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Fractal Fract., Volume 5, Issue 3 (September 2021) – 74 articles

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22 pages, 521 KiB  
Article
Finite Element Formulation of Fractional Constitutive Laws Using the Reformulated Infinite State Representation
by Matthias Hinze, André Schmidt and Remco I. Leine
Fractal Fract. 2021, 5(3), 132; https://doi.org/10.3390/fractalfract5030132 - 21 Sep 2021
Viewed by 1842
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Theory and Applications of 3D Fractional Models)
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13 pages, 512 KiB  
Article
Some Dynamical Models Involving Fractional-Order Derivatives with the Mittag-Leffler Type Kernels and Their Applications Based upon the Legendre Spectral Collocation Method
by Hari M. Srivastava, Abedel-Karrem N. Alomari, Khaled M. Saad and Waleed M. Hamanah
Fractal Fract. 2021, 5(3), 131; https://doi.org/10.3390/fractalfract5030131 - 20 Sep 2021
Cited by 21 | Viewed by 1996
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Fractional Calculus Operators and the Mittag-Leffler Function)
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12 pages, 316 KiB  
Article
Jafari Transformation for Solving a System of Ordinary Differential Equations with Medical Application
by Ahmed I. El-Mesady, Yasser S. Hamed and Abdullah M. Alsharif
Fractal Fract. 2021, 5(3), 130; https://doi.org/10.3390/fractalfract5030130 - 20 Sep 2021
Cited by 16 | Viewed by 2978
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Section General Mathematics, Analysis)
21 pages, 2857 KiB  
Article
Pseudo-Likelihood Estimation for Parameters of Stochastic Time-Fractional Diffusion Equations
by Guofei Pang and Wanrong Cao
Fractal Fract. 2021, 5(3), 129; https://doi.org/10.3390/fractalfract5030129 - 18 Sep 2021
Viewed by 1591
Abstract
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, [...] Read more.
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. Full article
(This article belongs to the Special Issue Fractional Deterministic and Stochastic Models and Their Calibration)
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17 pages, 4353 KiB  
Article
Dynamics of Fractional-Order Digital Manufacturing Supply Chain System and Its Control and Synchronization
by Yingjin He, Song Zheng and Liguo Yuan
Fractal Fract. 2021, 5(3), 128; https://doi.org/10.3390/fractalfract5030128 - 17 Sep 2021
Cited by 8 | Viewed by 1797
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Fractional Order Systems and Their Applications)
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14 pages, 4086 KiB  
Article
Unified Scale Theorem: A Mathematical Formulation of Scale in the Frame of Earth Observation Image Classification
by Christos G. Karydas
Fractal Fract. 2021, 5(3), 127; https://doi.org/10.3390/fractalfract5030127 - 17 Sep 2021
Cited by 5 | Viewed by 1902
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Fractals in Geosciences: Theory and Applications)
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17 pages, 342 KiB  
Article
A Note on Existence of Mild Solutions for Second-Order Neutral Integro-Differential Evolution Equations with State-Dependent Delay
by Shahram Rezapour, Hernán R. Henríquez, Velusamy Vijayakumar, Kottakkaran Sooppy Nisar and Anurag Shukla
Fractal Fract. 2021, 5(3), 126; https://doi.org/10.3390/fractalfract5030126 - 17 Sep 2021
Cited by 12 | Viewed by 2381
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Fractional Dynamical Systems: Applications and Theoretical Results)
16 pages, 2515 KiB  
Article
Design, Convergence and Stability of a Fourth-Order Class of Iterative Methods for Solving Nonlinear Vectorial Problems
by Alicia Cordero, Cristina Jordán, Esther Sanabria-Codesal and Juan R. Torregrosa
Fractal Fract. 2021, 5(3), 125; https://doi.org/10.3390/fractalfract5030125 - 17 Sep 2021
Cited by 6 | Viewed by 1595
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Section Numerical and Computational Methods)
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21 pages, 545 KiB  
Article
Thermophysical Investigation of Oldroyd-B Fluid with Functional Effects of Permeability: Memory Effect Study Using Non-Singular Kernel Derivative Approach
by Muhammad Bilal Riaz, Jan Awrejcewicz, Aziz-Ur Rehman and Ali Akgül
Fractal Fract. 2021, 5(3), 124; https://doi.org/10.3390/fractalfract5030124 - 15 Sep 2021
Cited by 23 | Viewed by 2447
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Recent Advances in Computational Physics with Fractional Application)
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17 pages, 308 KiB  
Article
Well Posedness of New Optimization Problems with Variational Inequality Constraints
by Savin Treanţă
Fractal Fract. 2021, 5(3), 123; https://doi.org/10.3390/fractalfract5030123 - 15 Sep 2021
Cited by 8 | Viewed by 1814
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Advances in Optimization and Nonlinear Analysis)
17 pages, 1613 KiB  
Article
CMOS OTA-Based Filters for Designing Fractional-Order Chaotic Oscillators
by Martín Alejandro Valencia-Ponce, Perla Rubí Castañeda-Aviña, Esteban Tlelo-Cuautle, Victor Hugo Carbajal-Gómez, Victor Rodolfo González-Díaz, Yuma Sandoval-Ibarra and Jose-Cruz Nuñez-Perez
Fractal Fract. 2021, 5(3), 122; https://doi.org/10.3390/fractalfract5030122 - 14 Sep 2021
Cited by 16 | Viewed by 3407
Abstract
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, [...] Read more.
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. Full article
(This article belongs to the Special Issue Fractional-Order Circuit Theory and Applications)
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13 pages, 497 KiB  
Article
On a New Modification of the Erdélyi–Kober Fractional Derivative
by Zaid Odibat and Dumitru Baleanu
Fractal Fract. 2021, 5(3), 121; https://doi.org/10.3390/fractalfract5030121 - 13 Sep 2021
Cited by 11 | Viewed by 2098
Abstract
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 [...] Read more.
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. Full article
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13 pages, 2426 KiB  
Article
Multi-Model Selection and Analysis for COVID-19
by Nuri Ma, Weiyuan Ma and Zhiming Li
Fractal Fract. 2021, 5(3), 120; https://doi.org/10.3390/fractalfract5030120 - 13 Sep 2021
Cited by 15 | Viewed by 2259
Abstract
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, [...] Read more.
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. Full article
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18 pages, 3544 KiB  
Article
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
by Yu-Ming Chu, Umar Nazir, Muhammad Sohail, Mahmoud M. Selim and Jung-Rye Lee
Fractal Fract. 2021, 5(3), 119; https://doi.org/10.3390/fractalfract5030119 - 13 Sep 2021
Cited by 450 | Viewed by 3880
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Section Mathematical Physics)
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18 pages, 338 KiB  
Article
On Weighted (k, s)-Riemann-Liouville Fractional Operators and Solution of Fractional Kinetic Equation
by Muhammad Samraiz, Muhammad Umer, Artion Kashuri, Thabet Abdeljawad, Sajid Iqbal and Nabil Mlaiki
Fractal Fract. 2021, 5(3), 118; https://doi.org/10.3390/fractalfract5030118 - 13 Sep 2021
Cited by 15 | Viewed by 1710
Abstract
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 [...] Read more.
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. Full article
18 pages, 355 KiB  
Article
On the General Solutions of Some Non-Homogeneous Div-Curl Systems with Riemann–Liouville and Caputo Fractional Derivatives
by Briceyda B. Delgado and Jorge E. Macías-Díaz
Fractal Fract. 2021, 5(3), 117; https://doi.org/10.3390/fractalfract5030117 - 10 Sep 2021
Cited by 16 | Viewed by 1602
Abstract
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 [...] Read more.
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. Full article
14 pages, 322 KiB  
Article
On Discrete Delta Caputo–Fabrizio Fractional Operators and Monotonicity Analysis
by Pshtiwan Othman Mohammed, Thabet Abdeljawad and Faraidun Kadir Hamasalh
Fractal Fract. 2021, 5(3), 116; https://doi.org/10.3390/fractalfract5030116 - 9 Sep 2021
Cited by 17 | Viewed by 1484
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Section General Mathematics, Analysis)
20 pages, 1283 KiB  
Article
Jacobi Spectral Collocation Technique for Time-Fractional Inverse Heat Equations
by Mohamed A. Abdelkawy, Ahmed Z. M. Amin, Mohammed M. Babatin, Abeer S. Alnahdi, Mahmoud A. Zaky and Ramy M. Hafez
Fractal Fract. 2021, 5(3), 115; https://doi.org/10.3390/fractalfract5030115 - 9 Sep 2021
Cited by 5 | Viewed by 1685
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Fractional Order Systems and Their Applications)
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17 pages, 2038 KiB  
Article
Estimating Conditional Power for Sequential Monitoring of Covariate Adaptive Randomized Designs: The Fractional Brownian Motion Approach
by Yiping Yang, Hongjian Zhu and Dejian Lai
Fractal Fract. 2021, 5(3), 114; https://doi.org/10.3390/fractalfract5030114 - 8 Sep 2021
Cited by 3 | Viewed by 1667
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Fractional Behavior in Nature 2021)
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31 pages, 1811 KiB  
Article
Analytic Fuzzy Formulation of a Time-Fractional Fornberg–Whitham Model with Power and Mittag–Leffler Kernels
by Saima Rashid, Rehana Ashraf, Ahmet Ocak Akdemir, Manar A. Alqudah, Thabet Abdeljawad and Mohamed S. Mohamed
Fractal Fract. 2021, 5(3), 113; https://doi.org/10.3390/fractalfract5030113 - 8 Sep 2021
Cited by 12 | Viewed by 1908
Abstract
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 [...] Read more.
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. Full article
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15 pages, 830 KiB  
Article
Controllability for Fuzzy Fractional Evolution Equations in Credibility Space
by Azmat Ullah Khan Niazi, Naveed Iqbal, Rasool Shah, Fongchan Wannalookkhee and Kamsing Nonlaopon
Fractal Fract. 2021, 5(3), 112; https://doi.org/10.3390/fractalfract5030112 - 8 Sep 2021
Cited by 27 | Viewed by 1709
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Fractional Order Systems and Their Applications)
14 pages, 424 KiB  
Article
Numerical Solutions of Fractional Differential Equations by Using Laplace Transformation Method and Quadrature Rule
by Samaneh Soradi-Zeid, Mehdi Mesrizadeh and Carlo Cattani
Fractal Fract. 2021, 5(3), 111; https://doi.org/10.3390/fractalfract5030111 - 7 Sep 2021
Cited by 3 | Viewed by 2038
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue 2021 Feature Papers by Fractal Fract's Editorial Board Members)
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25 pages, 460 KiB  
Article
A q-Gradient Descent Algorithm with Quasi-Fejér Convergence for Unconstrained Optimization Problems
by Shashi Kant Mishra, Predrag Rajković, Mohammad Esmael Samei, Suvra Kanti Chakraborty, Bhagwat Ram and Mohammed K. A. Kaabar
Fractal Fract. 2021, 5(3), 110; https://doi.org/10.3390/fractalfract5030110 - 3 Sep 2021
Cited by 11 | Viewed by 2464
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue The Materials Structure and Fractal Nature)
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21 pages, 362 KiB  
Article
On the Operator Method for Solving Linear Integro-Differential Equations with Fractional Conformable Derivatives
by Batirkhan Kh. Turmetov, Kairat I. Usmanov and Kulzina Zh. Nazarova
Fractal Fract. 2021, 5(3), 109; https://doi.org/10.3390/fractalfract5030109 - 2 Sep 2021
Cited by 2 | Viewed by 1359
Abstract
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 [...] Read more.
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. Full article
20 pages, 949 KiB  
Article
Qualitative Study on Solutions of a Hadamard Variable Order Boundary Problem via the Ulam–Hyers–Rassias Stability
by Amar Benkerrouche, Mohammed Said Souid, Sina Etemad, Ali Hakem, Praveen Agarwal, Shahram Rezapour, Sotiris K. Ntouyas and Jessada Tariboon
Fractal Fract. 2021, 5(3), 108; https://doi.org/10.3390/fractalfract5030108 - 2 Sep 2021
Cited by 25 | Viewed by 1915
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Fractional Dynamical Systems: Applications and Theoretical Results)
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15 pages, 4978 KiB  
Article
Influence of Fin Length on Magneto-Combined Convection Heat Transfer Performance in a Lid-Driven Wavy Cavity
by Md. Fayz-Al-Asad, Mehmet Yavuz, Md. Nur Alam, Md. Manirul Alam Sarker and Omar Bazighifan
Fractal Fract. 2021, 5(3), 107; https://doi.org/10.3390/fractalfract5030107 - 31 Aug 2021
Cited by 14 | Viewed by 2129
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Advanced Trends of Special Functions and Analysis of PDEs)
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15 pages, 2420 KiB  
Article
Approximate Solutions of Nonlinear Partial Differential Equations Using B-Polynomial Bases
by Muhammad I. Bhatti, Md. Habibur Rahman and Nicholas Dimakis
Fractal Fract. 2021, 5(3), 106; https://doi.org/10.3390/fractalfract5030106 - 31 Aug 2021
Cited by 4 | Viewed by 2069
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Frontiers in Fractional Schrödinger Equation)
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13 pages, 289 KiB  
Article
Uniqueness of Solutions of the Generalized Abel Integral Equations in Banach Spaces
by Chenkuan Li and Hari M. Srivastava
Fractal Fract. 2021, 5(3), 105; https://doi.org/10.3390/fractalfract5030105 - 31 Aug 2021
Cited by 7 | Viewed by 1679
Abstract
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 [...] Read more.
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. Full article
21 pages, 3444 KiB  
Article
An Experimental Approach towards Motion Modeling and Control of a Vehicle Transiting a Non-Newtonian Environment
by Isabela Birs, Cristina Muresan, Ovidiu Prodan, Silviu Folea and Clara Ionescu
Fractal Fract. 2021, 5(3), 104; https://doi.org/10.3390/fractalfract5030104 - 25 Aug 2021
Cited by 3 | Viewed by 1843
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Fractional Dynamics 2021)
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21 pages, 803 KiB  
Article
Numerical Solutions for Systems of Fractional and Classical Integro-Differential Equations via Finite Integration Method Based on Shifted Chebyshev Polynomials
by Ampol Duangpan, Ratinan Boonklurb and Matinee Juytai
Fractal Fract. 2021, 5(3), 103; https://doi.org/10.3390/fractalfract5030103 - 25 Aug 2021
Cited by 10 | Viewed by 2274
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Novel Numerical Solutions of Fractional PDEs)
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