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. 4, Pages 49: Numerical Simulation of the Fractal-Fractional Ebola Virus https://www.mdpi.com/2504-3110/4/4/49 In this work we present three new models of the fractal-fractional Ebola virus. We investigate the numerical solutions of the fractal-fractional Ebola virus in the sense of three different kernels based on the power law, the exponential decay and the generalized Mittag-Leffler function by using the concepts of the fractal differentiation and fractional differentiation. These operators have two parameters: The first parameter &amp;rho; is considered as the fractal dimension and the second parameter k is the fractional order. We evaluate the numerical solutions of the fractal-fractional Ebola virus for these operators with the theory of fractional calculus and the help of the Lagrange polynomial functions. In the case of &amp;rho;=k=1, all of the numerical solutions based on the power kernel, the exponential kernel and the generalized Mittag-Leffler kernel are found to be close to each other and, therefore, one of the kernels is compared with such numerical methods as the finite difference methods. This has led to an excellent agreement. For the effect of fractal-fractional on the behavior, we study the numerical solutions for different values of &amp;rho; and k. All calculations in this work are accomplished by using the Mathematica package. Fractal Fract, Vol. 4, Pages 49: Numerical Simulation of the Fractal-Fractional Ebola Virus

Fractal and Fractional doi: 10.3390/fractalfract4040049

Authors: H. M. Srivastava Khaled M. Saad

In this work we present three new models of the fractal-fractional Ebola virus. We investigate the numerical solutions of the fractal-fractional Ebola virus in the sense of three different kernels based on the power law, the exponential decay and the generalized Mittag-Leffler function by using the concepts of the fractal differentiation and fractional differentiation. These operators have two parameters: The first parameter &amp;rho; is considered as the fractal dimension and the second parameter k is the fractional order. We evaluate the numerical solutions of the fractal-fractional Ebola virus for these operators with the theory of fractional calculus and the help of the Lagrange polynomial functions. In the case of &amp;rho;=k=1, all of the numerical solutions based on the power kernel, the exponential kernel and the generalized Mittag-Leffler kernel are found to be close to each other and, therefore, one of the kernels is compared with such numerical methods as the finite difference methods. This has led to an excellent agreement. For the effect of fractal-fractional on the behavior, we study the numerical solutions for different values of &amp;rho; and k. All calculations in this work are accomplished by using the Mathematica package.

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Numerical Simulation of the Fractal-Fractional Ebola Virus H. M. Srivastava Khaled M. Saad doi: 10.3390/fractalfract4040049 Fractal and Fractional 2020-09-29 Fractal and Fractional 2020-09-29 4 4 Article 49 10.3390/fractalfract4040049 https://www.mdpi.com/2504-3110/4/4/49
Fractal Fract, Vol. 4, Pages 48: On Some Formulas for the k-Analogue of Appell Functions and Generating Relations via k-Fractional Derivative https://www.mdpi.com/2504-3110/4/4/48 Our present investigation is mainly based on the k-hypergeometric functions which are constructed by making use of the Pochhammer k-symbol Diaz et al. 2017, which are one of the vital generalizations of hypergeometric functions. We introduce k-analogues of F2and F3 Appell functions denoted by the symbols F2,kand F3,k,respectively, just like Mubeen et al. did for F1 in 2015. Meanwhile, we prove integral representations of the k-generalizations of F2and F3 which provide us with an opportunity to generalize widely used identities for Appell hypergeometric functions. In addition, we present some important transformation formulas and some reduction formulas which show close relation not only with k-Appell functions but also with k-hypergeometric functions. Finally, employing the theory of Riemann&amp;ndash;Liouville k-fractional derivative from Rahman et al. 2020, and using the relations which we consider in this paper, we acquire linear and bilinear generating relations for k-analogue of hypergeometric functions and Appell functions. Fractal Fract, Vol. 4, Pages 48: On Some Formulas for the k-Analogue of Appell Functions and Generating Relations via k-Fractional Derivative

Fractal and Fractional doi: 10.3390/fractalfract4040048

Authors: Övgü Gürel Yılmaz Rabia Aktaş Fatma Taşdelen

Our present investigation is mainly based on the k-hypergeometric functions which are constructed by making use of the Pochhammer k-symbol Diaz et al. 2017, which are one of the vital generalizations of hypergeometric functions. We introduce k-analogues of F2and F3 Appell functions denoted by the symbols F2,kand F3,k,respectively, just like Mubeen et al. did for F1 in 2015. Meanwhile, we prove integral representations of the k-generalizations of F2and F3 which provide us with an opportunity to generalize widely used identities for Appell hypergeometric functions. In addition, we present some important transformation formulas and some reduction formulas which show close relation not only with k-Appell functions but also with k-hypergeometric functions. Finally, employing the theory of Riemann&amp;ndash;Liouville k-fractional derivative from Rahman et al. 2020, and using the relations which we consider in this paper, we acquire linear and bilinear generating relations for k-analogue of hypergeometric functions and Appell functions.

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On Some Formulas for the k-Analogue of Appell Functions and Generating Relations via k-Fractional Derivative Övgü Gürel Yılmaz Rabia Aktaş Fatma Taşdelen doi: 10.3390/fractalfract4040048 Fractal and Fractional 2020-09-24 Fractal and Fractional 2020-09-24 4 4 Article 48 10.3390/fractalfract4040048 https://www.mdpi.com/2504-3110/4/4/48
Fractal Fract, Vol. 4, Pages 47: Integral Representation of Fractional Derivative of Delta Function https://www.mdpi.com/2504-3110/4/3/47 Delta function is a widely used generalized function in various fields, ranging from physics to mathematics. How to express its fractional derivative with integral representation is a tough problem. In this paper, we present an integral representation of the fractional derivative of the delta function. Moreover, we provide its application in representing the fractional Gaussian noise. Fractal Fract, Vol. 4, Pages 47: Integral Representation of Fractional Derivative of Delta Function

Fractal and Fractional doi: 10.3390/fractalfract4030047

Authors: Ming Li

Delta function is a widely used generalized function in various fields, ranging from physics to mathematics. How to express its fractional derivative with integral representation is a tough problem. In this paper, we present an integral representation of the fractional derivative of the delta function. Moreover, we provide its application in representing the fractional Gaussian noise.

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Integral Representation of Fractional Derivative of Delta Function Ming Li doi: 10.3390/fractalfract4030047 Fractal and Fractional 2020-09-20 Fractal and Fractional 2020-09-20 4 3 Article 47 10.3390/fractalfract4030047 https://www.mdpi.com/2504-3110/4/3/47
Fractal Fract, Vol. 4, Pages 46: Fractal and Fractional Derivative Modelling of Material Phase Change https://www.mdpi.com/2504-3110/4/3/46 An iterative approach is taken to develop a fractal topology that can describe the material structure of phase changing materials. Transfer functions and frequency response functions based on fractional calculus are used to describe this topology and then applied to model phase transformations in liquid/solid transitions in physical processes. Three types of transformation are tested experimentally, whipping of cream (rheopexy), solidification of gelatine and melting of ethyl vinyl acetate (EVA). A liquid-type model is used throughout the cream whipping process while liquid and solid models are required for gelatine and EVA to capture the yield characteristic of these materials. Fractal Fract, Vol. 4, Pages 46: Fractal and Fractional Derivative Modelling of Material Phase Change

Fractal and Fractional doi: 10.3390/fractalfract4030046

Authors: Harry Esmonde

An iterative approach is taken to develop a fractal topology that can describe the material structure of phase changing materials. Transfer functions and frequency response functions based on fractional calculus are used to describe this topology and then applied to model phase transformations in liquid/solid transitions in physical processes. Three types of transformation are tested experimentally, whipping of cream (rheopexy), solidification of gelatine and melting of ethyl vinyl acetate (EVA). A liquid-type model is used throughout the cream whipping process while liquid and solid models are required for gelatine and EVA to capture the yield characteristic of these materials.

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Fractal and Fractional Derivative Modelling of Material Phase Change Harry Esmonde doi: 10.3390/fractalfract4030046 Fractal and Fractional 2020-09-14 Fractal and Fractional 2020-09-14 4 3 Article 46 10.3390/fractalfract4030046 https://www.mdpi.com/2504-3110/4/3/46
Fractal Fract, Vol. 4, Pages 45: Modified Mittag-Leffler Functions with Applications in Complex Formulae for Fractional Calculus https://www.mdpi.com/2504-3110/4/3/45 Mittag-Leffler functions and their variations are a popular topic of study at the present time, mostly due to their applications in fractional calculus and fractional differential equations. Here we propose a modification of the usual Mittag-Leffler functions of one, two, or three parameters, which is ideally suited for extending certain fractional-calculus operators into the complex plane. Complex analysis has been underused in combination with fractional calculus, especially with newly developed operators like those with Mittag-Leffler kernels. Here we show the natural analytic continuations of these operators using the modified Mittag-Leffler functions defined in this paper. Fractal Fract, Vol. 4, Pages 45: Modified Mittag-Leffler Functions with Applications in Complex Formulae for Fractional Calculus

Fractal and Fractional doi: 10.3390/fractalfract4030045

Authors: Arran Fernandez Iftikhar Husain

Mittag-Leffler functions and their variations are a popular topic of study at the present time, mostly due to their applications in fractional calculus and fractional differential equations. Here we propose a modification of the usual Mittag-Leffler functions of one, two, or three parameters, which is ideally suited for extending certain fractional-calculus operators into the complex plane. Complex analysis has been underused in combination with fractional calculus, especially with newly developed operators like those with Mittag-Leffler kernels. Here we show the natural analytic continuations of these operators using the modified Mittag-Leffler functions defined in this paper.

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Modified Mittag-Leffler Functions with Applications in Complex Formulae for Fractional Calculus Arran Fernandez Iftikhar Husain doi: 10.3390/fractalfract4030045 Fractal and Fractional 2020-09-12 Fractal and Fractional 2020-09-12 4 3 Article 45 10.3390/fractalfract4030045 https://www.mdpi.com/2504-3110/4/3/45
Fractal Fract, Vol. 4, Pages 44: Fractional SIS Epidemic Models https://www.mdpi.com/2504-3110/4/3/44 In this paper, we consider the fractional SIS (susceptible-infectious-susceptible) epidemic model (&amp;alpha;-SIS model) in the case of constant population size. We provide a representation of the explicit solution to the fractional model and we illustrate the results by numerical schemes. A comparison with the limit case when the fractional order &amp;alpha; converges to 1 (the SIS model) is also given. We analyze the effects of the fractional derivatives by comparing the SIS and the &amp;alpha;-SIS models. Fractal Fract, Vol. 4, Pages 44: Fractional SIS Epidemic Models

Fractal and Fractional doi: 10.3390/fractalfract4030044

Authors: Caterina Balzotti Mirko D’Ovidio Paola Loreti

In this paper, we consider the fractional SIS (susceptible-infectious-susceptible) epidemic model (&amp;alpha;-SIS model) in the case of constant population size. We provide a representation of the explicit solution to the fractional model and we illustrate the results by numerical schemes. A comparison with the limit case when the fractional order &amp;alpha; converges to 1 (the SIS model) is also given. We analyze the effects of the fractional derivatives by comparing the SIS and the &amp;alpha;-SIS models.

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Fractional SIS Epidemic Models Caterina Balzotti Mirko D’Ovidio Paola Loreti doi: 10.3390/fractalfract4030044 Fractal and Fractional 2020-08-31 Fractal and Fractional 2020-08-31 4 3 Article 44 10.3390/fractalfract4030044 https://www.mdpi.com/2504-3110/4/3/44
Fractal Fract, Vol. 4, Pages 43: Analysis of Fractional Order Chaotic Financial Model with Minimum Interest Rate Impact https://www.mdpi.com/2504-3110/4/3/43 The main objective of this paper is to construct and test fractional order derivatives for the management and simulation of a fractional order disorderly finance system. In the developed system, we add the critical minimum interest rate d parameter in order to develop a new stable financial model. The new emerging paradigm increases the demand for innovation, which is the gateway to the knowledge economy. The derivatives are characterized in the Caputo fractional order derivative and Atangana-Baleanu derivative. We prove the existence and uniqueness of the solutions with fixed point theorem and an iterative scheme. The interest rate begins to rise according to initial conditions as investment demand and price exponent begin to fall, which shows the financial system&amp;rsquo;s actual macroeconomic behavior. Specifically component of its application to the large scale and smaller scale forms, just as the utilization of specific strategies and instruments such fractal stochastic procedures and expectation. Fractal Fract, Vol. 4, Pages 43: Analysis of Fractional Order Chaotic Financial Model with Minimum Interest Rate Impact

Fractal and Fractional doi: 10.3390/fractalfract4030043

The main objective of this paper is to construct and test fractional order derivatives for the management and simulation of a fractional order disorderly finance system. In the developed system, we add the critical minimum interest rate d parameter in order to develop a new stable financial model. The new emerging paradigm increases the demand for innovation, which is the gateway to the knowledge economy. The derivatives are characterized in the Caputo fractional order derivative and Atangana-Baleanu derivative. We prove the existence and uniqueness of the solutions with fixed point theorem and an iterative scheme. The interest rate begins to rise according to initial conditions as investment demand and price exponent begin to fall, which shows the financial system&amp;rsquo;s actual macroeconomic behavior. Specifically component of its application to the large scale and smaller scale forms, just as the utilization of specific strategies and instruments such fractal stochastic procedures and expectation.

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Analysis of Fractional Order Chaotic Financial Model with Minimum Interest Rate Impact Muhammad Farman Ali Akgül Dumitru Baleanu Sumaiyah Imtiaz Aqeel Ahmad doi: 10.3390/fractalfract4030043 Fractal and Fractional 2020-08-21 Fractal and Fractional 2020-08-21 4 3 Article 43 10.3390/fractalfract4030043 https://www.mdpi.com/2504-3110/4/3/43
Fractal Fract, Vol. 4, Pages 42: Dispersive Transport Described by the Generalized Fick Law with Different Fractional Operators https://www.mdpi.com/2504-3110/4/3/42 The approach based on fractional advection&amp;ndash;diffusion equations provides an effective and meaningful tool to describe the dispersive transport of charge carriers in disordered semiconductors. A fractional generalization of Fick&amp;rsquo;s law containing the Riemann&amp;ndash;Liouville fractional derivative is related to the well-known fractional Fokker&amp;ndash;Planck equation, and it is consistent with the universal characteristics of dispersive transport observed in the time-of-flight experiment (ToF). In the present paper, we consider the generalized Fick laws containing other forms of fractional time operators with singular and non-singular kernels and find out features of ToF transient currents that can indicate the presence of such fractional dynamics. Solutions of the corresponding fractional Fokker&amp;ndash;Planck equations are expressed through solutions of integer-order equation in terms of an integral with the subordinating function. This representation is used to calculate the ToF transient current curves. The physical reasons leading to the considered fractional generalizations are elucidated and discussed. Fractal Fract, Vol. 4, Pages 42: Dispersive Transport Described by the Generalized Fick Law with Different Fractional Operators

Fractal and Fractional doi: 10.3390/fractalfract4030042

Authors: Renat T. Sibatov HongGuang Sun

The approach based on fractional advection&amp;ndash;diffusion equations provides an effective and meaningful tool to describe the dispersive transport of charge carriers in disordered semiconductors. A fractional generalization of Fick&amp;rsquo;s law containing the Riemann&amp;ndash;Liouville fractional derivative is related to the well-known fractional Fokker&amp;ndash;Planck equation, and it is consistent with the universal characteristics of dispersive transport observed in the time-of-flight experiment (ToF). In the present paper, we consider the generalized Fick laws containing other forms of fractional time operators with singular and non-singular kernels and find out features of ToF transient currents that can indicate the presence of such fractional dynamics. Solutions of the corresponding fractional Fokker&amp;ndash;Planck equations are expressed through solutions of integer-order equation in terms of an integral with the subordinating function. This representation is used to calculate the ToF transient current curves. The physical reasons leading to the considered fractional generalizations are elucidated and discussed.

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Dispersive Transport Described by the Generalized Fick Law with Different Fractional Operators Renat T. Sibatov HongGuang Sun doi: 10.3390/fractalfract4030042 Fractal and Fractional 2020-08-17 Fractal and Fractional 2020-08-17 4 3 Article 42 10.3390/fractalfract4030042 https://www.mdpi.com/2504-3110/4/3/42
Fractal Fract, Vol. 4, Pages 41: Novel Complex Wave Solutions of the (2+1)-Dimensional Hyperbolic Nonlinear Schrödinger Equation https://www.mdpi.com/2504-3110/4/3/41 This manuscript focuses on the application of the (m+1/G&amp;prime;)-expansion method to the (2+1)-dimensional hyperbolic nonlinear Schr&amp;ouml;dinger equation. With the help of projected method, the periodic and singular complex wave solutions to the considered model are derived. Various figures such as 3D and 2D surfaces with the selecting the suitable of parameter values are plotted. Fractal Fract, Vol. 4, Pages 41: Novel Complex Wave Solutions of the (2+1)-Dimensional Hyperbolic Nonlinear Schrödinger Equation

Fractal and Fractional doi: 10.3390/fractalfract4030041

Authors: Hulya Durur Esin Ilhan Hasan Bulut

This manuscript focuses on the application of the (m+1/G&amp;prime;)-expansion method to the (2+1)-dimensional hyperbolic nonlinear Schr&amp;ouml;dinger equation. With the help of projected method, the periodic and singular complex wave solutions to the considered model are derived. Various figures such as 3D and 2D surfaces with the selecting the suitable of parameter values are plotted.

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Novel Complex Wave Solutions of the (2+1)-Dimensional Hyperbolic Nonlinear Schrödinger Equation Hulya Durur Esin Ilhan Hasan Bulut doi: 10.3390/fractalfract4030041 Fractal and Fractional 2020-08-16 Fractal and Fractional 2020-08-16 4 3 Article 41 10.3390/fractalfract4030041 https://www.mdpi.com/2504-3110/4/3/41
Fractal Fract, Vol. 4, Pages 40: Fractional-Order Derivatives Defined by Continuous Kernels: Are They Really Too Restrictive? https://www.mdpi.com/2504-3110/4/3/40 In the field of fractional calculus and applications, a current trend is to propose non-singular kernels for the definition of new fractional integration and differentiation operators. It was recently claimed that fractional-order derivatives defined by continuous (in the sense of non-singular) kernels are too restrictive. This note shows that this conclusion is wrong as it arises from considering the initial conditions incorrectly in (partial or not) fractional differential equations. Fractal Fract, Vol. 4, Pages 40: Fractional-Order Derivatives Defined by Continuous Kernels: Are They Really Too Restrictive?

Fractal and Fractional doi: 10.3390/fractalfract4030040

Authors: Jocelyn Sabatier

In the field of fractional calculus and applications, a current trend is to propose non-singular kernels for the definition of new fractional integration and differentiation operators. It was recently claimed that fractional-order derivatives defined by continuous (in the sense of non-singular) kernels are too restrictive. This note shows that this conclusion is wrong as it arises from considering the initial conditions incorrectly in (partial or not) fractional differential equations.

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Fractional-Order Derivatives Defined by Continuous Kernels: Are They Really Too Restrictive? Jocelyn Sabatier doi: 10.3390/fractalfract4030040 Fractal and Fractional 2020-08-11 Fractal and Fractional 2020-08-11 4 3 Viewpoint 40 10.3390/fractalfract4030040 https://www.mdpi.com/2504-3110/4/3/40
Fractal Fract, Vol. 4, Pages 39: Parameter Identification in the Two-Dimensional Riesz Space Fractional Diffusion Equation https://www.mdpi.com/2504-3110/4/3/39 This paper presents the application of the swarm intelligence algorithm for solving the inverse problem concerning the parameter identification. The paper examines the two-dimensional Riesz space fractional diffusion equation. Based on the values of the function (for the fixed points of the domain) which is the solution of the described differential equation, the order of the Riesz derivative and the diffusion coefficient are identified. The paper includes numerical examples illustrating the algorithm&amp;rsquo;s accuracy. Fractal Fract, Vol. 4, Pages 39: Parameter Identification in the Two-Dimensional Riesz Space Fractional Diffusion Equation

Fractal and Fractional doi: 10.3390/fractalfract4030039

Authors: Rafał Brociek Agata Chmielowska Damian Słota

This paper presents the application of the swarm intelligence algorithm for solving the inverse problem concerning the parameter identification. The paper examines the two-dimensional Riesz space fractional diffusion equation. Based on the values of the function (for the fixed points of the domain) which is the solution of the described differential equation, the order of the Riesz derivative and the diffusion coefficient are identified. The paper includes numerical examples illustrating the algorithm&amp;rsquo;s accuracy.

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Parameter Identification in the Two-Dimensional Riesz Space Fractional Diffusion Equation Rafał Brociek Agata Chmielowska Damian Słota doi: 10.3390/fractalfract4030039 Fractal and Fractional 2020-08-06 Fractal and Fractional 2020-08-06 4 3 Article 39 10.3390/fractalfract4030039 https://www.mdpi.com/2504-3110/4/3/39
Fractal Fract, Vol. 4, Pages 38: A Stochastic Fractional Calculus with Applications to Variational Principles https://www.mdpi.com/2504-3110/4/3/38 We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional Euler&amp;ndash;Lagrange equation is obtained, extending those available in the literature for the classical, fractional, and stochastic calculus of variations. To illustrate our main theoretical result, we discuss two examples: one derived from quantum mechanics, the second validated by an adequate numerical simulation. Fractal Fract, Vol. 4, Pages 38: A Stochastic Fractional Calculus with Applications to Variational Principles

Fractal and Fractional doi: 10.3390/fractalfract4030038

Authors: Houssine Zine Delfim F. M. Torres

We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional Euler&amp;ndash;Lagrange equation is obtained, extending those available in the literature for the classical, fractional, and stochastic calculus of variations. To illustrate our main theoretical result, we discuss two examples: one derived from quantum mechanics, the second validated by an adequate numerical simulation.

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A Stochastic Fractional Calculus with Applications to Variational Principles Houssine Zine Delfim F. M. Torres doi: 10.3390/fractalfract4030038 Fractal and Fractional 2020-08-01 Fractal and Fractional 2020-08-01 4 3 Article 38 10.3390/fractalfract4030038 https://www.mdpi.com/2504-3110/4/3/38

Fractal and Fractional doi: 10.3390/fractalfract4030037

Authors: Guido Maione

This paper concerns cascaded, shifted, fractional-order, lead compensators made by the serial connection of two stages introducing their respective phase leads in shifted adjacent frequency ranges. Adding up leads in these intervals gives a flat phase in a wide frequency range. Moreover, the simple elements of the cascade can be easily realized by rational transfer functions. On this basis, a method is proposed in order to design a robust controller for a class of benchmark plants that are difficult to compensate due to monotonically increasing lags. The simulation experiments show the efficiency, performance and robustness of the approach.

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Design of Cascaded and Shifted Fractional-Order Lead Compensators for Plants with Monotonically Increasing Lags Guido Maione doi: 10.3390/fractalfract4030037 Fractal and Fractional 2020-07-27 Fractal and Fractional 2020-07-27 4 3 Article 37 10.3390/fractalfract4030037 https://www.mdpi.com/2504-3110/4/3/37
Fractal Fract, Vol. 4, Pages 36: Mathematical Description of the Groundwater Flow and that of the Impurity Spread, which Use Temporal Caputo or Riemann–Liouville Fractional Partial Derivatives, Is Non-Objective https://www.mdpi.com/2504-3110/4/3/36 In this paper, it is shown that the mathematical description of the bulk fluid flow and that of content impurity spread, which uses temporal Caputo or temporal Riemann&amp;ndash;Liouville fractional order partial derivatives, having integral representation on a finite interval, in the case of a horizontal unconfined aquifer is non-objective. The basic idea is that different observers using this type of description obtain different results which cannot be reconciled, in other words, transformed into each other using only formulas that link the numbers representing a moment in time for two different choices from the origin of time measurement. This is not an academic curiosity; it is rather a problem to find which one of the obtained results is correct. Fractal Fract, Vol. 4, Pages 36: Mathematical Description of the Groundwater Flow and that of the Impurity Spread, which Use Temporal Caputo or Riemann–Liouville Fractional Partial Derivatives, Is Non-Objective

Fractal and Fractional doi: 10.3390/fractalfract4030036

Authors: Agneta M. Balint Stefan Balint

In this paper, it is shown that the mathematical description of the bulk fluid flow and that of content impurity spread, which uses temporal Caputo or temporal Riemann&amp;ndash;Liouville fractional order partial derivatives, having integral representation on a finite interval, in the case of a horizontal unconfined aquifer is non-objective. The basic idea is that different observers using this type of description obtain different results which cannot be reconciled, in other words, transformed into each other using only formulas that link the numbers representing a moment in time for two different choices from the origin of time measurement. This is not an academic curiosity; it is rather a problem to find which one of the obtained results is correct.

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Mathematical Description of the Groundwater Flow and that of the Impurity Spread, which Use Temporal Caputo or Riemann–Liouville Fractional Partial Derivatives, Is Non-Objective Agneta M. Balint Stefan Balint doi: 10.3390/fractalfract4030036 Fractal and Fractional 2020-07-21 Fractal and Fractional 2020-07-21 4 3 Article 36 10.3390/fractalfract4030036 https://www.mdpi.com/2504-3110/4/3/36
Fractal Fract, Vol. 4, Pages 35: Stability Analysis and Numerical Computation of the Fractional Predator–Prey Model with the Harvesting Rate https://www.mdpi.com/2504-3110/4/3/35 In this work, a fractional predator-prey model with the harvesting rate is considered. Besides the existence and uniqueness of the solution to the model, local stability and global stability are experienced. A novel discretization depending on the numerical discretization of the Riemann&amp;ndash;Liouville integral was introduced and the corresponding numerical discretization of the predator&amp;ndash;prey fractional model was obtained. The net reproduction number R 0 was obtained for the prediction and persistence of the disease. The dynamical behavior of the equilibria was examined by using the stability criteria. Furthermore, numerical simulations of the model were performed and their graphical representations are shown to support the numerical discretizations, to visualize the effectiveness of our theoretical results and to monitor the effect of arbitrary order derivative. In our investigations, the fractional operator is understood in the Caputo sense. Fractal Fract, Vol. 4, Pages 35: Stability Analysis and Numerical Computation of the Fractional Predator–Prey Model with the Harvesting Rate

Fractal and Fractional doi: 10.3390/fractalfract4030035

Authors: Mehmet Yavuz Ndolane Sene

In this work, a fractional predator-prey model with the harvesting rate is considered. Besides the existence and uniqueness of the solution to the model, local stability and global stability are experienced. A novel discretization depending on the numerical discretization of the Riemann&amp;ndash;Liouville integral was introduced and the corresponding numerical discretization of the predator&amp;ndash;prey fractional model was obtained. The net reproduction number R 0 was obtained for the prediction and persistence of the disease. The dynamical behavior of the equilibria was examined by using the stability criteria. Furthermore, numerical simulations of the model were performed and their graphical representations are shown to support the numerical discretizations, to visualize the effectiveness of our theoretical results and to monitor the effect of arbitrary order derivative. In our investigations, the fractional operator is understood in the Caputo sense.

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Stability Analysis and Numerical Computation of the Fractional Predator–Prey Model with the Harvesting Rate Mehmet Yavuz Ndolane Sene doi: 10.3390/fractalfract4030035 Fractal and Fractional 2020-07-16 Fractal and Fractional 2020-07-16 4 3 Article 35 10.3390/fractalfract4030035 https://www.mdpi.com/2504-3110/4/3/35
Fractal Fract, Vol. 4, Pages 34: Hardware Implementation and Performance Study of Analog PIλDμ Controllers on DC Motor https://www.mdpi.com/2504-3110/4/3/34 In this paper, the performance of an analog PI &amp;lambda; D &amp;mu; controller is done for speed regulation of a DC motor. The circuits for the fractional integrator and differentiator of PI &amp;lambda; D &amp;mu; controller are designed by optimal pole-zero interlacing algorithm. The performance of the controller is compared with another PI &amp;lambda; D &amp;mu; controller&amp;mdash;in which the fractional integrator circuit employs a solid-state fractional capacitor. It can be verified from the results that using PI &amp;lambda; D &amp;mu; controllers, the speed response of the DC motor has improved with reduction in settling time ( T s ), steady state error (SS error) and % overshoot (% M p ). Fractal Fract, Vol. 4, Pages 34: Hardware Implementation and Performance Study of Analog PIλDμ Controllers on DC Motor

Fractal and Fractional doi: 10.3390/fractalfract4030034

Authors: Dina A. John Saket Sehgal Karabi Biswas

In this paper, the performance of an analog PI &amp;lambda; D &amp;mu; controller is done for speed regulation of a DC motor. The circuits for the fractional integrator and differentiator of PI &amp;lambda; D &amp;mu; controller are designed by optimal pole-zero interlacing algorithm. The performance of the controller is compared with another PI &amp;lambda; D &amp;mu; controller&amp;mdash;in which the fractional integrator circuit employs a solid-state fractional capacitor. It can be verified from the results that using PI &amp;lambda; D &amp;mu; controllers, the speed response of the DC motor has improved with reduction in settling time ( T s ), steady state error (SS error) and % overshoot (% M p ).

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Hardware Implementation and Performance Study of Analog PIλDμ Controllers on DC Motor Dina A. John Saket Sehgal Karabi Biswas doi: 10.3390/fractalfract4030034 Fractal and Fractional 2020-07-15 Fractal and Fractional 2020-07-15 4 3 Article 34 10.3390/fractalfract4030034 https://www.mdpi.com/2504-3110/4/3/34
Fractal Fract, Vol. 4, Pages 33: On the Volterra-Type Fractional Integro-Differential Equations Pertaining to Special Functions https://www.mdpi.com/2504-3110/4/3/33 In this article, we apply an integral transform-based technique to solve the fractional order Volterra-type integro-differential equation (FVIDE) involving the generalized Lorenzo-Hartely function and generalized Lauricella confluent hypergeometric function in terms of several complex variables in the kernel. We also investigate and introduce the Elazki transform of Hilfer-derivative, generalized Lorenzo-Hartely function and generalized Lauricella confluent hypergeometric function. In this article, we have established three results that are present in the form of lemmas, which give us new results on the above mentioned three functions, and by using these results we have derived our main results that are given in the form of theorems. Our main results are very general in nature, which gives us some new and known results as a particular case of results established here. Fractal Fract, Vol. 4, Pages 33: On the Volterra-Type Fractional Integro-Differential Equations Pertaining to Special Functions

Fractal and Fractional doi: 10.3390/fractalfract4030033

Authors: Yudhveer Singh Vinod Gill Jagdev Singh Devendra Kumar Kottakkaran Sooppy Nisar

In this article, we apply an integral transform-based technique to solve the fractional order Volterra-type integro-differential equation (FVIDE) involving the generalized Lorenzo-Hartely function and generalized Lauricella confluent hypergeometric function in terms of several complex variables in the kernel. We also investigate and introduce the Elazki transform of Hilfer-derivative, generalized Lorenzo-Hartely function and generalized Lauricella confluent hypergeometric function. In this article, we have established three results that are present in the form of lemmas, which give us new results on the above mentioned three functions, and by using these results we have derived our main results that are given in the form of theorems. Our main results are very general in nature, which gives us some new and known results as a particular case of results established here.

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On the Volterra-Type Fractional Integro-Differential Equations Pertaining to Special Functions Yudhveer Singh Vinod Gill Jagdev Singh Devendra Kumar Kottakkaran Sooppy Nisar doi: 10.3390/fractalfract4030033 Fractal and Fractional 2020-07-09 Fractal and Fractional 2020-07-09 4 3 Article 33 10.3390/fractalfract4030033 https://www.mdpi.com/2504-3110/4/3/33
Fractal Fract, Vol. 4, Pages 32: Transition from Diffusion to Wave Propagation in Fractional Jeffreys-Type Heat Conduction Equation https://www.mdpi.com/2504-3110/4/3/32 The heat conduction equation with a fractional Jeffreys-type constitutive law is studied. Depending on the value of a characteristic parameter, two fundamentally different types of behavior are established: diffusion regime and propagation regime. In the first case, the considered equation is a generalized diffusion equation, while in the second it is a generalized wave equation. The corresponding memory kernels are expressed in both cases in terms of Mittag&amp;ndash;Leffler functions. Explicit representations for the one-dimensional fundamental solution and the mean squared displacement are provided and analyzed analytically and numerically. The one-dimensional fundamental solution is shown to be a spatial probability density function evolving in time, which is unimodal in the diffusion regime and bimodal in the propagation regime. The multi-dimensional fundamental solutions are probability densities only in the diffusion case, while in the propagation case they can have negative values. In addition, two different types of subordination principles are formulated for the two regimes. The Bernstein functions technique is extensively employed in the theoretical proofs. Fractal Fract, Vol. 4, Pages 32: Transition from Diffusion to Wave Propagation in Fractional Jeffreys-Type Heat Conduction Equation

Fractal and Fractional doi: 10.3390/fractalfract4030032

Authors: Emilia Bazhlekova Ivan Bazhlekov

The heat conduction equation with a fractional Jeffreys-type constitutive law is studied. Depending on the value of a characteristic parameter, two fundamentally different types of behavior are established: diffusion regime and propagation regime. In the first case, the considered equation is a generalized diffusion equation, while in the second it is a generalized wave equation. The corresponding memory kernels are expressed in both cases in terms of Mittag&amp;ndash;Leffler functions. Explicit representations for the one-dimensional fundamental solution and the mean squared displacement are provided and analyzed analytically and numerically. The one-dimensional fundamental solution is shown to be a spatial probability density function evolving in time, which is unimodal in the diffusion regime and bimodal in the propagation regime. The multi-dimensional fundamental solutions are probability densities only in the diffusion case, while in the propagation case they can have negative values. In addition, two different types of subordination principles are formulated for the two regimes. The Bernstein functions technique is extensively employed in the theoretical proofs.

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Transition from Diffusion to Wave Propagation in Fractional Jeffreys-Type Heat Conduction Equation Emilia Bazhlekova Ivan Bazhlekov doi: 10.3390/fractalfract4030032 Fractal and Fractional 2020-07-08 Fractal and Fractional 2020-07-08 4 3 Article 32 10.3390/fractalfract4030032 https://www.mdpi.com/2504-3110/4/3/32
Fractal Fract, Vol. 4, Pages 31: Existence Results for Fractional Order Single-Valued and Multi-Valued Problems with Integro-Multistrip-Multipoint Boundary Conditions https://www.mdpi.com/2504-3110/4/3/31 We study the existence of solutions for a new class of boundary value problems of arbitrary order fractional differential equations and inclusions, supplemented with integro-multistrip-multipoint boundary conditions. Suitable fixed point theorems are applied to prove some new existence results. The inclusion problem is discussed for convex valued as well as non-convex valued multi-valued map. Examples are also constructed to illustrate the main results. The results presented in this paper are not only new in the given configuration but also provide some interesting special cases. Fractal Fract, Vol. 4, Pages 31: Existence Results for Fractional Order Single-Valued and Multi-Valued Problems with Integro-Multistrip-Multipoint Boundary Conditions

Fractal and Fractional doi: 10.3390/fractalfract4030031

Authors: Sotiris K. Ntouyas Bashir Ahmad Ahmed Alsaedi

We study the existence of solutions for a new class of boundary value problems of arbitrary order fractional differential equations and inclusions, supplemented with integro-multistrip-multipoint boundary conditions. Suitable fixed point theorems are applied to prove some new existence results. The inclusion problem is discussed for convex valued as well as non-convex valued multi-valued map. Examples are also constructed to illustrate the main results. The results presented in this paper are not only new in the given configuration but also provide some interesting special cases.

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Existence Results for Fractional Order Single-Valued and Multi-Valued Problems with Integro-Multistrip-Multipoint Boundary Conditions Sotiris K. Ntouyas Bashir Ahmad Ahmed Alsaedi doi: 10.3390/fractalfract4030031 Fractal and Fractional 2020-07-05 Fractal and Fractional 2020-07-05 4 3 Article 31 10.3390/fractalfract4030031 https://www.mdpi.com/2504-3110/4/3/31
Fractal Fract, Vol. 4, Pages 30: Laplace Transform Method for Economic Models with Constant Proportional Caputo Derivative https://www.mdpi.com/2504-3110/4/3/30 In this study, we solved the economic models based on market equilibrium with constant proportional Caputo derivative using the Laplace transform. We proved the accuracy and efficiency of the method. We constructed the relations between the solutions of the problems and bivariate Mittag&amp;ndash;Leffler functions. Fractal Fract, Vol. 4, Pages 30: Laplace Transform Method for Economic Models with Constant Proportional Caputo Derivative

Fractal and Fractional doi: 10.3390/fractalfract4030030

Authors: Esra Karatas Akgül Ali Akgül Dumitru Baleanu

In this study, we solved the economic models based on market equilibrium with constant proportional Caputo derivative using the Laplace transform. We proved the accuracy and efficiency of the method. We constructed the relations between the solutions of the problems and bivariate Mittag&amp;ndash;Leffler functions.

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Laplace Transform Method for Economic Models with Constant Proportional Caputo Derivative Esra Karatas Akgül Ali Akgül Dumitru Baleanu doi: 10.3390/fractalfract4030030 Fractal and Fractional 2020-07-03 Fractal and Fractional 2020-07-03 4 3 Article 30 10.3390/fractalfract4030030 https://www.mdpi.com/2504-3110/4/3/30
Fractal Fract, Vol. 4, Pages 29: Functional Differential Equations Involving the ψ-Caputo Fractional Derivative https://www.mdpi.com/2504-3110/4/2/29 This paper is devoted to the study of existence and uniqueness of solutions for fractional functional differential equations, whose derivative operator depends on an arbitrary function. The introduction of such function allows generalization of some known results, and others can be also obtained. Fractal Fract, Vol. 4, Pages 29: Functional Differential Equations Involving the ψ-Caputo Fractional Derivative

Fractal and Fractional doi: 10.3390/fractalfract4020029

Authors: Ricardo Almeida

This paper is devoted to the study of existence and uniqueness of solutions for fractional functional differential equations, whose derivative operator depends on an arbitrary function. The introduction of such function allows generalization of some known results, and others can be also obtained.

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Functional Differential Equations Involving the ψ-Caputo Fractional Derivative Ricardo Almeida doi: 10.3390/fractalfract4020029 Fractal and Fractional 2020-06-23 Fractal and Fractional 2020-06-23 4 2 Article 29 10.3390/fractalfract4020029 https://www.mdpi.com/2504-3110/4/2/29
Fractal Fract, Vol. 4, Pages 28: Comb Model with Non-Static Stochastic Resetting and Anomalous Diffusion https://www.mdpi.com/2504-3110/4/2/28 Nowadays, the stochastic resetting process is an attractive research topic in stochastic process. At the same time, a series of researches on stochastic diffusion in complex structures introduced ways to understand the anomalous diffusion in complex systems. In this work, we propose a non-static stochastic resetting model in the context of comb structure that consists of a structure formed by backbone in x axis and branches in y axis. Then, we find the exact analytical solutions for marginal distribution concerning x and y axis. Moreover, we show the time evolution behavior to mean square displacements (MSD) in both directions. As a consequence, the model revels that until the system reaches the equilibrium, i.e., constant MSD, there is a Brownian diffusion in y direction, i.e., &amp;lang; ( &amp;Delta; y ) 2 &amp;rang; &amp;prop; t , and a crossover between sub and ballistic diffusion behaviors in x direction, i.e., &amp;lang; ( &amp;Delta; x ) 2 &amp;rang; &amp;prop; t 1 2 and &amp;lang; ( &amp;Delta; x ) 2 &amp;rang; &amp;prop; t 2 respectively. For static stochastic resetting, the ballistic regime vanishes. Also, we consider the idealized model according to the memory kernels to investigate the exponential and tempered power-law memory kernels effects on diffusive behaviors. In this way, we expose a rich class of anomalous diffusion process with crossovers among them. The proposal and the techniques applied in this work are useful to describe random walkers with non-static stochastic resetting on comb structure. Fractal Fract, Vol. 4, Pages 28: Comb Model with Non-Static Stochastic Resetting and Anomalous Diffusion

Fractal and Fractional doi: 10.3390/fractalfract4020028

Authors: Maike Antonio Faustino dos Santos

Nowadays, the stochastic resetting process is an attractive research topic in stochastic process. At the same time, a series of researches on stochastic diffusion in complex structures introduced ways to understand the anomalous diffusion in complex systems. In this work, we propose a non-static stochastic resetting model in the context of comb structure that consists of a structure formed by backbone in x axis and branches in y axis. Then, we find the exact analytical solutions for marginal distribution concerning x and y axis. Moreover, we show the time evolution behavior to mean square displacements (MSD) in both directions. As a consequence, the model revels that until the system reaches the equilibrium, i.e., constant MSD, there is a Brownian diffusion in y direction, i.e., &amp;lang; ( &amp;Delta; y ) 2 &amp;rang; &amp;prop; t , and a crossover between sub and ballistic diffusion behaviors in x direction, i.e., &amp;lang; ( &amp;Delta; x ) 2 &amp;rang; &amp;prop; t 1 2 and &amp;lang; ( &amp;Delta; x ) 2 &amp;rang; &amp;prop; t 2 respectively. For static stochastic resetting, the ballistic regime vanishes. Also, we consider the idealized model according to the memory kernels to investigate the exponential and tempered power-law memory kernels effects on diffusive behaviors. In this way, we expose a rich class of anomalous diffusion process with crossovers among them. The proposal and the techniques applied in this work are useful to describe random walkers with non-static stochastic resetting on comb structure.

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Comb Model with Non-Static Stochastic Resetting and Anomalous Diffusion Maike Antonio Faustino dos Santos doi: 10.3390/fractalfract4020028 Fractal and Fractional 2020-06-22 Fractal and Fractional 2020-06-22 4 2 Article 28 10.3390/fractalfract4020028 https://www.mdpi.com/2504-3110/4/2/28
Fractal Fract, Vol. 4, Pages 27: Numerical Solution of Fractional Order Burgers’ Equation with Dirichlet and Neumann Boundary Conditions by Reproducing Kernel Method https://www.mdpi.com/2504-3110/4/2/27 In this research, obtaining of approximate solution for fractional-order Burgers&amp;rsquo; equation will be presented in reproducing kernel Hilbert space (RKHS). Some special reproducing kernel spaces are identified according to inner products and norms. Then an iterative approach is constructed by using kernel functions. The convergence of this approach and its error estimates are given. The numerical algorithm of the method is presented. Furthermore, numerical outcomes are shown with tables and graphics for some examples. These outcomes demonstrate that the proposed method is convenient and effective. Fractal Fract, Vol. 4, Pages 27: Numerical Solution of Fractional Order Burgers’ Equation with Dirichlet and Neumann Boundary Conditions by Reproducing Kernel Method

Fractal and Fractional doi: 10.3390/fractalfract4020027

Authors: Onur Saldır Mehmet Giyas Sakar Fevzi Erdogan

In this research, obtaining of approximate solution for fractional-order Burgers&amp;rsquo; equation will be presented in reproducing kernel Hilbert space (RKHS). Some special reproducing kernel spaces are identified according to inner products and norms. Then an iterative approach is constructed by using kernel functions. The convergence of this approach and its error estimates are given. The numerical algorithm of the method is presented. Furthermore, numerical outcomes are shown with tables and graphics for some examples. These outcomes demonstrate that the proposed method is convenient and effective.

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Numerical Solution of Fractional Order Burgers’ Equation with Dirichlet and Neumann Boundary Conditions by Reproducing Kernel Method Onur Saldır Mehmet Giyas Sakar Fevzi Erdogan doi: 10.3390/fractalfract4020027 Fractal and Fractional 2020-06-19 Fractal and Fractional 2020-06-19 4 2 Article 27 10.3390/fractalfract4020027 https://www.mdpi.com/2504-3110/4/2/27
Fractal Fract, Vol. 4, Pages 26: On the Fractional Maximal Delta Integral Type Inequalities on Time Scales https://www.mdpi.com/2504-3110/4/2/26 Time scales have been the target of work of many mathematicians for more than a quarter century. Some of these studies are of inequalities and dynamic integrals. Inequalities and fractional maximal integrals have an important place in these studies. For example, inequalities and integrals contributed to the solution of many problems in various branches of science. In this paper, we will use fractional maximal integrals to establish integral inequalities on time scales. Moreover, our findings show that inequality is valid for discrete and continuous conditions. Fractal Fract, Vol. 4, Pages 26: On the Fractional Maximal Delta Integral Type Inequalities on Time Scales

Fractal and Fractional doi: 10.3390/fractalfract4020026

Authors: Lütfi Akın

Time scales have been the target of work of many mathematicians for more than a quarter century. Some of these studies are of inequalities and dynamic integrals. Inequalities and fractional maximal integrals have an important place in these studies. For example, inequalities and integrals contributed to the solution of many problems in various branches of science. In this paper, we will use fractional maximal integrals to establish integral inequalities on time scales. Moreover, our findings show that inequality is valid for discrete and continuous conditions.

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On the Fractional Maximal Delta Integral Type Inequalities on Time Scales Lütfi Akın doi: 10.3390/fractalfract4020026 Fractal and Fractional 2020-06-17 Fractal and Fractional 2020-06-17 4 2 Article 26 10.3390/fractalfract4020026 https://www.mdpi.com/2504-3110/4/2/26
Fractal Fract, Vol. 4, Pages 25: Miniaturization of a Koch-Type Fractal Antenna for Wi-Fi Applications https://www.mdpi.com/2504-3110/4/2/25 Koch-type wire dipole antennas are considered herein. In the case of a first-order prefractal, such antennas differ from a Koch-type dipole by the position of the central vertex of the dipole arm. Earlier, we investigated the dependence of the base frequency for different antenna scales for an arm in the form of a first-order prefractal. In this paper, dipoles for second-order prefractals are considered. The dependence of the base frequency and the reflection coefficient on the dipole wire length and scale is analyzed. It is shown that it is possible to distinguish a family of antennas operating at a given (identical) base frequency. The same length of a Koch-type curve can be obtained with different coordinates of the central vertex. This allows for obtaining numerous antennas with various scales and geometries of the arm. An algorithm for obtaining small antennas for Wi-Fi applications is proposed. Two antennas were obtained: an antenna with the smallest linear dimensions and a minimum antenna for a given reflection coefficient. Fractal Fract, Vol. 4, Pages 25: Miniaturization of a Koch-Type Fractal Antenna for Wi-Fi Applications

Fractal and Fractional doi: 10.3390/fractalfract4020025

Authors: Dmitrii Tumakov Dmitry Chikrin Petr Kokunin

Koch-type wire dipole antennas are considered herein. In the case of a first-order prefractal, such antennas differ from a Koch-type dipole by the position of the central vertex of the dipole arm. Earlier, we investigated the dependence of the base frequency for different antenna scales for an arm in the form of a first-order prefractal. In this paper, dipoles for second-order prefractals are considered. The dependence of the base frequency and the reflection coefficient on the dipole wire length and scale is analyzed. It is shown that it is possible to distinguish a family of antennas operating at a given (identical) base frequency. The same length of a Koch-type curve can be obtained with different coordinates of the central vertex. This allows for obtaining numerous antennas with various scales and geometries of the arm. An algorithm for obtaining small antennas for Wi-Fi applications is proposed. Two antennas were obtained: an antenna with the smallest linear dimensions and a minimum antenna for a given reflection coefficient.

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Miniaturization of a Koch-Type Fractal Antenna for Wi-Fi Applications Dmitrii Tumakov Dmitry Chikrin Petr Kokunin doi: 10.3390/fractalfract4020025 Fractal and Fractional 2020-06-04 Fractal and Fractional 2020-06-04 4 2 Article 25 10.3390/fractalfract4020025 https://www.mdpi.com/2504-3110/4/2/25
Fractal Fract, Vol. 4, Pages 24: Rotationally Symmetric Lacunary Functions and Products of Centered Polygonal Lacunary Functions https://www.mdpi.com/2504-3110/4/2/24 This work builds upon previous studies of centered polygonal lacunary functions by presenting proofs of theorems showing how rotational and dihedral mirror symmetry manifest in these lacunary functions at the modulus level. These theorems then provide a general framework for constructing other lacunary functions that exhibit the same symmetries. These investigations enable one to better explore the effects of the gap behavior on the qualitative features of the associated lacunary functions. Further, two renormalized products of centered polygonal lacunary functions are defined and a connection to Ramanunjan&amp;rsquo;s triangular lacunary series is made via several theorems. Fractal Fract, Vol. 4, Pages 24: Rotationally Symmetric Lacunary Functions and Products of Centered Polygonal Lacunary Functions

Fractal and Fractional doi: 10.3390/fractalfract4020024

Authors: L. K. Mork Keith Sullivan Trenton Vogt Darin J. Ulness

This work builds upon previous studies of centered polygonal lacunary functions by presenting proofs of theorems showing how rotational and dihedral mirror symmetry manifest in these lacunary functions at the modulus level. These theorems then provide a general framework for constructing other lacunary functions that exhibit the same symmetries. These investigations enable one to better explore the effects of the gap behavior on the qualitative features of the associated lacunary functions. Further, two renormalized products of centered polygonal lacunary functions are defined and a connection to Ramanunjan&amp;rsquo;s triangular lacunary series is made via several theorems.

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Rotationally Symmetric Lacunary Functions and Products of Centered Polygonal Lacunary Functions L. K. Mork Keith Sullivan Trenton Vogt Darin J. Ulness doi: 10.3390/fractalfract4020024 Fractal and Fractional 2020-05-26 Fractal and Fractional 2020-05-26 4 2 Article 24 10.3390/fractalfract4020024 https://www.mdpi.com/2504-3110/4/2/24
Fractal Fract, Vol. 4, Pages 23: Fractional State Space Description: A Particular Case of the Volterra Equations https://www.mdpi.com/2504-3110/4/2/23 To tackle several limitations recently highlighted in the field of fractional differentiation and fractional models, some authors have proposed new kernels for the definition of fractional integration/differentiation operators. Some limitations still remain, however, with these kernels, whereas solutions prior to the introduction of fractional models exist in the literature. This paper shows that the fractional pseudo state space description, a fractional model widely used in the literature, is a special case of the Volterra equations, equations introduced nearly a century ago. Volterra equations can thus be viewed as a serious alternative to fractional pseudo state space descriptions for modelling power law type long memory behaviours. This paper thus presents a new class of model involving a Volterra equation and several kernels associated with this equation capable of generating power law behaviours of various kinds. One is particularly interesting as it permits a power law behaviour in a given frequency band and, thus, a limited memory effect on a given time range (as the memory length is finite, the description does not exhibit infinitely slow and infinitely fast time constants as for pseudo state space descriptions). Fractal Fract, Vol. 4, Pages 23: Fractional State Space Description: A Particular Case of the Volterra Equations

Fractal and Fractional doi: 10.3390/fractalfract4020023

Authors: Jocelyn Sabatier

To tackle several limitations recently highlighted in the field of fractional differentiation and fractional models, some authors have proposed new kernels for the definition of fractional integration/differentiation operators. Some limitations still remain, however, with these kernels, whereas solutions prior to the introduction of fractional models exist in the literature. This paper shows that the fractional pseudo state space description, a fractional model widely used in the literature, is a special case of the Volterra equations, equations introduced nearly a century ago. Volterra equations can thus be viewed as a serious alternative to fractional pseudo state space descriptions for modelling power law type long memory behaviours. This paper thus presents a new class of model involving a Volterra equation and several kernels associated with this equation capable of generating power law behaviours of various kinds. One is particularly interesting as it permits a power law behaviour in a given frequency band and, thus, a limited memory effect on a given time range (as the memory length is finite, the description does not exhibit infinitely slow and infinitely fast time constants as for pseudo state space descriptions).

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Fractional State Space Description: A Particular Case of the Volterra Equations Jocelyn Sabatier doi: 10.3390/fractalfract4020023 Fractal and Fractional 2020-05-22 Fractal and Fractional 2020-05-22 4 2 Article 23 10.3390/fractalfract4020023 https://www.mdpi.com/2504-3110/4/2/23
Fractal Fract, Vol. 4, Pages 22: A Survey of Fractional Order Calculus Applications of Multiple-Input, Multiple-Output (MIMO) Process Control https://www.mdpi.com/2504-3110/4/2/22 Multiple-input multiple-output (MIMO) systems are usually present in process systems engineering. Due to the interaction among the variables and loops in the MIMO system, designing efficient control systems for both servo and regulatory scenarios remains a challenging task. The literature reports the use of several techniques mainly based on classical approaches, such as the proportional-integral-derivative (PID) controller, for single-input single-output (SISO) systems control. Furthermore, control system design approaches based on derivatives and integrals of non-integer order, also known as fractional control or fractional order (FO) control, are frequently used for SISO systems control. A natural consequence, already reported in the literature, is the application of these techniques to MIMO systems to address some inherent issues. Therefore, this work discusses the state-of-the-art of fractional control applied to MIMO systems. It outlines different types of applications, fractional controllers, controller tuning rules, experimental validation, software, and appropriate loop decoupling techniques, leading to literature gaps and research opportunities. The span of publications explored in this survey ranged from the years 1997 to 2019. Fractal Fract, Vol. 4, Pages 22: A Survey of Fractional Order Calculus Applications of Multiple-Input, Multiple-Output (MIMO) Process Control

Fractal and Fractional doi: 10.3390/fractalfract4020022

Authors: Alexandre Marques de Almeida Marcelo Kaminski Lenzi Ervin Kaminski Lenzi

Multiple-input multiple-output (MIMO) systems are usually present in process systems engineering. Due to the interaction among the variables and loops in the MIMO system, designing efficient control systems for both servo and regulatory scenarios remains a challenging task. The literature reports the use of several techniques mainly based on classical approaches, such as the proportional-integral-derivative (PID) controller, for single-input single-output (SISO) systems control. Furthermore, control system design approaches based on derivatives and integrals of non-integer order, also known as fractional control or fractional order (FO) control, are frequently used for SISO systems control. A natural consequence, already reported in the literature, is the application of these techniques to MIMO systems to address some inherent issues. Therefore, this work discusses the state-of-the-art of fractional control applied to MIMO systems. It outlines different types of applications, fractional controllers, controller tuning rules, experimental validation, software, and appropriate loop decoupling techniques, leading to literature gaps and research opportunities. The span of publications explored in this survey ranged from the years 1997 to 2019.

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A Survey of Fractional Order Calculus Applications of Multiple-Input, Multiple-Output (MIMO) Process Control Alexandre Marques de Almeida Marcelo Kaminski Lenzi Ervin Kaminski Lenzi doi: 10.3390/fractalfract4020022 Fractal and Fractional 2020-05-19 Fractal and Fractional 2020-05-19 4 2 Review 22 10.3390/fractalfract4020022 https://www.mdpi.com/2504-3110/4/2/22
Fractal Fract, Vol. 4, Pages 21: Exact Solution of Two-Dimensional Fractional Partial Differential Equations https://www.mdpi.com/2504-3110/4/2/21 In this study, we examine adapting and using the Sumudu decomposition method (SDM) as a way to find approximate solutions to two-dimensional fractional partial differential equations and propose a numerical algorithm for solving fractional Riccati equation. This method is a combination of the Sumudu transform method and decomposition method. The fractional derivative is described in the Caputo sense. The results obtained show that the approach is easy to implement and accurate when applied to various fractional differential equations. Fractal Fract, Vol. 4, Pages 21: Exact Solution of Two-Dimensional Fractional Partial Differential Equations

Fractal and Fractional doi: 10.3390/fractalfract4020021

Authors: Dumitru Baleanu Hassan Kamil Jassim

In this study, we examine adapting and using the Sumudu decomposition method (SDM) as a way to find approximate solutions to two-dimensional fractional partial differential equations and propose a numerical algorithm for solving fractional Riccati equation. This method is a combination of the Sumudu transform method and decomposition method. The fractional derivative is described in the Caputo sense. The results obtained show that the approach is easy to implement and accurate when applied to various fractional differential equations.

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Exact Solution of Two-Dimensional Fractional Partial Differential Equations Dumitru Baleanu Hassan Kamil Jassim doi: 10.3390/fractalfract4020021 Fractal and Fractional 2020-05-12 Fractal and Fractional 2020-05-12 4 2 Article 21 10.3390/fractalfract4020021 https://www.mdpi.com/2504-3110/4/2/21
Fractal Fract, Vol. 4, Pages 20: Simplified Mathematical Model for the Description of Anomalous Migration of Soluble Substances in Vertical Filtration Flow https://www.mdpi.com/2504-3110/4/2/20 Since the use of the fractional-differential mathematical model of anomalous geomigration process based on the MIM (mobile&amp;ndash;immoble media) approach in engineering practice significantly complicates simulations, a corresponding simplified mathematical model is constructed. For this model, we state a two-dimensional initial-boundary value problem of convective diffusion of soluble substances under the conditions of vertical steady-state filtration of groundwater with free surface from a reservoir to a coastal drain. To simplify the domain of simulation, we use the technique of transition into the domain of complex flow potential through a conformal mapping. In the case of averaging filtration velocity over the domain of complex flow potential, an analytical solution of the considered problem is obtained. In the general case of a variable filtration velocity, an algorithm has been developed to obtain numerical solutions. The results of process simulation using the presented algorithm shows that the constructed mathematical model can be efficiently used to simplify and accelerate modeling process. Fractal Fract, Vol. 4, Pages 20: Simplified Mathematical Model for the Description of Anomalous Migration of Soluble Substances in Vertical Filtration Flow

Fractal and Fractional doi: 10.3390/fractalfract4020020

Authors: Vsevolod Bohaienko Volodymyr Bulavatsky

Since the use of the fractional-differential mathematical model of anomalous geomigration process based on the MIM (mobile&amp;ndash;immoble media) approach in engineering practice significantly complicates simulations, a corresponding simplified mathematical model is constructed. For this model, we state a two-dimensional initial-boundary value problem of convective diffusion of soluble substances under the conditions of vertical steady-state filtration of groundwater with free surface from a reservoir to a coastal drain. To simplify the domain of simulation, we use the technique of transition into the domain of complex flow potential through a conformal mapping. In the case of averaging filtration velocity over the domain of complex flow potential, an analytical solution of the considered problem is obtained. In the general case of a variable filtration velocity, an algorithm has been developed to obtain numerical solutions. The results of process simulation using the presented algorithm shows that the constructed mathematical model can be efficiently used to simplify and accelerate modeling process.

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Simplified Mathematical Model for the Description of Anomalous Migration of Soluble Substances in Vertical Filtration Flow Vsevolod Bohaienko Volodymyr Bulavatsky doi: 10.3390/fractalfract4020020 Fractal and Fractional 2020-05-06 Fractal and Fractional 2020-05-06 4 2 Article 20 10.3390/fractalfract4020020 https://www.mdpi.com/2504-3110/4/2/20
Fractal Fract, Vol. 4, Pages 19: Fractional Kinetic Equations Associated with Incomplete I-Functions https://www.mdpi.com/2504-3110/4/2/19 In this paper, we investigate the solution of fractional kinetic equation (FKE) associated with the incomplete I-function (IIF) by using the well-known integral transform (Laplace transform). The FKE plays a great role in solving astrophysical problems. The solutions are represented in terms of IIF. Next, we present some interesting corollaries by specializing the parameters of IIF in the form of simpler special functions and also mention a few known results, which are very useful in solving physical or real-life problems. Finally, some graphical results are presented to demonstrate the influence of the order of the fractional integral operator on the reaction rate. Fractal Fract, Vol. 4, Pages 19: Fractional Kinetic Equations Associated with Incomplete I-Functions

Fractal and Fractional doi: 10.3390/fractalfract4020019

Authors: Manish Kumar Bansal Devendra Kumar Priyanka Harjule Jagdev Singh

In this paper, we investigate the solution of fractional kinetic equation (FKE) associated with the incomplete I-function (IIF) by using the well-known integral transform (Laplace transform). The FKE plays a great role in solving astrophysical problems. The solutions are represented in terms of IIF. Next, we present some interesting corollaries by specializing the parameters of IIF in the form of simpler special functions and also mention a few known results, which are very useful in solving physical or real-life problems. Finally, some graphical results are presented to demonstrate the influence of the order of the fractional integral operator on the reaction rate.

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Fractional Kinetic Equations Associated with Incomplete I-Functions Manish Kumar Bansal Devendra Kumar Priyanka Harjule Jagdev Singh doi: 10.3390/fractalfract4020019 Fractal and Fractional 2020-05-04 Fractal and Fractional 2020-05-04 4 2 Article 19 10.3390/fractalfract4020019 https://www.mdpi.com/2504-3110/4/2/19
Fractal Fract, Vol. 4, Pages 18: Multi-Strip and Multi-Point Boundary Conditions for Fractional Langevin Equation https://www.mdpi.com/2504-3110/4/2/18 In the present paper, we discuss a new boundary value problem for the nonlinear Langevin equation involving two distinct fractional derivative orders with multi-point and multi-nonlocal integral conditions. The fixed point theorems for Schauder and Krasnoselskii&amp;ndash;Zabreiko are applied to study the existence results. The uniqueness of the solution is given by implementing the Banach fixed point theorem. Some examples showing our basic results are provided. Fractal Fract, Vol. 4, Pages 18: Multi-Strip and Multi-Point Boundary Conditions for Fractional Langevin Equation

Fractal and Fractional doi: 10.3390/fractalfract4020018

Authors: Ahmed Salem Balqees Alghamdi

In the present paper, we discuss a new boundary value problem for the nonlinear Langevin equation involving two distinct fractional derivative orders with multi-point and multi-nonlocal integral conditions. The fixed point theorems for Schauder and Krasnoselskii&amp;ndash;Zabreiko are applied to study the existence results. The uniqueness of the solution is given by implementing the Banach fixed point theorem. Some examples showing our basic results are provided.

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Multi-Strip and Multi-Point Boundary Conditions for Fractional Langevin Equation Ahmed Salem Balqees Alghamdi doi: 10.3390/fractalfract4020018 Fractal and Fractional 2020-04-28 Fractal and Fractional 2020-04-28 4 2 Article 18 10.3390/fractalfract4020018 https://www.mdpi.com/2504-3110/4/2/18
Fractal Fract, Vol. 4, Pages 17: Kinetic Model for Drying in Frame of Generalized Fractional Derivatives https://www.mdpi.com/2504-3110/4/2/17 In this article, the Lewis model was considered for the soybean drying process by new fractional differential operators to analyze the estimated time in 50 ∘ C , 60 ∘ C , 70 ∘ C , and 80 ∘ C . Moreover, we used dimension parameters for the physical meaning of these fractional models within generalized and Caputo fractional derivatives. Results obtained with generalized fractional derivatives were analyzed comparatively with the Caputo fractional, integer order derivatives and Page model for the soybean drying process. All results for fractional derivatives are discussed and compared in detail. Fractal Fract, Vol. 4, Pages 17: Kinetic Model for Drying in Frame of Generalized Fractional Derivatives

Fractal and Fractional doi: 10.3390/fractalfract4020017

Authors: Ramazan Ozarslan Erdal Bas

In this article, the Lewis model was considered for the soybean drying process by new fractional differential operators to analyze the estimated time in 50 ∘ C , 60 ∘ C , 70 ∘ C , and 80 ∘ C . Moreover, we used dimension parameters for the physical meaning of these fractional models within generalized and Caputo fractional derivatives. Results obtained with generalized fractional derivatives were analyzed comparatively with the Caputo fractional, integer order derivatives and Page model for the soybean drying process. All results for fractional derivatives are discussed and compared in detail.

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Kinetic Model for Drying in Frame of Generalized Fractional Derivatives Ramazan Ozarslan Erdal Bas doi: 10.3390/fractalfract4020017 Fractal and Fractional 2020-04-24 Fractal and Fractional 2020-04-24 4 2 Article 17 10.3390/fractalfract4020017 https://www.mdpi.com/2504-3110/4/2/17
Fractal Fract, Vol. 4, Pages 16: Pricing Path-Independent Payoffs with Exotic Features in the Fractional Diffusion Model https://www.mdpi.com/2504-3110/4/2/16 We provide several practical formulas for pricing path-independent exotic instruments (log options and log contracts, digital options, gap options, power options with or without capped payoffs &amp;hellip;) in the context of the fractional diffusion model. This model combines a tail parameter governed by the space fractional derivative, and a subordination parameter governed by the time-fractional derivative. The pricing formulas we derive take the form of quickly convergent series of powers of the moneyness and of the convexity adjustment; they are obtained thanks to a factorized formula in the Mellin space valid for arbitrary payoffs, and by means of residue theory. We also discuss other aspects of option pricing such as volatility modeling, and provide comparisons of our results with other financial models. Fractal Fract, Vol. 4, Pages 16: Pricing Path-Independent Payoffs with Exotic Features in the Fractional Diffusion Model

Fractal and Fractional doi: 10.3390/fractalfract4020016

Authors: Jean-Philippe Aguilar

We provide several practical formulas for pricing path-independent exotic instruments (log options and log contracts, digital options, gap options, power options with or without capped payoffs &amp;hellip;) in the context of the fractional diffusion model. This model combines a tail parameter governed by the space fractional derivative, and a subordination parameter governed by the time-fractional derivative. The pricing formulas we derive take the form of quickly convergent series of powers of the moneyness and of the convexity adjustment; they are obtained thanks to a factorized formula in the Mellin space valid for arbitrary payoffs, and by means of residue theory. We also discuss other aspects of option pricing such as volatility modeling, and provide comparisons of our results with other financial models.

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Pricing Path-Independent Payoffs with Exotic Features in the Fractional Diffusion Model Jean-Philippe Aguilar doi: 10.3390/fractalfract4020016 Fractal and Fractional 2020-04-20 Fractal and Fractional 2020-04-20 4 2 Article 16 10.3390/fractalfract4020016 https://www.mdpi.com/2504-3110/4/2/16
Fractal Fract, Vol. 4, Pages 15: Fractional Model for a Class of Diffusion-Reaction Equation Represented by the Fractional-Order Derivative https://www.mdpi.com/2504-3110/4/2/15 This paper proposes the analytical solution for a class of the fractional diffusion equation represented by the fractional-order derivative. We mainly use the Grunwald&amp;ndash;Letnikov derivative in this paper. We are particularly interested in the application of the Laplace transform proposed for this fractional operator. We offer the analytical solution of the fractional model as the diffusion equation with a reaction term expressed by the Grunwald&amp;ndash;Letnikov derivative by using a double integration method. To illustrate our findings in this paper, we represent the analytical solutions for different values of the used fractional-order derivative. Fractal Fract, Vol. 4, Pages 15: Fractional Model for a Class of Diffusion-Reaction Equation Represented by the Fractional-Order Derivative

Fractal and Fractional doi: 10.3390/fractalfract4020015

Authors: Ndolane Sene

This paper proposes the analytical solution for a class of the fractional diffusion equation represented by the fractional-order derivative. We mainly use the Grunwald&amp;ndash;Letnikov derivative in this paper. We are particularly interested in the application of the Laplace transform proposed for this fractional operator. We offer the analytical solution of the fractional model as the diffusion equation with a reaction term expressed by the Grunwald&amp;ndash;Letnikov derivative by using a double integration method. To illustrate our findings in this paper, we represent the analytical solutions for different values of the used fractional-order derivative.

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Fractional Model for a Class of Diffusion-Reaction Equation Represented by the Fractional-Order Derivative Ndolane Sene doi: 10.3390/fractalfract4020015 Fractal and Fractional 2020-04-16 Fractal and Fractional 2020-04-16 4 2 Article 15 10.3390/fractalfract4020015 https://www.mdpi.com/2504-3110/4/2/15
Fractal Fract, Vol. 4, Pages 14: Fractional Derivatives and Dynamical Systems in Material Instability https://www.mdpi.com/2504-3110/4/2/14 Loss of stability is studied extensively in nonlinear investigations, and classified as generic bifurcations. It requires regularity, being connected with non-locality. Such behavior comes from gradient terms in constitutive equations. Most fractional derivatives are non-local, thus by using them in defining strain, a non-local strain appears. In such a way, various versions of non-localities are obtained by using various types of fractional derivatives. The study aims for constitutive modeling via instability phenomena, that is, by observing the way of loss of stability of material, we can be informed about the proper form of its mathematical model. Fractal Fract, Vol. 4, Pages 14: Fractional Derivatives and Dynamical Systems in Material Instability

Fractal and Fractional doi: 10.3390/fractalfract4020014

Authors: Peter B. Béda

Loss of stability is studied extensively in nonlinear investigations, and classified as generic bifurcations. It requires regularity, being connected with non-locality. Such behavior comes from gradient terms in constitutive equations. Most fractional derivatives are non-local, thus by using them in defining strain, a non-local strain appears. In such a way, various versions of non-localities are obtained by using various types of fractional derivatives. The study aims for constitutive modeling via instability phenomena, that is, by observing the way of loss of stability of material, we can be informed about the proper form of its mathematical model.

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Fractional Derivatives and Dynamical Systems in Material Instability Peter B. Béda doi: 10.3390/fractalfract4020014 Fractal and Fractional 2020-04-16 Fractal and Fractional 2020-04-16 4 2 Article 14 10.3390/fractalfract4020014 https://www.mdpi.com/2504-3110/4/2/14
Fractal Fract, Vol. 4, Pages 13: Existence and Uniqueness Results for a Coupled System of Caputo-Hadamard Fractional Differential Equations with Nonlocal Hadamard Type Integral Boundary Conditions https://www.mdpi.com/2504-3110/4/2/13 In this paper, we study a coupled system of Caputo-Hadamard type sequential fractional differential equations supplemented with nonlocal boundary conditions involving Hadamard fractional integrals. The sufficient criteria ensuring the existence and uniqueness of solutions for the given problem are obtained. We make use of the Leray-Schauder alternative and contraction mapping principle to derive the desired results. Illustrative examples for the main results are also presented. Fractal Fract, Vol. 4, Pages 13: Existence and Uniqueness Results for a Coupled System of Caputo-Hadamard Fractional Differential Equations with Nonlocal Hadamard Type Integral Boundary Conditions

Fractal and Fractional doi: 10.3390/fractalfract4020013

Authors: Shorog Aljoudi Bashir Ahmad Ahmed Alsaedi

In this paper, we study a coupled system of Caputo-Hadamard type sequential fractional differential equations supplemented with nonlocal boundary conditions involving Hadamard fractional integrals. The sufficient criteria ensuring the existence and uniqueness of solutions for the given problem are obtained. We make use of the Leray-Schauder alternative and contraction mapping principle to derive the desired results. Illustrative examples for the main results are also presented.

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Existence and Uniqueness Results for a Coupled System of Caputo-Hadamard Fractional Differential Equations with Nonlocal Hadamard Type Integral Boundary Conditions Shorog Aljoudi Bashir Ahmad Ahmed Alsaedi doi: 10.3390/fractalfract4020013 Fractal and Fractional 2020-04-12 Fractal and Fractional 2020-04-12 4 2 Article 13 10.3390/fractalfract4020013 https://www.mdpi.com/2504-3110/4/2/13
Fractal Fract, Vol. 4, Pages 12: Fractional-Order Models for Biochemical Processes https://www.mdpi.com/2504-3110/4/2/12 Biochemical processes present complex mechanisms and can be described by various computational models. Complex systems present a variety of problems, especially the loss of intuitive understanding. The present work uses fractional-order calculus to obtain mathematical models for erythritol and mannitol synthesis. The obtained models are useful for both prediction and process optimization. The models present the complex behavior of the process due to the fractional order, without losing the physical meaning of gain and time constants. To validate each obtained model, the simulation results were compared with experimental data. In order to highlight the advantages of fractional-order models, comparisons with the corresponding integer-order models are presented. Fractal Fract, Vol. 4, Pages 12: Fractional-Order Models for Biochemical Processes

Fractal and Fractional doi: 10.3390/fractalfract4020012

Authors: Eva-H. Dulf Dan C. Vodnar Alex Danku Cristina-I. Muresan Ovidiu Crisan

Biochemical processes present complex mechanisms and can be described by various computational models. Complex systems present a variety of problems, especially the loss of intuitive understanding. The present work uses fractional-order calculus to obtain mathematical models for erythritol and mannitol synthesis. The obtained models are useful for both prediction and process optimization. The models present the complex behavior of the process due to the fractional order, without losing the physical meaning of gain and time constants. To validate each obtained model, the simulation results were compared with experimental data. In order to highlight the advantages of fractional-order models, comparisons with the corresponding integer-order models are presented.

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Fractional-Order Models for Biochemical Processes Eva-H. Dulf Dan C. Vodnar Alex Danku Cristina-I. Muresan Ovidiu Crisan doi: 10.3390/fractalfract4020012 Fractal and Fractional 2020-04-10 Fractal and Fractional 2020-04-10 4 2 Article 12 10.3390/fractalfract4020012 https://www.mdpi.com/2504-3110/4/2/12
Fractal Fract, Vol. 4, Pages 11: Fractal, Scale Free Electromagnetic Resonance of a Single Brain Extracted Microtubule Nanowire, a Single Tubulin Protein and a Single Neuron https://www.mdpi.com/2504-3110/4/2/11 Biomaterials are primarily insulators. For nearly a century, electromagnetic resonance and antenna&amp;ndash;receiver properties have been measured and extensively theoretically modeled. The dielectric constituents of biomaterials&amp;mdash;if arranged in distinct symmetries, then each vibrational symmetry&amp;mdash;would lead to a distinct resonance frequency. While the literature is rich with data on the dielectric resonance of proteins, scale-free relationships of vibrational modes are scarce. Here, we report a self-similar triplet of triplet resonance frequency pattern for the four-4 nm-wide tubulin protein, for the 25-nm-wide microtubule nanowire and 1-&amp;mu;m-wide axon initial segment of a neuron. Thus, preserving the symmetry of vibrations was a fundamental integration feature of the three materials. There was no self-similarity in the physical appearance: the size varied by 106 orders, yet, when they vibrated, the ratios of the frequencies changed in such a way that each of the three resonance frequency bands held three more bands inside (triplet of triplet). This suggests that instead of symmetry, self-similarity lies in the principles of symmetry-breaking. This is why three elements, a protein, it&amp;rsquo;s complex and neuron resonated in 106 orders of different time domains, yet their vibrational frequencies grouped similarly. Our work supports already-existing hypotheses for the scale-free information integration in the brain from molecular scale to the cognition. Fractal Fract, Vol. 4, Pages 11: Fractal, Scale Free Electromagnetic Resonance of a Single Brain Extracted Microtubule Nanowire, a Single Tubulin Protein and a Single Neuron

Fractal and Fractional doi: 10.3390/fractalfract4020011

Authors: Komal Saxena Pushpendra Singh Pathik Sahoo Satyajit Sahu Subrata Ghosh Kanad Ray Daisuke Fujita Anirban Bandyopadhyay

Biomaterials are primarily insulators. For nearly a century, electromagnetic resonance and antenna&amp;ndash;receiver properties have been measured and extensively theoretically modeled. The dielectric constituents of biomaterials&amp;mdash;if arranged in distinct symmetries, then each vibrational symmetry&amp;mdash;would lead to a distinct resonance frequency. While the literature is rich with data on the dielectric resonance of proteins, scale-free relationships of vibrational modes are scarce. Here, we report a self-similar triplet of triplet resonance frequency pattern for the four-4 nm-wide tubulin protein, for the 25-nm-wide microtubule nanowire and 1-&amp;mu;m-wide axon initial segment of a neuron. Thus, preserving the symmetry of vibrations was a fundamental integration feature of the three materials. There was no self-similarity in the physical appearance: the size varied by 106 orders, yet, when they vibrated, the ratios of the frequencies changed in such a way that each of the three resonance frequency bands held three more bands inside (triplet of triplet). This suggests that instead of symmetry, self-similarity lies in the principles of symmetry-breaking. This is why three elements, a protein, it&amp;rsquo;s complex and neuron resonated in 106 orders of different time domains, yet their vibrational frequencies grouped similarly. Our work supports already-existing hypotheses for the scale-free information integration in the brain from molecular scale to the cognition.

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Fractal, Scale Free Electromagnetic Resonance of a Single Brain Extracted Microtubule Nanowire, a Single Tubulin Protein and a Single Neuron Komal Saxena Pushpendra Singh Pathik Sahoo Satyajit Sahu Subrata Ghosh Kanad Ray Daisuke Fujita Anirban Bandyopadhyay doi: 10.3390/fractalfract4020011 Fractal and Fractional 2020-04-06 Fractal and Fractional 2020-04-06 4 2 Article 11 10.3390/fractalfract4020011 https://www.mdpi.com/2504-3110/4/2/11
Fractal Fract, Vol. 4, Pages 10: Generalized Integral Inequalities of Chebyshev Type https://www.mdpi.com/2504-3110/4/2/10 In this paper, we present a number of Chebyshev type inequalities involving generalized integral operators, essentially motivated by the earlier works and their applications in diverse research subjects. Fractal Fract, Vol. 4, Pages 10: Generalized Integral Inequalities of Chebyshev Type

Fractal and Fractional doi: 10.3390/fractalfract4020010

Authors: Paulo M. Guzmán Péter Kórus Juan E. Nápoles Valdés

In this paper, we present a number of Chebyshev type inequalities involving generalized integral operators, essentially motivated by the earlier works and their applications in diverse research subjects.

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Generalized Integral Inequalities of Chebyshev Type Paulo M. Guzmán Péter Kórus Juan E. Nápoles Valdés doi: 10.3390/fractalfract4020010 Fractal and Fractional 2020-04-02 Fractal and Fractional 2020-04-02 4 2 Article 10 10.3390/fractalfract4020010 https://www.mdpi.com/2504-3110/4/2/10
Fractal Fract, Vol. 4, Pages 9: Homotopy Analysis Method to Solve Two-Dimensional Nonlinear Volterra-Fredholm Fuzzy Integral Equations https://www.mdpi.com/2504-3110/4/1/9 The main goal of the paper is to present an approximate method for solving of a two-dimensional nonlinear Volterra-Fredholm fuzzy integral equation (2D-NVFFIE). It is applied the homotopy analysis method (HAM). The studied equation is converted to a nonlinear system of Volterra-Fredholm integral equations in a crisp case. Approximate solutions of this system are obtained by the help with HAM and hence an approximation for the fuzzy solution of the nonlinear Volterra-Fredholm fuzzy integral equation is presented. The convergence of the proposed method is proved and the error estimate between the exact and the approximate solution is obtained. The validity and applicability of the proposed method is illustrated on a numerical example. Fractal Fract, Vol. 4, Pages 9: Homotopy Analysis Method to Solve Two-Dimensional Nonlinear Volterra-Fredholm Fuzzy Integral Equations

Fractal and Fractional doi: 10.3390/fractalfract4010009

The main goal of the paper is to present an approximate method for solving of a two-dimensional nonlinear Volterra-Fredholm fuzzy integral equation (2D-NVFFIE). It is applied the homotopy analysis method (HAM). The studied equation is converted to a nonlinear system of Volterra-Fredholm integral equations in a crisp case. Approximate solutions of this system are obtained by the help with HAM and hence an approximation for the fuzzy solution of the nonlinear Volterra-Fredholm fuzzy integral equation is presented. The convergence of the proposed method is proved and the error estimate between the exact and the approximate solution is obtained. The validity and applicability of the proposed method is illustrated on a numerical example.

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Homotopy Analysis Method to Solve Two-Dimensional Nonlinear Volterra-Fredholm Fuzzy Integral Equations Atanaska Georgieva Snezhana Hristova doi: 10.3390/fractalfract4010009 Fractal and Fractional 2020-03-18 Fractal and Fractional 2020-03-18 4 1 Article 9 10.3390/fractalfract4010009 https://www.mdpi.com/2504-3110/4/1/9
Fractal Fract, Vol. 4, Pages 8: Admissibility of Fractional Order Descriptor Systems Based on Complex Variables: An LMI Approach https://www.mdpi.com/2504-3110/4/1/8 This paper is devoted to the admissibility issue of singular fractional order systems with order &amp;alpha; &amp;isin; ( 0 , 1 ) based on complex variables. Firstly, with regard to admissibility, necessary and sufficient conditions are obtained by strict LMI in complex plane. Then, an observer-based controller is designed to ensure system admissible. Finally, numerical examples are given to reveal the validity of the theoretical conclusions. Fractal Fract, Vol. 4, Pages 8: Admissibility of Fractional Order Descriptor Systems Based on Complex Variables: An LMI Approach

Fractal and Fractional doi: 10.3390/fractalfract4010008

Authors: Xuefeng Zhang Yuqing Yan

This paper is devoted to the admissibility issue of singular fractional order systems with order &amp;alpha; &amp;isin; ( 0 , 1 ) based on complex variables. Firstly, with regard to admissibility, necessary and sufficient conditions are obtained by strict LMI in complex plane. Then, an observer-based controller is designed to ensure system admissible. Finally, numerical examples are given to reveal the validity of the theoretical conclusions.

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Admissibility of Fractional Order Descriptor Systems Based on Complex Variables: An LMI Approach Xuefeng Zhang Yuqing Yan doi: 10.3390/fractalfract4010008 Fractal and Fractional 2020-03-16 Fractal and Fractional 2020-03-16 4 1 Article 8 10.3390/fractalfract4010008 https://www.mdpi.com/2504-3110/4/1/8
Fractal Fract, Vol. 4, Pages 7: Kamenev-Type Asymptotic Criterion of Fourth-Order Delay Differential Equation https://www.mdpi.com/2504-3110/4/1/7 In this paper, we obtain necessary and sufficient conditions for a Kamenev-type oscillation criterion of a fourth order differential equation of the form r 3 t r 2 t r 1 t y &amp;prime; t &amp;prime; &amp;prime; &amp;prime; + q t f y &amp;sigma; t = 0 , where t &amp;ge; t 0 . The results presented here complement some of the known results reported in the literature. Moreover, the importance of the obtained conditions is illustrated via some examples. Fractal Fract, Vol. 4, Pages 7: Kamenev-Type Asymptotic Criterion of Fourth-Order Delay Differential Equation

Fractal and Fractional doi: 10.3390/fractalfract4010007

Authors: Omar Bazighifan

In this paper, we obtain necessary and sufficient conditions for a Kamenev-type oscillation criterion of a fourth order differential equation of the form r 3 t r 2 t r 1 t y &amp;prime; t &amp;prime; &amp;prime; &amp;prime; + q t f y &amp;sigma; t = 0 , where t &amp;ge; t 0 . The results presented here complement some of the known results reported in the literature. Moreover, the importance of the obtained conditions is illustrated via some examples.

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Kamenev-Type Asymptotic Criterion of Fourth-Order Delay Differential Equation Omar Bazighifan doi: 10.3390/fractalfract4010007 Fractal and Fractional 2020-03-10 Fractal and Fractional 2020-03-10 4 1 Article 7 10.3390/fractalfract4010007 https://www.mdpi.com/2504-3110/4/1/7
Fractal Fract, Vol. 4, Pages 6: Fractal Dimensions of Cell Wall in Growing Cotton Fibers https://www.mdpi.com/2504-3110/4/1/6 In this research, fractal properties of a cell wall in growing cotton fibers were studied. It was found that dependences of specific pore volume (P) and apparent density (&amp;rho;) on the scale factor, F = H/h, can be expressed by power-law equations: P = Po F(Dv&amp;minus;E) and &amp;rho; = &amp;rho;o F(E&amp;minus;D&amp;rho;), where h is minimum thickness of the microfibrilar network in the primary cell wall, H is total thickness of cell wall in growing cotton, Dv = 2.556 and D&amp;rho; = 2.988 are fractal dimensions. From the obtained results it follows that microfibrilar network of the primary cell wall in immature fibers is loose and disordered, and therefore it has an increased pore volume (Po = 0.037 cm3/g) and low density (&amp;rho;o = 1.47 g/cm3). With enhance days post anthesis of growing cotton fibers, the wall thickness and density increase, while the pore volume decreases, until dense structure of completely mature fibers is formed with maximum density (1.54 g/cm3) and minimum pore volume (0.006 cm3/g). The fractal dimension for specific pore volume, Dv = 2.556, evidences the mixed surface-volume sorption mechanism of sorbate vapor in the pores. On the other hand, the fractal dimension for apparent density, D&amp;rho; = 2.988, is very close to Euclidean volume dimension, E = 3, for the three-dimensional space. Fractal Fract, Vol. 4, Pages 6: Fractal Dimensions of Cell Wall in Growing Cotton Fibers

Fractal and Fractional doi: 10.3390/fractalfract4010006

Authors: Michael Ioelovich

In this research, fractal properties of a cell wall in growing cotton fibers were studied. It was found that dependences of specific pore volume (P) and apparent density (&amp;rho;) on the scale factor, F = H/h, can be expressed by power-law equations: P = Po F(Dv&amp;minus;E) and &amp;rho; = &amp;rho;o F(E&amp;minus;D&amp;rho;), where h is minimum thickness of the microfibrilar network in the primary cell wall, H is total thickness of cell wall in growing cotton, Dv = 2.556 and D&amp;rho; = 2.988 are fractal dimensions. From the obtained results it follows that microfibrilar network of the primary cell wall in immature fibers is loose and disordered, and therefore it has an increased pore volume (Po = 0.037 cm3/g) and low density (&amp;rho;o = 1.47 g/cm3). With enhance days post anthesis of growing cotton fibers, the wall thickness and density increase, while the pore volume decreases, until dense structure of completely mature fibers is formed with maximum density (1.54 g/cm3) and minimum pore volume (0.006 cm3/g). The fractal dimension for specific pore volume, Dv = 2.556, evidences the mixed surface-volume sorption mechanism of sorbate vapor in the pores. On the other hand, the fractal dimension for apparent density, D&amp;rho; = 2.988, is very close to Euclidean volume dimension, E = 3, for the three-dimensional space.

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Fractal Dimensions of Cell Wall in Growing Cotton Fibers Michael Ioelovich doi: 10.3390/fractalfract4010006 Fractal and Fractional 2020-03-09 Fractal and Fractional 2020-03-09 4 1 Article 6 10.3390/fractalfract4010006 https://www.mdpi.com/2504-3110/4/1/6
Fractal Fract, Vol. 4, Pages 5: Non-Differentiable Solution of Nonlinear Biological Population Model on Cantor Sets https://www.mdpi.com/2504-3110/4/1/5 The main objective of this study is to apply the local fractional homotopy analysis method (LFHAM) to obtain the non-differentiable solution of two nonlinear partial differential equations of the biological population model on Cantor sets. The derivative operator are taken in the local fractional sense. Two examples have been presented showing the effectiveness of this method in solving this model on Cantor sets. Fractal Fract, Vol. 4, Pages 5: Non-Differentiable Solution of Nonlinear Biological Population Model on Cantor Sets

Fractal and Fractional doi: 10.3390/fractalfract4010005

Authors: Djelloul Ziane Mountassir Hamdi Cherif Dumitru Baleanu Kacem Belghaba

The main objective of this study is to apply the local fractional homotopy analysis method (LFHAM) to obtain the non-differentiable solution of two nonlinear partial differential equations of the biological population model on Cantor sets. The derivative operator are taken in the local fractional sense. Two examples have been presented showing the effectiveness of this method in solving this model on Cantor sets.

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Non-Differentiable Solution of Nonlinear Biological Population Model on Cantor Sets Djelloul Ziane Mountassir Hamdi Cherif Dumitru Baleanu Kacem Belghaba doi: 10.3390/fractalfract4010005 Fractal and Fractional 2020-02-09 Fractal and Fractional 2020-02-09 4 1 Article 5 10.3390/fractalfract4010005 https://www.mdpi.com/2504-3110/4/1/5
Fractal Fract, Vol. 4, Pages 4: Acknowledgement to Reviewers of Fractal Fract in 2019 https://www.mdpi.com/2504-3110/4/1/4 The editorial team greatly appreciates the reviewers who have dedicated their considerable time and expertise to the journal&amp;rsquo;s rigorous editorial process over the past 12 months, regardless of whether the papers are finally published or not [...] Fractal Fract, Vol. 4, Pages 4: Acknowledgement to Reviewers of Fractal Fract in 2019

Fractal and Fractional doi: 10.3390/fractalfract4010004

Authors: Fractal and Fractional Editorial Office

The editorial team greatly appreciates the reviewers who have dedicated their considerable time and expertise to the journal&amp;rsquo;s rigorous editorial process over the past 12 months, regardless of whether the papers are finally published or not [...]

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Acknowledgement to Reviewers of Fractal Fract in 2019 Fractal and Fractional Editorial Office doi: 10.3390/fractalfract4010004 Fractal and Fractional 2020-01-23 Fractal and Fractional 2020-01-23 4 1 Editorial 4 10.3390/fractalfract4010004 https://www.mdpi.com/2504-3110/4/1/4
Fractal Fract, Vol. 4, Pages 3: Fractal Antennas: An Historical Perspective https://www.mdpi.com/2504-3110/4/1/3 Fractal geometry has been proven to be useful in several disciplines. In the field of antenna engineering, fractal geometry is useful to design small and multiband antenna and arrays, and high-directive elements. A historic overview of the most significant fractal mathematic pioneers is presented, at the same time showing how the fractal patterns inspired engineers to design antennas. Fractal Fract, Vol. 4, Pages 3: Fractal Antennas: An Historical Perspective

Fractal and Fractional doi: 10.3390/fractalfract4010003

Authors: Anguera Andújar Jayasinghe Chakravarthy Chowdary Pijoan Ali Cattani

Fractal geometry has been proven to be useful in several disciplines. In the field of antenna engineering, fractal geometry is useful to design small and multiband antenna and arrays, and high-directive elements. A historic overview of the most significant fractal mathematic pioneers is presented, at the same time showing how the fractal patterns inspired engineers to design antennas.

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Fractal Antennas: An Historical Perspective Anguera Andújar Jayasinghe Chakravarthy Chowdary Pijoan Ali Cattani doi: 10.3390/fractalfract4010003 Fractal and Fractional 2020-01-19 Fractal and Fractional 2020-01-19 4 1 Review 3 10.3390/fractalfract4010003 https://www.mdpi.com/2504-3110/4/1/3
Fractal Fract, Vol. 4, Pages 2: Parametric Identification of Nonlinear Fractional Hammerstein Models https://www.mdpi.com/2504-3110/4/1/2 In this paper, a system identification method for continuous fractional-order Hammerstein models is proposed. A block structured nonlinear system constituting a static nonlinear block followed by a fractional-order linear dynamic system is considered. The fractional differential operator is represented through the generalized operational matrix of block pulse functions to reduce computational complexity. A special test signal is developed to isolate the identification of the nonlinear static function from that of the fractional-order linear dynamic system. The merit of the proposed technique is indicated by concurrent identification of the fractional order with linear system coefficients, algebraic representation of the immeasurable nonlinear static function output, and permitting use of non-iterative procedures for identification of the nonlinearity. The efficacy of the proposed method is exhibited through simulation at various signal-to-noise ratios. Fractal Fract, Vol. 4, Pages 2: Parametric Identification of Nonlinear Fractional Hammerstein Models

Fractal and Fractional doi: 10.3390/fractalfract4010002

Authors: Vineet Prasad Kajal Kothari Utkal Mehta

In this paper, a system identification method for continuous fractional-order Hammerstein models is proposed. A block structured nonlinear system constituting a static nonlinear block followed by a fractional-order linear dynamic system is considered. The fractional differential operator is represented through the generalized operational matrix of block pulse functions to reduce computational complexity. A special test signal is developed to isolate the identification of the nonlinear static function from that of the fractional-order linear dynamic system. The merit of the proposed technique is indicated by concurrent identification of the fractional order with linear system coefficients, algebraic representation of the immeasurable nonlinear static function output, and permitting use of non-iterative procedures for identification of the nonlinearity. The efficacy of the proposed method is exhibited through simulation at various signal-to-noise ratios.

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Parametric Identification of Nonlinear Fractional Hammerstein Models Vineet Prasad Kajal Kothari Utkal Mehta doi: 10.3390/fractalfract4010002 Fractal and Fractional 2019-12-30 Fractal and Fractional 2019-12-30 4 1 Article 2 10.3390/fractalfract4010002 https://www.mdpi.com/2504-3110/4/1/2
Fractal Fract, Vol. 4, Pages 1: Power Law Type Long Memory Behaviors Modeled with Distributed Time Delay Systems https://www.mdpi.com/2504-3110/4/1/1 This paper studies a class of distributed time delay systems that exhibit power law type long memory behaviors. Such dynamical behaviors are present in multiple domains and it is therefore essential to have tools to model them. The literature is full of examples in which these behaviors are modeled by means of fractional models. However, several limitations of fractional models have recently been reported and other solutions must be found. In the literature, the analysis of distributed delay models and integro-differential equations in general is older than that of fractional models. In this paper, it is shown that particular delay distributions and conditions on the model coefficients make it possible to obtain power laws. The class of systems considered is then used to model the input-output behavior of a lithium-ion cell. Fractal Fract, Vol. 4, Pages 1: Power Law Type Long Memory Behaviors Modeled with Distributed Time Delay Systems

Fractal and Fractional doi: 10.3390/fractalfract4010001

Authors: Jocelyn Sabatier

This paper studies a class of distributed time delay systems that exhibit power law type long memory behaviors. Such dynamical behaviors are present in multiple domains and it is therefore essential to have tools to model them. The literature is full of examples in which these behaviors are modeled by means of fractional models. However, several limitations of fractional models have recently been reported and other solutions must be found. In the literature, the analysis of distributed delay models and integro-differential equations in general is older than that of fractional models. In this paper, it is shown that particular delay distributions and conditions on the model coefficients make it possible to obtain power laws. The class of systems considered is then used to model the input-output behavior of a lithium-ion cell.

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Power Law Type Long Memory Behaviors Modeled with Distributed Time Delay Systems Jocelyn Sabatier doi: 10.3390/fractalfract4010001 Fractal and Fractional 2019-12-27 Fractal and Fractional 2019-12-27 4 1 Article 1 10.3390/fractalfract4010001 https://www.mdpi.com/2504-3110/4/1/1
Fractal Fract, Vol. 3, Pages 54: Comb Model: Non-Markovian versus Markovian https://www.mdpi.com/2504-3110/3/4/54 Combs are a simple caricature of various types of natural branched structures, which belong to the category of loopless graphs and consist of a backbone and branches. We study two generalizations of comb models and present a generic method to obtain their transport properties. The first is a continuous time random walk on a many dimensional m + n comb, where m and n are the dimensions of the backbone and branches, respectively. We observe subdiffusion, ultra-slow diffusion and random localization as a function of n. The second deals with a quantum particle in the 1 + 1 comb. It turns out that the comb geometry leads to a power-law relaxation, described by a wave function in the framework of the Schr&amp;ouml;dinger equation. Fractal Fract, Vol. 3, Pages 54: Comb Model: Non-Markovian versus Markovian

Fractal and Fractional doi: 10.3390/fractalfract3040054

Authors: Alexander Iomin Vicenç Méndez Werner Horsthemke

Combs are a simple caricature of various types of natural branched structures, which belong to the category of loopless graphs and consist of a backbone and branches. We study two generalizations of comb models and present a generic method to obtain their transport properties. The first is a continuous time random walk on a many dimensional m + n comb, where m and n are the dimensions of the backbone and branches, respectively. We observe subdiffusion, ultra-slow diffusion and random localization as a function of n. The second deals with a quantum particle in the 1 + 1 comb. It turns out that the comb geometry leads to a power-law relaxation, described by a wave function in the framework of the Schr&amp;ouml;dinger equation.

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Comb Model: Non-Markovian versus Markovian Alexander Iomin Vicenç Méndez Werner Horsthemke doi: 10.3390/fractalfract3040054 Fractal and Fractional 2019-12-10 Fractal and Fractional 2019-12-10 3 4 Article 54 10.3390/fractalfract3040054 https://www.mdpi.com/2504-3110/3/4/54
Fractal Fract, Vol. 3, Pages 53: A Fractional Measles Model Having Monotonic Real Statistical Data for Constant Transmission Rate of the Disease https://www.mdpi.com/2504-3110/3/4/53 Non-Markovian effects have a vital role in modeling the processes related with natural phenomena such as epidemiology. Various infectious diseases have long-range memory characteristics and, thus, non-local operators are one of the best choices to be used to understand the transmission dynamics of such diseases and epidemics. In this paper, we study a fractional order epidemiological model of measles. Some relevant features, such as well-posedness and stability of the underlying Cauchy problem, are considered accompanying the proofs for a locally asymptotically stable equilibrium point for basic reproduction number R 0 &amp;lt; 1 , which is most sensitive to the fractional order parameter and to the percentage of vaccination. We show the efficiency of the model through a real life application of the spread of the epidemic in Pakistan, comparing the fractional and classical models, while assuming constant transmission rate of the epidemic with monotonically increasing and decreasing behavior of the infected population. Secondly, the fractional Caputo type model, based upon nonlinear least squares curve fitting technique, is found to have smaller residuals when compared with the classical model. Fractal Fract, Vol. 3, Pages 53: A Fractional Measles Model Having Monotonic Real Statistical Data for Constant Transmission Rate of the Disease

Fractal and Fractional doi: 10.3390/fractalfract3040053

Authors: Ricardo Almeida Sania Qureshi

Non-Markovian effects have a vital role in modeling the processes related with natural phenomena such as epidemiology. Various infectious diseases have long-range memory characteristics and, thus, non-local operators are one of the best choices to be used to understand the transmission dynamics of such diseases and epidemics. In this paper, we study a fractional order epidemiological model of measles. Some relevant features, such as well-posedness and stability of the underlying Cauchy problem, are considered accompanying the proofs for a locally asymptotically stable equilibrium point for basic reproduction number R 0 &amp;lt; 1 , which is most sensitive to the fractional order parameter and to the percentage of vaccination. We show the efficiency of the model through a real life application of the spread of the epidemic in Pakistan, comparing the fractional and classical models, while assuming constant transmission rate of the epidemic with monotonically increasing and decreasing behavior of the infected population. Secondly, the fractional Caputo type model, based upon nonlinear least squares curve fitting technique, is found to have smaller residuals when compared with the classical model.

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A Fractional Measles Model Having Monotonic Real Statistical Data for Constant Transmission Rate of the Disease Ricardo Almeida Sania Qureshi doi: 10.3390/fractalfract3040053 Fractal and Fractional 2019-11-21 Fractal and Fractional 2019-11-21 3 4 Article 53 10.3390/fractalfract3040053 https://www.mdpi.com/2504-3110/3/4/53
Fractal Fract, Vol. 3, Pages 52: Dynamic Analysis of the Viscoelastic Pipeline Conveying Fluid with an Improved Variable Fractional Order Model Based on Shifted Legendre Polynomials https://www.mdpi.com/2504-3110/3/4/52 Viscoelastic pipeline conveying fluid is analyzed with an improved variable fractional order model for researching its dynamic properties accurately in this study. After introducing the improved model, an involuted variable fractional order, which is an unknown piecewise nonlinear function for analytical solution, an equation is established as the governing equation for the dynamic displacement of the viscoelastic pipeline. In order to solve this class of equations, a numerical method based on shifted Legendre polynomials is presented for the first time. The method is effective and accurate after the numerical example verifying. Numerical results show that how dynamic properties are influenced by internal fluid velocity, force excitation, and variable fractional order through the proposed method. More importantly, the numerical method has shown great potentials for dynamic problems with the high precision model. Fractal Fract, Vol. 3, Pages 52: Dynamic Analysis of the Viscoelastic Pipeline Conveying Fluid with an Improved Variable Fractional Order Model Based on Shifted Legendre Polynomials

Fractal and Fractional doi: 10.3390/fractalfract3040052

Authors: Yuanhui Wang Yiming Chen

Viscoelastic pipeline conveying fluid is analyzed with an improved variable fractional order model for researching its dynamic properties accurately in this study. After introducing the improved model, an involuted variable fractional order, which is an unknown piecewise nonlinear function for analytical solution, an equation is established as the governing equation for the dynamic displacement of the viscoelastic pipeline. In order to solve this class of equations, a numerical method based on shifted Legendre polynomials is presented for the first time. The method is effective and accurate after the numerical example verifying. Numerical results show that how dynamic properties are influenced by internal fluid velocity, force excitation, and variable fractional order through the proposed method. More importantly, the numerical method has shown great potentials for dynamic problems with the high precision model.

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Dynamic Analysis of the Viscoelastic Pipeline Conveying Fluid with an Improved Variable Fractional Order Model Based on Shifted Legendre Polynomials Yuanhui Wang Yiming Chen doi: 10.3390/fractalfract3040052 Fractal and Fractional 2019-11-17 Fractal and Fractional 2019-11-17 3 4 Article 52 10.3390/fractalfract3040052 https://www.mdpi.com/2504-3110/3/4/52
Fractal Fract, Vol. 3, Pages 51: Multi-Point and Anti-Periodic Conditions for Generalized Langevin Equation with Two Fractional Orders https://www.mdpi.com/2504-3110/3/4/51 With anti-periodic and a new class of multi-point boundary conditions, we investigate, in this paper, the existence and uniqueness of solutions for the Langevin equation that has Caputo fractional derivatives of two different orders. Existence of solutions is obtained by applying Krasnoselskii&amp;ndash;Zabreiko&amp;rsquo;s and the Leray&amp;ndash;Schauder fixed point theorems. The Banach contraction mapping principle is used to investigate the uniqueness. Illustrative examples are provided to apply of the fundamental investigations. Fractal Fract, Vol. 3, Pages 51: Multi-Point and Anti-Periodic Conditions for Generalized Langevin Equation with Two Fractional Orders

Fractal and Fractional doi: 10.3390/fractalfract3040051

Authors: Ahmed Salem Balqees Alghamdi

With anti-periodic and a new class of multi-point boundary conditions, we investigate, in this paper, the existence and uniqueness of solutions for the Langevin equation that has Caputo fractional derivatives of two different orders. Existence of solutions is obtained by applying Krasnoselskii&amp;ndash;Zabreiko&amp;rsquo;s and the Leray&amp;ndash;Schauder fixed point theorems. The Banach contraction mapping principle is used to investigate the uniqueness. Illustrative examples are provided to apply of the fundamental investigations.

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Multi-Point and Anti-Periodic Conditions for Generalized Langevin Equation with Two Fractional Orders Ahmed Salem Balqees Alghamdi doi: 10.3390/fractalfract3040051 Fractal and Fractional 2019-11-08 Fractal and Fractional 2019-11-08 3 4 Article 51 10.3390/fractalfract3040051 https://www.mdpi.com/2504-3110/3/4/51
Fractal Fract, Vol. 3, Pages 50: Impulsive Fractional-Like Differential Equations: Practical Stability and Boundedness with Respect to h-Manifolds https://www.mdpi.com/2504-3110/3/4/50 In this paper, an impulsive fractional-like system of differential equations is introduced. The notions of practical stability and boundedness with respect to h-manifolds for fractional-like differential equations are generalized to the impulsive case. For the first time in the literature, Lyapunov-like functions and their derivatives with respect to impulsive fractional-like systems are defined. As an application, an impulsive fractional-like system of Lotka&amp;ndash;Volterra equations is considered and new criteria for practical exponential stability are proposed. In addition, the uncertain case is also investigated. Fractal Fract, Vol. 3, Pages 50: Impulsive Fractional-Like Differential Equations: Practical Stability and Boundedness with Respect to h-Manifolds

Fractal and Fractional doi: 10.3390/fractalfract3040050

Authors: Gani Stamov Anatoliy Martynyuk Ivanka Stamova

In this paper, an impulsive fractional-like system of differential equations is introduced. The notions of practical stability and boundedness with respect to h-manifolds for fractional-like differential equations are generalized to the impulsive case. For the first time in the literature, Lyapunov-like functions and their derivatives with respect to impulsive fractional-like systems are defined. As an application, an impulsive fractional-like system of Lotka&amp;ndash;Volterra equations is considered and new criteria for practical exponential stability are proposed. In addition, the uncertain case is also investigated.

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Impulsive Fractional-Like Differential Equations: Practical Stability and Boundedness with Respect to h-Manifolds Gani Stamov Anatoliy Martynyuk Ivanka Stamova doi: 10.3390/fractalfract3040050 Fractal and Fractional 2019-11-07 Fractal and Fractional 2019-11-07 3 4 Article 50 10.3390/fractalfract3040050 https://www.mdpi.com/2504-3110/3/4/50
Fractal Fract, Vol. 3, Pages 49: The Unexpected Fractal Signatures in Fibonacci Chains https://www.mdpi.com/2504-3110/3/4/49 In this paper, a new fractal signature possessing the cardioid shape in the Mandelbrot set is presented in the Fourier space of a Fibonacci chain with two lengths, L and S, where L / S = ϕ . The corresponding pointwise dimension is 1.7. Various modifications, such as truncation from the head or tail, scrambling the orders of the sequence and changing the ratio of the L and S, are done on the Fibonacci chain. The resulting patterns in the Fourier space show that that the fractal signature is very sensitive to changes in the Fibonacci order but not to the L / S ratio. Fractal Fract, Vol. 3, Pages 49: The Unexpected Fractal Signatures in Fibonacci Chains

Fractal and Fractional doi: 10.3390/fractalfract3040049

Authors: Fang Fang Raymond Aschheim Klee Irwin

In this paper, a new fractal signature possessing the cardioid shape in the Mandelbrot set is presented in the Fourier space of a Fibonacci chain with two lengths, L and S, where L / S = ϕ . The corresponding pointwise dimension is 1.7. Various modifications, such as truncation from the head or tail, scrambling the orders of the sequence and changing the ratio of the L and S, are done on the Fibonacci chain. The resulting patterns in the Fourier space show that that the fractal signature is very sensitive to changes in the Fibonacci order but not to the L / S ratio.

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The Unexpected Fractal Signatures in Fibonacci Chains Fang Fang Raymond Aschheim Klee Irwin doi: 10.3390/fractalfract3040049 Fractal and Fractional 2019-11-06 Fractal and Fractional 2019-11-06 3 4 Article 49 10.3390/fractalfract3040049 https://www.mdpi.com/2504-3110/3/4/49
Fractal Fract, Vol. 3, Pages 48: Cabinet of Curiosities: The Interesting Geometry of the Angle β = arccos ((3ϕ − 1)/4) https://www.mdpi.com/2504-3110/3/4/48 In this paper, we present the construction of several aggregates of tetrahedra. Each construction is obtained by performing rotations on an initial set of tetrahedra that either (1) contains gaps between adjacent tetrahedra, or (2) exhibits an aperiodic nature. Following this rotation, gaps of the former case are &amp;ldquo;closed&amp;rdquo; (in the sense that faces of adjacent tetrahedra are brought into contact to form a &amp;ldquo;face junction&amp;rdquo;), while translational and rotational symmetries are obtained in the latter case. In all cases, an angular displacement of &amp;beta; = arccos 3 ϕ &amp;minus; 1 / 4 (or a closely related angle), where ϕ = 1 + 5 / 2 is the golden ratio, is observed between faces of a junction. Additionally, the overall number of plane classes, defined as the number of distinct facial orientations in the collection of tetrahedra, is reduced following the transformation. Finally, we present several &amp;ldquo;curiosities&amp;rdquo; involving the structures discussed here with the goal of inspiring the reader&amp;rsquo;s interest in constructions of this nature and their attending, interesting properties. Fractal Fract, Vol. 3, Pages 48: Cabinet of Curiosities: The Interesting Geometry of the Angle β = arccos ((3ϕ − 1)/4)

Fractal and Fractional doi: 10.3390/fractalfract3040048

Authors: Fang Fang Klee Irwin Julio Kovacs Garrett Sadler

In this paper, we present the construction of several aggregates of tetrahedra. Each construction is obtained by performing rotations on an initial set of tetrahedra that either (1) contains gaps between adjacent tetrahedra, or (2) exhibits an aperiodic nature. Following this rotation, gaps of the former case are &amp;ldquo;closed&amp;rdquo; (in the sense that faces of adjacent tetrahedra are brought into contact to form a &amp;ldquo;face junction&amp;rdquo;), while translational and rotational symmetries are obtained in the latter case. In all cases, an angular displacement of &amp;beta; = arccos 3 ϕ &amp;minus; 1 / 4 (or a closely related angle), where ϕ = 1 + 5 / 2 is the golden ratio, is observed between faces of a junction. Additionally, the overall number of plane classes, defined as the number of distinct facial orientations in the collection of tetrahedra, is reduced following the transformation. Finally, we present several &amp;ldquo;curiosities&amp;rdquo; involving the structures discussed here with the goal of inspiring the reader&amp;rsquo;s interest in constructions of this nature and their attending, interesting properties.

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Cabinet of Curiosities: The Interesting Geometry of the Angle β = arccos ((3ϕ − 1)/4) Fang Fang Klee Irwin Julio Kovacs Garrett Sadler doi: 10.3390/fractalfract3040048 Fractal and Fractional 2019-10-30 Fractal and Fractional 2019-10-30 3 4 Article 48 10.3390/fractalfract3040048 https://www.mdpi.com/2504-3110/3/4/48
Fractal Fract, Vol. 3, Pages 47: Tempered Fractional Equations for Quantum Transport in Mesoscopic One-Dimensional Systems with Fractal Disorder https://www.mdpi.com/2504-3110/3/4/47 New aspects of electron transport in quantum wires with L&amp;eacute;vy-type disorder are described. We study the weak scattering and the incoherent sequential tunneling in one-dimensional quantum systems characterized by a tempered L&amp;eacute;vy stable distribution of spacing between scatterers or tunneling barriers. The generalized Dorokhov&amp;ndash;Mello&amp;ndash;Pereyra&amp;ndash;Kumar equation contains the tempered fractional derivative on wire length. The solution describes the evolution from the anomalous conductance distribution to the Dorokhov function for a long wire. For sequential tunneling, average values and relative fluctuations of conductance and resistance are calculated for different parameters of spatial distributions. A tempered L&amp;eacute;vy stable distribution of spacing between barriers leads to a transition in conductance scaling. Fractal Fract, Vol. 3, Pages 47: Tempered Fractional Equations for Quantum Transport in Mesoscopic One-Dimensional Systems with Fractal Disorder

Fractal and Fractional doi: 10.3390/fractalfract3040047

Authors: Renat T. Sibatov HongGuang Sun

New aspects of electron transport in quantum wires with L&amp;eacute;vy-type disorder are described. We study the weak scattering and the incoherent sequential tunneling in one-dimensional quantum systems characterized by a tempered L&amp;eacute;vy stable distribution of spacing between scatterers or tunneling barriers. The generalized Dorokhov&amp;ndash;Mello&amp;ndash;Pereyra&amp;ndash;Kumar equation contains the tempered fractional derivative on wire length. The solution describes the evolution from the anomalous conductance distribution to the Dorokhov function for a long wire. For sequential tunneling, average values and relative fluctuations of conductance and resistance are calculated for different parameters of spatial distributions. A tempered L&amp;eacute;vy stable distribution of spacing between barriers leads to a transition in conductance scaling.

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Tempered Fractional Equations for Quantum Transport in Mesoscopic One-Dimensional Systems with Fractal Disorder Renat T. Sibatov HongGuang Sun doi: 10.3390/fractalfract3040047 Fractal and Fractional 2019-10-19 Fractal and Fractional 2019-10-19 3 4 Article 47 10.3390/fractalfract3040047 https://www.mdpi.com/2504-3110/3/4/47
Fractal Fract, Vol. 3, Pages 46: A New Fractional Integration Operational Matrix of Chebyshev Wavelets in Fractional Delay Systems https://www.mdpi.com/2504-3110/3/3/46 Fractional integration operational matrix of Chebyshev wavelets based on the Riemann&amp;ndash;Liouville fractional integral operator is derived directly from Chebyshev wavelets for the first time. The formulation is accurate and can be applied for fractional orders or an integer order. Using the fractional integration operational matrix, new Chebyshev wavelet methods for finding solutions of linear-quadratic optimal control problems and analysis of linear fractional time-delay systems are presented. Different numerical examples are solved to show the accuracy and applicability of the new Chebyshev wavelet methods. Fractal Fract, Vol. 3, Pages 46: A New Fractional Integration Operational Matrix of Chebyshev Wavelets in Fractional Delay Systems

Fractal and Fractional doi: 10.3390/fractalfract3030046

Authors: Iman Malmir

Fractional integration operational matrix of Chebyshev wavelets based on the Riemann&amp;ndash;Liouville fractional integral operator is derived directly from Chebyshev wavelets for the first time. The formulation is accurate and can be applied for fractional orders or an integer order. Using the fractional integration operational matrix, new Chebyshev wavelet methods for finding solutions of linear-quadratic optimal control problems and analysis of linear fractional time-delay systems are presented. Different numerical examples are solved to show the accuracy and applicability of the new Chebyshev wavelet methods.

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A New Fractional Integration Operational Matrix of Chebyshev Wavelets in Fractional Delay Systems Iman Malmir doi: 10.3390/fractalfract3030046 Fractal and Fractional 2019-09-03 Fractal and Fractional 2019-09-03 3 3 Article 46 10.3390/fractalfract3030046 https://www.mdpi.com/2504-3110/3/3/46
Fractal Fract, Vol. 3, Pages 45: Characterization of the Local Growth of Two Cantor-Type Functions https://www.mdpi.com/2504-3110/3/3/45 The Cantor set and its homonymous function have been frequently utilized as examples for various physical phenomena occurring on discontinuous sets. This article characterizes the local growth of the Cantor&amp;rsquo;s singular function by means of its fractional velocity. It is demonstrated that the Cantor function has finite one-sided velocities, which are non-zero of the set of change of the function. In addition, a related singular function based on the Smith&amp;ndash;Volterra&amp;ndash;Cantor set is constructed. Its growth is characterized by one-sided derivatives. It is demonstrated that the continuity set of its derivative has a positive Lebesgue measure of 1/2. Fractal Fract, Vol. 3, Pages 45: Characterization of the Local Growth of Two Cantor-Type Functions

Fractal and Fractional doi: 10.3390/fractalfract3030045

Authors: Dimiter Prodanov

The Cantor set and its homonymous function have been frequently utilized as examples for various physical phenomena occurring on discontinuous sets. This article characterizes the local growth of the Cantor&amp;rsquo;s singular function by means of its fractional velocity. It is demonstrated that the Cantor function has finite one-sided velocities, which are non-zero of the set of change of the function. In addition, a related singular function based on the Smith&amp;ndash;Volterra&amp;ndash;Cantor set is constructed. Its growth is characterized by one-sided derivatives. It is demonstrated that the continuity set of its derivative has a positive Lebesgue measure of 1/2.

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Characterization of the Local Growth of Two Cantor-Type Functions Dimiter Prodanov doi: 10.3390/fractalfract3030045 Fractal and Fractional 2019-08-21 Fractal and Fractional 2019-08-21 3 3 Brief Report 45 10.3390/fractalfract3030045 https://www.mdpi.com/2504-3110/3/3/45
Fractal Fract, Vol. 3, Pages 44: Multi-Term Fractional Differential Equations with Generalized Integral Boundary Conditions https://www.mdpi.com/2504-3110/3/3/44 We discuss the existence of solutions for a Caputo type multi-term nonlinear fractional differential equation supplemented with generalized integral boundary conditions. The modern tools of functional analysis are applied to achieve the desired results. Examples are constructed for illustrating the obtained work. Some new results follow as spacial cases of the ones reported in this paper. Fractal Fract, Vol. 3, Pages 44: Multi-Term Fractional Differential Equations with Generalized Integral Boundary Conditions

Fractal and Fractional doi: 10.3390/fractalfract3030044

We discuss the existence of solutions for a Caputo type multi-term nonlinear fractional differential equation supplemented with generalized integral boundary conditions. The modern tools of functional analysis are applied to achieve the desired results. Examples are constructed for illustrating the obtained work. Some new results follow as spacial cases of the ones reported in this paper.

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Multi-Term Fractional Differential Equations with Generalized Integral Boundary Conditions Bashir Ahmad Madeaha Alghanmi Ahmed Alsaedi Sotiris K. Ntouyas doi: 10.3390/fractalfract3030044 Fractal and Fractional 2019-08-18 Fractal and Fractional 2019-08-18 3 3 Article 44 10.3390/fractalfract3030044 https://www.mdpi.com/2504-3110/3/3/44
Fractal Fract, Vol. 3, Pages 43: Solving Helmholtz Equation with Local Fractional Derivative Operators https://www.mdpi.com/2504-3110/3/3/43 The paper presents a new analytical method called the local fractional Laplace variational iteration method (LFLVIM), which is a combination of the local fractional Laplace transform (LFLT) and the local fractional variational iteration method (LFVIM), for solving the two-dimensional Helmholtz and coupled Helmholtz equations with local fractional derivative operators (LFDOs). The operators are taken in the local fractional sense. Two test problems are presented to demonstrate the efficiency and the accuracy of the proposed method. The approximate solutions obtained are compared with the results obtained by the local fractional Laplace decomposition method (LFLDM). The results reveal that the LFLVIM is very effective and convenient to solve linear and nonlinear PDEs. Fractal Fract, Vol. 3, Pages 43: Solving Helmholtz Equation with Local Fractional Derivative Operators

Fractal and Fractional doi: 10.3390/fractalfract3030043

Authors: Dumitru Baleanu Hassan Kamil Jassim Maysaa Al Qurashi

The paper presents a new analytical method called the local fractional Laplace variational iteration method (LFLVIM), which is a combination of the local fractional Laplace transform (LFLT) and the local fractional variational iteration method (LFVIM), for solving the two-dimensional Helmholtz and coupled Helmholtz equations with local fractional derivative operators (LFDOs). The operators are taken in the local fractional sense. Two test problems are presented to demonstrate the efficiency and the accuracy of the proposed method. The approximate solutions obtained are compared with the results obtained by the local fractional Laplace decomposition method (LFLDM). The results reveal that the LFLVIM is very effective and convenient to solve linear and nonlinear PDEs.

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Solving Helmholtz Equation with Local Fractional Derivative Operators Dumitru Baleanu Hassan Kamil Jassim Maysaa Al Qurashi doi: 10.3390/fractalfract3030043 Fractal and Fractional 2019-08-01 Fractal and Fractional 2019-08-01 3 3 Article 43 10.3390/fractalfract3030043 https://www.mdpi.com/2504-3110/3/3/43
Fractal Fract, Vol. 3, Pages 42: Exploration of Filled-In Julia Sets Arising from Centered Polygonal Lacunary Functions https://www.mdpi.com/2504-3110/3/3/42 Centered polygonal lacunary functions are a particular type of lacunary function that exhibit properties which set them apart from other lacunary functions. Primarily, centered polygonal lacunary functions have true rotational symmetry. This rotational symmetry is visually seen in the corresponding Julia and Mandelbrot sets. The features and characteristics of these related Julia and Mandelbrot sets are discussed and the parameter space, made with a phase rotation and offset shift, is intricately explored. Also studied in this work is the iterative dynamical map, its characteristics and its fixed points. Fractal Fract, Vol. 3, Pages 42: Exploration of Filled-In Julia Sets Arising from Centered Polygonal Lacunary Functions

Fractal and Fractional doi: 10.3390/fractalfract3030042

Authors: L.K. Mork Trenton Vogt Keith Sullivan Drew Rutherford Darin J. Ulness

Centered polygonal lacunary functions are a particular type of lacunary function that exhibit properties which set them apart from other lacunary functions. Primarily, centered polygonal lacunary functions have true rotational symmetry. This rotational symmetry is visually seen in the corresponding Julia and Mandelbrot sets. The features and characteristics of these related Julia and Mandelbrot sets are discussed and the parameter space, made with a phase rotation and offset shift, is intricately explored. Also studied in this work is the iterative dynamical map, its characteristics and its fixed points.

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Exploration of Filled-In Julia Sets Arising from Centered Polygonal Lacunary Functions L.K. Mork Trenton Vogt Keith Sullivan Drew Rutherford Darin J. Ulness doi: 10.3390/fractalfract3030042 Fractal and Fractional 2019-07-12 Fractal and Fractional 2019-07-12 3 3 Article 42 10.3390/fractalfract3030042 https://www.mdpi.com/2504-3110/3/3/42
Fractal Fract, Vol. 3, Pages 41: Fractal Logistic Equation https://www.mdpi.com/2504-3110/3/3/41 In this paper, we give difference equations on fractal sets and their corresponding fractal differential equations. An analogue of the classical Euler method in fractal calculus is defined. This fractal Euler method presets a numerical method for solving fractal differential equations and finding approximate analytical solutions. Fractal differential equations are solved by using the fractal Euler method. Furthermore, fractal logistic equations and functions are given, which are useful in modeling growth of elements in sciences including biology and economics. Fractal Fract, Vol. 3, Pages 41: Fractal Logistic Equation

Fractal and Fractional doi: 10.3390/fractalfract3030041

Authors: Alireza Khalili Golmankhaneh Carlo Cattani

In this paper, we give difference equations on fractal sets and their corresponding fractal differential equations. An analogue of the classical Euler method in fractal calculus is defined. This fractal Euler method presets a numerical method for solving fractal differential equations and finding approximate analytical solutions. Fractal differential equations are solved by using the fractal Euler method. Furthermore, fractal logistic equations and functions are given, which are useful in modeling growth of elements in sciences including biology and economics.

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Fractal Logistic Equation Alireza Khalili Golmankhaneh Carlo Cattani doi: 10.3390/fractalfract3030041 Fractal and Fractional 2019-07-11 Fractal and Fractional 2019-07-11 3 3 Article 41 10.3390/fractalfract3030041 https://www.mdpi.com/2504-3110/3/3/41
Fractal Fract, Vol. 3, Pages 40: Cornu Spirals and the Triangular Lacunary Trigonometric System https://www.mdpi.com/2504-3110/3/3/40 This work is intended to directly supplement the previous work by Coutsias and Kazarinoff on the foundational understanding of lacunary trigonometric systems and their relation to the Fresnel integrals, specifically the Cornu spirals [Physica 26D (1987) 295]. These systems are intimately related to incomplete Gaussian summations. The current work provides a focused look at the specific system built off of the triangular numbers. The special cyclic character of the triangular numbers modulo m carries through to triangular lacunary trigonometric systems. Specifically, this work characterizes the families of Cornu spirals arising from triangular lacunary trigonometric systems. Special features such as self-similarity, isometry, and symmetry are presented and discussed. Fractal Fract, Vol. 3, Pages 40: Cornu Spirals and the Triangular Lacunary Trigonometric System

Fractal and Fractional doi: 10.3390/fractalfract3030040

Authors: Trenton Vogt Darin J. Ulness

This work is intended to directly supplement the previous work by Coutsias and Kazarinoff on the foundational understanding of lacunary trigonometric systems and their relation to the Fresnel integrals, specifically the Cornu spirals [Physica 26D (1987) 295]. These systems are intimately related to incomplete Gaussian summations. The current work provides a focused look at the specific system built off of the triangular numbers. The special cyclic character of the triangular numbers modulo m carries through to triangular lacunary trigonometric systems. Specifically, this work characterizes the families of Cornu spirals arising from triangular lacunary trigonometric systems. Special features such as self-similarity, isometry, and symmetry are presented and discussed.

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Cornu Spirals and the Triangular Lacunary Trigonometric System Trenton Vogt Darin J. Ulness doi: 10.3390/fractalfract3030040 Fractal and Fractional 2019-07-10 Fractal and Fractional 2019-07-10 3 3 Article 40 10.3390/fractalfract3030040 https://www.mdpi.com/2504-3110/3/3/40
Fractal Fract, Vol. 3, Pages 39: Fractional Mass-Spring-Damper System Described by Generalized Fractional Order Derivatives https://www.mdpi.com/2504-3110/3/3/39 This paper proposes novel analytical solutions of the mass-spring-damper systems described by certain generalized fractional derivatives. The Liouville&amp;ndash;Caputo left generalized fractional derivative and the left generalized fractional derivative were used. The behaviors of the analytical solutions of the mass-spring-damper systems described by the left generalized fractional derivative and the Liouville&amp;ndash;Caputo left generalized fractional derivative were represented graphically and the effect of the orders of the fractional derivatives analyzed. We finish by analyzing the global asymptotic stability and the converging-input-converging-state of the unforced mass-damper system, the unforced spring-damper, the spring-damper system, and the mass-damper system. Fractal Fract, Vol. 3, Pages 39: Fractional Mass-Spring-Damper System Described by Generalized Fractional Order Derivatives

Fractal and Fractional doi: 10.3390/fractalfract3030039

Authors: Ndolane Sene José Francisco Gómez Aguilar

This paper proposes novel analytical solutions of the mass-spring-damper systems described by certain generalized fractional derivatives. The Liouville&amp;ndash;Caputo left generalized fractional derivative and the left generalized fractional derivative were used. The behaviors of the analytical solutions of the mass-spring-damper systems described by the left generalized fractional derivative and the Liouville&amp;ndash;Caputo left generalized fractional derivative were represented graphically and the effect of the orders of the fractional derivatives analyzed. We finish by analyzing the global asymptotic stability and the converging-input-converging-state of the unforced mass-damper system, the unforced spring-damper, the spring-damper system, and the mass-damper system.

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Fractional Mass-Spring-Damper System Described by Generalized Fractional Order Derivatives Ndolane Sene José Francisco Gómez Aguilar doi: 10.3390/fractalfract3030039 Fractal and Fractional 2019-07-07 Fractal and Fractional 2019-07-07 3 3 Article 39 10.3390/fractalfract3030039 https://www.mdpi.com/2504-3110/3/3/39
Fractal Fract, Vol. 3, Pages 38: k-Fractional Estimates of Hermite–Hadamard Type Inequalities Involving k-Appell’s Hypergeometric Functions and Applications https://www.mdpi.com/2504-3110/3/3/38 The main objective of this paper is to obtain certain new k-fractional estimates of Hermite&amp;ndash;Hadamard type inequalities via s-convex functions of Breckner type essentially involving k-Appell&amp;rsquo;s hypergeometric functions. We also present applications of the obtained results by considering particular examples. Fractal Fract, Vol. 3, Pages 38: k-Fractional Estimates of Hermite–Hadamard Type Inequalities Involving k-Appell’s Hypergeometric Functions and Applications

Fractal and Fractional doi: 10.3390/fractalfract3030038

Authors: Muhammad Uzair Awan Muhammad Aslam Noor Marcela V. Mihai Khalida Inayat Noor

The main objective of this paper is to obtain certain new k-fractional estimates of Hermite&amp;ndash;Hadamard type inequalities via s-convex functions of Breckner type essentially involving k-Appell&amp;rsquo;s hypergeometric functions. We also present applications of the obtained results by considering particular examples.

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k-Fractional Estimates of Hermite–Hadamard Type Inequalities Involving k-Appell’s Hypergeometric Functions and Applications Muhammad Uzair Awan Muhammad Aslam Noor Marcela V. Mihai Khalida Inayat Noor doi: 10.3390/fractalfract3030038 Fractal and Fractional 2019-07-03 Fractal and Fractional 2019-07-03 3 3 Article 38 10.3390/fractalfract3030038 https://www.mdpi.com/2504-3110/3/3/38
Fractal Fract, Vol. 3, Pages 37: Inequalities Pertaining Fractional Approach through Exponentially Convex Functions https://www.mdpi.com/2504-3110/3/3/37 In this article, certain Hermite-Hadamard-type inequalities are proven for an exponentially-convex function via Riemann-Liouville fractional integrals that generalize Hermite-Hadamard-type inequalities. These results have some relationships with the Hermite-Hadamard-type inequalities and related inequalities via Riemann-Liouville fractional integrals. Fractal Fract, Vol. 3, Pages 37: Inequalities Pertaining Fractional Approach through Exponentially Convex Functions

Fractal and Fractional doi: 10.3390/fractalfract3030037

Authors: Saima Rashid Muhammad Aslam Noor Khalida Inayat Noor

In this article, certain Hermite-Hadamard-type inequalities are proven for an exponentially-convex function via Riemann-Liouville fractional integrals that generalize Hermite-Hadamard-type inequalities. These results have some relationships with the Hermite-Hadamard-type inequalities and related inequalities via Riemann-Liouville fractional integrals.

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Inequalities Pertaining Fractional Approach through Exponentially Convex Functions Saima Rashid Muhammad Aslam Noor Khalida Inayat Noor doi: 10.3390/fractalfract3030037 Fractal and Fractional 2019-06-27 Fractal and Fractional 2019-06-27 3 3 Article 37 10.3390/fractalfract3030037 https://www.mdpi.com/2504-3110/3/3/37
Fractal Fract, Vol. 3, Pages 36: Green’s Function Estimates for Time-Fractional Evolution Equations https://www.mdpi.com/2504-3110/3/2/36 We look at estimates for the Green&amp;rsquo;s function of time-fractional evolution equations of the form D 0 + &amp;lowast; &amp;nu; u = L u , where D 0 + &amp;lowast; &amp;nu; is a Caputo-type time-fractional derivative, depending on a L&amp;eacute;vy kernel &amp;nu; with variable coefficients, which is comparable to y &amp;minus; 1 &amp;minus; &amp;beta; for &amp;beta; &amp;isin; ( 0 , 1 ) , and L is an operator acting on the spatial variable. First, we obtain global two-sided estimates for the Green&amp;rsquo;s function of D 0 &amp;beta; u = L u in the case that L is a second order elliptic operator in divergence form. Secondly, we obtain global upper bounds for the Green&amp;rsquo;s function of D 0 &amp;beta; u = &amp;Psi; ( &amp;minus; i &amp;nabla; ) u where &amp;Psi; is a pseudo-differential operator with constant coefficients that is homogeneous of order &amp;alpha; . Thirdly, we obtain local two-sided estimates for the Green&amp;rsquo;s function of D 0 &amp;beta; u = L u where L is a more general non-degenerate second order elliptic operator. Finally we look at the case of stable-like operator, extending the second result from a constant coefficient to variable coefficients. In each case, we also estimate the spatial derivatives of the Green&amp;rsquo;s functions. To obtain these bounds we use a particular form of the Mittag-Leffler functions, which allow us to use directly known estimates for the Green&amp;rsquo;s functions associated with L and &amp;Psi; , as well as estimates for stable densities. These estimates then allow us to estimate the solutions to a wide class of problems of the form D 0 ( &amp;nu; , t ) u = L u , where D ( &amp;nu; , t ) is a Caputo-type operator with variable coefficients. Fractal Fract, Vol. 3, Pages 36: Green’s Function Estimates for Time-Fractional Evolution Equations

Fractal and Fractional doi: 10.3390/fractalfract3020036

Authors: Ifan Johnston Vassili Kolokoltsov

We look at estimates for the Green&amp;rsquo;s function of time-fractional evolution equations of the form D 0 + &amp;lowast; &amp;nu; u = L u , where D 0 + &amp;lowast; &amp;nu; is a Caputo-type time-fractional derivative, depending on a L&amp;eacute;vy kernel &amp;nu; with variable coefficients, which is comparable to y &amp;minus; 1 &amp;minus; &amp;beta; for &amp;beta; &amp;isin; ( 0 , 1 ) , and L is an operator acting on the spatial variable. First, we obtain global two-sided estimates for the Green&amp;rsquo;s function of D 0 &amp;beta; u = L u in the case that L is a second order elliptic operator in divergence form. Secondly, we obtain global upper bounds for the Green&amp;rsquo;s function of D 0 &amp;beta; u = &amp;Psi; ( &amp;minus; i &amp;nabla; ) u where &amp;Psi; is a pseudo-differential operator with constant coefficients that is homogeneous of order &amp;alpha; . Thirdly, we obtain local two-sided estimates for the Green&amp;rsquo;s function of D 0 &amp;beta; u = L u where L is a more general non-degenerate second order elliptic operator. Finally we look at the case of stable-like operator, extending the second result from a constant coefficient to variable coefficients. In each case, we also estimate the spatial derivatives of the Green&amp;rsquo;s functions. To obtain these bounds we use a particular form of the Mittag-Leffler functions, which allow us to use directly known estimates for the Green&amp;rsquo;s functions associated with L and &amp;Psi; , as well as estimates for stable densities. These estimates then allow us to estimate the solutions to a wide class of problems of the form D 0 ( &amp;nu; , t ) u = L u , where D ( &amp;nu; , t ) is a Caputo-type operator with variable coefficients.

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Green’s Function Estimates for Time-Fractional Evolution Equations Ifan Johnston Vassili Kolokoltsov doi: 10.3390/fractalfract3020036 Fractal and Fractional 2019-06-25 Fractal and Fractional 2019-06-25 3 2 Article 36 10.3390/fractalfract3020036 https://www.mdpi.com/2504-3110/3/2/36
Fractal Fract, Vol. 3, Pages 35: A Criterion for Subfamilies of Multivalent Functions of Reciprocal Order with Respect to Symmetric Points https://www.mdpi.com/2504-3110/3/2/35 In the present research paper, our aim is to introduce a new subfamily of p-valent (multivalent) functions of reciprocal order. We investigate sufficiency criterion for such defined family. Fractal Fract, Vol. 3, Pages 35: A Criterion for Subfamilies of Multivalent Functions of Reciprocal Order with Respect to Symmetric Points

Fractal and Fractional doi: 10.3390/fractalfract3020035

Authors: Shahid Mahmood Hari Mohan Srivastava Muhammad Arif Fazal Ghani Eman S. A. AbuJarad

In the present research paper, our aim is to introduce a new subfamily of p-valent (multivalent) functions of reciprocal order. We investigate sufficiency criterion for such defined family.

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A Criterion for Subfamilies of Multivalent Functions of Reciprocal Order with Respect to Symmetric Points Shahid Mahmood Hari Mohan Srivastava Muhammad Arif Fazal Ghani Eman S. A. AbuJarad doi: 10.3390/fractalfract3020035 Fractal and Fractional 2019-06-25 Fractal and Fractional 2019-06-25 3 2 Article 35 10.3390/fractalfract3020035 https://www.mdpi.com/2504-3110/3/2/35
Fractal Fract, Vol. 3, Pages 34: Fractional Differential Equation Involving Mixed Nonlinearities with Nonlocal Multi-Point and Riemann-Stieltjes Integral-Multi-Strip Conditions https://www.mdpi.com/2504-3110/3/2/34 In this paper, we investigate a new class of boundary value problems involving fractional differential equations with mixed nonlinearities, and nonlocal multi-point and Riemann&amp;ndash;Stieltjes integral-multi-strip boundary conditions. Based on the standard tools of the fixed point theory, we obtain some existence and uniqueness results for the problem at hand, which are well illustrated with the aid of examples. Our results are not only in the given configuration but also yield several new results as special cases. Some variants of the given problem are also discussed. Fractal Fract, Vol. 3, Pages 34: Fractional Differential Equation Involving Mixed Nonlinearities with Nonlocal Multi-Point and Riemann-Stieltjes Integral-Multi-Strip Conditions

Fractal and Fractional doi: 10.3390/fractalfract3020034

Authors: Bashir Ahmad Ahmed Alsaedi Sara Salem Sotiris K. Ntouyas

In this paper, we investigate a new class of boundary value problems involving fractional differential equations with mixed nonlinearities, and nonlocal multi-point and Riemann&amp;ndash;Stieltjes integral-multi-strip boundary conditions. Based on the standard tools of the fixed point theory, we obtain some existence and uniqueness results for the problem at hand, which are well illustrated with the aid of examples. Our results are not only in the given configuration but also yield several new results as special cases. Some variants of the given problem are also discussed.

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Fractional Differential Equation Involving Mixed Nonlinearities with Nonlocal Multi-Point and Riemann-Stieltjes Integral-Multi-Strip Conditions Bashir Ahmad Ahmed Alsaedi Sara Salem Sotiris K. Ntouyas doi: 10.3390/fractalfract3020034 Fractal and Fractional 2019-06-21 Fractal and Fractional 2019-06-21 3 2 Article 34 10.3390/fractalfract3020034 https://www.mdpi.com/2504-3110/3/2/34
Fractal Fract, Vol. 3, Pages 33: A Novel Method for Solutions of Fourth-Order Fractional Boundary Value Problems https://www.mdpi.com/2504-3110/3/2/33 In this paper, we find the solutions of fourth order fractional boundary value problems by using the reproducing kernel Hilbert space method. Firstly, the reproducing kernel Hilbert space method is introduced and then the method is applied to this kind problems. The experiments are discussed and the approximate solutions are obtained to be more correct compared to the other obtained results in the literature. Fractal Fract, Vol. 3, Pages 33: A Novel Method for Solutions of Fourth-Order Fractional Boundary Value Problems

Fractal and Fractional doi: 10.3390/fractalfract3020033

Authors: Ali Akgül Esra Karatas Akgül

In this paper, we find the solutions of fourth order fractional boundary value problems by using the reproducing kernel Hilbert space method. Firstly, the reproducing kernel Hilbert space method is introduced and then the method is applied to this kind problems. The experiments are discussed and the approximate solutions are obtained to be more correct compared to the other obtained results in the literature.

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A Novel Method for Solutions of Fourth-Order Fractional Boundary Value Problems Ali Akgül Esra Karatas Akgül doi: 10.3390/fractalfract3020033 Fractal and Fractional 2019-06-18 Fractal and Fractional 2019-06-18 3 2 Article 33 10.3390/fractalfract3020033 https://www.mdpi.com/2504-3110/3/2/33
Fractal Fract, Vol. 3, Pages 32: On Extended General Mittag–Leffler Functions and Certain Inequalities https://www.mdpi.com/2504-3110/3/2/32 In this paper, we introduce and investigate generalized fractional integral operators containing the new generalized Mittag&amp;ndash;Leffler function of two variables. We establish several new refinements of Hermite&amp;ndash;Hadamard-like inequalities via co-ordinated convex functions. Fractal Fract, Vol. 3, Pages 32: On Extended General Mittag–Leffler Functions and Certain Inequalities

Fractal and Fractional doi: 10.3390/fractalfract3020032

Authors: Marcela V. Mihai Muhammad Uzair Awan Muhammad Aslam Noor Tingsong Du Artion Kashuri Khalida Inayat Noor

In this paper, we introduce and investigate generalized fractional integral operators containing the new generalized Mittag&amp;ndash;Leffler function of two variables. We establish several new refinements of Hermite&amp;ndash;Hadamard-like inequalities via co-ordinated convex functions.

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On Extended General Mittag–Leffler Functions and Certain Inequalities Marcela V. Mihai Muhammad Uzair Awan Muhammad Aslam Noor Tingsong Du Artion Kashuri Khalida Inayat Noor doi: 10.3390/fractalfract3020032 Fractal and Fractional 2019-06-18 Fractal and Fractional 2019-06-18 3 2 Article 32 10.3390/fractalfract3020032 https://www.mdpi.com/2504-3110/3/2/32
Fractal Fract, Vol. 3, Pages 31: Random Variables and Stable Distributions on Fractal Cantor Sets https://www.mdpi.com/2504-3110/3/2/31 In this paper, we introduce the concept of fractal random variables and their related distribution functions and statistical properties. Fractal calculus is a generalisation of standard calculus which includes function with fractal support. Here we combine this emerging field of study with probability theory, defining concepts such as Shannon entropy on fractal thin Cantor-like sets. Stable distributions on fractal sets are suggested and related physical models are presented. Our work is illustrated with graphs for clarity of the results. Fractal Fract, Vol. 3, Pages 31: Random Variables and Stable Distributions on Fractal Cantor Sets

Fractal and Fractional doi: 10.3390/fractalfract3020031

Authors: Alireza Khalili Golmankhaneh Arran Fernandez

In this paper, we introduce the concept of fractal random variables and their related distribution functions and statistical properties. Fractal calculus is a generalisation of standard calculus which includes function with fractal support. Here we combine this emerging field of study with probability theory, defining concepts such as Shannon entropy on fractal thin Cantor-like sets. Stable distributions on fractal sets are suggested and related physical models are presented. Our work is illustrated with graphs for clarity of the results.

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Random Variables and Stable Distributions on Fractal Cantor Sets Alireza Khalili Golmankhaneh Arran Fernandez doi: 10.3390/fractalfract3020031 Fractal and Fractional 2019-06-11 Fractal and Fractional 2019-06-11 3 2 Article 31 10.3390/fractalfract3020031 https://www.mdpi.com/2504-3110/3/2/31
Fractal Fract, Vol. 3, Pages 30: A Modification Fractional Homotopy Perturbation Method for Solving Helmholtz and Coupled Helmholtz Equations on Cantor Sets https://www.mdpi.com/2504-3110/3/2/30 In this paper, we apply a new technique, namely, the local fractional Laplace homotopy perturbation method (LFLHPM), on Helmholtz and coupled Helmholtz equations to obtain analytical approximate solutions. The iteration procedure is based on local fractional derivative operators (LFDOs). This method is a combination of the local fractional Laplace transform (LFLT) and the homotopy perturbation method (HPM). The method in general is easy to implement and yields good results. Illustrative examples are included to demonstrate the validity and applicability of the new technique. Fractal Fract, Vol. 3, Pages 30: A Modification Fractional Homotopy Perturbation Method for Solving Helmholtz and Coupled Helmholtz Equations on Cantor Sets

Fractal and Fractional doi: 10.3390/fractalfract3020030

Authors: Dumitru Baleanu Hassan Kamil Jassim

In this paper, we apply a new technique, namely, the local fractional Laplace homotopy perturbation method (LFLHPM), on Helmholtz and coupled Helmholtz equations to obtain analytical approximate solutions. The iteration procedure is based on local fractional derivative operators (LFDOs). This method is a combination of the local fractional Laplace transform (LFLT) and the homotopy perturbation method (HPM). The method in general is easy to implement and yields good results. Illustrative examples are included to demonstrate the validity and applicability of the new technique.

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A Modification Fractional Homotopy Perturbation Method for Solving Helmholtz and Coupled Helmholtz Equations on Cantor Sets Dumitru Baleanu Hassan Kamil Jassim doi: 10.3390/fractalfract3020030 Fractal and Fractional 2019-06-03 Fractal and Fractional 2019-06-03 3 2 Article 30 10.3390/fractalfract3020030 https://www.mdpi.com/2504-3110/3/2/30
Fractal Fract, Vol. 3, Pages 29: On Some Generalized Fractional Integral Inequalities for p-Convex Functions https://www.mdpi.com/2504-3110/3/2/29 In this paper, firstly we have established a new generalization of Hermite&amp;ndash;Hadamard inequality via p-convex function and fractional integral operators which generalize the Riemann&amp;ndash;Liouville fractional integral operators introduced by Raina, Lun and Agarwal. Secondly, we proved a new identity involving this generalized fractional integral operators. Then, by using this identity, a new generalization of Hermite&amp;ndash;Hadamard type inequalities for fractional integral are obtained. Fractal Fract, Vol. 3, Pages 29: On Some Generalized Fractional Integral Inequalities for p-Convex Functions

Fractal and Fractional doi: 10.3390/fractalfract3020029

Authors: Seren Salaş Yeter Erdaş Tekin Toplu Erhan Set

In this paper, firstly we have established a new generalization of Hermite&amp;ndash;Hadamard inequality via p-convex function and fractional integral operators which generalize the Riemann&amp;ndash;Liouville fractional integral operators introduced by Raina, Lun and Agarwal. Secondly, we proved a new identity involving this generalized fractional integral operators. Then, by using this identity, a new generalization of Hermite&amp;ndash;Hadamard type inequalities for fractional integral are obtained.

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On Some Generalized Fractional Integral Inequalities for p-Convex Functions Seren Salaş Yeter Erdaş Tekin Toplu Erhan Set doi: 10.3390/fractalfract3020029 Fractal and Fractional 2019-05-20 Fractal and Fractional 2019-05-20 3 2 Article 29 10.3390/fractalfract3020029 https://www.mdpi.com/2504-3110/3/2/29
Fractal Fract, Vol. 3, Pages 28: Caputo Fractional Differential Equations with Non-Instantaneous Random Erlang Distributed Impulses https://www.mdpi.com/2504-3110/3/2/28 The p-moment exponential stability of non-instantaneous impulsive Caputo fractional differential equations is studied. The impulses occur at random moments and their action continues on finite time intervals with initially given lengths. The time between two consecutive moments of impulses is the Erlang distributed random variable. The study is based on Lyapunov functions. The fractional Dini derivatives are applied. Fractal Fract, Vol. 3, Pages 28: Caputo Fractional Differential Equations with Non-Instantaneous Random Erlang Distributed Impulses

Fractal and Fractional doi: 10.3390/fractalfract3020028

Authors: Snezhana Hristova Krasimira Ivanova

The p-moment exponential stability of non-instantaneous impulsive Caputo fractional differential equations is studied. The impulses occur at random moments and their action continues on finite time intervals with initially given lengths. The time between two consecutive moments of impulses is the Erlang distributed random variable. The study is based on Lyapunov functions. The fractional Dini derivatives are applied.

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Caputo Fractional Differential Equations with Non-Instantaneous Random Erlang Distributed Impulses Snezhana Hristova Krasimira Ivanova doi: 10.3390/fractalfract3020028 Fractal and Fractional 2019-05-18 Fractal and Fractional 2019-05-18 3 2 Article 28 10.3390/fractalfract3020028 https://www.mdpi.com/2504-3110/3/2/28
Fractal Fract, Vol. 3, Pages 27: Nonlocal Cauchy Problem via a Fractional Operator Involving Power Kernel in Banach Spaces https://www.mdpi.com/2504-3110/3/2/27 We investigated existence and uniqueness conditions of solutions of a nonlinear differential equation containing the Caputo&amp;ndash;Fabrizio operator in Banach spaces. The mentioned derivative has been proposed by using the exponential decay law and hence it removed the computational complexities arising from the singular kernel functions inherit in the conventional fractional derivatives. The method used in this study is based on the Banach contraction mapping principle. Moreover, we gave a numerical example which shows the applicability of the obtained results. Fractal Fract, Vol. 3, Pages 27: Nonlocal Cauchy Problem via a Fractional Operator Involving Power Kernel in Banach Spaces

Fractal and Fractional doi: 10.3390/fractalfract3020027

Authors: Ayşegül Keten Mehmet Yavuz Dumitru Baleanu

We investigated existence and uniqueness conditions of solutions of a nonlinear differential equation containing the Caputo&amp;ndash;Fabrizio operator in Banach spaces. The mentioned derivative has been proposed by using the exponential decay law and hence it removed the computational complexities arising from the singular kernel functions inherit in the conventional fractional derivatives. The method used in this study is based on the Banach contraction mapping principle. Moreover, we gave a numerical example which shows the applicability of the obtained results.

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Nonlocal Cauchy Problem via a Fractional Operator Involving Power Kernel in Banach Spaces Ayşegül Keten Mehmet Yavuz Dumitru Baleanu doi: 10.3390/fractalfract3020027 Fractal and Fractional 2019-05-16 Fractal and Fractional 2019-05-16 3 2 Article 27 10.3390/fractalfract3020027 https://www.mdpi.com/2504-3110/3/2/27
Fractal Fract, Vol. 3, Pages 26: Approximate Solutions of the Damped Wave Equation and Dissipative Wave Equation in Fractal Strings https://www.mdpi.com/2504-3110/3/2/26 In this paper, we apply the local fractional Laplace variational iteration method (LFLVIM) and the local fractional Laplace decomposition method (LFLDM) to obtain approximate solutions for solving the damped wave equation and dissipative wave equation within local fractional derivative operators (LFDOs). The efficiency of the considered methods are illustrated by some examples. The results obtained by LFLVIM and LFLDM are compared with the results obtained by LFVIM. The results reveal that the suggested algorithms are very effective and simple, and can be applied for linear and nonlinear problems in sciences and engineering. Fractal Fract, Vol. 3, Pages 26: Approximate Solutions of the Damped Wave Equation and Dissipative Wave Equation in Fractal Strings

Fractal and Fractional doi: 10.3390/fractalfract3020026

Authors: Dumitru Baleanu Hassan Kamil Jassim

In this paper, we apply the local fractional Laplace variational iteration method (LFLVIM) and the local fractional Laplace decomposition method (LFLDM) to obtain approximate solutions for solving the damped wave equation and dissipative wave equation within local fractional derivative operators (LFDOs). The efficiency of the considered methods are illustrated by some examples. The results obtained by LFLVIM and LFLDM are compared with the results obtained by LFVIM. The results reveal that the suggested algorithms are very effective and simple, and can be applied for linear and nonlinear problems in sciences and engineering.

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Approximate Solutions of the Damped Wave Equation and Dissipative Wave Equation in Fractal Strings Dumitru Baleanu Hassan Kamil Jassim doi: 10.3390/fractalfract3020026 Fractal and Fractional 2019-05-11 Fractal and Fractional 2019-05-11 3 2 Article 26 10.3390/fractalfract3020026 https://www.mdpi.com/2504-3110/3/2/26
Fractal Fract, Vol. 3, Pages 25: Analogues to Lie Method and Noether’s Theorem in Fractal Calculus https://www.mdpi.com/2504-3110/3/2/25 In this manuscript, we study symmetries of fractal differential equations. We show that using symmetry properties, one of the solutions can map to another solution. We obtain canonical coordinate systems for differential equations on fractal sets, which makes them simpler to solve. An analogue for Noether’s Theorem on fractal sets is given, and a corresponding conservative quantity is suggested. Several examples are solved to illustrate the results. Fractal Fract, Vol. 3, Pages 25: Analogues to Lie Method and Noether’s Theorem in Fractal Calculus

Fractal and Fractional doi: 10.3390/fractalfract3020025

Authors: Alireza Khalili Golmankhaneh Cemil Tunç

In this manuscript, we study symmetries of fractal differential equations. We show that using symmetry properties, one of the solutions can map to another solution. We obtain canonical coordinate systems for differential equations on fractal sets, which makes them simpler to solve. An analogue for Noether’s Theorem on fractal sets is given, and a corresponding conservative quantity is suggested. Several examples are solved to illustrate the results.

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Analogues to Lie Method and Noether’s Theorem in Fractal Calculus Alireza Khalili Golmankhaneh Cemil Tunç doi: 10.3390/fractalfract3020025 Fractal and Fractional 2019-05-07 Fractal and Fractional 2019-05-07 3 2 Article 25 10.3390/fractalfract3020025 https://www.mdpi.com/2504-3110/3/2/25
Fractal Fract, Vol. 3, Pages 24: Some New Generalizations for Exponentially s-Convex Functions and Inequalities via Fractional Operators https://www.mdpi.com/2504-3110/3/2/24 The main objective of this paper is to obtain the Hermite&amp;ndash;Hadamard-type inequalities for exponentially s-convex functions via the Katugampola fractional integral. The Katugampola fractional integral is a generalization of Riemann&amp;ndash;Liouville fractional integral and Hadamard fractional integral. Some special cases and applications to special means are also discussed. Fractal Fract, Vol. 3, Pages 24: Some New Generalizations for Exponentially s-Convex Functions and Inequalities via Fractional Operators

Fractal and Fractional doi: 10.3390/fractalfract3020024

Authors: Saima Rashid Muhammad Aslam Noor Khalida Inayat Noor Ahmet Ocak Akdemir

The main objective of this paper is to obtain the Hermite&amp;ndash;Hadamard-type inequalities for exponentially s-convex functions via the Katugampola fractional integral. The Katugampola fractional integral is a generalization of Riemann&amp;ndash;Liouville fractional integral and Hadamard fractional integral. Some special cases and applications to special means are also discussed.

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Some New Generalizations for Exponentially s-Convex Functions and Inequalities via Fractional Operators Saima Rashid Muhammad Aslam Noor Khalida Inayat Noor Ahmet Ocak Akdemir doi: 10.3390/fractalfract3020024 Fractal and Fractional 2019-04-28 Fractal and Fractional 2019-04-28 3 2 Article 24 10.3390/fractalfract3020024 https://www.mdpi.com/2504-3110/3/2/24
Fractal Fract, Vol. 3, Pages 23: Partially Penetrated Well Solution of Fractal Single-Porosity Naturally Fractured Reservoirs https://www.mdpi.com/2504-3110/3/2/23 In the oil industry, many reservoirs produce from partially penetrated wells, either to postpone the arrival of undesirable fluids or to avoid problems during drilling operations. The majority of these reservoirs are heterogeneous and anisotropic, such as naturally fractured reservoirs. The analysis of pressure-transient tests is a very useful method to dynamically characterize both the heterogeneity and anisotropy existing in the reservoir. In this paper, a new analytical solution for a partially penetrated well based on a fractal approach to capture the distribution and connectivity of the fracture network is presented. This solution represents the complexity of the flow lines better than the traditional Euclidean flow models for single-porosity fractured reservoirs, i.e., for a tight matrix. The proposed solution takes into consideration the variations in fracture density throughout the reservoir, which have a direct influence on the porosity, permeability, and the size distribution of the matrix blocks as a result of the fracturing process. This solution generalizes previous solutions to model the pressure-transient behavior of partially penetrated wells as proposed in the technical literature for the classical Euclidean formulation, which considers a uniform distribution of fractures that are fully connected. Several synthetic cases obtained with the proposed solution are shown to illustrate the influence of different variables, including fractal parameters. Fractal Fract, Vol. 3, Pages 23: Partially Penetrated Well Solution of Fractal Single-Porosity Naturally Fractured Reservoirs

Fractal and Fractional doi: 10.3390/fractalfract3020023

Authors: Ricardo Posadas-Mondragón Rodolfo G. Camacho-Velázquez

In the oil industry, many reservoirs produce from partially penetrated wells, either to postpone the arrival of undesirable fluids or to avoid problems during drilling operations. The majority of these reservoirs are heterogeneous and anisotropic, such as naturally fractured reservoirs. The analysis of pressure-transient tests is a very useful method to dynamically characterize both the heterogeneity and anisotropy existing in the reservoir. In this paper, a new analytical solution for a partially penetrated well based on a fractal approach to capture the distribution and connectivity of the fracture network is presented. This solution represents the complexity of the flow lines better than the traditional Euclidean flow models for single-porosity fractured reservoirs, i.e., for a tight matrix. The proposed solution takes into consideration the variations in fracture density throughout the reservoir, which have a direct influence on the porosity, permeability, and the size distribution of the matrix blocks as a result of the fracturing process. This solution generalizes previous solutions to model the pressure-transient behavior of partially penetrated wells as proposed in the technical literature for the classical Euclidean formulation, which considers a uniform distribution of fractures that are fully connected. Several synthetic cases obtained with the proposed solution are shown to illustrate the influence of different variables, including fractal parameters.

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Partially Penetrated Well Solution of Fractal Single-Porosity Naturally Fractured Reservoirs Ricardo Posadas-Mondragón Rodolfo G. Camacho-Velázquez doi: 10.3390/fractalfract3020023 Fractal and Fractional 2019-04-24 Fractal and Fractional 2019-04-24 3 2 Article 23 10.3390/fractalfract3020023 https://www.mdpi.com/2504-3110/3/2/23
Fractal Fract, Vol. 3, Pages 22: Hankel Determinants of Non-Zero Modulus Dixon Elliptic Functions via Quasi C Fractions https://www.mdpi.com/2504-3110/3/2/22 The Sumudu transform of the Dixon elliptic function with non-zero modulus &amp;alpha; &amp;ne; 0 for arbitrary powers N is given by the product of quasi C fractions. Next, by assuming the denominators of quasi C fractions as one and applying the Heliermanncorrespondence relating formal power series (Maclaurin series of the Dixon elliptic function) and the regular C fraction, the Hankel determinants are calculated for the non-zero Dixon elliptic functions and shown by taking &amp;alpha; = 0 to give the Hankel determinants of the Dixon elliptic function with zero modulus. The derived results were back-tracked to the Laplace transform of Dixon elliptic functions. Fractal Fract, Vol. 3, Pages 22: Hankel Determinants of Non-Zero Modulus Dixon Elliptic Functions via Quasi C Fractions

Fractal and Fractional doi: 10.3390/fractalfract3020022

The Sumudu transform of the Dixon elliptic function with non-zero modulus &amp;alpha; &amp;ne; 0 for arbitrary powers N is given by the product of quasi C fractions. Next, by assuming the denominators of quasi C fractions as one and applying the Heliermanncorrespondence relating formal power series (Maclaurin series of the Dixon elliptic function) and the regular C fraction, the Hankel determinants are calculated for the non-zero Dixon elliptic functions and shown by taking &amp;alpha; = 0 to give the Hankel determinants of the Dixon elliptic function with zero modulus. The derived results were back-tracked to the Laplace transform of Dixon elliptic functions.

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Hankel Determinants of Non-Zero Modulus Dixon Elliptic Functions via Quasi C Fractions Rathinavel Silambarasan Adem Kılıçman doi: 10.3390/fractalfract3020022 Fractal and Fractional 2019-04-17 Fractal and Fractional 2019-04-17 3 2 Article 22 10.3390/fractalfract3020022 https://www.mdpi.com/2504-3110/3/2/22
Fractal Fract, Vol. 3, Pages 21: Existence Theorems for Mixed Riemann–Liouville and Caputo Fractional Differential Equations and Inclusions with Nonlocal Fractional Integro-Differential Boundary Conditions https://www.mdpi.com/2504-3110/3/2/21 In this paper, we discuss the existence and uniqueness of solutions for a new class of single and multi-valued boundary value problems involving both Riemann&amp;ndash;Liouville and Caputo fractional derivatives, and nonlocal fractional integro-differential boundary conditions. Our results rely on modern tools of functional analysis. We also demonstrate the application of the obtained results with the aid of examples. Fractal Fract, Vol. 3, Pages 21: Existence Theorems for Mixed Riemann–Liouville and Caputo Fractional Differential Equations and Inclusions with Nonlocal Fractional Integro-Differential Boundary Conditions

Fractal and Fractional doi: 10.3390/fractalfract3020021

Authors: Sotiris K. Ntouyas Ahmed Alsaedi Bashir Ahmad

In this paper, we discuss the existence and uniqueness of solutions for a new class of single and multi-valued boundary value problems involving both Riemann&amp;ndash;Liouville and Caputo fractional derivatives, and nonlocal fractional integro-differential boundary conditions. Our results rely on modern tools of functional analysis. We also demonstrate the application of the obtained results with the aid of examples.

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Existence Theorems for Mixed Riemann–Liouville and Caputo Fractional Differential Equations and Inclusions with Nonlocal Fractional Integro-Differential Boundary Conditions Sotiris K. Ntouyas Ahmed Alsaedi Bashir Ahmad doi: 10.3390/fractalfract3020021 Fractal and Fractional 2019-04-17 Fractal and Fractional 2019-04-17 3 2 Article 21 10.3390/fractalfract3020021 https://www.mdpi.com/2504-3110/3/2/21
Fractal Fract, Vol. 3, Pages 20: Statistical Mechanics Involving Fractal Temperature https://www.mdpi.com/2504-3110/3/2/20 In this paper, the Schr&amp;ouml;dinger equation involving a fractal time derivative is solved and corresponding eigenvalues and eigenfunctions are given. A partition function for fractal eigenvalues is defined. For generalizing thermodynamics, fractal temperature is considered, and adapted equations are defined. As an application, we present fractal Dulong-Petit, Debye, and Einstein solid models and corresponding fractal heat capacity. Furthermore, the density of states for fractal spaces with fractional dimension is obtained. Graphs and examples are given to show details. Fractal Fract, Vol. 3, Pages 20: Statistical Mechanics Involving Fractal Temperature

Fractal and Fractional doi: 10.3390/fractalfract3020020

Authors: Alireza Khalili Golmankhaneh

In this paper, the Schr&amp;ouml;dinger equation involving a fractal time derivative is solved and corresponding eigenvalues and eigenfunctions are given. A partition function for fractal eigenvalues is defined. For generalizing thermodynamics, fractal temperature is considered, and adapted equations are defined. As an application, we present fractal Dulong-Petit, Debye, and Einstein solid models and corresponding fractal heat capacity. Furthermore, the density of states for fractal spaces with fractional dimension is obtained. Graphs and examples are given to show details.

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Statistical Mechanics Involving Fractal Temperature Alireza Khalili Golmankhaneh doi: 10.3390/fractalfract3020020 Fractal and Fractional 2019-04-17 Fractal and Fractional 2019-04-17 3 2 Article 20 10.3390/fractalfract3020020 https://www.mdpi.com/2504-3110/3/2/20
Fractal Fract, Vol. 3, Pages 19: New Estimates for Exponentially Convex Functions via Conformable Fractional Operator https://www.mdpi.com/2504-3110/3/2/19 In this paper, we derive a new Hermite&amp;ndash;Hadamard inequality for exponentially convex functions via &amp;alpha;-fractional integral. We also prove a new integral identity. Using this identity, we establish several Hermite&amp;ndash;Hadamard type inequalities for exponentially convexity, which can be obtained from our results. Some special cases are also discussed. Fractal Fract, Vol. 3, Pages 19: New Estimates for Exponentially Convex Functions via Conformable Fractional Operator

Fractal and Fractional doi: 10.3390/fractalfract3020019

Authors: Saima Rashid Muhammad Aslam Noor Khalida Inayat Noor

In this paper, we derive a new Hermite&amp;ndash;Hadamard inequality for exponentially convex functions via &amp;alpha;-fractional integral. We also prove a new integral identity. Using this identity, we establish several Hermite&amp;ndash;Hadamard type inequalities for exponentially convexity, which can be obtained from our results. Some special cases are also discussed.

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New Estimates for Exponentially Convex Functions via Conformable Fractional Operator Saima Rashid Muhammad Aslam Noor Khalida Inayat Noor doi: 10.3390/fractalfract3020019 Fractal and Fractional 2019-04-15 Fractal and Fractional 2019-04-15 3 2 Article 19 10.3390/fractalfract3020019 https://www.mdpi.com/2504-3110/3/2/19
Fractal Fract, Vol. 3, Pages 18: Moment Bound of Solution to a Class of Conformable Time-Fractional Stochastic Equation https://www.mdpi.com/2504-3110/3/2/18 We study a class of conformable time-fractional stochastic equation T &amp;alpha; , t a u ( x , t ) = &amp;sigma; ( u ( x , t ) ) W ˙ t , x &amp;isin; R , t &amp;isin; [ a , T ] , T &amp;lt; &amp;infin; , 0 &amp;lt; &amp;alpha; &amp;lt; 1 . The initial condition u ( x , 0 ) = u 0 ( x ) , x &amp;isin; R is a non-random function assumed to be non-negative and bounded, T &amp;alpha; , t a is a conformable time-fractional derivative, &amp;sigma; : R &amp;rarr; R is Lipschitz continuous and W ˙ t a generalized derivative of Wiener process. Some precise condition for the existence and uniqueness of a solution of the class of equation is given and we also give an upper bound estimate on the growth moment of the solution. Unlike the growth moment of stochastic fractional heat equation with Riemann&amp;ndash;Liouville or Caputo&amp;ndash;Dzhrbashyan fractional derivative which grows in time like t c 1 exp ( c 2 t ) , c 1 , c 2 &amp;gt; 0 ; our result also shows that the energy of the solution (the second moment) grows exponentially in time for t &amp;isin; [ a , T ] , T &amp;lt; &amp;infin; but with at most c 1 exp ( c 2 ( t &amp;minus; a ) 2 &amp;alpha; &amp;minus; 1 ) for some constants c 1 , and c 2 . Fractal Fract, Vol. 3, Pages 18: Moment Bound of Solution to a Class of Conformable Time-Fractional Stochastic Equation

Fractal and Fractional doi: 10.3390/fractalfract3020018

Authors: McSylvester Ejighikeme Omaba Eze R. Nwaeze

We study a class of conformable time-fractional stochastic equation T &amp;alpha; , t a u ( x , t ) = &amp;sigma; ( u ( x , t ) ) W ˙ t , x &amp;isin; R , t &amp;isin; [ a , T ] , T &amp;lt; &amp;infin; , 0 &amp;lt; &amp;alpha; &amp;lt; 1 . The initial condition u ( x , 0 ) = u 0 ( x ) , x &amp;isin; R is a non-random function assumed to be non-negative and bounded, T &amp;alpha; , t a is a conformable time-fractional derivative, &amp;sigma; : R &amp;rarr; R is Lipschitz continuous and W ˙ t a generalized derivative of Wiener process. Some precise condition for the existence and uniqueness of a solution of the class of equation is given and we also give an upper bound estimate on the growth moment of the solution. Unlike the growth moment of stochastic fractional heat equation with Riemann&amp;ndash;Liouville or Caputo&amp;ndash;Dzhrbashyan fractional derivative which grows in time like t c 1 exp ( c 2 t ) , c 1 , c 2 &amp;gt; 0 ; our result also shows that the energy of the solution (the second moment) grows exponentially in time for t &amp;isin; [ a , T ] , T &amp;lt; &amp;infin; but with at most c 1 exp ( c 2 ( t &amp;minus; a ) 2 &amp;alpha; &amp;minus; 1 ) for some constants c 1 , and c 2 .

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Moment Bound of Solution to a Class of Conformable Time-Fractional Stochastic Equation McSylvester Ejighikeme Omaba Eze R. Nwaeze doi: 10.3390/fractalfract3020018 Fractal and Fractional 2019-04-09 Fractal and Fractional 2019-04-09 3 2 Article 18 10.3390/fractalfract3020018 https://www.mdpi.com/2504-3110/3/2/18
Fractal Fract, Vol. 3, Pages 17: Determination of the Fractal Dimension of the Fracture Network System Using Image Processing Technique https://www.mdpi.com/2504-3110/3/2/17 Fractal dimension (FD) is a critical parameter in the characterization of a rock fracture network system. This parameter represents the distribution pattern of fractures in rock media. Moreover, it can be used for the modeling of fracture networks when the spatial distribution of fractures is described by the distribution of power law. The main objective of this research is to propose an automatic method to determine the rock mass FD in MATLAB using digital image processing techniques. This method not only accelerates analysis and reduces human error, but also eliminates the access limitation to a rock face. In the proposed method, the intensity of image brightness is corrected using the histogram equalization process and applying smoothing filters to the image followed by revealing the edges using the Canny edge detection algorithm. In the next step, FD is calculated in the program using the box-counting method, which is applied randomly to the pixels detected as fractures. This algorithm was implemented in different geological images to calculate their FDs. The FD of the images was determined using a simple Canny edge detection algorithm, a manual calculation method, and an indirect approach based on spectral decay rate. The results showed that the proposed method is a reliable and fast approach for calculating FD in fractured geological media. Fractal Fract, Vol. 3, Pages 17: Determination of the Fractal Dimension of the Fracture Network System Using Image Processing Technique

Fractal and Fractional doi: 10.3390/fractalfract3020017

Authors: Rouhollah Basirat Kamran Goshtasbi Morteza Ahmadi

Fractal dimension (FD) is a critical parameter in the characterization of a rock fracture network system. This parameter represents the distribution pattern of fractures in rock media. Moreover, it can be used for the modeling of fracture networks when the spatial distribution of fractures is described by the distribution of power law. The main objective of this research is to propose an automatic method to determine the rock mass FD in MATLAB using digital image processing techniques. This method not only accelerates analysis and reduces human error, but also eliminates the access limitation to a rock face. In the proposed method, the intensity of image brightness is corrected using the histogram equalization process and applying smoothing filters to the image followed by revealing the edges using the Canny edge detection algorithm. In the next step, FD is calculated in the program using the box-counting method, which is applied randomly to the pixels detected as fractures. This algorithm was implemented in different geological images to calculate their FDs. The FD of the images was determined using a simple Canny edge detection algorithm, a manual calculation method, and an indirect approach based on spectral decay rate. The results showed that the proposed method is a reliable and fast approach for calculating FD in fractured geological media.

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Determination of the Fractal Dimension of the Fracture Network System Using Image Processing Technique Rouhollah Basirat Kamran Goshtasbi Morteza Ahmadi doi: 10.3390/fractalfract3020017 Fractal and Fractional 2019-04-08 Fractal and Fractional 2019-04-08 3 2 Article 17 10.3390/fractalfract3020017 https://www.mdpi.com/2504-3110/3/2/17
Fractal Fract, Vol. 3, Pages 16: An Application of the Sonine–Letnikov Fractional Derivative for the Radial Schrödinger Equation https://www.mdpi.com/2504-3110/3/2/16 The Sonine&amp;ndash;Letnikov fractional derivative provides the generalized Leibniz rule and, some singular differential equations with integer order can be transformed into the fractional differential equations. The solutions of these equations obtained by some transformations have the fractional forms, and these forms can be obtained as the explicit solutions of these singular equations by using the fractional calculus definitions of Riemann&amp;ndash;Liouville, Gr&amp;uuml;nwald&amp;ndash;Letnikov, Caputo, etc. Explicit solutions of the Schr&amp;ouml;dinger equation have an important position in quantum mechanics due to the fact that the wave function includes all essential information for the exact definition of a physical system. In this paper, our aim is to obtain fractional solutions of the radial Schr&amp;ouml;dinger equation which is a singular differential equation with second-order, via the Sonine&amp;ndash;Letnikov fractional derivative. Fractal Fract, Vol. 3, Pages 16: An Application of the Sonine–Letnikov Fractional Derivative for the Radial Schrödinger Equation

Fractal and Fractional doi: 10.3390/fractalfract3020016

Authors: Okkes Ozturk Resat Yilmazer

The Sonine&amp;ndash;Letnikov fractional derivative provides the generalized Leibniz rule and, some singular differential equations with integer order can be transformed into the fractional differential equations. The solutions of these equations obtained by some transformations have the fractional forms, and these forms can be obtained as the explicit solutions of these singular equations by using the fractional calculus definitions of Riemann&amp;ndash;Liouville, Gr&amp;uuml;nwald&amp;ndash;Letnikov, Caputo, etc. Explicit solutions of the Schr&amp;ouml;dinger equation have an important position in quantum mechanics due to the fact that the wave function includes all essential information for the exact definition of a physical system. In this paper, our aim is to obtain fractional solutions of the radial Schr&amp;ouml;dinger equation which is a singular differential equation with second-order, via the Sonine&amp;ndash;Letnikov fractional derivative.

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An Application of the Sonine–Letnikov Fractional Derivative for the Radial Schrödinger Equation Okkes Ozturk Resat Yilmazer doi: 10.3390/fractalfract3020016 Fractal and Fractional 2019-04-04 Fractal and Fractional 2019-04-04 3 2 Article 16 10.3390/fractalfract3020016 https://www.mdpi.com/2504-3110/3/2/16
Fractal Fract, Vol. 3, Pages 15: Novel Fractional Models Compatible with Real World Problems https://www.mdpi.com/2504-3110/3/2/15 In this paper, some real world modeling problems: vertical motion of a falling body problem in a resistant medium, and the Malthusian growth equation, are considered by the newly defined Liouville–Caputo fractional conformable derivative and the modified form of this new definition. We utilize the σ auxiliary parameter for preserving the dimension of physical quantities for newly defined fractional conformable vertical motion of a falling body problem in a resistant medium. The analytical solutions are obtained by iterating this new fractional integral and results are illustrated under different orders by comparison with the Liouville–Caputo fractional operator. Fractal Fract, Vol. 3, Pages 15: Novel Fractional Models Compatible with Real World Problems

Fractal and Fractional doi: 10.3390/fractalfract3020015

Authors: Ramazan Ozarslan Ahu Ercan Erdal Bas

In this paper, some real world modeling problems: vertical motion of a falling body problem in a resistant medium, and the Malthusian growth equation, are considered by the newly defined Liouville–Caputo fractional conformable derivative and the modified form of this new definition. We utilize the σ auxiliary parameter for preserving the dimension of physical quantities for newly defined fractional conformable vertical motion of a falling body problem in a resistant medium. The analytical solutions are obtained by iterating this new fractional integral and results are illustrated under different orders by comparison with the Liouville–Caputo fractional operator.

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Novel Fractional Models Compatible with Real World Problems Ramazan Ozarslan Ahu Ercan Erdal Bas doi: 10.3390/fractalfract3020015 Fractal and Fractional 2019-04-01 Fractal and Fractional 2019-04-01 3 2 Article 15 10.3390/fractalfract3020015 https://www.mdpi.com/2504-3110/3/2/15
Fractal Fract, Vol. 3, Pages 14: Homotopy Perturbation ρ-Laplace Transform Method and Its Application to the Fractional Diffusion Equation and the Fractional Diffusion-Reaction Equation https://www.mdpi.com/2504-3110/3/2/14 In this paper, the approximate solutions of the fractional diffusion equations described by the fractional derivative operator were investigated. The homotopy perturbation Laplace transform method of getting the approximate solution was proposed. The Caputo generalized fractional derivative was used. The effects of the orders &amp;alpha; and &amp;rho; in the diffusion processes was addressed. The graphical representations of the approximate solutions of the fractional diffusion equation and the fractional diffusion-reaction equation both described by the Caputo generalized fractional derivative were provided. Fractal Fract, Vol. 3, Pages 14: Homotopy Perturbation ρ-Laplace Transform Method and Its Application to the Fractional Diffusion Equation and the Fractional Diffusion-Reaction Equation

Fractal and Fractional doi: 10.3390/fractalfract3020014

Authors: Ndolane Sene Aliou Niang Fall

In this paper, the approximate solutions of the fractional diffusion equations described by the fractional derivative operator were investigated. The homotopy perturbation Laplace transform method of getting the approximate solution was proposed. The Caputo generalized fractional derivative was used. The effects of the orders &amp;alpha; and &amp;rho; in the diffusion processes was addressed. The graphical representations of the approximate solutions of the fractional diffusion equation and the fractional diffusion-reaction equation both described by the Caputo generalized fractional derivative were provided.

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Homotopy Perturbation ρ-Laplace Transform Method and Its Application to the Fractional Diffusion Equation and the Fractional Diffusion-Reaction Equation Ndolane Sene Aliou Niang Fall doi: 10.3390/fractalfract3020014 Fractal and Fractional 2019-03-27 Fractal and Fractional 2019-03-27 3 2 Article 14 10.3390/fractalfract3020014 https://www.mdpi.com/2504-3110/3/2/14
Fractal Fract, Vol. 3, Pages 13: Intrinsic Metric Formulas on Some Self-Similar Sets via the Code Representation https://www.mdpi.com/2504-3110/3/1/13 In recent years, intrinsic metrics have been described on various fractals with different formulas. The Sierpinski gasket is given as one of the fundamental models which defined the intrinsic metrics on them via the code representations of the points. In this paper, we obtain the explicit formulas of the intrinsic metrics on some self-similar sets (but not strictly self-similar), which are composed of different combinations of equilateral and right Sierpinski gaskets, respectively, by using the code representations of their points. We then express geometrical properties of these structures on their code sets and also give some illustrative examples. Fractal Fract, Vol. 3, Pages 13: Intrinsic Metric Formulas on Some Self-Similar Sets via the Code Representation

Fractal and Fractional doi: 10.3390/fractalfract3010013

Authors: Melis Güneri Mustafa Saltan

In recent years, intrinsic metrics have been described on various fractals with different formulas. The Sierpinski gasket is given as one of the fundamental models which defined the intrinsic metrics on them via the code representations of the points. In this paper, we obtain the explicit formulas of the intrinsic metrics on some self-similar sets (but not strictly self-similar), which are composed of different combinations of equilateral and right Sierpinski gaskets, respectively, by using the code representations of their points. We then express geometrical properties of these structures on their code sets and also give some illustrative examples.

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Intrinsic Metric Formulas on Some Self-Similar Sets via the Code Representation Melis Güneri Mustafa Saltan doi: 10.3390/fractalfract3010013 Fractal and Fractional 2019-03-25 Fractal and Fractional 2019-03-25 3 1 Article 13 10.3390/fractalfract3010013 https://www.mdpi.com/2504-3110/3/1/13
Fractal Fract, Vol. 3, Pages 12: Some New Fractional Trapezium-Type Inequalities for Preinvex Functions https://www.mdpi.com/2504-3110/3/1/12 In this paper, authors the present the discovery of an interesting identity regarding trapezium-type integral inequalities. By using the lemma as an auxiliary result, some new estimates with respect to trapezium-type integral inequalities via general fractional integrals are obtained. It is pointed out that some new special cases can be deduced from the main results. Some applications regarding special means for different real numbers are provided as well. The ideas and techniques described in this paper may stimulate further research. Fractal Fract, Vol. 3, Pages 12: Some New Fractional Trapezium-Type Inequalities for Preinvex Functions

Fractal and Fractional doi: 10.3390/fractalfract3010012

Authors: Artion Kashuri Erhan Set Rozana Liko

In this paper, authors the present the discovery of an interesting identity regarding trapezium-type integral inequalities. By using the lemma as an auxiliary result, some new estimates with respect to trapezium-type integral inequalities via general fractional integrals are obtained. It is pointed out that some new special cases can be deduced from the main results. Some applications regarding special means for different real numbers are provided as well. The ideas and techniques described in this paper may stimulate further research.

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Some New Fractional Trapezium-Type Inequalities for Preinvex Functions Artion Kashuri Erhan Set Rozana Liko doi: 10.3390/fractalfract3010012 Fractal and Fractional 2019-03-24 Fractal and Fractional 2019-03-24 3 1 Article 12 10.3390/fractalfract3010012 https://www.mdpi.com/2504-3110/3/1/12
Fractal Fract, Vol. 3, Pages 11: On the Fractal Langevin Equation https://www.mdpi.com/2504-3110/3/1/11 In this paper, fractal stochastic Langevin equations are suggested, providing a mathematical model for random walks on the middle- &amp;tau; Cantor set. The fractal mean square displacement of different random walks on the middle- &amp;tau; Cantor set are presented. Fractal under-damped and over-damped Langevin equations, fractal scaled Brownian motion, and ultra-slow fractal scaled Brownian motion are suggested and the corresponding fractal mean square displacements are obtained. The results are plotted to show the details. Fractal Fract, Vol. 3, Pages 11: On the Fractal Langevin Equation

Fractal and Fractional doi: 10.3390/fractalfract3010011

Authors: Alireza Khalili Golmankhaneh

In this paper, fractal stochastic Langevin equations are suggested, providing a mathematical model for random walks on the middle- &amp;tau; Cantor set. The fractal mean square displacement of different random walks on the middle- &amp;tau; Cantor set are presented. Fractal under-damped and over-damped Langevin equations, fractal scaled Brownian motion, and ultra-slow fractal scaled Brownian motion are suggested and the corresponding fractal mean square displacements are obtained. The results are plotted to show the details.

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On the Fractal Langevin Equation Alireza Khalili Golmankhaneh doi: 10.3390/fractalfract3010011 Fractal and Fractional 2019-03-13 Fractal and Fractional 2019-03-13 3 1 Article 11 10.3390/fractalfract3010011 https://www.mdpi.com/2504-3110/3/1/11
Fractal Fract, Vol. 3, Pages 10: On Analytic Functions Involving the q-Ruscheweyeh Derivative https://www.mdpi.com/2504-3110/3/1/10 In this paper, we use concepts of q-calculus to introduce a certain type of q-difference operator, and using it define some subclasses of analytic functions. Inclusion relations, coefficient result, and some other interesting properties of these classes are studied. Fractal Fract, Vol. 3, Pages 10: On Analytic Functions Involving the q-Ruscheweyeh Derivative

Fractal and Fractional doi: 10.3390/fractalfract3010010

Authors: Khalida Inayat Noor

In this paper, we use concepts of q-calculus to introduce a certain type of q-difference operator, and using it define some subclasses of analytic functions. Inclusion relations, coefficient result, and some other interesting properties of these classes are studied.

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On Analytic Functions Involving the q-Ruscheweyeh Derivative Khalida Inayat Noor doi: 10.3390/fractalfract3010010 Fractal and Fractional 2019-03-10 Fractal and Fractional 2019-03-10 3 1 Article 10 10.3390/fractalfract3010010 https://www.mdpi.com/2504-3110/3/1/10
Fractal Fract, Vol. 3, Pages 9: Residual Power Series Method for Fractional Swift–Hohenberg Equation https://www.mdpi.com/2504-3110/3/1/9 In this paper, the approximated analytical solution for the fractional Swift&amp;ndash;Hohenberg (S&amp;ndash;H) equation has been investigated with the help of the residual power series method (RPSM). To ensure the applicability and efficiency of the proposed technique, we consider a non-linear fractional order Swift&amp;ndash;Hohenberg equation in the presence and absence of dispersive terms. The effect of bifurcation and dispersive parameters with physical importance on the probability density function for distinct fractional Brownian and standard motions are studied and presented through plots. The results obtained show that the proposed technique is simple to implement and very effective for analyzing the complex problems that arise in connected areas of science and technology. Fractal Fract, Vol. 3, Pages 9: Residual Power Series Method for Fractional Swift–Hohenberg Equation

Fractal and Fractional doi: 10.3390/fractalfract3010009

Authors: D. G. Prakasha P. Veeresha Haci Mehmet Baskonus

In this paper, the approximated analytical solution for the fractional Swift&amp;ndash;Hohenberg (S&amp;ndash;H) equation has been investigated with the help of the residual power series method (RPSM). To ensure the applicability and efficiency of the proposed technique, we consider a non-linear fractional order Swift&amp;ndash;Hohenberg equation in the presence and absence of dispersive terms. The effect of bifurcation and dispersive parameters with physical importance on the probability density function for distinct fractional Brownian and standard motions are studied and presented through plots. The results obtained show that the proposed technique is simple to implement and very effective for analyzing the complex problems that arise in connected areas of science and technology.

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Residual Power Series Method for Fractional Swift–Hohenberg Equation D. G. Prakasha P. Veeresha Haci Mehmet Baskonus doi: 10.3390/fractalfract3010009 Fractal and Fractional 2019-03-07 Fractal and Fractional 2019-03-07 3 1 Article 9 10.3390/fractalfract3010009 https://www.mdpi.com/2504-3110/3/1/9
Fractal Fract, Vol. 3, Pages 8: The Fractal Calculus for Fractal Materials https://www.mdpi.com/2504-3110/3/1/8 The major problem in the process of mixing fluids (for instance liquid-liquid mixers) is turbulence, which is the outcome of the function of the equipment (engine). Fractal mixing is an alternative method that has symmetry and is predictable. Therefore, fractal structures and fractal reactors find importance. Using F &amp;alpha; -fractal calculus, in this paper, we derive exact F &amp;alpha; -differential forms of an ideal gas. Depending on the dimensionality of space, we should first obtain the integral staircase function and mass function of our geometry. When gases expand inside the fractal structure because of changes from the i + 1 iteration to the i iteration, in fact, we are faced with fluid mixing inside our fractal structure, which can be described by physical quantities P, V, and T. Finally, for the ideal gas equation, we calculate volume expansivity and isothermal compressibility. Fractal Fract, Vol. 3, Pages 8: The Fractal Calculus for Fractal Materials

Fractal and Fractional doi: 10.3390/fractalfract3010008

The major problem in the process of mixing fluids (for instance liquid-liquid mixers) is turbulence, which is the outcome of the function of the equipment (engine). Fractal mixing is an alternative method that has symmetry and is predictable. Therefore, fractal structures and fractal reactors find importance. Using F &amp;alpha; -fractal calculus, in this paper, we derive exact F &amp;alpha; -differential forms of an ideal gas. Depending on the dimensionality of space, we should first obtain the integral staircase function and mass function of our geometry. When gases expand inside the fractal structure because of changes from the i + 1 iteration to the i iteration, in fact, we are faced with fluid mixing inside our fractal structure, which can be described by physical quantities P, V, and T. Finally, for the ideal gas equation, we calculate volume expansivity and isothermal compressibility.

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The Fractal Calculus for Fractal Materials Fakhri Khajvand Jafari Mohammad Sadegh Asgari Amir Pishkoo doi: 10.3390/fractalfract3010008 Fractal and Fractional 2019-03-06 Fractal and Fractional 2019-03-06 3 1 Article 8 10.3390/fractalfract3010008 https://www.mdpi.com/2504-3110/3/1/8
Fractal Fract, Vol. 3, Pages 7: Fractal Image Interpolation: A Tutorial and New Result https://www.mdpi.com/2504-3110/3/1/7 This paper reviews the implementation of fractal based image interpolation, the associated visual artifacts of the interpolated images, and various techniques, including novel contributions, that alleviate these awkward visual artifacts to achieve visually pleasant interpolated image. The fractal interpolation methods considered in this paper are based on the plain Iterative Function System (IFS) in spatial domain without additional transformation, where we believe that the benefits of additional transformation can be added onto the presented study without complication. Simulation results are presented to demonstrate the discussed techniques, together with the pros and cons of each techniques. Finally, a novel spatial domain interleave layer has been proposed to add to the IFS image system for improving the performance of the system from image zooming to interpolation with the preservation of the pixel intensity from the original low resolution image. Fractal Fract, Vol. 3, Pages 7: Fractal Image Interpolation: A Tutorial and New Result

Fractal and Fractional doi: 10.3390/fractalfract3010007

Authors: Chi Wah Kok Wing Shan Tam

This paper reviews the implementation of fractal based image interpolation, the associated visual artifacts of the interpolated images, and various techniques, including novel contributions, that alleviate these awkward visual artifacts to achieve visually pleasant interpolated image. The fractal interpolation methods considered in this paper are based on the plain Iterative Function System (IFS) in spatial domain without additional transformation, where we believe that the benefits of additional transformation can be added onto the presented study without complication. Simulation results are presented to demonstrate the discussed techniques, together with the pros and cons of each techniques. Finally, a novel spatial domain interleave layer has been proposed to add to the IFS image system for improving the performance of the system from image zooming to interpolation with the preservation of the pixel intensity from the original low resolution image.

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Fractal Image Interpolation: A Tutorial and New Result Chi Wah Kok Wing Shan Tam doi: 10.3390/fractalfract3010007 Fractal and Fractional 2019-02-23 Fractal and Fractional 2019-02-23 3 1 Article 7 10.3390/fractalfract3010007 https://www.mdpi.com/2504-3110/3/1/7
Fractal Fract, Vol. 3, Pages 6: Julia and Mandelbrot Sets for Dynamics over the Hyperbolic Numbers https://www.mdpi.com/2504-3110/3/1/6 Julia and Mandelbrot sets, which characterize bounded orbits in dynamical systems over the complex numbers, are classic examples of fractal sets. We investigate the analogs of these sets for dynamical systems over the hyperbolic numbers. Hyperbolic numbers, which have the form x + &amp;tau; y for x , y &amp;isin; R , and &amp;tau; 2 = 1 but &amp;tau; &amp;ne; &amp;plusmn; 1 , are the natural number system in which to encode geometric properties of the Minkowski space R 1 , 1 . We show that the hyperbolic analog of the Mandelbrot set parameterizes the connectedness of hyperbolic Julia sets. We give a wall-and-chamber decomposition of the hyperbolic plane in terms of these Julia sets. Fractal Fract, Vol. 3, Pages 6: Julia and Mandelbrot Sets for Dynamics over the Hyperbolic Numbers

Fractal and Fractional doi: 10.3390/fractalfract3010006

Authors: Vance Blankers Tristan Rendfrey Aaron Shukert Patrick D. Shipman

Julia and Mandelbrot sets, which characterize bounded orbits in dynamical systems over the complex numbers, are classic examples of fractal sets. We investigate the analogs of these sets for dynamical systems over the hyperbolic numbers. Hyperbolic numbers, which have the form x + &amp;tau; y for x , y &amp;isin; R , and &amp;tau; 2 = 1 but &amp;tau; &amp;ne; &amp;plusmn; 1 , are the natural number system in which to encode geometric properties of the Minkowski space R 1 , 1 . We show that the hyperbolic analog of the Mandelbrot set parameterizes the connectedness of hyperbolic Julia sets. We give a wall-and-chamber decomposition of the hyperbolic plane in terms of these Julia sets.

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Julia and Mandelbrot Sets for Dynamics over the Hyperbolic Numbers Vance Blankers Tristan Rendfrey Aaron Shukert Patrick D. Shipman doi: 10.3390/fractalfract3010006 Fractal and Fractional 2019-02-20 Fractal and Fractional 2019-02-20 3 1 Brief Report 6 10.3390/fractalfract3010006 https://www.mdpi.com/2504-3110/3/1/6
Fractal Fract, Vol. 3, Pages 5: On q-Uniformly Mocanu Functions https://www.mdpi.com/2504-3110/3/1/5 Let f be analytic in open unit disc E = { z : | z | &amp;lt; 1 } with f ( 0 ) = 0 and f &amp;prime; ( 0 ) = 1 . The q-derivative of f is defined by: D q f ( z ) = f ( z ) &amp;minus; f ( q z ) ( 1 &amp;minus; q ) z , q &amp;isin; ( 0 , 1 ) , z &amp;isin; B &amp;minus; { 0 } , where B is a q-geometric subset of C . Using operator D q , q-analogue class k &amp;minus; U M q ( &amp;alpha; , &amp;beta; ) , k-uniformly Mocanu functions are defined as: For k = 0 and q &amp;rarr; 1 &amp;minus; , k &amp;minus; reduces to M ( &amp;alpha; ) of Mocanu functions. Subordination is used to investigate many important properties of these functions. Several interesting results are derived as special cases. Fractal Fract, Vol. 3, Pages 5: On q-Uniformly Mocanu Functions

Fractal and Fractional doi: 10.3390/fractalfract3010005

Authors: Rizwan S. Badar Khalida Inayat Noor

Let f be analytic in open unit disc E = { z : | z | &amp;lt; 1 } with f ( 0 ) = 0 and f &amp;prime; ( 0 ) = 1 . The q-derivative of f is defined by: D q f ( z ) = f ( z ) &amp;minus; f ( q z ) ( 1 &amp;minus; q ) z , q &amp;isin; ( 0 , 1 ) , z &amp;isin; B &amp;minus; { 0 } , where B is a q-geometric subset of C . Using operator D q , q-analogue class k &amp;minus; U M q ( &amp;alpha; , &amp;beta; ) , k-uniformly Mocanu functions are defined as: For k = 0 and q &amp;rarr; 1 &amp;minus; , k &amp;minus; reduces to M ( &amp;alpha; ) of Mocanu functions. Subordination is used to investigate many important properties of these functions. Several interesting results are derived as special cases.

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On q-Uniformly Mocanu Functions Rizwan S. Badar Khalida Inayat Noor doi: 10.3390/fractalfract3010005 Fractal and Fractional 2019-02-11 Fractal and Fractional 2019-02-11 3 1 Article 5 10.3390/fractalfract3010005 https://www.mdpi.com/2504-3110/3/1/5
Fractal Fract, Vol. 3, Pages 4: Integral Representations and Algebraic Decompositions of the Fox-Wright Type of Special Functions https://www.mdpi.com/2504-3110/3/1/4 The manuscript surveys the special functions of the Fox-Wright type. These functions are generalizations of the hypergeometric functions. Notable representatives of the type are the Mittag-Leffler functions and the Wright function. The integral representations of such functions are given and the conditions under which these function can be represented by simpler functions are demonstrated. The connection with generalized Erd&amp;eacute;lyi-Kober fractional differential and integral operators is demonstrated and discussed. Fractal Fract, Vol. 3, Pages 4: Integral Representations and Algebraic Decompositions of the Fox-Wright Type of Special Functions

Fractal and Fractional doi: 10.3390/fractalfract3010004

Authors: Dimiter Prodanov

The manuscript surveys the special functions of the Fox-Wright type. These functions are generalizations of the hypergeometric functions. Notable representatives of the type are the Mittag-Leffler functions and the Wright function. The integral representations of such functions are given and the conditions under which these function can be represented by simpler functions are demonstrated. The connection with generalized Erd&amp;eacute;lyi-Kober fractional differential and integral operators is demonstrated and discussed.

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Integral Representations and Algebraic Decompositions of the Fox-Wright Type of Special Functions Dimiter Prodanov doi: 10.3390/fractalfract3010004 Fractal and Fractional 2019-01-25 Fractal and Fractional 2019-01-25 3 1 Article 4 10.3390/fractalfract3010004 https://www.mdpi.com/2504-3110/3/1/4