to time which the former equation is firstly proposed in the form of the conformable integro-differential equation. To the best of the authors’ knowledge, the two equations have not been solved by means of the (G′/G2)-expansion method for their exact solutions. As a result, some exact solutions of the equations expressed in terms of trigonometric, exponential, and rational function solutions are reported here for the first time. Furthermore, graphical representations of some selected solutions, plotted using some specific sets of the parameter values and the fractional orders, reveal certain physical features such as a singular single-soliton solution and a doubly periodic wave solution. These kinds of the solutions are usually discovered in natural phenomena. In particular, the soliton solution, which is a solitary wave whose amplitude, velocity, and shape are conserved after a collision with another soliton for a nondissipative system, arises ubiquitously in fluid mechanics, fiber optics, atomic physics, water waves, and plasmas. The method, along with the help of symbolic software packages, can be efficiently and simply used to solve the proposed problems for trustworthy and accurate exact solutions. Consequently, the method could be employed to determine some new exact solutions for other nonlinear conformable evolution equations.

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Applications of the (G′/G2)-Expansion Method for Solving Certain Nonlinear Conformable Evolution Equations Supaporn Kaewta Sekson Sirisubtawee Sanoe Koonprasert Surattana Sungnul doi: 10.3390/fractalfract5030088 Fractal and Fractional 2021-08-04 Fractal and Fractional 2021-08-04 5 3 Article 88 10.3390/fractalfract5030088 https://www.mdpi.com/2504-3110/5/3/88
Fractal Fract, Vol. 5, Pages 87: A Mathematical Study of a Coronavirus Model with the Caputo Fractional-Order Derivative https://www.mdpi.com/2504-3110/5/3/87 In this work, we introduce a minimal model for the current pandemic. It incorporates the basic compartments: exposed, and both symptomatic and asymptomatic infected. The dynamical system is formulated by means of fractional operators. The model equilibria are analyzed. The theoretical results indicate that their stability behavior is the same as for the corresponding system formulated via standard derivatives. This suggests that, at least in this case for the model presented here, the memory effects contained in the fractional operators apparently do not seem to play a relevant role. The numerical simulations instead reveal that the order of the fractional derivative has a definite influence on both the equilibrium population levels and the speed at which they are attained. 2021-08-03 Fractal Fract, Vol. 5, Pages 87: A Mathematical Study of a Coronavirus Model with the Caputo Fractional-Order Derivative

Fractal and Fractional doi: 10.3390/fractalfract5030087

Authors: Youcef Belgaid Mohamed Helal Abdelkader Lakmeche Ezio Venturino

In this work, we introduce a minimal model for the current pandemic. It incorporates the basic compartments: exposed, and both symptomatic and asymptomatic infected. The dynamical system is formulated by means of fractional operators. The model equilibria are analyzed. The theoretical results indicate that their stability behavior is the same as for the corresponding system formulated via standard derivatives. This suggests that, at least in this case for the model presented here, the memory effects contained in the fractional operators apparently do not seem to play a relevant role. The numerical simulations instead reveal that the order of the fractional derivative has a definite influence on both the equilibrium population levels and the speed at which they are attained.

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A Mathematical Study of a Coronavirus Model with the Caputo Fractional-Order Derivative Youcef Belgaid Mohamed Helal Abdelkader Lakmeche Ezio Venturino doi: 10.3390/fractalfract5030087 Fractal and Fractional 2021-08-03 Fractal and Fractional 2021-08-03 5 3 Article 87 10.3390/fractalfract5030087 https://www.mdpi.com/2504-3110/5/3/87
Fractal Fract, Vol. 5, Pages 86: A Grey System Approach for Estimating the Hölderian Regularity with an Application to Algerian Well Log Data https://www.mdpi.com/2504-3110/5/3/86 The Hölderian regularity is an important mathematical feature of a signal, connected with the physical nature of the measured parameter. Many algorithms have been proposed in literature for estimating the local Hölder exponent value, but all of them lead to biased estimates. This paper attempts to apply the grey system theory (GST) on the raw signal for improving the accuracy of Hölderian regularity estimation. First, synthetic logs data are generated by the successive random additions (SRA) method with different types of Hölder functions. The application on these simulated signals shows that the Hölder functions estimated by the GST are more precise than those derived from the raw data. Additionally, noisy signals are considered for the same experiment, and more accurate regularity is obtained using signals processed using GST. Second, the proposed technique is implemented on well log data measured at an Algerian exploration borehole. It is demonstrated that the regularity determined from the well logs analyzed by the GST is more reliable than that inferred from the raw data. In addition, the obtained Hölder functions almost reflect the lithological discontinuities encountered by the well. To conclude, the GST is a powerful tool for enhancing the estimation of the Hölderian regularity of signals. 2021-08-02 Fractal Fract, Vol. 5, Pages 86: A Grey System Approach for Estimating the Hölderian Regularity with an Application to Algerian Well Log Data

Fractal and Fractional doi: 10.3390/fractalfract5030086

Authors: Said Gaci Orietta Nicolis

The Hölderian regularity is an important mathematical feature of a signal, connected with the physical nature of the measured parameter. Many algorithms have been proposed in literature for estimating the local Hölder exponent value, but all of them lead to biased estimates. This paper attempts to apply the grey system theory (GST) on the raw signal for improving the accuracy of Hölderian regularity estimation. First, synthetic logs data are generated by the successive random additions (SRA) method with different types of Hölder functions. The application on these simulated signals shows that the Hölder functions estimated by the GST are more precise than those derived from the raw data. Additionally, noisy signals are considered for the same experiment, and more accurate regularity is obtained using signals processed using GST. Second, the proposed technique is implemented on well log data measured at an Algerian exploration borehole. It is demonstrated that the regularity determined from the well logs analyzed by the GST is more reliable than that inferred from the raw data. In addition, the obtained Hölder functions almost reflect the lithological discontinuities encountered by the well. To conclude, the GST is a powerful tool for enhancing the estimation of the Hölderian regularity of signals.

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A Grey System Approach for Estimating the Hölderian Regularity with an Application to Algerian Well Log Data Said Gaci Orietta Nicolis doi: 10.3390/fractalfract5030086 Fractal and Fractional 2021-08-02 Fractal and Fractional 2021-08-02 5 3 Article 86 10.3390/fractalfract5030086 https://www.mdpi.com/2504-3110/5/3/86
Fractal Fract, Vol. 5, Pages 85: A Numerical Study of Nonlinear Fractional Order Partial Integro-Differential Equation with a Weakly Singular Kernel https://www.mdpi.com/2504-3110/5/3/85 Fractional differential equations can present the physical pathways with the storage and inherited properties due to the memory factor of fractional order. The purpose of this work is to interpret the collocation approach for tackling the fractional partial integro-differential equation (FPIDE) by employing the extended cubic B-spline (ECBS). To determine the time approximation, we utilize the Caputo approach. The stability and convergence analysis have also been analyzed. The efficiency and reliability of the suggested technique are demonstrated by two numerical applications, which support the theoretical results and the effectiveness of the implemented algorithm. 2021-08-02 Fractal Fract, Vol. 5, Pages 85: A Numerical Study of Nonlinear Fractional Order Partial Integro-Differential Equation with a Weakly Singular Kernel

Fractal and Fractional doi: 10.3390/fractalfract5030085

Authors: Tayyaba Akram Zeeshan Ali Faranak Rabiei Kamal Shah Poom Kumam

Fractional differential equations can present the physical pathways with the storage and inherited properties due to the memory factor of fractional order. The purpose of this work is to interpret the collocation approach for tackling the fractional partial integro-differential equation (FPIDE) by employing the extended cubic B-spline (ECBS). To determine the time approximation, we utilize the Caputo approach. The stability and convergence analysis have also been analyzed. The efficiency and reliability of the suggested technique are demonstrated by two numerical applications, which support the theoretical results and the effectiveness of the implemented algorithm.

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A Numerical Study of Nonlinear Fractional Order Partial Integro-Differential Equation with a Weakly Singular Kernel Tayyaba Akram Zeeshan Ali Faranak Rabiei Kamal Shah Poom Kumam doi: 10.3390/fractalfract5030085 Fractal and Fractional 2021-08-02 Fractal and Fractional 2021-08-02 5 3 Article 85 10.3390/fractalfract5030085 https://www.mdpi.com/2504-3110/5/3/85
Fractal Fract, Vol. 5, Pages 84: A Modified Leslie–Gower Model Incorporating Beddington–DeAngelis Functional Response, Double Allee Effect and Memory Effect https://www.mdpi.com/2504-3110/5/3/84 In this paper, a modified Leslie–Gower predator-prey model with Beddington–DeAngelis functional response and double Allee effect in the growth rate of a predator population is proposed. In order to consider memory effect on the proposed model, we employ the Caputo fractional-order derivative. We investigate the dynamic behaviors of the proposed model for both strong and weak Allee effect cases. The existence, uniqueness, non-negativity, and boundedness of the solution are discussed. Then, we determine the existing condition and local stability analysis of all possible equilibrium points. Necessary conditions for the existence of the Hopf bifurcation driven by the order of the fractional derivative are also determined analytically. Furthermore, by choosing a suitable Lyapunov function, we derive the sufficient conditions to ensure the global asymptotic stability for the predator extinction point for the strong Allee effect case as well as for the prey extinction point and the interior point for the weak Allee effect case. Finally, numerical simulations are shown to confirm the theoretical results and can explore more dynamical behaviors of the system, such as the bi-stability and forward bifurcation. 2021-08-01 Fractal Fract, Vol. 5, Pages 84: A Modified Leslie–Gower Model Incorporating Beddington–DeAngelis Functional Response, Double Allee Effect and Memory Effect

Fractal and Fractional doi: 10.3390/fractalfract5030084

Authors: Emli Rahmi Isnani Darti Agus Suryanto Trisilowati

In this paper, a modified Leslie–Gower predator-prey model with Beddington–DeAngelis functional response and double Allee effect in the growth rate of a predator population is proposed. In order to consider memory effect on the proposed model, we employ the Caputo fractional-order derivative. We investigate the dynamic behaviors of the proposed model for both strong and weak Allee effect cases. The existence, uniqueness, non-negativity, and boundedness of the solution are discussed. Then, we determine the existing condition and local stability analysis of all possible equilibrium points. Necessary conditions for the existence of the Hopf bifurcation driven by the order of the fractional derivative are also determined analytically. Furthermore, by choosing a suitable Lyapunov function, we derive the sufficient conditions to ensure the global asymptotic stability for the predator extinction point for the strong Allee effect case as well as for the prey extinction point and the interior point for the weak Allee effect case. Finally, numerical simulations are shown to confirm the theoretical results and can explore more dynamical behaviors of the system, such as the bi-stability and forward bifurcation.

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A Modified Leslie–Gower Model Incorporating Beddington–DeAngelis Functional Response, Double Allee Effect and Memory Effect Emli Rahmi Isnani Darti Agus Suryanto Trisilowati doi: 10.3390/fractalfract5030084 Fractal and Fractional 2021-08-01 Fractal and Fractional 2021-08-01 5 3 Article 84 10.3390/fractalfract5030084 https://www.mdpi.com/2504-3110/5/3/84
Fractal Fract, Vol. 5, Pages 83: Fractional Differential Equation-Based Instantaneous Frequency Estimation for Signal Reconstruction https://www.mdpi.com/2504-3110/5/3/83 In this paper, we propose a fractional differential equation (FDE)-based approach for the estimation of instantaneous frequencies for windowed signals as a part of signal reconstruction. This approach is based on modeling bandpass filter results around the peaks of a windowed signal as fractional differential equations and linking differ-integrator parameters, thereby determining the long-range dependence on estimated instantaneous frequencies. We investigated the performance of the proposed approach with two evaluation measures and compared it to a benchmark noniterative signal reconstruction method (SPSI). The comparison was provided with different overlap parameters to investigate the performance of the proposed model concerning resolution. An additional comparison was provided by applying the proposed method and benchmark method outputs to iterative signal reconstruction algorithms. The proposed FDE method received better evaluation results in high resolution for the noniterative case and comparable results with SPSI with an increasing iteration number of iterative methods, regardless of the overlap parameter. 2021-07-30 Fractal Fract, Vol. 5, Pages 83: Fractional Differential Equation-Based Instantaneous Frequency Estimation for Signal Reconstruction

Fractal and Fractional doi: 10.3390/fractalfract5030083

Authors: Bilgi Görkem Yazgaç Mürvet Kırcı

In this paper, we propose a fractional differential equation (FDE)-based approach for the estimation of instantaneous frequencies for windowed signals as a part of signal reconstruction. This approach is based on modeling bandpass filter results around the peaks of a windowed signal as fractional differential equations and linking differ-integrator parameters, thereby determining the long-range dependence on estimated instantaneous frequencies. We investigated the performance of the proposed approach with two evaluation measures and compared it to a benchmark noniterative signal reconstruction method (SPSI). The comparison was provided with different overlap parameters to investigate the performance of the proposed model concerning resolution. An additional comparison was provided by applying the proposed method and benchmark method outputs to iterative signal reconstruction algorithms. The proposed FDE method received better evaluation results in high resolution for the noniterative case and comparable results with SPSI with an increasing iteration number of iterative methods, regardless of the overlap parameter.

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Fractional Differential Equation-Based Instantaneous Frequency Estimation for Signal Reconstruction Bilgi Görkem Yazgaç Mürvet Kırcı doi: 10.3390/fractalfract5030083 Fractal and Fractional 2021-07-30 Fractal and Fractional 2021-07-30 5 3 Article 83 10.3390/fractalfract5030083 https://www.mdpi.com/2504-3110/5/3/83
Fractal Fract, Vol. 5, Pages 82: On the Nonlinear Integro-Differential Equations https://www.mdpi.com/2504-3110/5/3/82 The goal of this paper is to study the uniqueness of solutions of several nonlinear Liouville–Caputo integro-differential equations with variable coefficients and initial conditions, as well as an associated coupled system in Banach spaces. The results derived are new and based on Banach’s contractive principle, the multivariate Mittag–Leffler function and Babenko’s approach. We also provide a few examples to demonstrate the use of our main theorems by convolutions and the gamma function. 2021-07-30 Fractal Fract, Vol. 5, Pages 82: On the Nonlinear Integro-Differential Equations

Fractal and Fractional doi: 10.3390/fractalfract5030082

Authors: Chenkuan Li Joshua Beaudin

The goal of this paper is to study the uniqueness of solutions of several nonlinear Liouville–Caputo integro-differential equations with variable coefficients and initial conditions, as well as an associated coupled system in Banach spaces. The results derived are new and based on Banach’s contractive principle, the multivariate Mittag–Leffler function and Babenko’s approach. We also provide a few examples to demonstrate the use of our main theorems by convolutions and the gamma function.

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On the Nonlinear Integro-Differential Equations Chenkuan Li Joshua Beaudin doi: 10.3390/fractalfract5030082 Fractal and Fractional 2021-07-30 Fractal and Fractional 2021-07-30 5 3 Article 82 10.3390/fractalfract5030082 https://www.mdpi.com/2504-3110/5/3/82
Fractal Fract, Vol. 5, Pages 81: Monotone Iterative Method for ψ-Caputo Fractional Differential Equation with Nonlinear Boundary Conditions https://www.mdpi.com/2504-3110/5/3/81 The main contribution of this paper is to prove the existence of extremal solutions for a novel class of ψ-Caputo fractional differential equation with nonlinear boundary conditions. For this purpose, we utilize the well-known monotone iterative technique together with the method of upper and lower solutions. Finally, we provide an example along with graphical representations to confirm the validity of our main results. 2021-07-29 Fractal Fract, Vol. 5, Pages 81: Monotone Iterative Method for ψ-Caputo Fractional Differential Equation with Nonlinear Boundary Conditions

Fractal and Fractional doi: 10.3390/fractalfract5030081

Authors: Zidane Baitiche Choukri Derbazi Jehad Alzabut Mohammad Esmael Samei Mohammed K. A. Kaabar Zailan Siri

The main contribution of this paper is to prove the existence of extremal solutions for a novel class of ψ-Caputo fractional differential equation with nonlinear boundary conditions. For this purpose, we utilize the well-known monotone iterative technique together with the method of upper and lower solutions. Finally, we provide an example along with graphical representations to confirm the validity of our main results.

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Monotone Iterative Method for ψ-Caputo Fractional Differential Equation with Nonlinear Boundary Conditions Zidane Baitiche Choukri Derbazi Jehad Alzabut Mohammad Esmael Samei Mohammed K. A. Kaabar Zailan Siri doi: 10.3390/fractalfract5030081 Fractal and Fractional 2021-07-29 Fractal and Fractional 2021-07-29 5 3 Article 81 10.3390/fractalfract5030081 https://www.mdpi.com/2504-3110/5/3/81
Fractal Fract, Vol. 5, Pages 80: Fractional Integral Inequalities for Exponentially Nonconvex Functions and Their Applications https://www.mdpi.com/2504-3110/5/3/80 In this paper, the authors define a new generic class of functions involving a certain modified Fox–Wright function. A useful identity using fractional integrals and this modified Fox–Wright function with two parameters is also found. Applying this as an auxiliary result, we establish some Hermite–Hadamard-type integral inequalities by using the above-mentioned class of functions. Some special cases are derived with relevant details. Moreover, in order to show the efficiency of our main results, an application for error estimation is obtained as well. 2021-07-29 Fractal Fract, Vol. 5, Pages 80: Fractional Integral Inequalities for Exponentially Nonconvex Functions and Their Applications

Fractal and Fractional doi: 10.3390/fractalfract5030080

Authors: Hari Mohan Srivastava Artion Kashuri Pshtiwan Othman Mohammed Dumitru Baleanu Y. S. Hamed

In this paper, the authors define a new generic class of functions involving a certain modified Fox–Wright function. A useful identity using fractional integrals and this modified Fox–Wright function with two parameters is also found. Applying this as an auxiliary result, we establish some Hermite–Hadamard-type integral inequalities by using the above-mentioned class of functions. Some special cases are derived with relevant details. Moreover, in order to show the efficiency of our main results, an application for error estimation is obtained as well.

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Fractional Integral Inequalities for Exponentially Nonconvex Functions and Their Applications Hari Mohan Srivastava Artion Kashuri Pshtiwan Othman Mohammed Dumitru Baleanu Y. S. Hamed doi: 10.3390/fractalfract5030080 Fractal and Fractional 2021-07-29 Fractal and Fractional 2021-07-29 5 3 Article 80 10.3390/fractalfract5030080 https://www.mdpi.com/2504-3110/5/3/80
Fractal Fract, Vol. 5, Pages 79: Influence of Different Alkali Sulfates on the Shrinkage, Hydration, Pore Structure, Fractal Dimension and Microstructure of Low-Heat Portland Cement, Medium-Heat Portland Cement and Ordinary Portland Cement https://www.mdpi.com/2504-3110/5/3/79 In cement-based materials, alkalis mainly exist in the form of different alkali sulfates. In this study, the impacts of different alkali sulfates on the shrinkage, hydration, pore structure, fractal dimension and microstructure of low-heat Portland cement (LHPC), medium-heat Portland cement (MHPC) and ordinary Portland cement (OPC) are investigated. The results indicate that alkali sulfates magnify the autogenous shrinkage and drying shrinkage of cement-based materials with different mineral compositions, which are mainly related to different pore structures and hydration processes. LHPC has the lowest shrinkage. Otherwise, the effect of alkali sulfates on the autogenous shrinkage is more profound than that of drying shrinkage. Compared with the pore size distribution, the fractal dimension can better characterize the shrinkage properties of cement-based materials. It is noted that the contribution of K2SO4 (K alkali) to the promotion effect of shrinkage on cement-based materials is more significant than that of Na2SO4 (Na alkali), which cannot be ignored. The microstructure investigation of different cement-based materials by means of nuclear magnetic resonance (NMR), mercury intrusion porosimetry (MIP) and scanning electron microscope (SEM) shows that this effect may be related to the different pore structures, crystal forms and morphologies of hydration products of cement-based materials. 2021-07-27 Fractal Fract, Vol. 5, Pages 79: Influence of Different Alkali Sulfates on the Shrinkage, Hydration, Pore Structure, Fractal Dimension and Microstructure of Low-Heat Portland Cement, Medium-Heat Portland Cement and Ordinary Portland Cement

Fractal and Fractional doi: 10.3390/fractalfract5030079

Authors: Yang Li Hui Zhang Minghui Huang Haibo Yin Ke Jiang Kaitao Xiao Shengwen Tang

In cement-based materials, alkalis mainly exist in the form of different alkali sulfates. In this study, the impacts of different alkali sulfates on the shrinkage, hydration, pore structure, fractal dimension and microstructure of low-heat Portland cement (LHPC), medium-heat Portland cement (MHPC) and ordinary Portland cement (OPC) are investigated. The results indicate that alkali sulfates magnify the autogenous shrinkage and drying shrinkage of cement-based materials with different mineral compositions, which are mainly related to different pore structures and hydration processes. LHPC has the lowest shrinkage. Otherwise, the effect of alkali sulfates on the autogenous shrinkage is more profound than that of drying shrinkage. Compared with the pore size distribution, the fractal dimension can better characterize the shrinkage properties of cement-based materials. It is noted that the contribution of K2SO4 (K alkali) to the promotion effect of shrinkage on cement-based materials is more significant than that of Na2SO4 (Na alkali), which cannot be ignored. The microstructure investigation of different cement-based materials by means of nuclear magnetic resonance (NMR), mercury intrusion porosimetry (MIP) and scanning electron microscope (SEM) shows that this effect may be related to the different pore structures, crystal forms and morphologies of hydration products of cement-based materials.

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Influence of Different Alkali Sulfates on the Shrinkage, Hydration, Pore Structure, Fractal Dimension and Microstructure of Low-Heat Portland Cement, Medium-Heat Portland Cement and Ordinary Portland Cement Yang Li Hui Zhang Minghui Huang Haibo Yin Ke Jiang Kaitao Xiao Shengwen Tang doi: 10.3390/fractalfract5030079 Fractal and Fractional 2021-07-27 Fractal and Fractional 2021-07-27 5 3 Article 79 10.3390/fractalfract5030079 https://www.mdpi.com/2504-3110/5/3/79
Fractal Fract, Vol. 5, Pages 78: Impulsive Fractional Cohen-Grossberg Neural Networks: Almost Periodicity Analysis https://www.mdpi.com/2504-3110/5/3/78 In this paper, a fractional-order Cohen–Grossberg-type neural network with Caputo fractional derivatives is investigated. The notion of almost periodicity is adapted to the impulsive generalization of the model. General types of impulsive perturbations not necessarily at fixed moments are considered. Criteria for the existence and uniqueness of almost periodic waves are proposed. Furthermore, the global perfect Mittag–Leffler stability notion for the almost periodic solution is defined and studied. In addition, a robust global perfect Mittag–Leffler stability analysis is proposed. Lyapunov-type functions and fractional inequalities are applied in the proof. Since the type of Cohen–Grossberg neural networks generalizes several basic neural network models, this research contributes to the development of the investigations on numerous fractional neural network models. 2021-07-27 Fractal Fract, Vol. 5, Pages 78: Impulsive Fractional Cohen-Grossberg Neural Networks: Almost Periodicity Analysis

Fractal and Fractional doi: 10.3390/fractalfract5030078

Authors: Ivanka Stamova Sotir Sotirov Evdokia Sotirova Gani Stamov

In this paper, a fractional-order Cohen–Grossberg-type neural network with Caputo fractional derivatives is investigated. The notion of almost periodicity is adapted to the impulsive generalization of the model. General types of impulsive perturbations not necessarily at fixed moments are considered. Criteria for the existence and uniqueness of almost periodic waves are proposed. Furthermore, the global perfect Mittag–Leffler stability notion for the almost periodic solution is defined and studied. In addition, a robust global perfect Mittag–Leffler stability analysis is proposed. Lyapunov-type functions and fractional inequalities are applied in the proof. Since the type of Cohen–Grossberg neural networks generalizes several basic neural network models, this research contributes to the development of the investigations on numerous fractional neural network models.

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Impulsive Fractional Cohen-Grossberg Neural Networks: Almost Periodicity Analysis Ivanka Stamova Sotir Sotirov Evdokia Sotirova Gani Stamov doi: 10.3390/fractalfract5030078 Fractal and Fractional 2021-07-27 Fractal and Fractional 2021-07-27 5 3 Article 78 10.3390/fractalfract5030078 https://www.mdpi.com/2504-3110/5/3/78
Fractal Fract, Vol. 5, Pages 77: On Solvability of the Sonin–Abel Equation in the Weighted Lebesgue Space https://www.mdpi.com/2504-3110/5/3/77 In this paper we present a method of studying a convolution operator under the Sonin conditions imposed on the kernel. The particular case of the Sonin kernel is a kernel of the fractional integral Riemman–Liouville operator, other various types of the Sonin kernels are a Bessel-type function, functions with power-logarithmic singularities at the origin e.t.c. We pay special attention to study kernels close to power type functions. The main our aim is to study the Sonin–Abel equation in the weighted Lebesgue space, the used method allows us to formulate a criterion of existence and uniqueness of the solution and classify a solution, due to the asymptotics of the Jacobi series coefficients of the right-hand side. 2021-07-26 Fractal Fract, Vol. 5, Pages 77: On Solvability of the Sonin–Abel Equation in the Weighted Lebesgue Space

Fractal and Fractional doi: 10.3390/fractalfract5030077

Authors: Maksim V. Kukushkin

In this paper we present a method of studying a convolution operator under the Sonin conditions imposed on the kernel. The particular case of the Sonin kernel is a kernel of the fractional integral Riemman–Liouville operator, other various types of the Sonin kernels are a Bessel-type function, functions with power-logarithmic singularities at the origin e.t.c. We pay special attention to study kernels close to power type functions. The main our aim is to study the Sonin–Abel equation in the weighted Lebesgue space, the used method allows us to formulate a criterion of existence and uniqueness of the solution and classify a solution, due to the asymptotics of the Jacobi series coefficients of the right-hand side.

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On Solvability of the Sonin–Abel Equation in the Weighted Lebesgue Space Maksim V. Kukushkin doi: 10.3390/fractalfract5030077 Fractal and Fractional 2021-07-26 Fractal and Fractional 2021-07-26 5 3 Article 77 10.3390/fractalfract5030077 https://www.mdpi.com/2504-3110/5/3/77
Fractal Fract, Vol. 5, Pages 76: A Nonlocal Fractional Peridynamic Diffusion Model https://www.mdpi.com/2504-3110/5/3/76 This paper proposes a nonlocal fractional peridynamic (FPD) model to characterize the nonlocality of physical processes or systems, based on analysis with the fractional derivative model (FDM) and the peridynamic (PD) model. The main idea is to use the fractional Euler–Lagrange formula to establish a peridynamic anomalous diffusion model, in which the classical exponential kernel function is replaced by using a power-law kernel function. Fractional Taylor series expansion was used to construct a fractional peridynamic differential operator method to complete the above model. To explore the properties of the FPD model, the FDM, the PD model and the FPD model are dissected via numerical analysis on a diffusion process in complex media. The FPD model provides a generalized model connecting a local model and a nonlocal model for physical systems. The fractional peridynamic differential operator (FPDDO) method provides a simple and efficient numerical method for solving fractional derivative equations. 2021-07-23 Fractal Fract, Vol. 5, Pages 76: A Nonlocal Fractional Peridynamic Diffusion Model

Fractal and Fractional doi: 10.3390/fractalfract5030076

Authors: Yuanyuan Wang HongGuang Sun Siyuan Fan Yan Gu Xiangnan Yu

This paper proposes a nonlocal fractional peridynamic (FPD) model to characterize the nonlocality of physical processes or systems, based on analysis with the fractional derivative model (FDM) and the peridynamic (PD) model. The main idea is to use the fractional Euler–Lagrange formula to establish a peridynamic anomalous diffusion model, in which the classical exponential kernel function is replaced by using a power-law kernel function. Fractional Taylor series expansion was used to construct a fractional peridynamic differential operator method to complete the above model. To explore the properties of the FPD model, the FDM, the PD model and the FPD model are dissected via numerical analysis on a diffusion process in complex media. The FPD model provides a generalized model connecting a local model and a nonlocal model for physical systems. The fractional peridynamic differential operator (FPDDO) method provides a simple and efficient numerical method for solving fractional derivative equations.

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A Nonlocal Fractional Peridynamic Diffusion Model Yuanyuan Wang HongGuang Sun Siyuan Fan Yan Gu Xiangnan Yu doi: 10.3390/fractalfract5030076 Fractal and Fractional 2021-07-23 Fractal and Fractional 2021-07-23 5 3 Article 76 10.3390/fractalfract5030076 https://www.mdpi.com/2504-3110/5/3/76
Fractal Fract, Vol. 5, Pages 75: Numerical Solution of Fractional Elliptic Problems with Inhomogeneous Boundary Conditions https://www.mdpi.com/2504-3110/5/3/75 The numerical solution of fractional-order elliptic problems is investigated in bounded domains. According to real-life situations, we assumed inhomogeneous boundary terms, while the underlying equations contain the full-space fractional Laplacian operator. The basis of the convergence analysis for a lower-order boundary element approximation is the theory for the corresponding continuous problem. In particular, we need continuity results for Riesz potentials and the fractional-order extension of the theory for boundary integral equations with the Laplacian operator. Accordingly, the convergence is stated in fractional-order Sobolev norms. The results were confirmed in a numerical experiment. 2021-07-21 Fractal Fract, Vol. 5, Pages 75: Numerical Solution of Fractional Elliptic Problems with Inhomogeneous Boundary Conditions

Fractal and Fractional doi: 10.3390/fractalfract5030075

Authors: Gábor Maros Ferenc Izsák

The numerical solution of fractional-order elliptic problems is investigated in bounded domains. According to real-life situations, we assumed inhomogeneous boundary terms, while the underlying equations contain the full-space fractional Laplacian operator. The basis of the convergence analysis for a lower-order boundary element approximation is the theory for the corresponding continuous problem. In particular, we need continuity results for Riesz potentials and the fractional-order extension of the theory for boundary integral equations with the Laplacian operator. Accordingly, the convergence is stated in fractional-order Sobolev norms. The results were confirmed in a numerical experiment.

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Numerical Solution of Fractional Elliptic Problems with Inhomogeneous Boundary Conditions Gábor Maros Ferenc Izsák doi: 10.3390/fractalfract5030075 Fractal and Fractional 2021-07-21 Fractal and Fractional 2021-07-21 5 3 Article 75 10.3390/fractalfract5030075 https://www.mdpi.com/2504-3110/5/3/75
Fractal Fract, Vol. 5, Pages 74: Modeling and Application of Fractional-Order Economic Growth Model with Time Delay https://www.mdpi.com/2504-3110/5/3/74 This paper proposes a fractional-order economic growth model with time delay based on the Solow model to describe the economic growth path and explore the underlying growth factors. It effectively captures memory characteristics in economic operations by adding a time lag to the capital stock. The proposed model is presented in the form of a fractional differential equations system, and the sufficient conditions for the local stability are obtained. In the simulation, the theoretical results are verified and the sensitivity analysis is performed on individual parameters. Based on the proposed model, we predict China’s GDP in the next thirty years through optimization and find medium-to-high-speed growth in the short term. Furthermore, the application results indicate that China is facing the disappearance of demographic dividend and the deceleration of capital accumulation. Therefore, it is urgent for China to increase the total factor productivity (TFP) and transform its economic growth into a trajectory dependent on TFP growth. 2021-07-21 Fractal Fract, Vol. 5, Pages 74: Modeling and Application of Fractional-Order Economic Growth Model with Time Delay

Fractal and Fractional doi: 10.3390/fractalfract5030074

Authors: Ziyi Lin Hu Wang

This paper proposes a fractional-order economic growth model with time delay based on the Solow model to describe the economic growth path and explore the underlying growth factors. It effectively captures memory characteristics in economic operations by adding a time lag to the capital stock. The proposed model is presented in the form of a fractional differential equations system, and the sufficient conditions for the local stability are obtained. In the simulation, the theoretical results are verified and the sensitivity analysis is performed on individual parameters. Based on the proposed model, we predict China’s GDP in the next thirty years through optimization and find medium-to-high-speed growth in the short term. Furthermore, the application results indicate that China is facing the disappearance of demographic dividend and the deceleration of capital accumulation. Therefore, it is urgent for China to increase the total factor productivity (TFP) and transform its economic growth into a trajectory dependent on TFP growth.

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Modeling and Application of Fractional-Order Economic Growth Model with Time Delay Ziyi Lin Hu Wang doi: 10.3390/fractalfract5030074 Fractal and Fractional 2021-07-21 Fractal and Fractional 2021-07-21 5 3 Article 74 10.3390/fractalfract5030074 https://www.mdpi.com/2504-3110/5/3/74
Fractal Fract, Vol. 5, Pages 73: Visualization of Mandelbrot and Julia Sets of Möbius Transformations https://www.mdpi.com/2504-3110/5/3/73 This work reports on a study of the Mandelbrot set and Julia set for a generalization of the well-explored function η(z)=z2+λ. The generalization consists of composing with a fixed Möbius transformation at each iteration step. In particular, affine and inverse Möbius transformations are explored. This work offers a new way of visualizing the Mandelbrot and filled-in Julia sets. An interesting and unexpected appearance of hyperbolic triangles occurs in the structure of the Mandelbrot sets for the case of inverse Möbius transforms. Several lemmas and theorems associated with these types of fractal sets are presented. 2021-07-17 Fractal Fract, Vol. 5, Pages 73: Visualization of Mandelbrot and Julia Sets of Möbius Transformations

Fractal and Fractional doi: 10.3390/fractalfract5030073

Authors: Leah K. Mork Darin J. Ulness

This work reports on a study of the Mandelbrot set and Julia set for a generalization of the well-explored function η(z)=z2+λ. The generalization consists of composing with a fixed Möbius transformation at each iteration step. In particular, affine and inverse Möbius transformations are explored. This work offers a new way of visualizing the Mandelbrot and filled-in Julia sets. An interesting and unexpected appearance of hyperbolic triangles occurs in the structure of the Mandelbrot sets for the case of inverse Möbius transforms. Several lemmas and theorems associated with these types of fractal sets are presented.

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Visualization of Mandelbrot and Julia Sets of Möbius Transformations Leah K. Mork Darin J. Ulness doi: 10.3390/fractalfract5030073 Fractal and Fractional 2021-07-17 Fractal and Fractional 2021-07-17 5 3 Article 73 10.3390/fractalfract5030073 https://www.mdpi.com/2504-3110/5/3/73
Fractal Fract, Vol. 5, Pages 72: Lévy Processes Linked to the Lower-Incomplete Gamma Function https://www.mdpi.com/2504-3110/5/3/72 We start by defining a subordinator by means of the lower-incomplete gamma function. This can be considered as an approximation of the stable subordinator, easier to be handled in view of its finite activity. A tempered version is also considered in order to overcome the drawback of infinite moments. Then, we study Lévy processes that are time-changed by these subordinators with particular attention to the Brownian case. An approximation of the fractional derivative (as well as of the fractional power of operators) arises from the analysis of governing equations. Finally, we show that time-changing the fractional Brownian motion produces a model of anomalous diffusion, which exhibits a sub-diffusive behavior. 2021-07-17 Fractal Fract, Vol. 5, Pages 72: Lévy Processes Linked to the Lower-Incomplete Gamma Function

Fractal and Fractional doi: 10.3390/fractalfract5030072

Authors: Luisa Beghin Costantino Ricciuti

We start by defining a subordinator by means of the lower-incomplete gamma function. This can be considered as an approximation of the stable subordinator, easier to be handled in view of its finite activity. A tempered version is also considered in order to overcome the drawback of infinite moments. Then, we study Lévy processes that are time-changed by these subordinators with particular attention to the Brownian case. An approximation of the fractional derivative (as well as of the fractional power of operators) arises from the analysis of governing equations. Finally, we show that time-changing the fractional Brownian motion produces a model of anomalous diffusion, which exhibits a sub-diffusive behavior.

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Lévy Processes Linked to the Lower-Incomplete Gamma Function Luisa Beghin Costantino Ricciuti doi: 10.3390/fractalfract5030072 Fractal and Fractional 2021-07-17 Fractal and Fractional 2021-07-17 5 3 Article 72 10.3390/fractalfract5030072 https://www.mdpi.com/2504-3110/5/3/72
Fractal Fract, Vol. 5, Pages 71: Approximation of Space-Time Fractional Equations https://www.mdpi.com/2504-3110/5/3/71 The aim of this paper is to provide approximation results for space-time non-local equations with general non-local (and fractional) operators in space and time. We consider a general Markov process time changed with general subordinators or inverses to general subordinators. Our analysis is based on Bernstein symbols and Dirichlet forms, where the symbols characterize the time changes, and the Dirichlet forms characterize the Markov processes. 2021-07-17 Fractal Fract, Vol. 5, Pages 71: Approximation of Space-Time Fractional Equations

Fractal and Fractional doi: 10.3390/fractalfract5030071

Authors: Raffaela Capitanelli Mirko D’Ovidio

The aim of this paper is to provide approximation results for space-time non-local equations with general non-local (and fractional) operators in space and time. We consider a general Markov process time changed with general subordinators or inverses to general subordinators. Our analysis is based on Bernstein symbols and Dirichlet forms, where the symbols characterize the time changes, and the Dirichlet forms characterize the Markov processes.

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Approximation of Space-Time Fractional Equations Raffaela Capitanelli Mirko D’Ovidio doi: 10.3390/fractalfract5030071 Fractal and Fractional 2021-07-17 Fractal and Fractional 2021-07-17 5 3 Article 71 10.3390/fractalfract5030071 https://www.mdpi.com/2504-3110/5/3/71
Fractal Fract, Vol. 5, Pages 70: Solving a System of Fractional-Order Volterra-Fredholm Integro-Differential Equations with Weakly Singular Kernels via the Second Chebyshev Wavelets Method https://www.mdpi.com/2504-3110/5/3/70 In this paper, by means of the second Chebyshev wavelet and its operational matrix, we solve a system of fractional-order Volterra–Fredholm integro-differential equations with weakly singular kernels. We estimate the functions by using the wavelet basis and then obtain the approximate solutions from the algebraic system corresponding to the main system. Moreover, the implementation of our scheme is presented, and the error bounds of approximations are analyzed. Finally, we evaluate the efficiency of the method through a numerical example. 2021-07-14 Fractal Fract, Vol. 5, Pages 70: Solving a System of Fractional-Order Volterra-Fredholm Integro-Differential Equations with Weakly Singular Kernels via the Second Chebyshev Wavelets Method

Fractal and Fractional doi: 10.3390/fractalfract5030070

In this paper, by means of the second Chebyshev wavelet and its operational matrix, we solve a system of fractional-order Volterra–Fredholm integro-differential equations with weakly singular kernels. We estimate the functions by using the wavelet basis and then obtain the approximate solutions from the algebraic system corresponding to the main system. Moreover, the implementation of our scheme is presented, and the error bounds of approximations are analyzed. Finally, we evaluate the efficiency of the method through a numerical example.

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Solving a System of Fractional-Order Volterra-Fredholm Integro-Differential Equations with Weakly Singular Kernels via the Second Chebyshev Wavelets Method Esmail Bargamadi Leila Torkzadeh Kazem Nouri Amin Jajarmi doi: 10.3390/fractalfract5030070 Fractal and Fractional 2021-07-14 Fractal and Fractional 2021-07-14 5 3 Article 70 10.3390/fractalfract5030070 https://www.mdpi.com/2504-3110/5/3/70
Fractal Fract, Vol. 5, Pages 69: Iterated Functions Systems Composed of Generalized θ-Contractions https://www.mdpi.com/2504-3110/5/3/69 The theory of iterated function systems (IFSs) has been an active area of research on fractals and various types of self-similarity in nature. The basic theoretical work on IFSs has been proposed by Hutchinson. In this paper, we introduce a new generalization of Hutchinson IFS, namely generalized θ-contraction IFS, which is a finite collection of generalized θ-contraction functions T1,…,TN from finite Cartesian product space X×⋯×X into X, where (X,d) is a complete metric space. We prove the existence of attractor for this generalized IFS. We show that the Hutchinson operators for countable and multivalued θ-contraction IFSs are Picard. Finally, when the map θ is continuous, we show the relation between the code space and the attractor of θ-contraction IFS. 2021-07-14 Fractal Fract, Vol. 5, Pages 69: Iterated Functions Systems Composed of Generalized θ-Contractions

Fractal and Fractional doi: 10.3390/fractalfract5030069

Authors: Pasupathi Rajan María A. Navascués Arya Kumar Bedabrata Chand

The theory of iterated function systems (IFSs) has been an active area of research on fractals and various types of self-similarity in nature. The basic theoretical work on IFSs has been proposed by Hutchinson. In this paper, we introduce a new generalization of Hutchinson IFS, namely generalized θ-contraction IFS, which is a finite collection of generalized θ-contraction functions T1,…,TN from finite Cartesian product space X×⋯×X into X, where (X,d) is a complete metric space. We prove the existence of attractor for this generalized IFS. We show that the Hutchinson operators for countable and multivalued θ-contraction IFSs are Picard. Finally, when the map θ is continuous, we show the relation between the code space and the attractor of θ-contraction IFS.

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Iterated Functions Systems Composed of Generalized θ-Contractions Pasupathi Rajan María A. Navascués Arya Kumar Bedabrata Chand doi: 10.3390/fractalfract5030069 Fractal and Fractional 2021-07-14 Fractal and Fractional 2021-07-14 5 3 Article 69 10.3390/fractalfract5030069 https://www.mdpi.com/2504-3110/5/3/69
Fractal Fract, Vol. 5, Pages 68: Some New Results on Hermite–Hadamard–Mercer-Type Inequalities Using a General Family of Fractional Integral Operators https://www.mdpi.com/2504-3110/5/3/68 The aim of this article is to obtain new Hermite–Hadamard–Mercer-type inequalities using Raina’s fractional integral operators. We present some distinct and novel fractional Hermite–Hadamard–Mercer-type inequalities for the functions whose absolute value of derivatives are convex. Our main findings are generalizations and extensions of some results that existed in the literature. 2021-07-13 Fractal Fract, Vol. 5, Pages 68: Some New Results on Hermite–Hadamard–Mercer-Type Inequalities Using a General Family of Fractional Integral Operators

Fractal and Fractional doi: 10.3390/fractalfract5030068

Authors: Erhan Set Barış Çelik M. Emin Özdemir Mücahit Aslan

The aim of this article is to obtain new Hermite–Hadamard–Mercer-type inequalities using Raina’s fractional integral operators. We present some distinct and novel fractional Hermite–Hadamard–Mercer-type inequalities for the functions whose absolute value of derivatives are convex. Our main findings are generalizations and extensions of some results that existed in the literature.

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Some New Results on Hermite–Hadamard–Mercer-Type Inequalities Using a General Family of Fractional Integral Operators Erhan Set Barış Çelik M. Emin Özdemir Mücahit Aslan doi: 10.3390/fractalfract5030068 Fractal and Fractional 2021-07-13 Fractal and Fractional 2021-07-13 5 3 Article 68 10.3390/fractalfract5030068 https://www.mdpi.com/2504-3110/5/3/68
Fractal Fract, Vol. 5, Pages 67: Vibration Systems with Fractional-Order and Distributed-Order Derivatives Characterizing Viscoinertia https://www.mdpi.com/2504-3110/5/3/67 We considered forced harmonic vibration systems with the Liouville–Weyl fractional derivative where the order is between 1 and 2 and with a distributed-order derivative where the Liouville–Weyl fractional derivatives are integrated on the interval [1, 2] with respect to the order. Both types of derivatives enhance the viscosity and inertia of the system and contribute to damping and mass, respectively. Hence, such types of derivatives characterize the viscoinertia and represent an “inerter-pot” element. For such vibration systems, we derived the equivalent damping and equivalent mass and gave the equivalent integer-order vibration systems. Particularly, for the distributed-order vibration model where the weight function was taken as an exponential function that involved a parameter, we gave detailed analyses for the weight function, the damping contribution, and the mass contribution. Frequency–amplitude curves and frequency-phase curves were plotted for various coefficients and parameters for the comparison of the two types of vibration models. In the distributed-order vibration system, the weight function of the order enables us to simultaneously involve different orders, whilst the fractional-order model has a single order. Thus, the distributed-order vibration model is more general and flexible than the fractional vibration system. 2021-07-12 Fractal Fract, Vol. 5, Pages 67: Vibration Systems with Fractional-Order and Distributed-Order Derivatives Characterizing Viscoinertia

Fractal and Fractional doi: 10.3390/fractalfract5030067

Authors: Jun-Sheng Duan Di-Chen Hu

We considered forced harmonic vibration systems with the Liouville–Weyl fractional derivative where the order is between 1 and 2 and with a distributed-order derivative where the Liouville–Weyl fractional derivatives are integrated on the interval [1, 2] with respect to the order. Both types of derivatives enhance the viscosity and inertia of the system and contribute to damping and mass, respectively. Hence, such types of derivatives characterize the viscoinertia and represent an “inerter-pot” element. For such vibration systems, we derived the equivalent damping and equivalent mass and gave the equivalent integer-order vibration systems. Particularly, for the distributed-order vibration model where the weight function was taken as an exponential function that involved a parameter, we gave detailed analyses for the weight function, the damping contribution, and the mass contribution. Frequency–amplitude curves and frequency-phase curves were plotted for various coefficients and parameters for the comparison of the two types of vibration models. In the distributed-order vibration system, the weight function of the order enables us to simultaneously involve different orders, whilst the fractional-order model has a single order. Thus, the distributed-order vibration model is more general and flexible than the fractional vibration system.

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Vibration Systems with Fractional-Order and Distributed-Order Derivatives Characterizing Viscoinertia Jun-Sheng Duan Di-Chen Hu doi: 10.3390/fractalfract5030067 Fractal and Fractional 2021-07-12 Fractal and Fractional 2021-07-12 5 3 Article 67 10.3390/fractalfract5030067 https://www.mdpi.com/2504-3110/5/3/67
Fractal Fract, Vol. 5, Pages 66: Existence, Uniqueness, and Eq–Ulam-Type Stability of Fuzzy Fractional Differential Equation https://www.mdpi.com/2504-3110/5/3/66 This paper concerns with the existence and uniqueness of the Cauchy problem for a system of fuzzy fractional differential equation with Caputo derivative of order q∈(1,2], 0cD0+qu(t)=λu(t)⊕f(t,u(t))⊕B(t)C(t),t∈[0,T] with initial conditions u(0)=u0,u′(0)=u1. Moreover, by using direct analytic methods, the Eq–Ulam-type results are also presented. In addition, several examples are given which show the applicability of fuzzy fractional differential equations. 2021-07-11 Fractal Fract, Vol. 5, Pages 66: Existence, Uniqueness, and Eq–Ulam-Type Stability of Fuzzy Fractional Differential Equation

Fractal and Fractional doi: 10.3390/fractalfract5030066

Authors: Azmat Ullah Khan Niazi Jiawei He Ramsha Shafqat Bilal Ahmed

This paper concerns with the existence and uniqueness of the Cauchy problem for a system of fuzzy fractional differential equation with Caputo derivative of order q∈(1,2], 0cD0+qu(t)=λu(t)⊕f(t,u(t))⊕B(t)C(t),t∈[0,T] with initial conditions u(0)=u0,u′(0)=u1. Moreover, by using direct analytic methods, the Eq–Ulam-type results are also presented. In addition, several examples are given which show the applicability of fuzzy fractional differential equations.

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Existence, Uniqueness, and Eq–Ulam-Type Stability of Fuzzy Fractional Differential Equation Azmat Ullah Khan Niazi Jiawei He Ramsha Shafqat Bilal Ahmed doi: 10.3390/fractalfract5030066 Fractal and Fractional 2021-07-11 Fractal and Fractional 2021-07-11 5 3 Article 66 10.3390/fractalfract5030066 https://www.mdpi.com/2504-3110/5/3/66
Fractal Fract, Vol. 5, Pages 65: Adsorption on Fractal Surfaces: A Non Linear Modeling Approach of a Fractional Behavior https://www.mdpi.com/2504-3110/5/3/65 This article deals with the random sequential adsorption (RSA) of 2D disks of the same size on fractal surfaces with a Hausdorff dimension 1&amp;lt;d&amp;lt;2. According to the literature and confirmed by numerical simulations in the paper, the high coverage regime exhibits fractional dynamics, i.e., dynamics in t−1/d where d is the fractal dimension of the surface. The main contribution this paper is that it proposes to capture this behavior with a particular class of nonlinear model: a driftless control affine model. 2021-07-10 Fractal Fract, Vol. 5, Pages 65: Adsorption on Fractal Surfaces: A Non Linear Modeling Approach of a Fractional Behavior

Fractal and Fractional doi: 10.3390/fractalfract5030065

Authors: Vincent Tartaglione Jocelyn Sabatier Christophe Farges

This article deals with the random sequential adsorption (RSA) of 2D disks of the same size on fractal surfaces with a Hausdorff dimension 1&amp;lt;d&amp;lt;2. According to the literature and confirmed by numerical simulations in the paper, the high coverage regime exhibits fractional dynamics, i.e., dynamics in t−1/d where d is the fractal dimension of the surface. The main contribution this paper is that it proposes to capture this behavior with a particular class of nonlinear model: a driftless control affine model.

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Adsorption on Fractal Surfaces: A Non Linear Modeling Approach of a Fractional Behavior Vincent Tartaglione Jocelyn Sabatier Christophe Farges doi: 10.3390/fractalfract5030065 Fractal and Fractional 2021-07-10 Fractal and Fractional 2021-07-10 5 3 Article 65 10.3390/fractalfract5030065 https://www.mdpi.com/2504-3110/5/3/65
Fractal Fract, Vol. 5, Pages 64: Numerical Solution of Nonlinear Fractional Diffusion Equation in Framework of the Yang–Abdel–Cattani Derivative Operator https://www.mdpi.com/2504-3110/5/3/64 In this manuscript, the time-fractional diffusion equation in the framework of the Yang–Abdel–Cattani derivative operator is taken into account. A detailed proof for the existence, as well as the uniqueness of the solution of the time-fractional diffusion equation, in the sense of YAC derivative operator, is explained, and, using the method of α-HATM, we find the analytical solution of the time-fractional diffusion equation. Three cases are considered to exhibit the convergence and fidelity of the aforementioned α-HATM. The analytical solutions obtained for the diffusion equation using the Yang–Abdel–Cattani derivative operator are compared with the analytical solutions obtained using the Riemann–Liouville (RL) derivative operator for the fractional order γ=0.99 (nearby 1) and with the exact solution at different values of t to verify the efficiency of the YAC derivative operator. 2021-07-02 Fractal Fract, Vol. 5, Pages 64: Numerical Solution of Nonlinear Fractional Diffusion Equation in Framework of the Yang–Abdel–Cattani Derivative Operator

Fractal and Fractional doi: 10.3390/fractalfract5030064

Authors: Igor V. Malyk Mykola Gorbatenko Arun Chaudhary Shivani Sharma Ravi Shanker Dubey

In this manuscript, the time-fractional diffusion equation in the framework of the Yang–Abdel–Cattani derivative operator is taken into account. A detailed proof for the existence, as well as the uniqueness of the solution of the time-fractional diffusion equation, in the sense of YAC derivative operator, is explained, and, using the method of α-HATM, we find the analytical solution of the time-fractional diffusion equation. Three cases are considered to exhibit the convergence and fidelity of the aforementioned α-HATM. The analytical solutions obtained for the diffusion equation using the Yang–Abdel–Cattani derivative operator are compared with the analytical solutions obtained using the Riemann–Liouville (RL) derivative operator for the fractional order γ=0.99 (nearby 1) and with the exact solution at different values of t to verify the efficiency of the YAC derivative operator.

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Numerical Solution of Nonlinear Fractional Diffusion Equation in Framework of the Yang–Abdel–Cattani Derivative Operator Igor V. Malyk Mykola Gorbatenko Arun Chaudhary Shivani Sharma Ravi Shanker Dubey doi: 10.3390/fractalfract5030064 Fractal and Fractional 2021-07-02 Fractal and Fractional 2021-07-02 5 3 Article 64 10.3390/fractalfract5030064 https://www.mdpi.com/2504-3110/5/3/64
Fractal Fract, Vol. 5, Pages 63: An Inverse Source Problem for the Generalized Subdiffusion Equation with Nonclassical Boundary Conditions https://www.mdpi.com/2504-3110/5/3/63 An initial-boundary-value problem is considered for the one-dimensional diffusion equation with a general convolutional derivative in time and nonclassical boundary conditions. We are concerned with the inverse source problem of recovery of a space-dependent source term from given final time data. Generalized eigenfunction expansions are used with respect to a biorthogonal pair of bases. Existence, uniqueness and stability estimates in Sobolev spaces are established. 2021-06-30 Fractal Fract, Vol. 5, Pages 63: An Inverse Source Problem for the Generalized Subdiffusion Equation with Nonclassical Boundary Conditions

Fractal and Fractional doi: 10.3390/fractalfract5030063

Authors: Emilia Bazhlekova

An initial-boundary-value problem is considered for the one-dimensional diffusion equation with a general convolutional derivative in time and nonclassical boundary conditions. We are concerned with the inverse source problem of recovery of a space-dependent source term from given final time data. Generalized eigenfunction expansions are used with respect to a biorthogonal pair of bases. Existence, uniqueness and stability estimates in Sobolev spaces are established.

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An Inverse Source Problem for the Generalized Subdiffusion Equation with Nonclassical Boundary Conditions Emilia Bazhlekova doi: 10.3390/fractalfract5030063 Fractal and Fractional 2021-06-30 Fractal and Fractional 2021-06-30 5 3 Article 63 10.3390/fractalfract5030063 https://www.mdpi.com/2504-3110/5/3/63
Fractal Fract, Vol. 5, Pages 62: Alternating Inertial and Overrelaxed Algorithms for Distributed Generalized Nash Equilibrium Seeking in Multi-Player Games https://www.mdpi.com/2504-3110/5/3/62 This paper investigates the distributed computation issue of generalized Nash equilibrium (GNE) in a multi-player game with shared coupling constraints. Two kinds of relatively fast distributed algorithms are constructed with alternating inertia and overrelaxation in the partial-decision information setting. We prove their convergence to GNE with fixed step-sizes by resorting to the operator splitting technique under the assumptions of Lipschitz continuity of the extended pseudo-gradient mappings. Finally, one numerical simulation is given to illustrate the efficiency and performance of the algorithm. 2021-06-28 Fractal Fract, Vol. 5, Pages 62: Alternating Inertial and Overrelaxed Algorithms for Distributed Generalized Nash Equilibrium Seeking in Multi-Player Games

Fractal and Fractional doi: 10.3390/fractalfract5030062

Authors: Zhangcheng Feng Wenying Xu Jinde Cao

This paper investigates the distributed computation issue of generalized Nash equilibrium (GNE) in a multi-player game with shared coupling constraints. Two kinds of relatively fast distributed algorithms are constructed with alternating inertia and overrelaxation in the partial-decision information setting. We prove their convergence to GNE with fixed step-sizes by resorting to the operator splitting technique under the assumptions of Lipschitz continuity of the extended pseudo-gradient mappings. Finally, one numerical simulation is given to illustrate the efficiency and performance of the algorithm.

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Alternating Inertial and Overrelaxed Algorithms for Distributed Generalized Nash Equilibrium Seeking in Multi-Player Games Zhangcheng Feng Wenying Xu Jinde Cao doi: 10.3390/fractalfract5030062 Fractal and Fractional 2021-06-28 Fractal and Fractional 2021-06-28 5 3 Article 62 10.3390/fractalfract5030062 https://www.mdpi.com/2504-3110/5/3/62
Fractal Fract, Vol. 5, Pages 61: Reduced Multiplicative (BURA-MR) and Additive (BURA-AR) Best Uniform Rational Approximation Methods and Algorithms for Fractional Elliptic Equations https://www.mdpi.com/2504-3110/5/3/61 Numerical methods for spectral space-fractional elliptic equations are studied. The boundary value problem is defined in a bounded domain of general geometry, Ω⊂Rd, d∈{1,2,3}. Assuming that the finite difference method (FDM) or the finite element method (FEM) is applied for discretization in space, the approximate solution is described by the system of linear algebraic equations Aαu=f, α∈(0,1). Although matrix A∈RN×N is sparse, symmetric and positive definite (SPD), matrix Aα is dense. The recent achievements in the field are determined by methods that reduce the original non-local problem to solving k auxiliary linear systems with sparse SPD matrices that can be expressed as positive diagonal perturbations of A. The present study is in the spirit of the BURA method, based on the best uniform rational approximation rα,k(t) of degree k of tα in the interval [0,1]. The introduced additive BURA-AR and multiplicative BURA-MR methods follow the observation that the matrices of part of the auxiliary systems possess very different properties. As a result, solution methods with substantially improved computational complexity are developed. In this paper, we present new theoretical characterizations of the BURA parameters, which gives a theoretical justification for the new methods. The theoretical estimates are supported by a set of representative numerical tests. The new theoretical and experimental results raise the question of whether the almost optimal estimate of the computational complexity of the BURA method in the form O(Nlog2N) can be improved. 2021-06-28 Fractal Fract, Vol. 5, Pages 61: Reduced Multiplicative (BURA-MR) and Additive (BURA-AR) Best Uniform Rational Approximation Methods and Algorithms for Fractional Elliptic Equations

Fractal and Fractional doi: 10.3390/fractalfract5030061

Authors: Stanislav Harizanov Nikola Kosturski Ivan Lirkov Svetozar Margenov Yavor Vutov

Numerical methods for spectral space-fractional elliptic equations are studied. The boundary value problem is defined in a bounded domain of general geometry, Ω⊂Rd, d∈{1,2,3}. Assuming that the finite difference method (FDM) or the finite element method (FEM) is applied for discretization in space, the approximate solution is described by the system of linear algebraic equations Aαu=f, α∈(0,1). Although matrix A∈RN×N is sparse, symmetric and positive definite (SPD), matrix Aα is dense. The recent achievements in the field are determined by methods that reduce the original non-local problem to solving k auxiliary linear systems with sparse SPD matrices that can be expressed as positive diagonal perturbations of A. The present study is in the spirit of the BURA method, based on the best uniform rational approximation rα,k(t) of degree k of tα in the interval [0,1]. The introduced additive BURA-AR and multiplicative BURA-MR methods follow the observation that the matrices of part of the auxiliary systems possess very different properties. As a result, solution methods with substantially improved computational complexity are developed. In this paper, we present new theoretical characterizations of the BURA parameters, which gives a theoretical justification for the new methods. The theoretical estimates are supported by a set of representative numerical tests. The new theoretical and experimental results raise the question of whether the almost optimal estimate of the computational complexity of the BURA method in the form O(Nlog2N) can be improved.

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Reduced Multiplicative (BURA-MR) and Additive (BURA-AR) Best Uniform Rational Approximation Methods and Algorithms for Fractional Elliptic Equations Stanislav Harizanov Nikola Kosturski Ivan Lirkov Svetozar Margenov Yavor Vutov doi: 10.3390/fractalfract5030061 Fractal and Fractional 2021-06-28 Fractal and Fractional 2021-06-28 5 3 Article 61 10.3390/fractalfract5030061 https://www.mdpi.com/2504-3110/5/3/61
Fractal Fract, Vol. 5, Pages 60: On Iterative Methods for Solving Nonlinear Equations in Quantum Calculus https://www.mdpi.com/2504-3110/5/3/60 Quantum calculus (also known as the q-calculus) is a technique that is similar to traditional calculus, but focuses on the concept of deriving q-analogous results without the use of the limits. In this paper, we suggest and analyze some new q-iterative methods by using the q-analogue of the Taylor’s series and the coupled system technique. In the domain of q-calculus, we determine the convergence of our proposed q-algorithms. Numerical examples demonstrate that the new q-iterative methods can generate solutions to the nonlinear equations with acceptable accuracy. These newly established methods also exhibit predictability. Furthermore, an analogy is settled between the well known classical methods and our proposed q-Iterative methods. 2021-06-25 Fractal Fract, Vol. 5, Pages 60: On Iterative Methods for Solving Nonlinear Equations in Quantum Calculus

Fractal and Fractional doi: 10.3390/fractalfract5030060

Authors: Gul Sana Pshtiwan Othman Mohammed Dong Yun Shin Muhmmad Aslam Noor Mohammad Salem Oudat

Quantum calculus (also known as the q-calculus) is a technique that is similar to traditional calculus, but focuses on the concept of deriving q-analogous results without the use of the limits. In this paper, we suggest and analyze some new q-iterative methods by using the q-analogue of the Taylor’s series and the coupled system technique. In the domain of q-calculus, we determine the convergence of our proposed q-algorithms. Numerical examples demonstrate that the new q-iterative methods can generate solutions to the nonlinear equations with acceptable accuracy. These newly established methods also exhibit predictability. Furthermore, an analogy is settled between the well known classical methods and our proposed q-Iterative methods.

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On Iterative Methods for Solving Nonlinear Equations in Quantum Calculus Gul Sana Pshtiwan Othman Mohammed Dong Yun Shin Muhmmad Aslam Noor Mohammad Salem Oudat doi: 10.3390/fractalfract5030060 Fractal and Fractional 2021-06-25 Fractal and Fractional 2021-06-25 5 3 Article 60 10.3390/fractalfract5030060 https://www.mdpi.com/2504-3110/5/3/60
Fractal Fract, Vol. 5, Pages 59: Degenerate Derangement Polynomials and Numbers https://www.mdpi.com/2504-3110/5/3/59 In this paper, we consider a new type of degenerate derangement polynomial and number, which shall be called the degenerate derangement polynomials and numbers of the second kind. These concepts are motivated by Kim et al.’s work on degenerate derangement polynomials and numbers. We investigate some properties of these new degenerate derangement polynomials and numbers and explore their connections with the degenerate gamma distributions for the case λ∈(−1,0). In more detail, we derive their explicit expressions, recurrence relations, and some identities involving our degenerate derangement polynomials and numbers and other special polynomials and numbers, which include the fully degenerate Bell polynomials, the degenerate Fubini polynomials, and the degenerate Stirling numbers of the first and the second kinds. We also show that those polynomials and numbers are connected with the moments of some variants of the degenerate gamma distributions. Moreover, we compare the degenerate derangement polynomials and numbers of the second kind to those of Kim et al. 2021-06-22 Fractal Fract, Vol. 5, Pages 59: Degenerate Derangement Polynomials and Numbers

Fractal and Fractional doi: 10.3390/fractalfract5030059

Authors: Minyoung Ma Dongkyu Lim

In this paper, we consider a new type of degenerate derangement polynomial and number, which shall be called the degenerate derangement polynomials and numbers of the second kind. These concepts are motivated by Kim et al.’s work on degenerate derangement polynomials and numbers. We investigate some properties of these new degenerate derangement polynomials and numbers and explore their connections with the degenerate gamma distributions for the case λ∈(−1,0). In more detail, we derive their explicit expressions, recurrence relations, and some identities involving our degenerate derangement polynomials and numbers and other special polynomials and numbers, which include the fully degenerate Bell polynomials, the degenerate Fubini polynomials, and the degenerate Stirling numbers of the first and the second kinds. We also show that those polynomials and numbers are connected with the moments of some variants of the degenerate gamma distributions. Moreover, we compare the degenerate derangement polynomials and numbers of the second kind to those of Kim et al.

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Degenerate Derangement Polynomials and Numbers Minyoung Ma Dongkyu Lim doi: 10.3390/fractalfract5030059 Fractal and Fractional 2021-06-22 Fractal and Fractional 2021-06-22 5 3 Article 59 10.3390/fractalfract5030059 https://www.mdpi.com/2504-3110/5/3/59
Fractal Fract, Vol. 5, Pages 58: Inverse Problem for a Partial Differential Equation with Gerasimov–Caputo-Type Operator and Degeneration https://www.mdpi.com/2504-3110/5/2/58 In the three-dimensional open rectangular domain, the problem of the identification of the redefinition function for a partial differential equation with Gerasimov–Caputo-type fractional operator, degeneration, and integral form condition is considered in the case of the 0&amp;lt;α≤1 order. A positive parameter is present in the mixed derivatives. The solution of this fractional differential equation is studied in the class of regular functions. The Fourier series method is used, and a countable system of ordinary fractional differential equations with degeneration is obtained. The presentation for the redefinition function is obtained using a given additional condition. Using the Cauchy–Schwarz inequality and the Bessel inequality, the absolute and uniform convergence of the obtained Fourier series is proven. 2021-06-21 Fractal Fract, Vol. 5, Pages 58: Inverse Problem for a Partial Differential Equation with Gerasimov–Caputo-Type Operator and Degeneration

Fractal and Fractional doi: 10.3390/fractalfract5020058

Authors: Tursun K. Yuldashev Bakhtiyar J. Kadirkulov

In the three-dimensional open rectangular domain, the problem of the identification of the redefinition function for a partial differential equation with Gerasimov–Caputo-type fractional operator, degeneration, and integral form condition is considered in the case of the 0&amp;lt;α≤1 order. A positive parameter is present in the mixed derivatives. The solution of this fractional differential equation is studied in the class of regular functions. The Fourier series method is used, and a countable system of ordinary fractional differential equations with degeneration is obtained. The presentation for the redefinition function is obtained using a given additional condition. Using the Cauchy–Schwarz inequality and the Bessel inequality, the absolute and uniform convergence of the obtained Fourier series is proven.

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Inverse Problem for a Partial Differential Equation with Gerasimov–Caputo-Type Operator and Degeneration Tursun K. Yuldashev Bakhtiyar J. Kadirkulov doi: 10.3390/fractalfract5020058 Fractal and Fractional 2021-06-21 Fractal and Fractional 2021-06-21 5 2 Article 58 10.3390/fractalfract5020058 https://www.mdpi.com/2504-3110/5/2/58
Fractal Fract, Vol. 5, Pages 57: Solutions of Bernoulli Equations in the Fractional Setting https://www.mdpi.com/2504-3110/5/2/57 We present a general series representation formula for the local solution of the Bernoulli equation with Caputo fractional derivatives. We then focus on a generalization of the fractional logistic equation and present some related numerical simulations. 2021-06-17 Fractal Fract, Vol. 5, Pages 57: Solutions of Bernoulli Equations in the Fractional Setting

Fractal and Fractional doi: 10.3390/fractalfract5020057

Authors: Mirko D’Ovidio Anna Chiara Lai Paola Loreti

We present a general series representation formula for the local solution of the Bernoulli equation with Caputo fractional derivatives. We then focus on a generalization of the fractional logistic equation and present some related numerical simulations.

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Solutions of Bernoulli Equations in the Fractional Setting Mirko D’Ovidio Anna Chiara Lai Paola Loreti doi: 10.3390/fractalfract5020057 Fractal and Fractional 2021-06-17 Fractal and Fractional 2021-06-17 5 2 Article 57 10.3390/fractalfract5020057 https://www.mdpi.com/2504-3110/5/2/57
Fractal Fract, Vol. 5, Pages 56: Numerical Simulation for the Treatment of Nonlinear Predator–Prey Equations by Using the Finite Element Optimization Method https://www.mdpi.com/2504-3110/5/2/56 This article aims to introduce an efficient simulation to obtain the solution for a dynamical–biological system, which is called the Lotka–Volterra system, involving predator–prey equations. The finite element method (FEM) is employed to solve this problem. This technique is based mainly upon the appropriate conversion of the proposed model to a system of algebraic equations. The resulting system is then constructed as a constrained optimization problem and optimized in order to get the unknown coefficients and, consequently, the solution itself. We call this combination of the two well-known methods the finite element optimization method (FEOM). We compare the obtained results with the solutions obtained by using the fourth-order Runge–Kutta method (RK4 method). The residual error function is evaluated, which supports the efficiency and the accuracy of the presented procedure. From the given results, we can say that the presented procedure provides an easy and efficient tool to investigate the solution for such models as those investigated in this paper. 2021-06-16 Fractal Fract, Vol. 5, Pages 56: Numerical Simulation for the Treatment of Nonlinear Predator–Prey Equations by Using the Finite Element Optimization Method

Fractal and Fractional doi: 10.3390/fractalfract5020056

Authors: H. M. Srivastava M. M. Khader

This article aims to introduce an efficient simulation to obtain the solution for a dynamical–biological system, which is called the Lotka–Volterra system, involving predator–prey equations. The finite element method (FEM) is employed to solve this problem. This technique is based mainly upon the appropriate conversion of the proposed model to a system of algebraic equations. The resulting system is then constructed as a constrained optimization problem and optimized in order to get the unknown coefficients and, consequently, the solution itself. We call this combination of the two well-known methods the finite element optimization method (FEOM). We compare the obtained results with the solutions obtained by using the fourth-order Runge–Kutta method (RK4 method). The residual error function is evaluated, which supports the efficiency and the accuracy of the presented procedure. From the given results, we can say that the presented procedure provides an easy and efficient tool to investigate the solution for such models as those investigated in this paper.

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Numerical Simulation for the Treatment of Nonlinear Predator–Prey Equations by Using the Finite Element Optimization Method H. M. Srivastava M. M. Khader doi: 10.3390/fractalfract5020056 Fractal and Fractional 2021-06-16 Fractal and Fractional 2021-06-16 5 2 Article 56 10.3390/fractalfract5020056 https://www.mdpi.com/2504-3110/5/2/56
Fractal Fract, Vol. 5, Pages 55: On the Connectivity Measurement of the Fractal Julia Sets Generated from Polynomial Maps: A Novel Escape-Time Algorithm https://www.mdpi.com/2504-3110/5/2/55 In this paper, a novel escape-time algorithm is proposed to calculate the connectivity’s degree of Julia sets generated from polynomial maps. The proposed algorithm contains both quantitative analysis and visual display to measure the connectivity of Julia sets. For the quantitative part, a connectivity criterion method is designed by exploring the distribution rule of the connected regions, with an output value Co in the range of [0,1]. The smaller the Co value outputs, the better the connectivity is. For the visual part, we modify the classical escape-time algorithm by highlighting and separating the initial point of each connected area. Finally, the Julia set is drawn into different brightnesses according to different Co values. The darker the color, the better the connectivity of the Julia set. Numerical results are included to assess the efficiency of the algorithm. 2021-06-13 Fractal Fract, Vol. 5, Pages 55: On the Connectivity Measurement of the Fractal Julia Sets Generated from Polynomial Maps: A Novel Escape-Time Algorithm

Fractal and Fractional doi: 10.3390/fractalfract5020055

Authors: Yang Zhao Shicun Zhao Yi Zhang Da Wang

In this paper, a novel escape-time algorithm is proposed to calculate the connectivity’s degree of Julia sets generated from polynomial maps. The proposed algorithm contains both quantitative analysis and visual display to measure the connectivity of Julia sets. For the quantitative part, a connectivity criterion method is designed by exploring the distribution rule of the connected regions, with an output value Co in the range of [0,1]. The smaller the Co value outputs, the better the connectivity is. For the visual part, we modify the classical escape-time algorithm by highlighting and separating the initial point of each connected area. Finally, the Julia set is drawn into different brightnesses according to different Co values. The darker the color, the better the connectivity of the Julia set. Numerical results are included to assess the efficiency of the algorithm.

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On the Connectivity Measurement of the Fractal Julia Sets Generated from Polynomial Maps: A Novel Escape-Time Algorithm Yang Zhao Shicun Zhao Yi Zhang Da Wang doi: 10.3390/fractalfract5020055 Fractal and Fractional 2021-06-13 Fractal and Fractional 2021-06-13 5 2 Article 55 10.3390/fractalfract5020055 https://www.mdpi.com/2504-3110/5/2/55
Fractal Fract, Vol. 5, Pages 54: Some New Harmonically Convex Function Type Generalized Fractional Integral Inequalities https://www.mdpi.com/2504-3110/5/2/54 In this article, we established a new version of generalized fractional Hadamard and Fejér–Hadamard type integral inequalities. A fractional integral operator (FIO) with a non-singular function (multi-index Bessel function) as its kernel and monotone increasing functions is utilized to obtain the new version of such fractional inequalities. Our derived results are a generalized form of several proven inequalities already existing in the literature. The proven inequalities are useful for studying the stability and control of corresponding fractional dynamic equations. 2021-06-07 Fractal Fract, Vol. 5, Pages 54: Some New Harmonically Convex Function Type Generalized Fractional Integral Inequalities

Fractal and Fractional doi: 10.3390/fractalfract5020054

Authors: Rana Safdar Ali Aiman Mukheimer Thabet Abdeljawad Shahid Mubeen Sabila Ali Gauhar Rahman Kottakkaran Sooppy Nisar

In this article, we established a new version of generalized fractional Hadamard and Fejér–Hadamard type integral inequalities. A fractional integral operator (FIO) with a non-singular function (multi-index Bessel function) as its kernel and monotone increasing functions is utilized to obtain the new version of such fractional inequalities. Our derived results are a generalized form of several proven inequalities already existing in the literature. The proven inequalities are useful for studying the stability and control of corresponding fractional dynamic equations.

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Some New Harmonically Convex Function Type Generalized Fractional Integral Inequalities Rana Safdar Ali Aiman Mukheimer Thabet Abdeljawad Shahid Mubeen Sabila Ali Gauhar Rahman Kottakkaran Sooppy Nisar doi: 10.3390/fractalfract5020054 Fractal and Fractional 2021-06-07 Fractal and Fractional 2021-06-07 5 2 Article 54 10.3390/fractalfract5020054 https://www.mdpi.com/2504-3110/5/2/54
Fractal Fract, Vol. 5, Pages 53: Novel Expressions for the Derivatives of Sixth Kind Chebyshev Polynomials: Spectral Solution of the Non-Linear One-Dimensional Burgers’ Equation https://www.mdpi.com/2504-3110/5/2/53 This paper is concerned with establishing novel expressions that express the derivative of any order of the orthogonal polynomials, namely, Chebyshev polynomials of the sixth kind in terms of Chebyshev polynomials themselves. We will prove that these expressions involve certain terminating hypergeometric functions of the type 4F3(1) that can be reduced in some specific cases. The derived expressions along with the linearization formula of Chebyshev polynomials of the sixth kind serve in obtaining a numerical solution of the non-linear one-dimensional Burgers’ equation based on the application of the spectral tau method. Convergence analysis of the proposed double shifted Chebyshev expansion of the sixth kind is investigated. Numerical results are displayed aiming to show the efficiency and applicability of the proposed algorithm. 2021-06-06 Fractal Fract, Vol. 5, Pages 53: Novel Expressions for the Derivatives of Sixth Kind Chebyshev Polynomials: Spectral Solution of the Non-Linear One-Dimensional Burgers’ Equation

Fractal and Fractional doi: 10.3390/fractalfract5020053

Authors: Waleed Mohamed Abd-Elhameed

This paper is concerned with establishing novel expressions that express the derivative of any order of the orthogonal polynomials, namely, Chebyshev polynomials of the sixth kind in terms of Chebyshev polynomials themselves. We will prove that these expressions involve certain terminating hypergeometric functions of the type 4F3(1) that can be reduced in some specific cases. The derived expressions along with the linearization formula of Chebyshev polynomials of the sixth kind serve in obtaining a numerical solution of the non-linear one-dimensional Burgers’ equation based on the application of the spectral tau method. Convergence analysis of the proposed double shifted Chebyshev expansion of the sixth kind is investigated. Numerical results are displayed aiming to show the efficiency and applicability of the proposed algorithm.

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Novel Expressions for the Derivatives of Sixth Kind Chebyshev Polynomials: Spectral Solution of the Non-Linear One-Dimensional Burgers’ Equation Waleed Mohamed Abd-Elhameed doi: 10.3390/fractalfract5020053 Fractal and Fractional 2021-06-06 Fractal and Fractional 2021-06-06 5 2 Article 53 10.3390/fractalfract5020053 https://www.mdpi.com/2504-3110/5/2/53
Fractal Fract, Vol. 5, Pages 52: Study on the Existence of Solutions for a Class of Nonlinear Neutral Hadamard-Type Fractional Integro-Differential Equation with Infinite Delay https://www.mdpi.com/2504-3110/5/2/52 The existence of solutions for a class of nonlinear neutral Hadamard-type fractional integro-differential equations with infinite delay is researched in this paper. By constructing an appropriate normed space and utilizing the Banach contraction principle, Krasnoselskii’s fixed point theorem, we obtain some sufficient conditions for the existence of solutions. Finally, we provide an example to illustrate the validity of our main results. 2021-06-04 Fractal Fract, Vol. 5, Pages 52: Study on the Existence of Solutions for a Class of Nonlinear Neutral Hadamard-Type Fractional Integro-Differential Equation with Infinite Delay

Fractal and Fractional doi: 10.3390/fractalfract5020052

Authors: Kaihong Zhao Yue Ma

The existence of solutions for a class of nonlinear neutral Hadamard-type fractional integro-differential equations with infinite delay is researched in this paper. By constructing an appropriate normed space and utilizing the Banach contraction principle, Krasnoselskii’s fixed point theorem, we obtain some sufficient conditions for the existence of solutions. Finally, we provide an example to illustrate the validity of our main results.

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Study on the Existence of Solutions for a Class of Nonlinear Neutral Hadamard-Type Fractional Integro-Differential Equation with Infinite Delay Kaihong Zhao Yue Ma doi: 10.3390/fractalfract5020052 Fractal and Fractional 2021-06-04 Fractal and Fractional 2021-06-04 5 2 Article 52 10.3390/fractalfract5020052 https://www.mdpi.com/2504-3110/5/2/52
Fractal Fract, Vol. 5, Pages 51: Subdiffusive Reaction Model of Molecular Species in Liquid Layers: Fractional Reaction-Telegraph Approach https://www.mdpi.com/2504-3110/5/2/51 In recent years, different experimental works with molecular simulation techniques have been developed to study the transport of plasma-generated reactive species in liquid layers. Here, we improve the classical transport model that describes the molecular species movement in liquid layers via considering the fractional reaction–telegraph equation. We have considered the fractional equation to describe a non-Brownian motion of molecular species in a liquid layer, which have different diffusivities. The analytical solution of the fractional reaction–telegraph equation, which is defined in terms of the Caputo fractional derivative, is obtained by using the Laplace–Fourier technique. The profiles of species density with the mean square displacement are discussed in each case for different values of the time-fractional order and relaxation time. 2021-06-03 Fractal Fract, Vol. 5, Pages 51: Subdiffusive Reaction Model of Molecular Species in Liquid Layers: Fractional Reaction-Telegraph Approach

Fractal and Fractional doi: 10.3390/fractalfract5020051

Authors: Ashraf M. Tawfik Mohamed Mokhtar Hefny

In recent years, different experimental works with molecular simulation techniques have been developed to study the transport of plasma-generated reactive species in liquid layers. Here, we improve the classical transport model that describes the molecular species movement in liquid layers via considering the fractional reaction–telegraph equation. We have considered the fractional equation to describe a non-Brownian motion of molecular species in a liquid layer, which have different diffusivities. The analytical solution of the fractional reaction–telegraph equation, which is defined in terms of the Caputo fractional derivative, is obtained by using the Laplace–Fourier technique. The profiles of species density with the mean square displacement are discussed in each case for different values of the time-fractional order and relaxation time.

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Subdiffusive Reaction Model of Molecular Species in Liquid Layers: Fractional Reaction-Telegraph Approach Ashraf M. Tawfik Mohamed Mokhtar Hefny doi: 10.3390/fractalfract5020051 Fractal and Fractional 2021-06-03 Fractal and Fractional 2021-06-03 5 2 Article 51 10.3390/fractalfract5020051 https://www.mdpi.com/2504-3110/5/2/51
Fractal Fract, Vol. 5, Pages 50: Analytic Solution of the Langevin Differential Equations Dominated by a Multibrot Fractal Set https://www.mdpi.com/2504-3110/5/2/50 We present an analytic solvability of a class of Langevin differential equations (LDEs) in the asset of geometric function theory. The analytic solutions of the LDEs are presented by utilizing a special kind of fractal function in a complex domain, linked with the subordination theory. The fractal functions are suggested for the multi-parametric coefficients type motorboat fractal set. We obtain different formulas of fractal analytic solutions of LDEs. Moreover, we determine the maximum value of the fractal coefficients to obtain the optimal solution. Through the subordination inequality, we determined the upper boundary determination of a class of fractal functions holding multibrot function ϑ(z)=1+3κz+z3. 2021-05-25 Fractal Fract, Vol. 5, Pages 50: Analytic Solution of the Langevin Differential Equations Dominated by a Multibrot Fractal Set

Fractal and Fractional doi: 10.3390/fractalfract5020050

Authors: Rabha W. Ibrahim Dumitru Baleanu

We present an analytic solvability of a class of Langevin differential equations (LDEs) in the asset of geometric function theory. The analytic solutions of the LDEs are presented by utilizing a special kind of fractal function in a complex domain, linked with the subordination theory. The fractal functions are suggested for the multi-parametric coefficients type motorboat fractal set. We obtain different formulas of fractal analytic solutions of LDEs. Moreover, we determine the maximum value of the fractal coefficients to obtain the optimal solution. Through the subordination inequality, we determined the upper boundary determination of a class of fractal functions holding multibrot function ϑ(z)=1+3κz+z3.

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Analytic Solution of the Langevin Differential Equations Dominated by a Multibrot Fractal Set Rabha W. Ibrahim Dumitru Baleanu doi: 10.3390/fractalfract5020050 Fractal and Fractional 2021-05-25 Fractal and Fractional 2021-05-25 5 2 Article 50 10.3390/fractalfract5020050 https://www.mdpi.com/2504-3110/5/2/50
Fractal Fract, Vol. 5, Pages 49: Fat Tail in the Phytoplankton Movement Patterns and Swimming Behavior: New Insights into the Prey-Predator Interactions https://www.mdpi.com/2504-3110/5/2/49 Phytoplankton movement patterns and swimming behavior are important and basic topics in aquatic biology. Heavy tail distribution exists in diverse taxa and shows theoretical advantages in environments. The fat tails in the movement patterns and swimming behavior of phytoplankton in response to the food supply were studied. The log-normal distribution was used for fitting the probability density values of the movement data of Oxyrrhis marina. Results showed that obvious fat tails exist in the movement patterns of O. marina without and with positive stimulations of food supply. The algal cells tended to show a more chaotic and disorderly movement, with shorter and neat steps after adding the food source. At the same time, the randomness of turning rate, path curvature and swimming speed increased in O. marina cells with food supply. Generally, the responses of phytoplankton movement were stronger when supplied with direct prey cells rather than the cell-free filtrate. The scale-free random movements are considered to benefit the adaption of the entire phytoplankton population to varied environmental conditions. Inferentially, the movement pattern of O. marina should also have the characteristics of long-range dependence, local self-similarity and a system of fractional order. 2021-05-25 Fractal Fract, Vol. 5, Pages 49: Fat Tail in the Phytoplankton Movement Patterns and Swimming Behavior: New Insights into the Prey-Predator Interactions

Fractal and Fractional doi: 10.3390/fractalfract5020049

Authors: Xi Xiao Caicai Xu Yan Yu Junyu He Ming Li Carlo Cattani

Phytoplankton movement patterns and swimming behavior are important and basic topics in aquatic biology. Heavy tail distribution exists in diverse taxa and shows theoretical advantages in environments. The fat tails in the movement patterns and swimming behavior of phytoplankton in response to the food supply were studied. The log-normal distribution was used for fitting the probability density values of the movement data of Oxyrrhis marina. Results showed that obvious fat tails exist in the movement patterns of O. marina without and with positive stimulations of food supply. The algal cells tended to show a more chaotic and disorderly movement, with shorter and neat steps after adding the food source. At the same time, the randomness of turning rate, path curvature and swimming speed increased in O. marina cells with food supply. Generally, the responses of phytoplankton movement were stronger when supplied with direct prey cells rather than the cell-free filtrate. The scale-free random movements are considered to benefit the adaption of the entire phytoplankton population to varied environmental conditions. Inferentially, the movement pattern of O. marina should also have the characteristics of long-range dependence, local self-similarity and a system of fractional order.

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Fat Tail in the Phytoplankton Movement Patterns and Swimming Behavior: New Insights into the Prey-Predator Interactions Xi Xiao Caicai Xu Yan Yu Junyu He Ming Li Carlo Cattani doi: 10.3390/fractalfract5020049 Fractal and Fractional 2021-05-25 Fractal and Fractional 2021-05-25 5 2 Article 49 10.3390/fractalfract5020049 https://www.mdpi.com/2504-3110/5/2/49
Fractal Fract, Vol. 5, Pages 48: Hadamard-Type Fractional Heat Equations and Ultra-Slow Diffusions https://www.mdpi.com/2504-3110/5/2/48 In this paper, we study diffusion equations involving Hadamard-type time-fractional derivatives related to ultra-slow random models. We start our analysis using the abstract fractional Cauchy problem, replacing the classical time derivative with the Hadamard operator. The stochastic meaning of the introduced abstract differential equation is provided, and the application to the particular case of the fractional heat equation is then discussed in detail. The ultra-slow behaviour emerges from the explicit form of the variance of the random process arising from our analysis. Finally, we obtain a particular solution for the nonlinear Hadamard-diffusive equation. 2021-05-23 Fractal Fract, Vol. 5, Pages 48: Hadamard-Type Fractional Heat Equations and Ultra-Slow Diffusions

Fractal and Fractional doi: 10.3390/fractalfract5020048

Authors: Alessandro De Gregorio Roberto Garra

In this paper, we study diffusion equations involving Hadamard-type time-fractional derivatives related to ultra-slow random models. We start our analysis using the abstract fractional Cauchy problem, replacing the classical time derivative with the Hadamard operator. The stochastic meaning of the introduced abstract differential equation is provided, and the application to the particular case of the fractional heat equation is then discussed in detail. The ultra-slow behaviour emerges from the explicit form of the variance of the random process arising from our analysis. Finally, we obtain a particular solution for the nonlinear Hadamard-diffusive equation.

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Hadamard-Type Fractional Heat Equations and Ultra-Slow Diffusions Alessandro De Gregorio Roberto Garra doi: 10.3390/fractalfract5020048 Fractal and Fractional 2021-05-23 Fractal and Fractional 2021-05-23 5 2 Article 48 10.3390/fractalfract5020048 https://www.mdpi.com/2504-3110/5/2/48
Fractal Fract, Vol. 5, Pages 47: Structure, Fractality, Mechanics and Durability of Calcium Silicate Hydrates https://www.mdpi.com/2504-3110/5/2/47 Cement-based materials are widely utilized in infrastructure. The main product of hydrated products of cement-based materials is calcium silicate hydrate (C-S-H) gels that are considered as the binding phase of cement paste. C-S-H gels in Portland cement paste account for 60–70% of hydrated products by volume, which has profound influence on the mechanical properties and durability of cement-based materials. The preparation method of C-S-H gels has been well documented, but the quality of the prepared C-S-H affects experimental results; therefore, this review studies the preparation method of C-S-H under different conditions and materials. The progress related to C-S-H microstructure is explored from the theoretical and computational point of view. The fractality of C-S-H is discussed. An evaluation of the mechanical properties of C-S-H has also been included in this review. Finally, there is a discussion of the durability of C-S-H, with special reference to the carbonization and chloride/sulfate attacks. 2021-05-17 Fractal Fract, Vol. 5, Pages 47: Structure, Fractality, Mechanics and Durability of Calcium Silicate Hydrates

Fractal and Fractional doi: 10.3390/fractalfract5020047

Authors: Shengwen Tang Yang Wang Zhicheng Geng Xiaofei Xu Wenzhi Yu Hubao A Jingtao Chen

Cement-based materials are widely utilized in infrastructure. The main product of hydrated products of cement-based materials is calcium silicate hydrate (C-S-H) gels that are considered as the binding phase of cement paste. C-S-H gels in Portland cement paste account for 60–70% of hydrated products by volume, which has profound influence on the mechanical properties and durability of cement-based materials. The preparation method of C-S-H gels has been well documented, but the quality of the prepared C-S-H affects experimental results; therefore, this review studies the preparation method of C-S-H under different conditions and materials. The progress related to C-S-H microstructure is explored from the theoretical and computational point of view. The fractality of C-S-H is discussed. An evaluation of the mechanical properties of C-S-H has also been included in this review. Finally, there is a discussion of the durability of C-S-H, with special reference to the carbonization and chloride/sulfate attacks.

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Structure, Fractality, Mechanics and Durability of Calcium Silicate Hydrates Shengwen Tang Yang Wang Zhicheng Geng Xiaofei Xu Wenzhi Yu Hubao A Jingtao Chen doi: 10.3390/fractalfract5020047 Fractal and Fractional 2021-05-17 Fractal and Fractional 2021-05-17 5 2 Review 47 10.3390/fractalfract5020047 https://www.mdpi.com/2504-3110/5/2/47
Fractal Fract, Vol. 5, Pages 46: Design of Fractional-Order Lead Compensator for a Car Suspension System Based on Curve-Fitting Approximation https://www.mdpi.com/2504-3110/5/2/46 An alternative procedure for the implementation of fractional-order compensators is presented in this work. The employment of a curve-fitting-based approximation technique for the approximation of the compensator transfer function offers improved accuracy compared to the Oustaloup and Padé methods. As a design example, a lead compensator intended for usage in car suspension systems is realized. The open-loop and closed-loop behavior of the system is evaluated by post-layout simulation results obtained using the Cadence IC design suite and the Metal Oxide Semiconductor (MOS) transistor models provided by the Austria Mikro Systeme 0.35 μm Complementary Metal Oxide Semiconductor (CMOS) process. The derived results verify the efficient performance of the introduced implementation. 2021-05-15 Fractal Fract, Vol. 5, Pages 46: Design of Fractional-Order Lead Compensator for a Car Suspension System Based on Curve-Fitting Approximation

Fractal and Fractional doi: 10.3390/fractalfract5020046

Authors: Evisa Memlikai Stavroula Kapoulea Costas Psychalinos Jerzy Baranowski Waldemar Bauer Andrzej Tutaj Paweł Piątek

An alternative procedure for the implementation of fractional-order compensators is presented in this work. The employment of a curve-fitting-based approximation technique for the approximation of the compensator transfer function offers improved accuracy compared to the Oustaloup and Padé methods. As a design example, a lead compensator intended for usage in car suspension systems is realized. The open-loop and closed-loop behavior of the system is evaluated by post-layout simulation results obtained using the Cadence IC design suite and the Metal Oxide Semiconductor (MOS) transistor models provided by the Austria Mikro Systeme 0.35 μm Complementary Metal Oxide Semiconductor (CMOS) process. The derived results verify the efficient performance of the introduced implementation.

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Design of Fractional-Order Lead Compensator for a Car Suspension System Based on Curve-Fitting Approximation Evisa Memlikai Stavroula Kapoulea Costas Psychalinos Jerzy Baranowski Waldemar Bauer Andrzej Tutaj Paweł Piątek doi: 10.3390/fractalfract5020046 Fractal and Fractional 2021-05-15 Fractal and Fractional 2021-05-15 5 2 Article 46 10.3390/fractalfract5020046 https://www.mdpi.com/2504-3110/5/2/46
Fractal Fract, Vol. 5, Pages 45: On a Five-Parameter Mittag-Leffler Function and the Corresponding Bivariate Fractional Operators https://www.mdpi.com/2504-3110/5/2/45 Several extensions of the classical Mittag-Leffler function, including multi-parameter and multivariate versions, have been used to define fractional integral and derivative operators. In this paper, we consider a function of one variable with five parameters, a special case of the Fox–Wright function. It turns out that the most natural way to define a fractional integral based on this function requires considering it as a function of two variables. This gives rise to a model of bivariate fractional calculus, which is useful in understanding fractional differential equations involving mixed partial derivatives. 2021-05-14 Fractal Fract, Vol. 5, Pages 45: On a Five-Parameter Mittag-Leffler Function and the Corresponding Bivariate Fractional Operators

Fractal and Fractional doi: 10.3390/fractalfract5020045

Authors: Mehmet Ali Özarslan Arran Fernandez

Several extensions of the classical Mittag-Leffler function, including multi-parameter and multivariate versions, have been used to define fractional integral and derivative operators. In this paper, we consider a function of one variable with five parameters, a special case of the Fox–Wright function. It turns out that the most natural way to define a fractional integral based on this function requires considering it as a function of two variables. This gives rise to a model of bivariate fractional calculus, which is useful in understanding fractional differential equations involving mixed partial derivatives.

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On a Five-Parameter Mittag-Leffler Function and the Corresponding Bivariate Fractional Operators Mehmet Ali Özarslan Arran Fernandez doi: 10.3390/fractalfract5020045 Fractal and Fractional 2021-05-14 Fractal and Fractional 2021-05-14 5 2 Article 45 10.3390/fractalfract5020045 https://www.mdpi.com/2504-3110/5/2/45
Fractal Fract, Vol. 5, Pages 44: Elastic Rough Surface Contact and the Root Mean Square Slope of Measured Surfaces over Multiple Scales https://www.mdpi.com/2504-3110/5/2/44 This study investigates the predictions of the real contact area for perfectly elastic rough surfaces using a boundary element method (BEM). Sample surface measurements were used in the BEM to predict the real contact area as a function of load. The surfaces were normalized by the root-mean-square (RMS) slope to evaluate if contact area measurements would collapse onto one master curve. If so, this would confirm that the contact areas of manufactured, real measured surfaces are directly proportional to the root mean square slope and the applied load, which is predicted by fractal diffusion-based rough surface contact theory. The data predicts a complex response that deviates from this behavior. The variation in the RMS slope and the spectrum of the system related to the features in contact are further evaluated to illuminate why this property is seen in some types of surfaces and not others. 2021-05-14 Fractal Fract, Vol. 5, Pages 44: Elastic Rough Surface Contact and the Root Mean Square Slope of Measured Surfaces over Multiple Scales

Fractal and Fractional doi: 10.3390/fractalfract5020044

Authors: Robert Jackson Yang Xu Swarna Saha Kyle Schulze

This study investigates the predictions of the real contact area for perfectly elastic rough surfaces using a boundary element method (BEM). Sample surface measurements were used in the BEM to predict the real contact area as a function of load. The surfaces were normalized by the root-mean-square (RMS) slope to evaluate if contact area measurements would collapse onto one master curve. If so, this would confirm that the contact areas of manufactured, real measured surfaces are directly proportional to the root mean square slope and the applied load, which is predicted by fractal diffusion-based rough surface contact theory. The data predicts a complex response that deviates from this behavior. The variation in the RMS slope and the spectrum of the system related to the features in contact are further evaluated to illuminate why this property is seen in some types of surfaces and not others.

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Elastic Rough Surface Contact and the Root Mean Square Slope of Measured Surfaces over Multiple Scales Robert Jackson Yang Xu Swarna Saha Kyle Schulze doi: 10.3390/fractalfract5020044 Fractal and Fractional 2021-05-14 Fractal and Fractional 2021-05-14 5 2 Article 44 10.3390/fractalfract5020044 https://www.mdpi.com/2504-3110/5/2/44
Fractal Fract, Vol. 5, Pages 43: Sinc Based Inverse Laplace Transforms, Mittag-Leffler Functions and Their Approximation for Fractional Calculus https://www.mdpi.com/2504-3110/5/2/43 We shall discuss three methods of inverse Laplace transforms. A Sinc-Thiele approximation, a pure Sinc, and a Sinc-Gaussian based method. The two last Sinc related methods are exact methods of inverse Laplace transforms which allow us a numerical approximation using Sinc methods. The inverse Laplace transform converges exponentially and does not use Bromwich contours for computations. We apply the three methods to Mittag-Leffler functions incorporating one, two, and three parameters. The three parameter Mittag-Leffler function represents Prabhakar’s function. The exact Sinc methods are used to solve fractional differential equations of constant and variable differentiation order. 2021-05-10 Fractal Fract, Vol. 5, Pages 43: Sinc Based Inverse Laplace Transforms, Mittag-Leffler Functions and Their Approximation for Fractional Calculus

Fractal and Fractional doi: 10.3390/fractalfract5020043

Authors: Gerd Baumann

We shall discuss three methods of inverse Laplace transforms. A Sinc-Thiele approximation, a pure Sinc, and a Sinc-Gaussian based method. The two last Sinc related methods are exact methods of inverse Laplace transforms which allow us a numerical approximation using Sinc methods. The inverse Laplace transform converges exponentially and does not use Bromwich contours for computations. We apply the three methods to Mittag-Leffler functions incorporating one, two, and three parameters. The three parameter Mittag-Leffler function represents Prabhakar’s function. The exact Sinc methods are used to solve fractional differential equations of constant and variable differentiation order.

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Sinc Based Inverse Laplace Transforms, Mittag-Leffler Functions and Their Approximation for Fractional Calculus Gerd Baumann doi: 10.3390/fractalfract5020043 Fractal and Fractional 2021-05-10 Fractal and Fractional 2021-05-10 5 2 Article 43 10.3390/fractalfract5020043 https://www.mdpi.com/2504-3110/5/2/43
Fractal Fract, Vol. 5, Pages 42: Fractal Frames of Functions on the Rectangle https://www.mdpi.com/2504-3110/5/2/42 In this paper, we define fractal bases and fractal frames of L2(I×J), where I and J are real compact intervals, in order to approximate two-dimensional square-integrable maps whose domain is a rectangle, using the identification of L2(I×J) with the tensor product space L2(I)⨂L2(J). First, we recall the procedure of constructing a fractal perturbation of a continuous or integrable function. Then, we define fractal frames and bases of L2(I×J) composed of product of such fractal functions. We also obtain weaker families as Bessel, Riesz and Schauder sequences for the same space. Additionally, we study some properties of the tensor product of the fractal operators associated with the maps corresponding to each variable. 2021-05-08 Fractal Fract, Vol. 5, Pages 42: Fractal Frames of Functions on the Rectangle

Fractal and Fractional doi: 10.3390/fractalfract5020042

Authors: María A. Navascués Ram Mohapatra Md. Nasim Akhtar

In this paper, we define fractal bases and fractal frames of L2(I×J), where I and J are real compact intervals, in order to approximate two-dimensional square-integrable maps whose domain is a rectangle, using the identification of L2(I×J) with the tensor product space L2(I)⨂L2(J). First, we recall the procedure of constructing a fractal perturbation of a continuous or integrable function. Then, we define fractal frames and bases of L2(I×J) composed of product of such fractal functions. We also obtain weaker families as Bessel, Riesz and Schauder sequences for the same space. Additionally, we study some properties of the tensor product of the fractal operators associated with the maps corresponding to each variable.

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Fractal Frames of Functions on the Rectangle María A. Navascués Ram Mohapatra Md. Nasim Akhtar doi: 10.3390/fractalfract5020042 Fractal and Fractional 2021-05-08 Fractal and Fractional 2021-05-08 5 2 Article 42 10.3390/fractalfract5020042 https://www.mdpi.com/2504-3110/5/2/42
Fractal Fract, Vol. 5, Pages 41: Continuity Result on the Order of a Nonlinear Fractional Pseudo-Parabolic Equation with Caputo Derivative https://www.mdpi.com/2504-3110/5/2/41 In this paper, we consider a problem of continuity fractional-order for pseudo-parabolic equations with the fractional derivative of Caputo. Here, we investigate the stability of the problem with respect to derivative parameters and initial data. We also show that uω′→uω in an appropriate sense as ω′→ω, where ω is the fractional order. Moreover, to test the continuity fractional-order, we present several numerical examples to illustrate this property. 2021-05-05 Fractal Fract, Vol. 5, Pages 41: Continuity Result on the Order of a Nonlinear Fractional Pseudo-Parabolic Equation with Caputo Derivative

Fractal and Fractional doi: 10.3390/fractalfract5020041

Authors: Ho Duy Binh Luc Nguyen Hoang Dumitru Baleanu Ho Thi Kim Van

In this paper, we consider a problem of continuity fractional-order for pseudo-parabolic equations with the fractional derivative of Caputo. Here, we investigate the stability of the problem with respect to derivative parameters and initial data. We also show that uω′→uω in an appropriate sense as ω′→ω, where ω is the fractional order. Moreover, to test the continuity fractional-order, we present several numerical examples to illustrate this property.

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Continuity Result on the Order of a Nonlinear Fractional Pseudo-Parabolic Equation with Caputo Derivative Ho Duy Binh Luc Nguyen Hoang Dumitru Baleanu Ho Thi Kim Van doi: 10.3390/fractalfract5020041 Fractal and Fractional 2021-05-05 Fractal and Fractional 2021-05-05 5 2 Article 41 10.3390/fractalfract5020041 https://www.mdpi.com/2504-3110/5/2/41
Fractal Fract, Vol. 5, Pages 40: Utilizing Fractals for Modeling and 3D Printing of Porous Structures https://www.mdpi.com/2504-3110/5/2/40 Porous structures exhibiting randomly sized and distributed pores are required in biomedical applications (producing implants), materials science (developing cermet-based materials with desired properties), engineering applications (objects having controlled mass and energy transfer properties), and smart agriculture (devices for soilless cultivation). In most cases, a scaffold-based method is used to design porous structures. This approach fails to produce randomly sized and distributed pores, which is a pressing need as far as the aforementioned application areas are concerned. Thus, more effective porous structure design methods are required. This article presents how to utilize fractal geometry to model porous structures and then print them using 3D printing technology. A mathematical procedure was developed to create stochastic point clouds using the affine maps of a predefined Iterative Function Systems (IFS)-based fractal. In addition, a method is developed to modify a given IFS fractal-generated point cloud. The modification process controls the self-similarity levels of the fractal and ultimately results in a model of porous structure exhibiting randomly sized and distributed pores. The model can be transformed into a 3D Computer-Aided Design (CAD) model using voxel-based modeling or other means for digitization and 3D printing. The efficacy of the proposed method is demonstrated by transforming the Sierpinski Carpet (an IFS-based fractal) into 3D-printed porous structures with randomly sized and distributed pores. Other IFS-based fractals than the Sierpinski Carpet can be used to model and fabricate porous structures effectively. This issue remains open for further research. 2021-04-30 Fractal Fract, Vol. 5, Pages 40: Utilizing Fractals for Modeling and 3D Printing of Porous Structures

Fractal and Fractional doi: 10.3390/fractalfract5020040

Authors: AMM Sharif Ullah Doriana Marilena D’Addona Yusuke Seto Shota Yonehara Akihiko Kubo

Porous structures exhibiting randomly sized and distributed pores are required in biomedical applications (producing implants), materials science (developing cermet-based materials with desired properties), engineering applications (objects having controlled mass and energy transfer properties), and smart agriculture (devices for soilless cultivation). In most cases, a scaffold-based method is used to design porous structures. This approach fails to produce randomly sized and distributed pores, which is a pressing need as far as the aforementioned application areas are concerned. Thus, more effective porous structure design methods are required. This article presents how to utilize fractal geometry to model porous structures and then print them using 3D printing technology. A mathematical procedure was developed to create stochastic point clouds using the affine maps of a predefined Iterative Function Systems (IFS)-based fractal. In addition, a method is developed to modify a given IFS fractal-generated point cloud. The modification process controls the self-similarity levels of the fractal and ultimately results in a model of porous structure exhibiting randomly sized and distributed pores. The model can be transformed into a 3D Computer-Aided Design (CAD) model using voxel-based modeling or other means for digitization and 3D printing. The efficacy of the proposed method is demonstrated by transforming the Sierpinski Carpet (an IFS-based fractal) into 3D-printed porous structures with randomly sized and distributed pores. Other IFS-based fractals than the Sierpinski Carpet can be used to model and fabricate porous structures effectively. This issue remains open for further research.

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Utilizing Fractals for Modeling and 3D Printing of Porous Structures AMM Sharif Ullah Doriana Marilena D’Addona Yusuke Seto Shota Yonehara Akihiko Kubo doi: 10.3390/fractalfract5020040 Fractal and Fractional 2021-04-30 Fractal and Fractional 2021-04-30 5 2 Article 40 10.3390/fractalfract5020040 https://www.mdpi.com/2504-3110/5/2/40
Fractal Fract, Vol. 5, Pages 39: Fractals Parrondo’s Paradox in Alternated Superior Complex System https://www.mdpi.com/2504-3110/5/2/39 This work focuses on a kind of fractals Parrondo’s paradoxial phenomenon “deiconnected+diconnected=connected” in an alternated superior complex system zn+1=β(zn2+ci)+(1−β)zn,i=1,2. On the one hand, the connectivity variation in superior Julia sets is explored by analyzing the connectivity loci. On the other hand, we graphically investigate the position relation between superior Mandelbrot set and the Connectivity Loci, which results in the conclusion that two totally disconnected superior Julia sets can originate a new, connected, superior Julia set. Moreover, we present some graphical examples obtained by the use of the escape-time algorithm and the derived criteria. 2021-04-28 Fractal Fract, Vol. 5, Pages 39: Fractals Parrondo’s Paradox in Alternated Superior Complex System

Fractal and Fractional doi: 10.3390/fractalfract5020039

Authors: Yi Zhang Da Wang

This work focuses on a kind of fractals Parrondo’s paradoxial phenomenon “deiconnected+diconnected=connected” in an alternated superior complex system zn+1=β(zn2+ci)+(1−β)zn,i=1,2. On the one hand, the connectivity variation in superior Julia sets is explored by analyzing the connectivity loci. On the other hand, we graphically investigate the position relation between superior Mandelbrot set and the Connectivity Loci, which results in the conclusion that two totally disconnected superior Julia sets can originate a new, connected, superior Julia set. Moreover, we present some graphical examples obtained by the use of the escape-time algorithm and the derived criteria.

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Fractals Parrondo’s Paradox in Alternated Superior Complex System Yi Zhang Da Wang doi: 10.3390/fractalfract5020039 Fractal and Fractional 2021-04-28 Fractal and Fractional 2021-04-28 5 2 Article 39 10.3390/fractalfract5020039 https://www.mdpi.com/2504-3110/5/2/39
Fractal Fract, Vol. 5, Pages 38: Generalized Cauchy Process: Difference Iterative Forecasting Model https://www.mdpi.com/2504-3110/5/2/38 The contribution of this article is mainly to develop a new stochastic sequence forecasting model, which is also called the difference iterative forecasting model based on the Generalized Cauchy (GC) process. The GC process is a Long-Range Dependent (LRD) process described by two independent parameters: Hurst parameter H and fractal dimension D. Compared with the fractional Brownian motion (fBm) with a linear relationship between H and D, the GC process can more flexibly describe various LRD processes. Before building the forecasting model, this article demonstrates the GC process using H and D to describe the LRD and fractal properties of stochastic sequences, respectively. The GC process is taken as the diffusion term to establish a differential iterative forecasting model, where the incremental distribution of the GC process is obtained by statistics. The parameters of the forecasting model are estimated by the box dimension, the rescaled range, and the maximum likelihood methods. Finally, a real wind speed data set is used to verify the performance of the GC difference iterative forecasting model. 2021-04-23 Fractal Fract, Vol. 5, Pages 38: Generalized Cauchy Process: Difference Iterative Forecasting Model

Fractal and Fractional doi: 10.3390/fractalfract5020038

Authors: Jie Xing Wanqing Song Francesco Villecco

The contribution of this article is mainly to develop a new stochastic sequence forecasting model, which is also called the difference iterative forecasting model based on the Generalized Cauchy (GC) process. The GC process is a Long-Range Dependent (LRD) process described by two independent parameters: Hurst parameter H and fractal dimension D. Compared with the fractional Brownian motion (fBm) with a linear relationship between H and D, the GC process can more flexibly describe various LRD processes. Before building the forecasting model, this article demonstrates the GC process using H and D to describe the LRD and fractal properties of stochastic sequences, respectively. The GC process is taken as the diffusion term to establish a differential iterative forecasting model, where the incremental distribution of the GC process is obtained by statistics. The parameters of the forecasting model are estimated by the box dimension, the rescaled range, and the maximum likelihood methods. Finally, a real wind speed data set is used to verify the performance of the GC difference iterative forecasting model.

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Generalized Cauchy Process: Difference Iterative Forecasting Model Jie Xing Wanqing Song Francesco Villecco doi: 10.3390/fractalfract5020038 Fractal and Fractional 2021-04-23 Fractal and Fractional 2021-04-23 5 2 Article 38 10.3390/fractalfract5020038 https://www.mdpi.com/2504-3110/5/2/38
Fractal Fract, Vol. 5, Pages 37: Lipschitz Stability in Time for Riemann–Liouville Fractional Differential Equations https://www.mdpi.com/2504-3110/5/2/37 A system of nonlinear fractional differential equations with the Riemann–Liouville fractional derivative is considered. Lipschitz stability in time for the studied equations is defined and studied. This stability is connected with the singularity of the Riemann–Liouville fractional derivative at the initial point. Two types of derivatives of Lyapunov functions among the studied fractional equations are applied to obtain sufficient conditions for the defined stability property. Some examples illustrate the results. 2021-04-21 Fractal Fract, Vol. 5, Pages 37: Lipschitz Stability in Time for Riemann–Liouville Fractional Differential Equations

Fractal and Fractional doi: 10.3390/fractalfract5020037

Authors: Snezhana Hristova Stepan Tersian Radoslava Terzieva

A system of nonlinear fractional differential equations with the Riemann–Liouville fractional derivative is considered. Lipschitz stability in time for the studied equations is defined and studied. This stability is connected with the singularity of the Riemann–Liouville fractional derivative at the initial point. Two types of derivatives of Lyapunov functions among the studied fractional equations are applied to obtain sufficient conditions for the defined stability property. Some examples illustrate the results.

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Lipschitz Stability in Time for Riemann–Liouville Fractional Differential Equations Snezhana Hristova Stepan Tersian Radoslava Terzieva doi: 10.3390/fractalfract5020037 Fractal and Fractional 2021-04-21 Fractal and Fractional 2021-04-21 5 2 Article 37 10.3390/fractalfract5020037 https://www.mdpi.com/2504-3110/5/2/37
Fractal Fract, Vol. 5, Pages 36: Simultaneous Characterization of Relaxation, Creep, Dissipation, and Hysteresis by Fractional-Order Constitutive Models https://www.mdpi.com/2504-3110/5/2/36 We considered relaxation, creep, dissipation, and hysteresis resulting from a six-parameter fractional constitutive model and its particular cases. The storage modulus, loss modulus, and loss factor, as well as their characteristics based on the thermodynamic requirements, were investigated. It was proved that for the fractional Maxwell model, the storage modulus increases monotonically, while the loss modulus has symmetrical peaks for its curve against the logarithmic scale log(ω), and for the fractional Zener model, the storage modulus monotonically increases while the loss modulus and the loss factor have symmetrical peaks for their curves against the logarithmic scale log(ω). The peak values and corresponding stationary points were analytically given. The relaxation modulus and the creep compliance for the six-parameter fractional constitutive model were given in terms of the Mittag–Leffler functions. Finally, the stress–strain hysteresis loops were simulated by making use of the derived creep compliance for the fractional Zener model. These results show that the fractional constitutive models could characterize the relaxation, creep, dissipation, and hysteresis phenomena of viscoelastic bodies, and fractional orders α and β could be used to model real-world physical properties well. 2021-04-20 Fractal Fract, Vol. 5, Pages 36: Simultaneous Characterization of Relaxation, Creep, Dissipation, and Hysteresis by Fractional-Order Constitutive Models

Fractal and Fractional doi: 10.3390/fractalfract5020036

Authors: Jun-Sheng Duan Di-Chen Hu Yang-Quan Chen

We considered relaxation, creep, dissipation, and hysteresis resulting from a six-parameter fractional constitutive model and its particular cases. The storage modulus, loss modulus, and loss factor, as well as their characteristics based on the thermodynamic requirements, were investigated. It was proved that for the fractional Maxwell model, the storage modulus increases monotonically, while the loss modulus has symmetrical peaks for its curve against the logarithmic scale log(ω), and for the fractional Zener model, the storage modulus monotonically increases while the loss modulus and the loss factor have symmetrical peaks for their curves against the logarithmic scale log(ω). The peak values and corresponding stationary points were analytically given. The relaxation modulus and the creep compliance for the six-parameter fractional constitutive model were given in terms of the Mittag–Leffler functions. Finally, the stress–strain hysteresis loops were simulated by making use of the derived creep compliance for the fractional Zener model. These results show that the fractional constitutive models could characterize the relaxation, creep, dissipation, and hysteresis phenomena of viscoelastic bodies, and fractional orders α and β could be used to model real-world physical properties well.

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Simultaneous Characterization of Relaxation, Creep, Dissipation, and Hysteresis by Fractional-Order Constitutive Models Jun-Sheng Duan Di-Chen Hu Yang-Quan Chen doi: 10.3390/fractalfract5020036 Fractal and Fractional 2021-04-20 Fractal and Fractional 2021-04-20 5 2 Article 36 10.3390/fractalfract5020036 https://www.mdpi.com/2504-3110/5/2/36
Fractal Fract, Vol. 5, Pages 35: New Challenges Arising in Engineering Problems with Fractional and Integer Order https://www.mdpi.com/2504-3110/5/2/35 Mathematical models have been frequently studied in recent decades in order to obtain the deeper properties of real-world problems [...] 2021-04-19 Fractal Fract, Vol. 5, Pages 35: New Challenges Arising in Engineering Problems with Fractional and Integer Order

Fractal and Fractional doi: 10.3390/fractalfract5020035

Authors: Haci Mehmet Baskonus Luis Manuel Sánchez Ruiz Armando Ciancio

Mathematical models have been frequently studied in recent decades in order to obtain the deeper properties of real-world problems [...]

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New Challenges Arising in Engineering Problems with Fractional and Integer Order Haci Mehmet Baskonus Luis Manuel Sánchez Ruiz Armando Ciancio doi: 10.3390/fractalfract5020035 Fractal and Fractional 2021-04-19 Fractal and Fractional 2021-04-19 5 2 Editorial 35 10.3390/fractalfract5020035 https://www.mdpi.com/2504-3110/5/2/35
Fractal Fract, Vol. 5, Pages 34: Some New Results on F-Contractions in 0-Complete Partial Metric Spaces and 0-Complete Metric-Like Spaces https://www.mdpi.com/2504-3110/5/2/34 Within this manuscript we generalize the two recently obtained results of O. Popescu and G. Stan, regarding the F-contractions in complete, ordinary metric space to 0-complete partial metric space and 0-complete metric-like space. As Popescu and Stan we use less conditions than D. Wardovski did in his paper from 2012, and we introduce, with the help of one of our lemmas, a new method of proving the results in fixed point theory. Requiring that the function F only be strictly increasing, we obtain for consequence new families of contractive conditions that cannot be found in the existing literature. Note that our results generalize and complement many well-known results in the fixed point theory. Also, at the end of the paper, we have stated an application of our theoretical results for solving fractional differential equations. 2021-04-19 Fractal Fract, Vol. 5, Pages 34: Some New Results on F-Contractions in 0-Complete Partial Metric Spaces and 0-Complete Metric-Like Spaces

Fractal and Fractional doi: 10.3390/fractalfract5020034

Authors: Stojan Radenović Nikola Mirkov Ljiljana R. Paunović

Within this manuscript we generalize the two recently obtained results of O. Popescu and G. Stan, regarding the F-contractions in complete, ordinary metric space to 0-complete partial metric space and 0-complete metric-like space. As Popescu and Stan we use less conditions than D. Wardovski did in his paper from 2012, and we introduce, with the help of one of our lemmas, a new method of proving the results in fixed point theory. Requiring that the function F only be strictly increasing, we obtain for consequence new families of contractive conditions that cannot be found in the existing literature. Note that our results generalize and complement many well-known results in the fixed point theory. Also, at the end of the paper, we have stated an application of our theoretical results for solving fractional differential equations.

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Some New Results on F-Contractions in 0-Complete Partial Metric Spaces and 0-Complete Metric-Like Spaces Stojan Radenović Nikola Mirkov Ljiljana R. Paunović doi: 10.3390/fractalfract5020034 Fractal and Fractional 2021-04-19 Fractal and Fractional 2021-04-19 5 2 Article 34 10.3390/fractalfract5020034 https://www.mdpi.com/2504-3110/5/2/34
Fractal Fract, Vol. 5, Pages 33: A New Approach for Dynamic Stochastic Fractal Search with Fuzzy Logic for Parameter Adaptation https://www.mdpi.com/2504-3110/5/2/33 Stochastic fractal search (SFS) is a novel method inspired by the process of stochastic growth in nature and the use of the fractal mathematical concept. Considering the chaotic stochastic diffusion property, an improved dynamic stochastic fractal search (DSFS) optimization algorithm is presented. The DSFS algorithm was tested with benchmark functions, such as the multimodal, hybrid, and composite functions, to evaluate the performance of the algorithm with dynamic parameter adaptation with type-1 and type-2 fuzzy inference models. The main contribution of the article is the utilization of fuzzy logic in the adaptation of the diffusion parameter in a dynamic fashion. This parameter is in charge of creating new fractal particles, and the diversity and iteration are the input information used in the fuzzy system to control the values of diffusion. 2021-04-17 Fractal Fract, Vol. 5, Pages 33: A New Approach for Dynamic Stochastic Fractal Search with Fuzzy Logic for Parameter Adaptation

Fractal and Fractional doi: 10.3390/fractalfract5020033

Authors: Marylu L. Lagunes Oscar Castillo Fevrier Valdez Jose Soria Patricia Melin

Stochastic fractal search (SFS) is a novel method inspired by the process of stochastic growth in nature and the use of the fractal mathematical concept. Considering the chaotic stochastic diffusion property, an improved dynamic stochastic fractal search (DSFS) optimization algorithm is presented. The DSFS algorithm was tested with benchmark functions, such as the multimodal, hybrid, and composite functions, to evaluate the performance of the algorithm with dynamic parameter adaptation with type-1 and type-2 fuzzy inference models. The main contribution of the article is the utilization of fuzzy logic in the adaptation of the diffusion parameter in a dynamic fashion. This parameter is in charge of creating new fractal particles, and the diversity and iteration are the input information used in the fuzzy system to control the values of diffusion.

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A New Approach for Dynamic Stochastic Fractal Search with Fuzzy Logic for Parameter Adaptation Marylu L. Lagunes Oscar Castillo Fevrier Valdez Jose Soria Patricia Melin doi: 10.3390/fractalfract5020033 Fractal and Fractional 2021-04-17 Fractal and Fractional 2021-04-17 5 2 Article 33 10.3390/fractalfract5020033 https://www.mdpi.com/2504-3110/5/2/33