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. 3, Pages 46: A New Fractional Integration Operational Matrix of Chebyshev Wavelets in Fractional Delay Systems http://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 http://www.mdpi.com/2504-3110/3/3/46
Fractal Fract, Vol. 3, Pages 45: Characterization of the Local Growth of Two Cantor-Type Functions http://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 http://www.mdpi.com/2504-3110/3/3/45
Fractal Fract, Vol. 3, Pages 44: Multi-Term Fractional Differential Equations with Generalized Integral Boundary Conditions http://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

Authors: Bashir Ahmad Madeaha Alghanmi Ahmed Alsaedi Sotiris K. Ntouyas

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 http://www.mdpi.com/2504-3110/3/3/44
Fractal Fract, Vol. 3, Pages 43: Solving Helmholtz Equation with Local Fractional Derivative Operators http://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 http://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 http://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 http://www.mdpi.com/2504-3110/3/3/42
Fractal Fract, Vol. 3, Pages 41: Fractal Logistic Equation http://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 http://www.mdpi.com/2504-3110/3/3/41
Fractal Fract, Vol. 3, Pages 40: Cornu Spirals and the Triangular Lacunary Trigonometric System http://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 http://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 http://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 http://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 http://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 http://www.mdpi.com/2504-3110/3/3/38
Fractal Fract, Vol. 3, Pages 37: Inequalities Pertaining Fractional Approach through Exponentially Convex Functions http://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 http://www.mdpi.com/2504-3110/3/3/37
Fractal Fract, Vol. 3, Pages 36: Green’s Function Estimates for Time-Fractional Evolution Equations http://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 http://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 http://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 http://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 http://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 http://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 http://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 http://www.mdpi.com/2504-3110/3/2/33
Fractal Fract, Vol. 3, Pages 32: On Extended General Mittag–Leffler Functions and Certain Inequalities http://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 http://www.mdpi.com/2504-3110/3/2/32
Fractal Fract, Vol. 3, Pages 31: Random Variables and Stable Distributions on Fractal Cantor Sets http://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 http://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 http://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 http://www.mdpi.com/2504-3110/3/2/30
Fractal Fract, Vol. 3, Pages 29: On Some Generalized Fractional Integral Inequalities for p-Convex Functions http://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 http://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 http://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 http://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 http://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 http://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 http://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 http://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 http://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 http://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 http://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 http://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 http://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 http://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 http://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

Authors: Rathinavel Silambarasan Adem Kılıçman

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 http://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 http://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 http://www.mdpi.com/2504-3110/3/2/21
Fractal Fract, Vol. 3, Pages 20: Statistical Mechanics Involving Fractal Temperature http://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 http://www.mdpi.com/2504-3110/3/2/20
Fractal Fract, Vol. 3, Pages 19: New Estimates for Exponentially Convex Functions via Conformable Fractional Operator http://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 http://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 http://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 http://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 http://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 http://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 http://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 http://www.mdpi.com/2504-3110/3/2/16
Fractal Fract, Vol. 3, Pages 15: Novel Fractional Models Compatible with Real World Problems http://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 http://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 http://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 http://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 http://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 http://www.mdpi.com/2504-3110/3/1/13
Fractal Fract, Vol. 3, Pages 12: Some New Fractional Trapezium-Type Inequalities for Preinvex Functions http://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 http://www.mdpi.com/2504-3110/3/1/12
Fractal Fract, Vol. 3, Pages 11: On the Fractal Langevin Equation http://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 http://www.mdpi.com/2504-3110/3/1/11
Fractal Fract, Vol. 3, Pages 10: On Analytic Functions Involving the q-Ruscheweyeh Derivative http://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 http://www.mdpi.com/2504-3110/3/1/10
Fractal Fract, Vol. 3, Pages 9: Residual Power Series Method for Fractional Swift–Hohenberg Equation http://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 http://www.mdpi.com/2504-3110/3/1/9
Fractal Fract, Vol. 3, Pages 8: The Fractal Calculus for Fractal Materials http://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

Authors: Fakhri Khajvand Jafari Mohammad Sadegh Asgari Amir Pishkoo

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 http://www.mdpi.com/2504-3110/3/1/8
Fractal Fract, Vol. 3, Pages 7: Fractal Image Interpolation: A Tutorial and New Result http://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 http://www.mdpi.com/2504-3110/3/1/7
Fractal Fract, Vol. 3, Pages 6: Julia and Mandelbrot Sets for Dynamics over the Hyperbolic Numbers http://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 http://www.mdpi.com/2504-3110/3/1/6
Fractal Fract, Vol. 3, Pages 5: On q-Uniformly Mocanu Functions http://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 http://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 http://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 http://www.mdpi.com/2504-3110/3/1/4
Fractal Fract, Vol. 3, Pages 3: Study of Fractal Dimensions of Microcrystalline Cellulose Obtained by the Spray-Drying Method http://www.mdpi.com/2504-3110/3/1/3 In this research, the fractal structure of beads of different sizes obtained by the spray-drying of aqueous dispersions of microcrystalline cellulose (MCC) was studied. These beads were formed as a result of the aggregation of rod-shaped cellulose nanocrystalline particles (CNP). It was found that increasing the average radius (R) of the formed MCC beads resulted in increased specific pore volume (P) and reduced apparent density (&amp;rho;). The dependences of P and &amp;rho; on the scale factor (R/r) can be expressed by power-law equations: P = Po (R/r)E&amp;minus;Dp and &amp;rho; = d (R/r)Dd&amp;minus;E, where the fractal dimensions Dp = 2.887 and Dd = 2.986 are close to the Euclidean dimension E = 3 for three-dimensional space; r = 3 nm is the radius of the cellulose nanocrystalline particles, Po = 0.03 cm3/g is the specific pore volume, and d = 1.585 g/cm3 is the true density (specific gravity) of the CNP, respectively. With the increase in the size of the formed MCC beads, the order in the packing of the beads was distorted, conforming to theory of the diffusion-limited aggregation process. Fractal Fract, Vol. 3, Pages 3: Study of Fractal Dimensions of Microcrystalline Cellulose Obtained by the Spray-Drying Method

Fractal and Fractional doi: 10.3390/fractalfract3010003

Authors: Michael Ioelovich

In this research, the fractal structure of beads of different sizes obtained by the spray-drying of aqueous dispersions of microcrystalline cellulose (MCC) was studied. These beads were formed as a result of the aggregation of rod-shaped cellulose nanocrystalline particles (CNP). It was found that increasing the average radius (R) of the formed MCC beads resulted in increased specific pore volume (P) and reduced apparent density (&amp;rho;). The dependences of P and &amp;rho; on the scale factor (R/r) can be expressed by power-law equations: P = Po (R/r)E&amp;minus;Dp and &amp;rho; = d (R/r)Dd&amp;minus;E, where the fractal dimensions Dp = 2.887 and Dd = 2.986 are close to the Euclidean dimension E = 3 for three-dimensional space; r = 3 nm is the radius of the cellulose nanocrystalline particles, Po = 0.03 cm3/g is the specific pore volume, and d = 1.585 g/cm3 is the true density (specific gravity) of the CNP, respectively. With the increase in the size of the formed MCC beads, the order in the packing of the beads was distorted, conforming to theory of the diffusion-limited aggregation process.

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Study of Fractal Dimensions of Microcrystalline Cellulose Obtained by the Spray-Drying Method Michael Ioelovich doi: 10.3390/fractalfract3010003 Fractal and Fractional 2019-01-24 Fractal and Fractional 2019-01-24 3 1 Article 3 10.3390/fractalfract3010003 http://www.mdpi.com/2504-3110/3/1/3
Fractal Fract, Vol. 3, Pages 2: Acknowledgement to Reviewers of Fractal Fract in 2018 http://www.mdpi.com/2504-3110/3/1/2 Rigorous peer-review is the corner-stone of high-quality academic publishing [...] Fractal Fract, Vol. 3, Pages 2: Acknowledgement to Reviewers of Fractal Fract in 2018

Fractal and Fractional doi: 10.3390/fractalfract3010002

Authors: Fractal Fract Editorial Office

Rigorous peer-review is the corner-stone of high-quality academic publishing [...]

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Acknowledgement to Reviewers of Fractal Fract in 2018 Fractal Fract Editorial Office doi: 10.3390/fractalfract3010002 Fractal and Fractional 2019-01-16 Fractal and Fractional 2019-01-16 3 1 Editorial 2 10.3390/fractalfract3010002 http://www.mdpi.com/2504-3110/3/1/2
Fractal Fract, Vol. 3, Pages 1: Regularized Integral Representations of the Reciprocal Gamma Function http://www.mdpi.com/2504-3110/3/1/1 This paper establishes a real integral representation of the reciprocal Gamma function in terms of a regularized hypersingular integral along the real line. A regularized complex representation along the Hankel path is derived. The equivalence with the Heine&amp;rsquo;s complex representation is demonstrated. For both real and complex integrals, the regularized representation can be expressed in terms of the two-parameter Mittag-Leffler function. Reference numerical implementations in the Computer Algebra System Maxima are provided. Fractal Fract, Vol. 3, Pages 1: Regularized Integral Representations of the Reciprocal Gamma Function

Fractal and Fractional doi: 10.3390/fractalfract3010001

Authors: Dimiter Prodanov

This paper establishes a real integral representation of the reciprocal Gamma function in terms of a regularized hypersingular integral along the real line. A regularized complex representation along the Hankel path is derived. The equivalence with the Heine&amp;rsquo;s complex representation is demonstrated. For both real and complex integrals, the regularized representation can be expressed in terms of the two-parameter Mittag-Leffler function. Reference numerical implementations in the Computer Algebra System Maxima are provided.

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Regularized Integral Representations of the Reciprocal Gamma Function Dimiter Prodanov doi: 10.3390/fractalfract3010001 Fractal and Fractional 2019-01-12 Fractal and Fractional 2019-01-12 3 1 Article 1 10.3390/fractalfract3010001 http://www.mdpi.com/2504-3110/3/1/1
Fractal Fract, Vol. 2, Pages 30: Fractal Calculus of Functions on Cantor Tartan Spaces http://www.mdpi.com/2504-3110/2/4/30 In this manuscript, integrals and derivatives of functions on Cantor tartan spaces are defined. The generalisation of standard calculus, which is called F &amp;eta; -calculus, is utilised to obtain definitions of the integral and derivative of functions on Cantor tartan spaces of different dimensions. Differential equations involving the new derivatives are solved. Illustrative examples are presented to check the details. Fractal Fract, Vol. 2, Pages 30: Fractal Calculus of Functions on Cantor Tartan Spaces

Fractal and Fractional doi: 10.3390/fractalfract2040030

Authors: Alireza Khalili Golmankhaneh Arran Fernandez

In this manuscript, integrals and derivatives of functions on Cantor tartan spaces are defined. The generalisation of standard calculus, which is called F &amp;eta; -calculus, is utilised to obtain definitions of the integral and derivative of functions on Cantor tartan spaces of different dimensions. Differential equations involving the new derivatives are solved. Illustrative examples are presented to check the details.

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Fractal Calculus of Functions on Cantor Tartan Spaces Alireza Khalili Golmankhaneh Arran Fernandez doi: 10.3390/fractalfract2040030 Fractal and Fractional 2018-12-18 Fractal and Fractional 2018-12-18 2 4 Article 30 10.3390/fractalfract2040030 http://www.mdpi.com/2504-3110/2/4/30
Fractal Fract, Vol. 2, Pages 29: Approximate Controllability of Semilinear Stochastic Integrodifferential System with Nonlocal Conditions http://www.mdpi.com/2504-3110/2/4/29 The objective of this paper is to analyze the approximate controllability of a semilinear stochastic integrodifferential system with nonlocal conditions in Hilbert spaces. The nonlocal initial condition is a generalization of the classical initial condition and is motivated by physical phenomena. The results are obtained by using Sadovskii&amp;rsquo;s fixed point theorem. At the end, an example is given to show the effectiveness of the result. Fractal Fract, Vol. 2, Pages 29: Approximate Controllability of Semilinear Stochastic Integrodifferential System with Nonlocal Conditions

Fractal and Fractional doi: 10.3390/fractalfract2040029

Authors: Annamalai Anguraj K. Ramkumar

The objective of this paper is to analyze the approximate controllability of a semilinear stochastic integrodifferential system with nonlocal conditions in Hilbert spaces. The nonlocal initial condition is a generalization of the classical initial condition and is motivated by physical phenomena. The results are obtained by using Sadovskii&amp;rsquo;s fixed point theorem. At the end, an example is given to show the effectiveness of the result.

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Approximate Controllability of Semilinear Stochastic Integrodifferential System with Nonlocal Conditions Annamalai Anguraj K. Ramkumar doi: 10.3390/fractalfract2040029 Fractal and Fractional 2018-11-20 Fractal and Fractional 2018-11-20 2 4 Article 29 10.3390/fractalfract2040029 http://www.mdpi.com/2504-3110/2/4/29
Fractal Fract, Vol. 2, Pages 28: Mathematical Modeling of Solutes Migration under the Conditions of Groundwater Filtration by the Model with the k-Caputo Fractional Derivative http://www.mdpi.com/2504-3110/2/4/28 Within the framework of a new mathematical model of convective diffusion with the k-Caputo derivative, we simulate the dynamics of anomalous soluble substances migration under the conditions of two-dimensional steady-state plane-vertical filtration with a free surface. As a corresponding filtration scheme, we consider the scheme for the spread of pollution from rivers, canals, or storages of industrial wastes. On the base of a locally one-dimensional finite-difference scheme, we develop a numerical method for obtaining solutions of boundary value problem for fractional differential equation with k-Caputo derivative with respect to the time variable that describes the convective diffusion of salt solution. The results of numerical experiments on modeling the dynamics of the considered process are presented. The results that show an existence of a time lag in the process of diffusion field formation are presented. Fractal Fract, Vol. 2, Pages 28: Mathematical Modeling of Solutes Migration under the Conditions of Groundwater Filtration by the Model with the k-Caputo Fractional Derivative

Fractal and Fractional doi: 10.3390/fractalfract2040028

Authors: Vsevolod Bohaienko Volodymyr Bulavatsky

Within the framework of a new mathematical model of convective diffusion with the k-Caputo derivative, we simulate the dynamics of anomalous soluble substances migration under the conditions of two-dimensional steady-state plane-vertical filtration with a free surface. As a corresponding filtration scheme, we consider the scheme for the spread of pollution from rivers, canals, or storages of industrial wastes. On the base of a locally one-dimensional finite-difference scheme, we develop a numerical method for obtaining solutions of boundary value problem for fractional differential equation with k-Caputo derivative with respect to the time variable that describes the convective diffusion of salt solution. The results of numerical experiments on modeling the dynamics of the considered process are presented. The results that show an existence of a time lag in the process of diffusion field formation are presented.

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Mathematical Modeling of Solutes Migration under the Conditions of Groundwater Filtration by the Model with the k-Caputo Fractional Derivative Vsevolod Bohaienko Volodymyr Bulavatsky doi: 10.3390/fractalfract2040028 Fractal and Fractional 2018-10-24 Fractal and Fractional 2018-10-24 2 4 Article 28 10.3390/fractalfract2040028 http://www.mdpi.com/2504-3110/2/4/28
Fractal Fract, Vol. 2, Pages 27: Fatigue-Induced Cole Electrical Impedance Model Changes of Biceps Tissue Bioimpedance http://www.mdpi.com/2504-3110/2/4/27 Bioimpedance, or the electrical impedance of biological tissues, describes the passive electrical properties of these materials. To simplify bioimpedance datasets, fractional-order equivalent circuit presentations are often used, with the Cole-impedance model being one of the most widely used fractional-order circuits for this purpose. In this work, bioimpedance measurements from 10 kHz to 100 kHz were collected from participants biceps tissues immediately prior and immediately post completion of a fatiguing exercise protocol. The Cole-impedance parameters that best fit these datasets were determined using numerical optimization procedures, with relative errors of within approximately &amp;plusmn; 0.5 % and &amp;plusmn; 2 % for the simulated resistance and reactance compared to the experimental data. Comparison between the pre and post fatigue Cole-impedance parameters shows that the R &amp;infin; , R 1 , and f p components exhibited statistically significant mean differences as a result of the fatigue induced changes in the study participants. Fractal Fract, Vol. 2, Pages 27: Fatigue-Induced Cole Electrical Impedance Model Changes of Biceps Tissue Bioimpedance

Fractal and Fractional doi: 10.3390/fractalfract2040027

Authors: Todd J. Freeborn Bo Fu

Bioimpedance, or the electrical impedance of biological tissues, describes the passive electrical properties of these materials. To simplify bioimpedance datasets, fractional-order equivalent circuit presentations are often used, with the Cole-impedance model being one of the most widely used fractional-order circuits for this purpose. In this work, bioimpedance measurements from 10 kHz to 100 kHz were collected from participants biceps tissues immediately prior and immediately post completion of a fatiguing exercise protocol. The Cole-impedance parameters that best fit these datasets were determined using numerical optimization procedures, with relative errors of within approximately &amp;plusmn; 0.5 % and &amp;plusmn; 2 % for the simulated resistance and reactance compared to the experimental data. Comparison between the pre and post fatigue Cole-impedance parameters shows that the R &amp;infin; , R 1 , and f p components exhibited statistically significant mean differences as a result of the fatigue induced changes in the study participants.

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Fatigue-Induced Cole Electrical Impedance Model Changes of Biceps Tissue Bioimpedance Todd J. Freeborn Bo Fu doi: 10.3390/fractalfract2040027 Fractal and Fractional 2018-10-24 Fractal and Fractional 2018-10-24 2 4 Article 27 10.3390/fractalfract2040027 http://www.mdpi.com/2504-3110/2/4/27
Fractal Fract, Vol. 2, Pages 26: Minkowski Dimension and Explicit Tube Formulas for p-Adic Fractal Strings http://www.mdpi.com/2504-3110/2/4/26 The theory of complex dimensions describes the oscillations in the geometry (spectra and dynamics) of fractal strings. Such geometric oscillations can be seen most clearly in the explicit volume formula for the tubular neighborhoods of a p-adic fractal string L p , expressed in terms of the underlying complex dimensions. The general fractal tube formula obtained in this paper is illustrated by several examples, including the nonarchimedean Cantor and Euler strings. Moreover, we show that the Minkowski dimension of a p-adic fractal string coincides with the abscissa of convergence of the geometric zeta function associated with the string, as well as with the asymptotic growth rate of the corresponding geometric counting function. The proof of this new result can be applied to both real and p-adic fractal strings and hence, yields a unifying explanation of a key result in the theory of complex dimensions for fractal strings, even in the archimedean (or real) case. Fractal Fract, Vol. 2, Pages 26: Minkowski Dimension and Explicit Tube Formulas for p-Adic Fractal Strings

Fractal and Fractional doi: 10.3390/fractalfract2040026

Authors: Michel L. Lapidus Hùng Lũ’ Machiel Van Frankenhuijsen

The theory of complex dimensions describes the oscillations in the geometry (spectra and dynamics) of fractal strings. Such geometric oscillations can be seen most clearly in the explicit volume formula for the tubular neighborhoods of a p-adic fractal string L p , expressed in terms of the underlying complex dimensions. The general fractal tube formula obtained in this paper is illustrated by several examples, including the nonarchimedean Cantor and Euler strings. Moreover, we show that the Minkowski dimension of a p-adic fractal string coincides with the abscissa of convergence of the geometric zeta function associated with the string, as well as with the asymptotic growth rate of the corresponding geometric counting function. The proof of this new result can be applied to both real and p-adic fractal strings and hence, yields a unifying explanation of a key result in the theory of complex dimensions for fractal strings, even in the archimedean (or real) case.

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Minkowski Dimension and Explicit Tube Formulas for p-Adic Fractal Strings Michel L. Lapidus Hùng Lũ’ Machiel Van Frankenhuijsen doi: 10.3390/fractalfract2040026 Fractal and Fractional 2018-10-04 Fractal and Fractional 2018-10-04 2 4 Article 26 10.3390/fractalfract2040026 http://www.mdpi.com/2504-3110/2/4/26
Fractal Fract, Vol. 2, Pages 25: Dynamic Fractional Inequalities Amplified on Time Scale Calculus Revealing Coalition of Discreteness and Continuity http://www.mdpi.com/2504-3110/2/4/25 In this paper, we present a generalization of Radon&amp;rsquo;s inequality on dynamic time scale calculus, which is widely studied by many authors and an intrinsic inequality. Further, we present the classical Bergstr&amp;ouml;m&amp;rsquo;s inequality and refinement of Nesbitt&amp;rsquo;s inequality unified on dynamic time scale calculus in extended form. Fractal Fract, Vol. 2, Pages 25: Dynamic Fractional Inequalities Amplified on Time Scale Calculus Revealing Coalition of Discreteness and Continuity

Fractal and Fractional doi: 10.3390/fractalfract2040025

In this paper, we present a generalization of Radon&amp;rsquo;s inequality on dynamic time scale calculus, which is widely studied by many authors and an intrinsic inequality. Further, we present the classical Bergstr&amp;ouml;m&amp;rsquo;s inequality and refinement of Nesbitt&amp;rsquo;s inequality unified on dynamic time scale calculus in extended form.

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Dynamic Fractional Inequalities Amplified on Time Scale Calculus Revealing Coalition of Discreteness and Continuity Muhammad Sahir doi: 10.3390/fractalfract2040025 Fractal and Fractional 2018-09-29 Fractal and Fractional 2018-09-29 2 4 Article 25 10.3390/fractalfract2040025 http://www.mdpi.com/2504-3110/2/4/25
Fractal Fract, Vol. 2, Pages 24: Power Laws in Fractionally Electronic Elements http://www.mdpi.com/2504-3110/2/4/24 The highlight presented in this short article is about the power laws with respect to fractional capacitance and fractional inductance in terms of frequency. Fractal Fract, Vol. 2, Pages 24: Power Laws in Fractionally Electronic Elements

Fractal and Fractional doi: 10.3390/fractalfract2040024

Authors: Ming Li

The highlight presented in this short article is about the power laws with respect to fractional capacitance and fractional inductance in terms of frequency.

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Power Laws in Fractionally Electronic Elements Ming Li doi: 10.3390/fractalfract2040024 Fractal and Fractional 2018-09-26 Fractal and Fractional 2018-09-26 2 4 Article 24 10.3390/fractalfract2040024 http://www.mdpi.com/2504-3110/2/4/24
Fractal Fract, Vol. 2, Pages 23: Generalized Memory: Fractional Calculus Approach http://www.mdpi.com/2504-3110/2/4/23 The memory means an existence of output (response, endogenous variable) at the present time that depends on the history of the change of the input (impact, exogenous variable) on a finite (or infinite) time interval. The memory can be described by the function that is called the memory function, which is a kernel of the integro-differential operator. The main purpose of the paper is to answer the question of the possibility of using the fractional calculus, when the memory function does not have a power-law form. Using the generalized Taylor series in the Trujillo-Rivero-Bonilla (TRB) form for the memory function, we represent the integro-differential equations with memory functions by fractional integral and differential equations with derivatives and integrals of non-integer orders. This allows us to describe general economic dynamics with memory by the methods of fractional calculus. We prove that equation of the generalized accelerator with the TRB memory function can be represented by as a composition of actions of the accelerator with simplest power-law memory and the multi-parametric power-law multiplier. As an example of application of the suggested approach, we consider a generalization of the Harrod-Domar growth model with continuous time. Fractal Fract, Vol. 2, Pages 23: Generalized Memory: Fractional Calculus Approach

Fractal and Fractional doi: 10.3390/fractalfract2040023

Authors: Vasily E. Tarasov

The memory means an existence of output (response, endogenous variable) at the present time that depends on the history of the change of the input (impact, exogenous variable) on a finite (or infinite) time interval. The memory can be described by the function that is called the memory function, which is a kernel of the integro-differential operator. The main purpose of the paper is to answer the question of the possibility of using the fractional calculus, when the memory function does not have a power-law form. Using the generalized Taylor series in the Trujillo-Rivero-Bonilla (TRB) form for the memory function, we represent the integro-differential equations with memory functions by fractional integral and differential equations with derivatives and integrals of non-integer orders. This allows us to describe general economic dynamics with memory by the methods of fractional calculus. We prove that equation of the generalized accelerator with the TRB memory function can be represented by as a composition of actions of the accelerator with simplest power-law memory and the multi-parametric power-law multiplier. As an example of application of the suggested approach, we consider a generalization of the Harrod-Domar growth model with continuous time.

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Generalized Memory: Fractional Calculus Approach Vasily E. Tarasov doi: 10.3390/fractalfract2040023 Fractal and Fractional 2018-09-24 Fractal and Fractional 2018-09-24 2 4 Article 23 10.3390/fractalfract2040023 http://www.mdpi.com/2504-3110/2/4/23
Fractal Fract, Vol. 2, Pages 22: Implementation and Convergence Analysis of Homotopy Perturbation Coupled With Sumudu Transform to Construct Solutions of Local-Fractional PDEs http://www.mdpi.com/2504-3110/2/3/22 In the present paper, the explicit solutions of some local fractional partial differential equations are constructed through the integration of local fractional Sumudu transform and homotopy perturbation such as local fractional dissipative and damped wave equations. The convergence aspect of this technique is also discussed and presented. The obtained results prove that the employed method is very simple and effective for treating analytically various kinds of problems comprising local fractional derivatives. Fractal Fract, Vol. 2, Pages 22: Implementation and Convergence Analysis of Homotopy Perturbation Coupled With Sumudu Transform to Construct Solutions of Local-Fractional PDEs

Fractal and Fractional doi: 10.3390/fractalfract2030022

Authors: Kamal Ait Touchent Zakia Hammouch Toufik Mekkaoui Fethi B. M. Belgacem

In the present paper, the explicit solutions of some local fractional partial differential equations are constructed through the integration of local fractional Sumudu transform and homotopy perturbation such as local fractional dissipative and damped wave equations. The convergence aspect of this technique is also discussed and presented. The obtained results prove that the employed method is very simple and effective for treating analytically various kinds of problems comprising local fractional derivatives.

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Implementation and Convergence Analysis of Homotopy Perturbation Coupled With Sumudu Transform to Construct Solutions of Local-Fractional PDEs Kamal Ait Touchent Zakia Hammouch Toufik Mekkaoui Fethi B. M. Belgacem doi: 10.3390/fractalfract2030022 Fractal and Fractional 2018-09-07 Fractal and Fractional 2018-09-07 2 3 Article 22 10.3390/fractalfract2030022 http://www.mdpi.com/2504-3110/2/3/22
Fractal Fract, Vol. 2, Pages 21: Nonlinear Vibration of a Nonlocal Nanobeam Resting on Fractional-Order Viscoelastic Pasternak Foundations http://www.mdpi.com/2504-3110/2/3/21 In the present study, the nonlinear vibration of a nanobeam resting on the fractional order viscoelastic Winkler&amp;ndash;Pasternak foundation is studied using nonlocal elasticity theory. The D&amp;rsquo;Alembert principle is used to derive the governing equation and the associated boundary conditions. The approximate analytical solution is obtained by applying the multiple scales method. A detailed parametric study is conducted, and the effects of the variation of different parameters belonging to the application problems on the system are calculated numerically and depicted. We remark that the order and the coefficient of the fractional derivative have a significant effect on the natural frequency and the amplitude of vibrations. Fractal Fract, Vol. 2, Pages 21: Nonlinear Vibration of a Nonlocal Nanobeam Resting on Fractional-Order Viscoelastic Pasternak Foundations

Fractal and Fractional doi: 10.3390/fractalfract2030021

Authors: Guy Joseph Eyebe Gambo Betchewe Alidou Mohamadou Timoleon Crepin Kofane

In the present study, the nonlinear vibration of a nanobeam resting on the fractional order viscoelastic Winkler&amp;ndash;Pasternak foundation is studied using nonlocal elasticity theory. The D&amp;rsquo;Alembert principle is used to derive the governing equation and the associated boundary conditions. The approximate analytical solution is obtained by applying the multiple scales method. A detailed parametric study is conducted, and the effects of the variation of different parameters belonging to the application problems on the system are calculated numerically and depicted. We remark that the order and the coefficient of the fractional derivative have a significant effect on the natural frequency and the amplitude of vibrations.

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Nonlinear Vibration of a Nonlocal Nanobeam Resting on Fractional-Order Viscoelastic Pasternak Foundations Guy Joseph Eyebe Gambo Betchewe Alidou Mohamadou Timoleon Crepin Kofane doi: 10.3390/fractalfract2030021 Fractal and Fractional 2018-08-05 Fractal and Fractional 2018-08-05 2 3 Article 21 10.3390/fractalfract2030021 http://www.mdpi.com/2504-3110/2/3/21
Fractal Fract, Vol. 2, Pages 20: Non-Gaussian Distributions to Random Walk in the Context of Memory Kernels http://www.mdpi.com/2504-3110/2/3/20 The investigation of diffusive process in nature presents a complexity associated with memory effects. Thereby, it is necessary new mathematical models to involve memory concept in diffusion. In the following, I approach the continuous time random walks in the context of generalised diffusion equations. To do this, I investigate the diffusion equation with exponential and Mittag-Leffler memory-kernels in the context of Caputo-Fabrizio and Atangana-Baleanu fractional operators on Caputo sense. Thus, exact expressions for the probability distributions are obtained, in that non-Gaussian distributions emerge. I connect the distribution obtained with a rich class of diffusive behaviour. Moreover, I propose a generalised model to describe the random walk process with resetting on memory kernel context. Fractal Fract, Vol. 2, Pages 20: Non-Gaussian Distributions to Random Walk in the Context of Memory Kernels

Fractal and Fractional doi: 10.3390/fractalfract2030020

Authors: Maike A. F. Dos Santos

The investigation of diffusive process in nature presents a complexity associated with memory effects. Thereby, it is necessary new mathematical models to involve memory concept in diffusion. In the following, I approach the continuous time random walks in the context of generalised diffusion equations. To do this, I investigate the diffusion equation with exponential and Mittag-Leffler memory-kernels in the context of Caputo-Fabrizio and Atangana-Baleanu fractional operators on Caputo sense. Thus, exact expressions for the probability distributions are obtained, in that non-Gaussian distributions emerge. I connect the distribution obtained with a rich class of diffusive behaviour. Moreover, I propose a generalised model to describe the random walk process with resetting on memory kernel context.

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Non-Gaussian Distributions to Random Walk in the Context of Memory Kernels Maike A. F. Dos Santos doi: 10.3390/fractalfract2030020 Fractal and Fractional 2018-07-29 Fractal and Fractional 2018-07-29 2 3 Article 20 10.3390/fractalfract2030020 http://www.mdpi.com/2504-3110/2/3/20
Fractal Fract, Vol. 2, Pages 19: Fractional Dynamics http://www.mdpi.com/2504-3110/2/2/19 n/a Fractal Fract, Vol. 2, Pages 19: Fractional Dynamics

Fractal and Fractional doi: 10.3390/fractalfract2020019

Authors: Carlo Cattani Renato Spigler

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Fractional Dynamics Carlo Cattani Renato Spigler doi: 10.3390/fractalfract2020019 Fractal and Fractional 2018-06-17 Fractal and Fractional 2018-06-17 2 2 Editorial 19 10.3390/fractalfract2020019 http://www.mdpi.com/2504-3110/2/2/19
Fractal Fract, Vol. 2, Pages 18: Analytical Solutions to Fractional Fluid Flow and Oscillatory Process Models http://www.mdpi.com/2504-3110/2/2/18 In this paper, we provide solutions to the general fractional Caputo-type differential equation models for the dynamics of a sphere immersed in an incompressible viscous fluid and oscillatory process with fractional damping using Laplace transform method. We study the effects of fixing one of the fractional indices while varying the other as particular examples. We conclude this article by explaining the dynamics of the solutions of the models. Fractal Fract, Vol. 2, Pages 18: Analytical Solutions to Fractional Fluid Flow and Oscillatory Process Models

Fractal and Fractional doi: 10.3390/fractalfract2020018

Authors: Yusuf Zakariya Yusuf Afolabi Rahmatullah Nuruddeen Ibrahim Sarumi

In this paper, we provide solutions to the general fractional Caputo-type differential equation models for the dynamics of a sphere immersed in an incompressible viscous fluid and oscillatory process with fractional damping using Laplace transform method. We study the effects of fixing one of the fractional indices while varying the other as particular examples. We conclude this article by explaining the dynamics of the solutions of the models.

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Analytical Solutions to Fractional Fluid Flow and Oscillatory Process Models Yusuf Zakariya Yusuf Afolabi Rahmatullah Nuruddeen Ibrahim Sarumi doi: 10.3390/fractalfract2020018 Fractal and Fractional 2018-05-27 Fractal and Fractional 2018-05-27 2 2 Article 18 10.3390/fractalfract2020018 http://www.mdpi.com/2504-3110/2/2/18
Fractal Fract, Vol. 2, Pages 17: Lyapunov Characterization of the Fractional Nonlinear Systems with Exogenous Input http://www.mdpi.com/2504-3110/2/2/17 This paper deals with a Lyapunov characterization of the conditional Mittag-Leffler stability and conditional asymptotic stability of the fractional nonlinear systems with exogenous input. A particular class of the fractional nonlinear systems is studied. The paper contributes to giving in particular the Lyapunov characterization of fractional linear systems and fractional bilinear systems with exogenous input. Fractal Fract, Vol. 2, Pages 17: Lyapunov Characterization of the Fractional Nonlinear Systems with Exogenous Input

Fractal and Fractional doi: 10.3390/fractalfract2020017

Authors: Ndolane Sene

This paper deals with a Lyapunov characterization of the conditional Mittag-Leffler stability and conditional asymptotic stability of the fractional nonlinear systems with exogenous input. A particular class of the fractional nonlinear systems is studied. The paper contributes to giving in particular the Lyapunov characterization of fractional linear systems and fractional bilinear systems with exogenous input.

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Lyapunov Characterization of the Fractional Nonlinear Systems with Exogenous Input Ndolane Sene doi: 10.3390/fractalfract2020017 Fractal and Fractional 2018-05-02 Fractal and Fractional 2018-05-02 2 2 Article 17 10.3390/fractalfract2020017 http://www.mdpi.com/2504-3110/2/2/17
Fractal Fract, Vol. 2, Pages 16: The Craft of Fractional Modeling in Science and Engineering 2017 http://www.mdpi.com/2504-3110/2/2/16 n/a Fractal Fract, Vol. 2, Pages 16: The Craft of Fractional Modeling in Science and Engineering 2017

Fractal and Fractional doi: 10.3390/fractalfract2020016

Authors: Jordan Hristov

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The Craft of Fractional Modeling in Science and Engineering 2017 Jordan Hristov doi: 10.3390/fractalfract2020016 Fractal and Fractional 2018-04-15 Fractal and Fractional 2018-04-15 2 2 Editorial 16 10.3390/fractalfract2020016 http://www.mdpi.com/2504-3110/2/2/16
Fractal Fract, Vol. 2, Pages 15: Option Pricing Models Driven by the Space-Time Fractional Diffusion: Series Representation and Applications http://www.mdpi.com/2504-3110/2/1/15 In this paper, we focus on option pricing models based on space-time fractional diffusion. We briefly revise recent results which show that the option price can be represented in the terms of rapidly converging double-series and apply these results to the data from real markets. We focus on estimation of model parameters from the market data and estimation of implied volatility within the space-time fractional option pricing models. Fractal Fract, Vol. 2, Pages 15: Option Pricing Models Driven by the Space-Time Fractional Diffusion: Series Representation and Applications

Fractal and Fractional doi: 10.3390/fractalfract2010015

Authors: Jean-Philippe Aguilar Jan Korbel

In this paper, we focus on option pricing models based on space-time fractional diffusion. We briefly revise recent results which show that the option price can be represented in the terms of rapidly converging double-series and apply these results to the data from real markets. We focus on estimation of model parameters from the market data and estimation of implied volatility within the space-time fractional option pricing models.

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Option Pricing Models Driven by the Space-Time Fractional Diffusion: Series Representation and Applications Jean-Philippe Aguilar Jan Korbel doi: 10.3390/fractalfract2010015 Fractal and Fractional 2018-03-16 Fractal and Fractional 2018-03-16 2 1 Article 15 10.3390/fractalfract2010015 http://www.mdpi.com/2504-3110/2/1/15
Fractal Fract, Vol. 2, Pages 14: Identifying the Fractional Orders in Anomalous Diffusion Models from Real Data http://www.mdpi.com/2504-3110/2/1/14 An attempt is made to identify the orders of the fractional derivatives in a simple anomalous diffusion model, starting from real data. We consider experimental data taken at the Columbus Air Force Base in Mississippi. Using as a model a one-dimensional fractional diffusion equation in both space and time, we fit the data by choosing several values of the fractional orders and computing the infinite-norm “errors”, representing the discrepancy between the numerical solution to the model equation and the experimental data. Data were also filtered before being used, to see possible improvements. The minimal discrepancy is attained correspondingly to a fractional order in time around 0 . 6 and a fractional order in space near 2. These results may describe well the memory properties of the porous medium that can be observed. Fractal Fract, Vol. 2, Pages 14: Identifying the Fractional Orders in Anomalous Diffusion Models from Real Data

Fractal and Fractional doi: 10.3390/fractalfract2010014

Authors: Moreno Concezzi Renato Spigler

An attempt is made to identify the orders of the fractional derivatives in a simple anomalous diffusion model, starting from real data. We consider experimental data taken at the Columbus Air Force Base in Mississippi. Using as a model a one-dimensional fractional diffusion equation in both space and time, we fit the data by choosing several values of the fractional orders and computing the infinite-norm “errors”, representing the discrepancy between the numerical solution to the model equation and the experimental data. Data were also filtered before being used, to see possible improvements. The minimal discrepancy is attained correspondingly to a fractional order in time around 0 . 6 and a fractional order in space near 2. These results may describe well the memory properties of the porous medium that can be observed.

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Identifying the Fractional Orders in Anomalous Diffusion Models from Real Data Moreno Concezzi Renato Spigler doi: 10.3390/fractalfract2010014 Fractal and Fractional 2018-02-24 Fractal and Fractional 2018-02-24 2 1 Article 14 10.3390/fractalfract2010014 http://www.mdpi.com/2504-3110/2/1/14
Fractal Fract, Vol. 2, Pages 13: A Fractional B-spline Collocation Method for the Numerical Solution of Fractional Predator-Prey Models http://www.mdpi.com/2504-3110/2/1/13 We present a collocation method based on fractional B-splines for the solution of fractional differential problems. The key-idea is to use the space generated by the fractional B-splines, i.e., piecewise polynomials of noninteger degree, as approximating space. Then, in the collocation step the fractional derivative of the approximating function is approximated accurately and efficiently by an exact differentiation rule that involves the generalized finite difference operator. To show the effectiveness of the method for the solution of nonlinear dynamical systems of fractional order, we solved the fractional Lotka-Volterra model and a fractional predator-pray model with variable coefficients. The numerical tests show that the method we proposed is accurate while keeping a low computational cost. Fractal Fract, Vol. 2, Pages 13: A Fractional B-spline Collocation Method for the Numerical Solution of Fractional Predator-Prey Models

Fractal and Fractional doi: 10.3390/fractalfract2010013

Authors: Francesca Pitolli

We present a collocation method based on fractional B-splines for the solution of fractional differential problems. The key-idea is to use the space generated by the fractional B-splines, i.e., piecewise polynomials of noninteger degree, as approximating space. Then, in the collocation step the fractional derivative of the approximating function is approximated accurately and efficiently by an exact differentiation rule that involves the generalized finite difference operator. To show the effectiveness of the method for the solution of nonlinear dynamical systems of fractional order, we solved the fractional Lotka-Volterra model and a fractional predator-pray model with variable coefficients. The numerical tests show that the method we proposed is accurate while keeping a low computational cost.

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A Fractional B-spline Collocation Method for the Numerical Solution of Fractional Predator-Prey Models Francesca Pitolli doi: 10.3390/fractalfract2010013 Fractal and Fractional 2018-02-17 Fractal and Fractional 2018-02-17 2 1 Article 13 10.3390/fractalfract2010013 http://www.mdpi.com/2504-3110/2/1/13
Fractal Fract, Vol. 2, Pages 12: Does a Fractal Microstructure Require a Fractional Viscoelastic Model? http://www.mdpi.com/2504-3110/2/1/12 The question addressed by this paper is tackled through a continuum micromechanics model of a 2D random checkerboard, in which one phase is linear elastic and another linear viscoelastic of integer-order. The spatial homogeneity and ergodicity of the material statistics justify homogenization in the vein of the Hill–Mandel condition for viscoelastic media. Thus, uniform kinematic- or traction-controlled boundary conditions, applied to sufficiently large domains, provide macroscopic (RVE level) responses. With computational mechanics, this strategy is applied over the entire range of the relative content of both phases. Setting the volume fraction of either the elastic phase or the viscoelastic phase at the critical value (≃0.59) results in fractal patterns of site-percolation. Extensive simulations of boundary value problems show that, for a viscoelastic composite having such a fractal structure, the integer (not fractional) calculus model is adequate. In other words, the spatial randomness of the composite material—even in the fractal regime—is not necessarily the cause of the fractional order viscoelasticity. Fractal Fract, Vol. 2, Pages 12: Does a Fractal Microstructure Require a Fractional Viscoelastic Model?

Fractal and Fractional doi: 10.3390/fractalfract2010012

Authors: Martin Ostoja-Starzewski Jun Zhang

The question addressed by this paper is tackled through a continuum micromechanics model of a 2D random checkerboard, in which one phase is linear elastic and another linear viscoelastic of integer-order. The spatial homogeneity and ergodicity of the material statistics justify homogenization in the vein of the Hill–Mandel condition for viscoelastic media. Thus, uniform kinematic- or traction-controlled boundary conditions, applied to sufficiently large domains, provide macroscopic (RVE level) responses. With computational mechanics, this strategy is applied over the entire range of the relative content of both phases. Setting the volume fraction of either the elastic phase or the viscoelastic phase at the critical value (≃0.59) results in fractal patterns of site-percolation. Extensive simulations of boundary value problems show that, for a viscoelastic composite having such a fractal structure, the integer (not fractional) calculus model is adequate. In other words, the spatial randomness of the composite material—even in the fractal regime—is not necessarily the cause of the fractional order viscoelasticity.

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Does a Fractal Microstructure Require a Fractional Viscoelastic Model? Martin Ostoja-Starzewski Jun Zhang doi: 10.3390/fractalfract2010012 Fractal and Fractional 2018-02-13 Fractal and Fractional 2018-02-13 2 1 Article 12 10.3390/fractalfract2010012 http://www.mdpi.com/2504-3110/2/1/12
Fractal Fract, Vol. 2, Pages 11: Monitoring Liquid-Liquid Mixtures Using Fractional Calculus and Image Analysis http://www.mdpi.com/2504-3110/2/1/11 A fractional-calculus-based model is used to analyze the data obtained from the image analysis of mixtures of olive and soybean oil, which were quantified with the RGB color system. The model consists in a linear fractional differential equation, containing one fractional derivative of order α and an additional term multiplied by a parameter k. Using a hybrid parameter estimation scheme (genetic algorithm and a simplex-based algorithm), the model parameters were estimated as k = 3.42 ± 0.12 and α = 1.196 ± 0.027, while a correlation coefficient value of 0.997 was obtained. For the sake of comparison, parameter α was set equal to 1 and an integer order model was also studied, resulting in a one-parameter model with k = 3.11 ± 0.28. Joint confidence regions are calculated for the fractional order model, showing that the derivative order is statistically different from 1. Finally, an independent validation sample of color component B equal to 96 obtained from a sample with olive oil mass fraction equal to 0.25 is used for prediction purposes. The fractional model predicted the color B value equal to 93.1 ± 6.6. Fractal Fract, Vol. 2, Pages 11: Monitoring Liquid-Liquid Mixtures Using Fractional Calculus and Image Analysis

Fractal and Fractional doi: 10.3390/fractalfract2010011

Authors: Ervin Lenzi Andrea Ryba Marcelo Lenzi

A fractional-calculus-based model is used to analyze the data obtained from the image analysis of mixtures of olive and soybean oil, which were quantified with the RGB color system. The model consists in a linear fractional differential equation, containing one fractional derivative of order α and an additional term multiplied by a parameter k. Using a hybrid parameter estimation scheme (genetic algorithm and a simplex-based algorithm), the model parameters were estimated as k = 3.42 ± 0.12 and α = 1.196 ± 0.027, while a correlation coefficient value of 0.997 was obtained. For the sake of comparison, parameter α was set equal to 1 and an integer order model was also studied, resulting in a one-parameter model with k = 3.11 ± 0.28. Joint confidence regions are calculated for the fractional order model, showing that the derivative order is statistically different from 1. Finally, an independent validation sample of color component B equal to 96 obtained from a sample with olive oil mass fraction equal to 0.25 is used for prediction purposes. The fractional model predicted the color B value equal to 93.1 ± 6.6.

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Monitoring Liquid-Liquid Mixtures Using Fractional Calculus and Image Analysis Ervin Lenzi Andrea Ryba Marcelo Lenzi doi: 10.3390/fractalfract2010011 Fractal and Fractional 2018-02-11 Fractal and Fractional 2018-02-11 2 1 Article 11 10.3390/fractalfract2010011 http://www.mdpi.com/2504-3110/2/1/11
Fractal Fract, Vol. 2, Pages 10: Fractional Derivatives with the Power-Law and the Mittag–Leffler Kernel Applied to the Nonlinear Baggs–Freedman Model http://www.mdpi.com/2504-3110/2/1/10 This paper considers the Freedman model using the Liouville–Caputo fractional-order derivative and the fractional-order derivative with Mittag–Leffler kernel in the Liouville–Caputo sense. Alternative solutions via Laplace transform, Sumudu–Picard and Adams–Moulton rules were obtained. We prove the uniqueness and existence of the solutions for the alternative model. Numerical simulations for the prediction and interaction between a unilingual and a bilingual population were obtained for different values of the fractional order. Fractal Fract, Vol. 2, Pages 10: Fractional Derivatives with the Power-Law and the Mittag–Leffler Kernel Applied to the Nonlinear Baggs–Freedman Model

Fractal and Fractional doi: 10.3390/fractalfract2010010

Authors: José Francisco Gómez-Aguilar Abdon Atangana

This paper considers the Freedman model using the Liouville–Caputo fractional-order derivative and the fractional-order derivative with Mittag–Leffler kernel in the Liouville–Caputo sense. Alternative solutions via Laplace transform, Sumudu–Picard and Adams–Moulton rules were obtained. We prove the uniqueness and existence of the solutions for the alternative model. Numerical simulations for the prediction and interaction between a unilingual and a bilingual population were obtained for different values of the fractional order.

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Fractional Derivatives with the Power-Law and the Mittag–Leffler Kernel Applied to the Nonlinear Baggs–Freedman Model José Francisco Gómez-Aguilar Abdon Atangana doi: 10.3390/fractalfract2010010 Fractal and Fractional 2018-02-09 Fractal and Fractional 2018-02-09 2 1 Article 10 10.3390/fractalfract2010010 http://www.mdpi.com/2504-3110/2/1/10
Fractal Fract, Vol. 2, Pages 9: Poiseuille Flow of a Non-Local Non-Newtonian Fluid with Wall Slip: A First Step in Modeling Cerebral Microaneurysms http://www.mdpi.com/2504-3110/2/1/9 Cerebral aneurysms and microaneurysms are abnormal vascular dilatations with high risk of rupture. An aneurysmal rupture could cause permanent disability and even death. Finding and treating aneurysms before their rupture is very difficult since symptoms can be easily attributed mistakenly to other common brain diseases. Mathematical models could highlight possible mechanisms of aneurysmal development and suggest specialized biomarkers for aneurysms. Existing mathematical models of intracranial aneurysms focus on mechanical interactions between blood flow and arteries. However, these models cannot be applied to microaneurysms since the anatomy and physiology at the length scale of cerebral microcirculation are different. In this paper, we propose a mechanism for the formation of microaneurysms that involves the chemo-mechanical coupling of blood and endothelial and neuroglial cells. We model the blood as a non-local non-Newtonian incompressible fluid and solve analytically the Poiseuille flow of such a fluid through an axi-symmetric circular rigid and impermeable pipe in the presence of wall slip. The spatial derivatives of the proposed generalization of the rate of deformation tensor are expressed using Caputo fractional derivatives. The wall slip is represented by the classic Navier law and a generalization of this law involving fractional derivatives. Numerical simulations suggest that hypertension could contribute to microaneurysmal formation. Fractal Fract, Vol. 2, Pages 9: Poiseuille Flow of a Non-Local Non-Newtonian Fluid with Wall Slip: A First Step in Modeling Cerebral Microaneurysms

Fractal and Fractional doi: 10.3390/fractalfract2010009

Authors: Corina S. Drapaca

Cerebral aneurysms and microaneurysms are abnormal vascular dilatations with high risk of rupture. An aneurysmal rupture could cause permanent disability and even death. Finding and treating aneurysms before their rupture is very difficult since symptoms can be easily attributed mistakenly to other common brain diseases. Mathematical models could highlight possible mechanisms of aneurysmal development and suggest specialized biomarkers for aneurysms. Existing mathematical models of intracranial aneurysms focus on mechanical interactions between blood flow and arteries. However, these models cannot be applied to microaneurysms since the anatomy and physiology at the length scale of cerebral microcirculation are different. In this paper, we propose a mechanism for the formation of microaneurysms that involves the chemo-mechanical coupling of blood and endothelial and neuroglial cells. We model the blood as a non-local non-Newtonian incompressible fluid and solve analytically the Poiseuille flow of such a fluid through an axi-symmetric circular rigid and impermeable pipe in the presence of wall slip. The spatial derivatives of the proposed generalization of the rate of deformation tensor are expressed using Caputo fractional derivatives. The wall slip is represented by the classic Navier law and a generalization of this law involving fractional derivatives. Numerical simulations suggest that hypertension could contribute to microaneurysmal formation.

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Poiseuille Flow of a Non-Local Non-Newtonian Fluid with Wall Slip: A First Step in Modeling Cerebral Microaneurysms Corina S. Drapaca doi: 10.3390/fractalfract2010009 Fractal and Fractional 2018-02-06 Fractal and Fractional 2018-02-06 2 1 Article 9 10.3390/fractalfract2010009 http://www.mdpi.com/2504-3110/2/1/9
Fractal Fract, Vol. 2, Pages 8: Towards a Generalized Beer-Lambert Law http://www.mdpi.com/2504-3110/2/1/8 Anomalous deviations from the Beer-Lambert law have been observed for a long time in a wide range of application. Despite all the attempts, a reliable and accepted model has not been provided so far. In addition, in some cases the attenuation of radiation seems to follow a hyperbolic more than an exponential extinction law. Starting from a probabilistic interpretation of the Beer-Lambert law based on Poissonian distribution of extinction events, in this paper we consider deviations from the classical exponential extinction introducing a weighted version of the classical law. The generalized law is able to account for both sub or super-exponential extinction of radiation, and can be extended to the case of inhomogeneous media. Focusing on this case, we consider a generalized Beer-Lambert law based on an inhomogeneous weighted Poisson distribution involving a Mittag-Leffler function, and show how it can be directly related to hyperbolic decay laws observed in some applications particularly relevant to microbiology and pharmacology. Fractal Fract, Vol. 2, Pages 8: Towards a Generalized Beer-Lambert Law

Fractal and Fractional doi: 10.3390/fractalfract2010008

Authors: Giampietro Casasanta Roberto Garra

Anomalous deviations from the Beer-Lambert law have been observed for a long time in a wide range of application. Despite all the attempts, a reliable and accepted model has not been provided so far. In addition, in some cases the attenuation of radiation seems to follow a hyperbolic more than an exponential extinction law. Starting from a probabilistic interpretation of the Beer-Lambert law based on Poissonian distribution of extinction events, in this paper we consider deviations from the classical exponential extinction introducing a weighted version of the classical law. The generalized law is able to account for both sub or super-exponential extinction of radiation, and can be extended to the case of inhomogeneous media. Focusing on this case, we consider a generalized Beer-Lambert law based on an inhomogeneous weighted Poisson distribution involving a Mittag-Leffler function, and show how it can be directly related to hyperbolic decay laws observed in some applications particularly relevant to microbiology and pharmacology.

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Towards a Generalized Beer-Lambert Law Giampietro Casasanta Roberto Garra doi: 10.3390/fractalfract2010008 Fractal and Fractional 2018-01-31 Fractal and Fractional 2018-01-31 2 1 Article 8 10.3390/fractalfract2010008 http://www.mdpi.com/2504-3110/2/1/8
Fractal Fract, Vol. 2, Pages 7: Acknowledgement to Reviewers of Fractal and Fractional in 2017 http://www.mdpi.com/2504-3110/2/1/7 Peer review is an essential part in the publication process, ensuring that Fractal and Fractional maintains high quality standards for its published papers. In 2017, a total of 17 papers were published in the journal. Thanks to the cooperation of our reviewers, the median time to first decision was 14 days and the median time to publication was 24 days. The editors would like to express their sincere gratitude to the reviewers for their time and dedication in 2017. Fractal Fract, Vol. 2, Pages 7: Acknowledgement to Reviewers of Fractal and Fractional in 2017

Fractal and Fractional doi: 10.3390/fractalfract2010007

Authors: Fractal and Fractional Editorial Office

Peer review is an essential part in the publication process, ensuring that Fractal and Fractional maintains high quality standards for its published papers. In 2017, a total of 17 papers were published in the journal. Thanks to the cooperation of our reviewers, the median time to first decision was 14 days and the median time to publication was 24 days. The editors would like to express their sincere gratitude to the reviewers for their time and dedication in 2017.

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Acknowledgement to Reviewers of Fractal and Fractional in 2017 Fractal and Fractional Editorial Office doi: 10.3390/fractalfract2010007 Fractal and Fractional 2018-01-30 Fractal and Fractional 2018-01-30 2 1 Editorial 7 10.3390/fractalfract2010007 http://www.mdpi.com/2504-3110/2/1/7
Fractal Fract, Vol. 2, Pages 6: Emergence of Fractional Kinetics in Spiny Dendrites http://www.mdpi.com/2504-3110/2/1/6 Fractional extensions of the cable equation have been proposed in the literature to describe transmembrane potential in spiny dendrites. The anomalous behavior has been related in the literature to the geometrical properties of the system, in particular, the density of spines, by experiments, computer simulations, and in comb-like models. The same PDE can be related to more than one stochastic process leading to anomalous diffusion behavior. The time-fractional diffusion equation can be associated to a continuous time random walk (CTRW) with power-law waiting time probability or to a special case of the Erdély-Kober fractional diffusion, described by the ggBm. In this work, we show that time fractional generalization of the cable equation arises naturally in the CTRW by considering a superposition of Markovian processes and in a ggBm-like construction of the random variable. Fractal Fract, Vol. 2, Pages 6: Emergence of Fractional Kinetics in Spiny Dendrites

Fractal and Fractional doi: 10.3390/fractalfract2010006

Authors: Silvia Vitali Francesco Mainardi Gastone Castellani

Fractional extensions of the cable equation have been proposed in the literature to describe transmembrane potential in spiny dendrites. The anomalous behavior has been related in the literature to the geometrical properties of the system, in particular, the density of spines, by experiments, computer simulations, and in comb-like models. The same PDE can be related to more than one stochastic process leading to anomalous diffusion behavior. The time-fractional diffusion equation can be associated to a continuous time random walk (CTRW) with power-law waiting time probability or to a special case of the Erdély-Kober fractional diffusion, described by the ggBm. In this work, we show that time fractional generalization of the cable equation arises naturally in the CTRW by considering a superposition of Markovian processes and in a ggBm-like construction of the random variable.

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Emergence of Fractional Kinetics in Spiny Dendrites Silvia Vitali Francesco Mainardi Gastone Castellani doi: 10.3390/fractalfract2010006 Fractal and Fractional 2018-01-25 Fractal and Fractional 2018-01-25 2 1 Article 6 10.3390/fractalfract2010006 http://www.mdpi.com/2504-3110/2/1/6
Fractal Fract, Vol. 2, Pages 5: Comparison between the Second and Third Generations of the CRONE Controller: Application to a Thermal Diffusive Interface Medium http://www.mdpi.com/2504-3110/2/1/5 The control of thermal interfaces has gained importance in recent years because of the high cost of heating and cooling materials in many applications. Thus, the main focus in this work is to compare the second and third generations of the CRONE controller (French acronym of Commande Robuste d’Ordre Non Entier), which means a non-integer order robust controller, and to synthesize a robust controller that can fit several types of systems. For this study, the plant consists of a rectangular homogeneous bar of length L, where the heating element in applied on one boundary, and a temperature sensor is placed at distance x from that boundary (x is considered very small with respect to L). The type of material used is the third parameter, which may help in analyzing the robustness of the synthesized controller. The originality of this work resides in controlling a non-integer plant using a fractional order controller, as, so far, almost all of the systems where the CRONE controller has been implemented were of integer order. Three case studies were defined in order to show how and where each CRONE generation controller can be applied. These case studies were chosen in such a way as to influence the asymptotic behavior of the open-loop transfer function in the Black–Nichols diagram in order to point out the importance of respecting the conditions of the applications of the CRONE generations. Results show that the second generation performs well when the parametric uncertainties do not affect the phase of the plant, whereas the third generation is the most robust, even when both the phase and the gain variations are encountered. However, it also has some limitations, especially when the temperature to be controlled is far from the interface when the density of flux is applied. Fractal Fract, Vol. 2, Pages 5: Comparison between the Second and Third Generations of the CRONE Controller: Application to a Thermal Diffusive Interface Medium

Fractal and Fractional doi: 10.3390/fractalfract2010005

Authors: Xavier Moreau Roy Abi Zeid Daou Fady Christophy

The control of thermal interfaces has gained importance in recent years because of the high cost of heating and cooling materials in many applications. Thus, the main focus in this work is to compare the second and third generations of the CRONE controller (French acronym of Commande Robuste d’Ordre Non Entier), which means a non-integer order robust controller, and to synthesize a robust controller that can fit several types of systems. For this study, the plant consists of a rectangular homogeneous bar of length L, where the heating element in applied on one boundary, and a temperature sensor is placed at distance x from that boundary (x is considered very small with respect to L). The type of material used is the third parameter, which may help in analyzing the robustness of the synthesized controller. The originality of this work resides in controlling a non-integer plant using a fractional order controller, as, so far, almost all of the systems where the CRONE controller has been implemented were of integer order. Three case studies were defined in order to show how and where each CRONE generation controller can be applied. These case studies were chosen in such a way as to influence the asymptotic behavior of the open-loop transfer function in the Black–Nichols diagram in order to point out the importance of respecting the conditions of the applications of the CRONE generations. Results show that the second generation performs well when the parametric uncertainties do not affect the phase of the plant, whereas the third generation is the most robust, even when both the phase and the gain variations are encountered. However, it also has some limitations, especially when the temperature to be controlled is far from the interface when the density of flux is applied.

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Comparison between the Second and Third Generations of the CRONE Controller: Application to a Thermal Diffusive Interface Medium Xavier Moreau Roy Abi Zeid Daou Fady Christophy doi: 10.3390/fractalfract2010005 Fractal and Fractional 2018-01-17 Fractal and Fractional 2018-01-17 2 1 Article 5 10.3390/fractalfract2010005 http://www.mdpi.com/2504-3110/2/1/5
Fractal Fract, Vol. 2, Pages 4: Fractional Velocity as a Tool for the Study of Non-Linear Problems http://www.mdpi.com/2504-3110/2/1/4 Singular functions and, in general, Hölder functions represent conceptual models of nonlinear physical phenomena. The purpose of this survey is to demonstrate the applicability of fractional velocities as tools to characterize Hölder and singular functions, in particular. Fractional velocities are defined as limits of the difference quotients of a fractional power and they generalize the local notion of a derivative. On the other hand, their properties contrast some of the usual properties of derivatives. One of the most peculiar properties of these operators is that the set of their non trivial values is disconnected. This can be used for example to model instantaneous interactions, for example Langevin dynamics. Examples are given by the De Rham and Neidinger’s singular functions, represented by limits of iterative function systems. Finally, the conditions for equivalence with the Kolwankar-Gangal local fractional derivative are investigated. Fractal Fract, Vol. 2, Pages 4: Fractional Velocity as a Tool for the Study of Non-Linear Problems

Fractal and Fractional doi: 10.3390/fractalfract2010004

Authors: Dimiter Prodanov

Singular functions and, in general, Hölder functions represent conceptual models of nonlinear physical phenomena. The purpose of this survey is to demonstrate the applicability of fractional velocities as tools to characterize Hölder and singular functions, in particular. Fractional velocities are defined as limits of the difference quotients of a fractional power and they generalize the local notion of a derivative. On the other hand, their properties contrast some of the usual properties of derivatives. One of the most peculiar properties of these operators is that the set of their non trivial values is disconnected. This can be used for example to model instantaneous interactions, for example Langevin dynamics. Examples are given by the De Rham and Neidinger’s singular functions, represented by limits of iterative function systems. Finally, the conditions for equivalence with the Kolwankar-Gangal local fractional derivative are investigated.

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Fractional Velocity as a Tool for the Study of Non-Linear Problems Dimiter Prodanov doi: 10.3390/fractalfract2010004 Fractal and Fractional 2018-01-17 Fractal and Fractional 2018-01-17 2 1 Article 4 10.3390/fractalfract2010004 http://www.mdpi.com/2504-3110/2/1/4
Fractal Fract, Vol. 2, Pages 3: European Vanilla Option Pricing Model of Fractional Order without Singular Kernel http://www.mdpi.com/2504-3110/2/1/3 Recently, fractional differential equations (FDEs) have attracted much more attention in modeling real-life problems. Since most FDEs do not have exact solutions, numerical solution methods are used commonly. Therefore, in this study, we have demonstrated a novel approximate-analytical solution method, which is called the Laplace homotopy analysis method (LHAM) using the Caputo–Fabrizio (CF) fractional derivative operator. The recommended method is obtained by combining Laplace transform (LT) and the homotopy analysis method (HAM). We have used the fractional operator suggested by Caputo and Fabrizio in 2015 based on the exponential kernel. We have considered the LHAM with this derivative in order to obtain the solutions of the fractional Black–Scholes equations (FBSEs) with the initial conditions. In addition to this, the convergence and stability analysis of the model have been constructed. According to the results of this study, it can be concluded that the LHAM in the sense of the CF fractional derivative is an effective and accurate method, which is computable in the series easily in a short time. Fractal Fract, Vol. 2, Pages 3: European Vanilla Option Pricing Model of Fractional Order without Singular Kernel

Fractal and Fractional doi: 10.3390/fractalfract2010003

Authors: Mehmet Yavuz Necati Özdemir

Recently, fractional differential equations (FDEs) have attracted much more attention in modeling real-life problems. Since most FDEs do not have exact solutions, numerical solution methods are used commonly. Therefore, in this study, we have demonstrated a novel approximate-analytical solution method, which is called the Laplace homotopy analysis method (LHAM) using the Caputo–Fabrizio (CF) fractional derivative operator. The recommended method is obtained by combining Laplace transform (LT) and the homotopy analysis method (HAM). We have used the fractional operator suggested by Caputo and Fabrizio in 2015 based on the exponential kernel. We have considered the LHAM with this derivative in order to obtain the solutions of the fractional Black–Scholes equations (FBSEs) with the initial conditions. In addition to this, the convergence and stability analysis of the model have been constructed. According to the results of this study, it can be concluded that the LHAM in the sense of the CF fractional derivative is an effective and accurate method, which is computable in the series easily in a short time.

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European Vanilla Option Pricing Model of Fractional Order without Singular Kernel Mehmet Yavuz Necati Özdemir doi: 10.3390/fractalfract2010003 Fractal and Fractional 2018-01-16 Fractal and Fractional 2018-01-16 2 1 Article 3 10.3390/fractalfract2010003 http://www.mdpi.com/2504-3110/2/1/3
Fractal Fract, Vol. 2, Pages 2: Fractal Curves from Prime Trigonometric Series http://www.mdpi.com/2504-3110/2/1/2 We study the convergence of the parameter family of series: V α , β ( t ) = ∑ p p − α exp ( 2 π i p β t ) , α , β ∈ R &amp;gt; 0 , t ∈ [ 0 , 1 ) defined over prime numbers p and, subsequently, their differentiability properties. The visible fractal nature of the graphs as a function of α , β is analyzed in terms of Hölder continuity, self-similarity and fractal dimension, backed with numerical results. Although this series is not a lacunary series, it has properties in common, such that we also discuss the link of this series with random walks and, consequently, explore its random properties numerically. Fractal Fract, Vol. 2, Pages 2: Fractal Curves from Prime Trigonometric Series

Fractal and Fractional doi: 10.3390/fractalfract2010002

Authors: Dimitris Vartziotis Doris Bohnet

We study the convergence of the parameter family of series: V α , β ( t ) = ∑ p p − α exp ( 2 π i p β t ) , α , β ∈ R &amp;gt; 0 , t ∈ [ 0 , 1 ) defined over prime numbers p and, subsequently, their differentiability properties. The visible fractal nature of the graphs as a function of α , β is analyzed in terms of Hölder continuity, self-similarity and fractal dimension, backed with numerical results. Although this series is not a lacunary series, it has properties in common, such that we also discuss the link of this series with random walks and, consequently, explore its random properties numerically.

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Fractal Curves from Prime Trigonometric Series Dimitris Vartziotis Doris Bohnet doi: 10.3390/fractalfract2010002 Fractal and Fractional 2018-01-03 Fractal and Fractional 2018-01-03 2 1 Article 2 10.3390/fractalfract2010002 http://www.mdpi.com/2504-3110/2/1/2
Fractal Fract, Vol. 2, Pages 1: Fractional Diffusion Models for the Atmosphere of Mars http://www.mdpi.com/2504-3110/2/1/1 The dust aerosols floating in the atmosphere of Mars cause an attenuation of the solar radiation traversing the atmosphere that cannot be modeled through the use of classical diffusion processes. However, the definition of a type of fractional diffusion equation offers a more accurate model for this dynamic and the second order moment of this equation allows one to establish a connection between the fractional equation and the Ångstrom law that models the attenuation of the solar radiation. In this work we consider both one and three dimensional wavelength-fractional diffusion equations, and we obtain the analytical solutions and numerical methods using two different approaches of the fractional derivative. Fractal Fract, Vol. 2, Pages 1: Fractional Diffusion Models for the Atmosphere of Mars

Fractal and Fractional doi: 10.3390/fractalfract2010001

Authors: Salvador Jiménez David Usero Luis Vázquez Maria Velasco

The dust aerosols floating in the atmosphere of Mars cause an attenuation of the solar radiation traversing the atmosphere that cannot be modeled through the use of classical diffusion processes. However, the definition of a type of fractional diffusion equation offers a more accurate model for this dynamic and the second order moment of this equation allows one to establish a connection between the fractional equation and the Ångstrom law that models the attenuation of the solar radiation. In this work we consider both one and three dimensional wavelength-fractional diffusion equations, and we obtain the analytical solutions and numerical methods using two different approaches of the fractional derivative.

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Fractional Diffusion Models for the Atmosphere of Mars Salvador Jiménez David Usero Luis Vázquez Maria Velasco doi: 10.3390/fractalfract2010001 Fractal and Fractional 2017-12-28 Fractal and Fractional 2017-12-28 2 1 Article 1 10.3390/fractalfract2010001 http://www.mdpi.com/2504-3110/2/1/1
Fractal Fract, Vol. 1, Pages 17: Modeling of Heat Distribution in Porous Aluminum Using Fractional Differential Equation http://www.mdpi.com/2504-3110/1/1/17 The authors present a model of heat conduction using the Caputo fractional derivative with respect to time. The presented model was used to reconstruct the thermal conductivity coefficient, heat transfer coefficient, initial condition and order of fractional derivative in the fractional heat conduction inverse problem. Additional information for the inverse problem was the temperature measurements obtained from porous aluminum. In this paper, the authors used a finite difference method to solve direct problems and the Real Ant Colony Optimization algorithm to find a minimum of certain functional (solve the inverse problem). Finally, the authors present the temperature values computed from the model and compare them with the measured data from real objects. Fractal Fract, Vol. 1, Pages 17: Modeling of Heat Distribution in Porous Aluminum Using Fractional Differential Equation

Fractal and Fractional doi: 10.3390/fractalfract1010017

Authors: Rafał Brociek Damian Słota Mariusz Król Grzegorz Matula Waldemar Kwaśny

The authors present a model of heat conduction using the Caputo fractional derivative with respect to time. The presented model was used to reconstruct the thermal conductivity coefficient, heat transfer coefficient, initial condition and order of fractional derivative in the fractional heat conduction inverse problem. Additional information for the inverse problem was the temperature measurements obtained from porous aluminum. In this paper, the authors used a finite difference method to solve direct problems and the Real Ant Colony Optimization algorithm to find a minimum of certain functional (solve the inverse problem). Finally, the authors present the temperature values computed from the model and compare them with the measured data from real objects.

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Modeling of Heat Distribution in Porous Aluminum Using Fractional Differential Equation Rafał Brociek Damian Słota Mariusz Król Grzegorz Matula Waldemar Kwaśny doi: 10.3390/fractalfract1010017 Fractal and Fractional 2017-12-12 Fractal and Fractional 2017-12-12 1 1 Article 17 10.3390/fractalfract1010017 http://www.mdpi.com/2504-3110/1/1/17
Fractal Fract, Vol. 1, Pages 16: Series Solution of the Pantograph Equation and Its Properties http://www.mdpi.com/2504-3110/1/1/16 In this paper, we discuss the classical pantograph equation and its generalizations to include fractional order and the higher order case. The special functions are obtained from the series solution of these equations. We study different properties of these special functions and establish the relation with other functions. Further, we discuss some contiguous relations for these special functions. Fractal Fract, Vol. 1, Pages 16: Series Solution of the Pantograph Equation and Its Properties

Fractal and Fractional doi: 10.3390/fractalfract1010016

Authors: Sachin Bhalekar Jayvant Patade

In this paper, we discuss the classical pantograph equation and its generalizations to include fractional order and the higher order case. The special functions are obtained from the series solution of these equations. We study different properties of these special functions and establish the relation with other functions. Further, we discuss some contiguous relations for these special functions.

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Series Solution of the Pantograph Equation and Its Properties Sachin Bhalekar Jayvant Patade doi: 10.3390/fractalfract1010016 Fractal and Fractional 2017-12-08 Fractal and Fractional 2017-12-08 1 1 Article 16 10.3390/fractalfract1010016 http://www.mdpi.com/2504-3110/1/1/16
Fractal Fract, Vol. 1, Pages 15: Some Nonlocal Operators in the First Heisenberg Group http://www.mdpi.com/2504-3110/1/1/15 In this paper we construct some nonlocal operators in the Heisenberg group. Specifically, starting from the Grünwald-Letnikov derivative and Marchaud derivative in the Euclidean setting, we revisit those definitions with respect to the one of the fractional Laplace operator. Then, we define some nonlocal operators in the non-commutative structure of the first Heisenberg group adapting the approach applied in the Euclidean case to the new framework. Fractal Fract, Vol. 1, Pages 15: Some Nonlocal Operators in the First Heisenberg Group

Fractal and Fractional doi: 10.3390/fractalfract1010015

Authors: Fausto Ferrari

In this paper we construct some nonlocal operators in the Heisenberg group. Specifically, starting from the Grünwald-Letnikov derivative and Marchaud derivative in the Euclidean setting, we revisit those definitions with respect to the one of the fractional Laplace operator. Then, we define some nonlocal operators in the non-commutative structure of the first Heisenberg group adapting the approach applied in the Euclidean case to the new framework.

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Some Nonlocal Operators in the First Heisenberg Group Fausto Ferrari doi: 10.3390/fractalfract1010015 Fractal and Fractional 2017-11-27 Fractal and Fractional 2017-11-27 1 1 Article 15 10.3390/fractalfract1010015 http://www.mdpi.com/2504-3110/1/1/15
Fractal Fract, Vol. 1, Pages 14: A Solution to the Time-Scale Fractional Puzzle in the Implied Volatility http://www.mdpi.com/2504-3110/1/1/14 In the option pricing literature, it is well known that (i) the decrease in the smile amplitude is much slower than the standard stochastic volatility models and (ii) the term structure of the at-the-money volatility skew is approximated by a power-law function with the exponent close to zero. These stylized facts cannot be captured by standard models, and while (i) has been explained by using a fractional volatility model with Hurst index H &amp;gt; 1 / 2 , (ii) is proven to be satisfied by a rough volatility model with H &amp;lt; 1 / 2 under a risk-neutral measure. This paper provides a solution to this fractional puzzle in the implied volatility. Namely, we construct a two-factor fractional volatility model and develop an approximation formula for European option prices. It is shown through numerical examples that our model can resolve the fractional puzzle, when the correlations between the underlying asset process and the factors of rough volatility and persistence belong to a certain range. More specifically, depending on the three correlation values, the implied volatility surface is classified into four types: (1) the roughness exists, but the persistence does not; (2) the persistence exists, but the roughness does not; (3) both the roughness and the persistence exist; and (4) neither the roughness nor the persistence exist. Fractal Fract, Vol. 1, Pages 14: A Solution to the Time-Scale Fractional Puzzle in the Implied Volatility

Fractal and Fractional doi: 10.3390/fractalfract1010014

Authors: Hideharu Funahashi Masaaki Kijima

In the option pricing literature, it is well known that (i) the decrease in the smile amplitude is much slower than the standard stochastic volatility models and (ii) the term structure of the at-the-money volatility skew is approximated by a power-law function with the exponent close to zero. These stylized facts cannot be captured by standard models, and while (i) has been explained by using a fractional volatility model with Hurst index H &amp;gt; 1 / 2 , (ii) is proven to be satisfied by a rough volatility model with H &amp;lt; 1 / 2 under a risk-neutral measure. This paper provides a solution to this fractional puzzle in the implied volatility. Namely, we construct a two-factor fractional volatility model and develop an approximation formula for European option prices. It is shown through numerical examples that our model can resolve the fractional puzzle, when the correlations between the underlying asset process and the factors of rough volatility and persistence belong to a certain range. More specifically, depending on the three correlation values, the implied volatility surface is classified into four types: (1) the roughness exists, but the persistence does not; (2) the persistence exists, but the roughness does not; (3) both the roughness and the persistence exist; and (4) neither the roughness nor the persistence exist.

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A Solution to the Time-Scale Fractional Puzzle in the Implied Volatility Hideharu Funahashi Masaaki Kijima doi: 10.3390/fractalfract1010014 Fractal and Fractional 2017-11-25 Fractal and Fractional 2017-11-25 1 1 Article 14 10.3390/fractalfract1010014 http://www.mdpi.com/2504-3110/1/1/14
Fractal Fract, Vol. 1, Pages 13: The Kernel of the Distributed Order Fractional Derivatives with an Application to Complex Materials http://www.mdpi.com/2504-3110/1/1/13 The extension of the fractional order derivative to the distributed order fractional derivative (DOFD) is somewhat simple from a formal point of view, but it does not yet have a simple, obvious analytic form that allows its fast numerical calculation, which is necessary when solving differential equations with DOFD. In this paper, we supply a simple analytic kernel for the Caputo DOFD and the Caputo-Fabrizio DOFD, which may be used for numerical calculation in cases where the weight function is unity. This, in turn, could potentially allow faster solution of differential equations containing DOFD. Utilizing an analytical formulation of simple physical systems with phenomenological equations that include a DOFD, we show the relevant differences between the Caputo DOFD and the Caputo-Fabrizio DOFD. Finally, we propose a model based on DOFD for modeling composed materials that comprise different constituents, and show its compatibility with thermodynamics. Fractal Fract, Vol. 1, Pages 13: The Kernel of the Distributed Order Fractional Derivatives with an Application to Complex Materials

Fractal and Fractional doi: 10.3390/fractalfract1010013

Authors: Michele Caputo Mauro Fabrizio

The extension of the fractional order derivative to the distributed order fractional derivative (DOFD) is somewhat simple from a formal point of view, but it does not yet have a simple, obvious analytic form that allows its fast numerical calculation, which is necessary when solving differential equations with DOFD. In this paper, we supply a simple analytic kernel for the Caputo DOFD and the Caputo-Fabrizio DOFD, which may be used for numerical calculation in cases where the weight function is unity. This, in turn, could potentially allow faster solution of differential equations containing DOFD. Utilizing an analytical formulation of simple physical systems with phenomenological equations that include a DOFD, we show the relevant differences between the Caputo DOFD and the Caputo-Fabrizio DOFD. Finally, we propose a model based on DOFD for modeling composed materials that comprise different constituents, and show its compatibility with thermodynamics.

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The Kernel of the Distributed Order Fractional Derivatives with an Application to Complex Materials Michele Caputo Mauro Fabrizio doi: 10.3390/fractalfract1010013 Fractal and Fractional 2017-11-21 Fractal and Fractional 2017-11-21 1 1 Article 13 10.3390/fractalfract1010013 http://www.mdpi.com/2504-3110/1/1/13
Fractal Fract, Vol. 1, Pages 12: Fractal Simulation of Flocculation Processes Using a Diffusion-Limited Aggregation Model http://www.mdpi.com/2504-3110/1/1/12 In flocculation processes, particulates randomly collide and coagulate with each other, leading to the formation and sedimention of aggregates exhibiting fractal characteristics. The diffusion-limited aggregation (DLA) model is extensively employed to describe and study flocculation processes. To more accurately simulate flocculation processes with the DLA model, the effects of particle number (denoting flocculation time), motion step length (denoting water temperature), launch radius (representing initial particulate concentration), and finite motion step (representing the motion energy of the particles) on the morphology and structure of the two-dimensional (2D) as well as three-dimensional (3D) DLA aggregates are studied. The results show that the 2D DLA aggregates possess conspicuous fractal features when the particle number is above 1000, motion step length is 1.5–3.5, launch radius is 1–10, and finite motion step is more than 3000; the 3D DLA aggregates present clear fractal characteristics when the particle number is above 500, the motion step length is 1.5–3.5, the launch radius is 1–10, and the finite motion step exceeds 200. The fractal dimensions of 3D DLA aggregates are appreciably higher than those of 2D DLA aggregates. Fractal Fract, Vol. 1, Pages 12: Fractal Simulation of Flocculation Processes Using a Diffusion-Limited Aggregation Model

Fractal and Fractional doi: 10.3390/fractalfract1010012

Authors: Dongjing Liu Weiguo Zhou Xu Song Zumin Qiu

In flocculation processes, particulates randomly collide and coagulate with each other, leading to the formation and sedimention of aggregates exhibiting fractal characteristics. The diffusion-limited aggregation (DLA) model is extensively employed to describe and study flocculation processes. To more accurately simulate flocculation processes with the DLA model, the effects of particle number (denoting flocculation time), motion step length (denoting water temperature), launch radius (representing initial particulate concentration), and finite motion step (representing the motion energy of the particles) on the morphology and structure of the two-dimensional (2D) as well as three-dimensional (3D) DLA aggregates are studied. The results show that the 2D DLA aggregates possess conspicuous fractal features when the particle number is above 1000, motion step length is 1.5–3.5, launch radius is 1–10, and finite motion step is more than 3000; the 3D DLA aggregates present clear fractal characteristics when the particle number is above 500, the motion step length is 1.5–3.5, the launch radius is 1–10, and the finite motion step exceeds 200. The fractal dimensions of 3D DLA aggregates are appreciably higher than those of 2D DLA aggregates.

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Fractal Simulation of Flocculation Processes Using a Diffusion-Limited Aggregation Model Dongjing Liu Weiguo Zhou Xu Song Zumin Qiu doi: 10.3390/fractalfract1010012 Fractal and Fractional 2017-11-18 Fractal and Fractional 2017-11-18 1 1 Article 12 10.3390/fractalfract1010012 http://www.mdpi.com/2504-3110/1/1/12
Fractal Fract, Vol. 1, Pages 11: A Fractional-Order Infectivity and Recovery SIR Model http://www.mdpi.com/2504-3110/1/1/11 The introduction of fractional-order derivatives to epidemiological compartment models, such as SIR models, has attracted much attention. When this introduction is done in an ad hoc manner, it is difficult to reconcile parameters in the resulting fractional-order equations with the dynamics of individuals. This issue is circumvented by deriving fractional-order models from an underlying stochastic process. Here, we derive a fractional-order infectivity and recovery Susceptible Infectious Recovered (SIR) model from the stochastic process of a continuous-time random walk (CTRW) that incorporates a time-since-infection dependence on both the infectivity and the recovery of the population. By considering a power-law dependence in the infectivity and recovery, fractional-order derivatives appear in the generalised master equations that govern the evolution of the SIR populations. Under the appropriate limits, this fractional-order infectivity and recovery model reduces to both the standard SIR model and the fractional recovery SIR model. Fractal Fract, Vol. 1, Pages 11: A Fractional-Order Infectivity and Recovery SIR Model

Fractal and Fractional doi: 10.3390/fractalfract1010011

Authors: Christopher N. Angstmann Bruce I. Henry Anna V. McGann

The introduction of fractional-order derivatives to epidemiological compartment models, such as SIR models, has attracted much attention. When this introduction is done in an ad hoc manner, it is difficult to reconcile parameters in the resulting fractional-order equations with the dynamics of individuals. This issue is circumvented by deriving fractional-order models from an underlying stochastic process. Here, we derive a fractional-order infectivity and recovery Susceptible Infectious Recovered (SIR) model from the stochastic process of a continuous-time random walk (CTRW) that incorporates a time-since-infection dependence on both the infectivity and the recovery of the population. By considering a power-law dependence in the infectivity and recovery, fractional-order derivatives appear in the generalised master equations that govern the evolution of the SIR populations. Under the appropriate limits, this fractional-order infectivity and recovery model reduces to both the standard SIR model and the fractional recovery SIR model.

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A Fractional-Order Infectivity and Recovery SIR Model Christopher N. Angstmann Bruce I. Henry Anna V. McGann doi: 10.3390/fractalfract1010011 Fractal and Fractional 2017-11-17 Fractal and Fractional 2017-11-17 1 1 Article 11 10.3390/fractalfract1010011 http://www.mdpi.com/2504-3110/1/1/11
Fractal Fract, Vol. 1, Pages 10: The Fractal Nature of an Approximate Prime Counting Function http://www.mdpi.com/2504-3110/1/1/10 Prime number related fractal polygons and curves are derived by combining two different aspects. One is an approximation of the prime counting function build on an additive function. The other is prime number indexed basis entities taken from the discrete or continuous Fourier basis. Fractal Fract, Vol. 1, Pages 10: The Fractal Nature of an Approximate Prime Counting Function

Fractal and Fractional doi: 10.3390/fractalfract1010010

Authors: Dimitris Vartziotis Joachim Wipper

Prime number related fractal polygons and curves are derived by combining two different aspects. One is an approximation of the prime counting function build on an additive function. The other is prime number indexed basis entities taken from the discrete or continuous Fourier basis.

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The Fractal Nature of an Approximate Prime Counting Function Dimitris Vartziotis Joachim Wipper doi: 10.3390/fractalfract1010010 Fractal and Fractional 2017-11-08 Fractal and Fractional 2017-11-08 1 1 Article 10 10.3390/fractalfract1010010 http://www.mdpi.com/2504-3110/1/1/10
Fractal Fract, Vol. 1, Pages 9: From Circular to Bessel Functions: A Transition through the Umbral Method http://www.mdpi.com/2504-3110/1/1/9 A common environment in which to place Bessel and circular functions is envisaged. We show, by the use of operational methods, that the Gaussian provides the umbral image of these functions. We emphasize the role of the spherical Bessel functions and a family of associated auxiliary polynomials, as transition elements between these families of functions. The consequences of this point of view and the relevant impact on the study of the properties of special functions is carefully discussed. Fractal Fract, Vol. 1, Pages 9: From Circular to Bessel Functions: A Transition through the Umbral Method

Fractal and Fractional doi: 10.3390/fractalfract1010009

Authors: Giuseppe Dattoli Emanuele Di Palma Silvia Licciardi Elio Sabia

A common environment in which to place Bessel and circular functions is envisaged. We show, by the use of operational methods, that the Gaussian provides the umbral image of these functions. We emphasize the role of the spherical Bessel functions and a family of associated auxiliary polynomials, as transition elements between these families of functions. The consequences of this point of view and the relevant impact on the study of the properties of special functions is carefully discussed.

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From Circular to Bessel Functions: A Transition through the Umbral Method Giuseppe Dattoli Emanuele Di Palma Silvia Licciardi Elio Sabia doi: 10.3390/fractalfract1010009 Fractal and Fractional 2017-11-08 Fractal and Fractional 2017-11-08 1 1 Article 9 10.3390/fractalfract1010009 http://www.mdpi.com/2504-3110/1/1/9
Fractal Fract, Vol. 1, Pages 8: Fractional Divergence of Probability Densities http://www.mdpi.com/2504-3110/1/1/8 The divergence or relative entropy between probability densities is examined. Solutions that minimise the divergence between two distributions are usually “trivial” or unique. By using a fractional-order formulation for the divergence with respect to the parameters, the distance between probability densities can be minimised so that multiple non-trivial solutions can be obtained. As a result, the fractional divergence approach reduces the divergence to zero even when this is not possible via the conventional method. This allows replacement of a more complicated probability density with one that has a simpler mathematical form for more general cases. Fractal Fract, Vol. 1, Pages 8: Fractional Divergence of Probability Densities

Fractal and Fractional doi: 10.3390/fractalfract1010008

Authors: Aris Alexopoulos

The divergence or relative entropy between probability densities is examined. Solutions that minimise the divergence between two distributions are usually “trivial” or unique. By using a fractional-order formulation for the divergence with respect to the parameters, the distance between probability densities can be minimised so that multiple non-trivial solutions can be obtained. As a result, the fractional divergence approach reduces the divergence to zero even when this is not possible via the conventional method. This allows replacement of a more complicated probability density with one that has a simpler mathematical form for more general cases.

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Fractional Divergence of Probability Densities Aris Alexopoulos doi: 10.3390/fractalfract1010008 Fractal and Fractional 2017-10-25 Fractal and Fractional 2017-10-25 1 1 Article 8 10.3390/fractalfract1010008 http://www.mdpi.com/2504-3110/1/1/8
Fractal Fract, Vol. 1, Pages 7: Stokes’ First Problem for Viscoelastic Fluids with a Fractional Maxwell Model http://www.mdpi.com/2504-3110/1/1/7 Stokes’ first problem for a class of viscoelastic fluids with the generalized fractional Maxwell constitutive model is considered. The constitutive equation is obtained from the classical Maxwell stress–strain relation by substituting the first-order derivatives of stress and strain by derivatives of non-integer orders in the interval ( 0 , 1 ] . Explicit integral representation of the solution is derived and some of its characteristics are discussed: non-negativity and monotonicity, asymptotic behavior, analyticity, finite/infinite propagation speed, and absence of wave front. To illustrate analytical findings, numerical results for different values of the parameters are presented. Fractal Fract, Vol. 1, Pages 7: Stokes’ First Problem for Viscoelastic Fluids with a Fractional Maxwell Model

Fractal and Fractional doi: 10.3390/fractalfract1010007

Authors: Emilia Bazhlekova Ivan Bazhlekov

Stokes’ first problem for a class of viscoelastic fluids with the generalized fractional Maxwell constitutive model is considered. The constitutive equation is obtained from the classical Maxwell stress–strain relation by substituting the first-order derivatives of stress and strain by derivatives of non-integer orders in the interval ( 0 , 1 ] . Explicit integral representation of the solution is derived and some of its characteristics are discussed: non-negativity and monotonicity, asymptotic behavior, analyticity, finite/infinite propagation speed, and absence of wave front. To illustrate analytical findings, numerical results for different values of the parameters are presented.

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Stokes’ First Problem for Viscoelastic Fluids with a Fractional Maxwell Model Emilia Bazhlekova Ivan Bazhlekov doi: 10.3390/fractalfract1010007 Fractal and Fractional 2017-10-24 Fractal and Fractional 2017-10-24 1 1 Article 7 10.3390/fractalfract1010007 http://www.mdpi.com/2504-3110/1/1/7
Fractal Fract, Vol. 1, Pages 6: Exact Discretization of an Economic Accelerator and Multiplier with Memory http://www.mdpi.com/2504-3110/1/1/6 Fractional differential equations of macroeconomics, which allow us to take into account power-law memory effects, are considered. We describe an economic accelerator and multiplier with fading memory in the framework of discrete-time and continuous-time approaches. A relationship of the continuous- and discrete-time fractional-order equations is considered. We propose equations of the accelerator and multiplier for economic processes with power-law memory. Exact discrete analogs of these equations are suggested by using the exact fractional differences of integer and non-integer orders. Exact correspondence between the equations with finite differences and differential equations lies not so much in the limiting condition, when the step of discretization tends to zero, as in the fact that mathematical operations, which are used in these equations, satisfy in many cases the same mathematical laws. Fractal Fract, Vol. 1, Pages 6: Exact Discretization of an Economic Accelerator and Multiplier with Memory

Fractal and Fractional doi: 10.3390/fractalfract1010006

Authors: Valentina Tarasova Vasily Tarasov

Fractional differential equations of macroeconomics, which allow us to take into account power-law memory effects, are considered. We describe an economic accelerator and multiplier with fading memory in the framework of discrete-time and continuous-time approaches. A relationship of the continuous- and discrete-time fractional-order equations is considered. We propose equations of the accelerator and multiplier for economic processes with power-law memory. Exact discrete analogs of these equations are suggested by using the exact fractional differences of integer and non-integer orders. Exact correspondence between the equations with finite differences and differential equations lies not so much in the limiting condition, when the step of discretization tends to zero, as in the fact that mathematical operations, which are used in these equations, satisfy in many cases the same mathematical laws.

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Exact Discretization of an Economic Accelerator and Multiplier with Memory Valentina Tarasova Vasily Tarasov doi: 10.3390/fractalfract1010006 Fractal and Fractional 2017-09-11 Fractal and Fractional 2017-09-11 1 1 Article 6 10.3390/fractalfract1010006 http://www.mdpi.com/2504-3110/1/1/6
Fractal Fract, Vol. 1, Pages 5: Dynamics and Stability Results for Hilfer Fractional Type Thermistor Problem http://www.mdpi.com/2504-3110/1/1/5 In this paper, we study the dynamics and stability of thermistor problem for Hilfer fractional type. Classical fixed point theorems are utilized in deriving the results. Fractal Fract, Vol. 1, Pages 5: Dynamics and Stability Results for Hilfer Fractional Type Thermistor Problem

Fractal and Fractional doi: 10.3390/fractalfract1010005

Authors: D. Vivek K. Kanagarajan Seenith Sivasundaram

In this paper, we study the dynamics and stability of thermistor problem for Hilfer fractional type. Classical fixed point theorems are utilized in deriving the results.

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Dynamics and Stability Results for Hilfer Fractional Type Thermistor Problem D. Vivek K. Kanagarajan Seenith Sivasundaram doi: 10.3390/fractalfract1010005 Fractal and Fractional 2017-09-09 Fractal and Fractional 2017-09-09 1 1 Article 5 10.3390/fractalfract1010005 http://www.mdpi.com/2504-3110/1/1/5
Fractal Fract, Vol. 1, Pages 4: A Fractional Complex Permittivity Model of Media with Dielectric Relaxation http://www.mdpi.com/2504-3110/1/1/4 In this work, we propose a fractional complex permittivity model of dielectric media with memory. Debye’s generalized equation, expressed in terms of the phenomenological coefficients, is replaced with the corresponding differential equation by applying Caputo’s fractional derivative. We observe how fractional order depends on the frequency band of excitation energy in accordance with the 2nd Principle of Thermodynamics. The model obtained is validated with respect to the measurements made on the biological tissues and in particular on the human aorta. Fractal Fract, Vol. 1, Pages 4: A Fractional Complex Permittivity Model of Media with Dielectric Relaxation

Fractal and Fractional doi: 10.3390/fractalfract1010004

Authors: Armando Ciancio Bruno Flora

In this work, we propose a fractional complex permittivity model of dielectric media with memory. Debye’s generalized equation, expressed in terms of the phenomenological coefficients, is replaced with the corresponding differential equation by applying Caputo’s fractional derivative. We observe how fractional order depends on the frequency band of excitation energy in accordance with the 2nd Principle of Thermodynamics. The model obtained is validated with respect to the measurements made on the biological tissues and in particular on the human aorta.

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A Fractional Complex Permittivity Model of Media with Dielectric Relaxation Armando Ciancio Bruno Flora doi: 10.3390/fractalfract1010004 Fractal and Fractional 2017-08-29 Fractal and Fractional 2017-08-29 1 1 Article 4 10.3390/fractalfract1010004 http://www.mdpi.com/2504-3110/1/1/4
Fractal Fract, Vol. 1, Pages 3: Which Derivative? http://www.mdpi.com/2504-3110/1/1/3 The actual state of interplay between Fractional Calculus, Signal Processing, and Applied Sciences is discussed in this paper. A framework for compatible integer and fractional derivatives/integrals in signals and systems context is described. It is shown how suitable fractional formulations are really extensions of the integer order definitions currently used in Signal Processing. The particular case of fractional linear systems is considered and the problem of initial conditions is tackled. Fractal Fract, Vol. 1, Pages 3: Which Derivative?

Fractal and Fractional doi: 10.3390/fractalfract1010003

Authors: Manuel Ortigueira José Machado

The actual state of interplay between Fractional Calculus, Signal Processing, and Applied Sciences is discussed in this paper. A framework for compatible integer and fractional derivatives/integrals in signals and systems context is described. It is shown how suitable fractional formulations are really extensions of the integer order definitions currently used in Signal Processing. The particular case of fractional linear systems is considered and the problem of initial conditions is tackled.

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Which Derivative? Manuel Ortigueira José Machado doi: 10.3390/fractalfract1010003 Fractal and Fractional 2017-07-25 Fractal and Fractional 2017-07-25 1 1 Article 3 10.3390/fractalfract1010003 http://www.mdpi.com/2504-3110/1/1/3
Fractal Fract, Vol. 1, Pages 2: Fractional Definite Integral http://www.mdpi.com/2504-3110/1/1/2 This paper proposes the definition of fractional definite integral and analyses the corresponding fundamental theorem of fractional calculus. In this context, we studied the relevant properties of the fractional derivatives that lead to such a definition. Finally, integrals on R2 R 2 and R3 R 3 are also proposed. Fractal Fract, Vol. 1, Pages 2: Fractional Definite Integral

Fractal and Fractional doi: 10.3390/fractalfract1010002

Authors: Manuel Ortigueira José Machado

This paper proposes the definition of fractional definite integral and analyses the corresponding fundamental theorem of fractional calculus. In this context, we studied the relevant properties of the fractional derivatives that lead to such a definition. Finally, integrals on R2 R 2 and R3 R 3 are also proposed.

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Fractional Definite Integral Manuel Ortigueira José Machado doi: 10.3390/fractalfract1010002 Fractal and Fractional 2017-07-02 Fractal and Fractional 2017-07-02 1 1 Article 2 10.3390/fractalfract1010002 http://www.mdpi.com/2504-3110/1/1/2
Fractal Fract, Vol. 1, Pages 1: Fractal and Fractional http://www.mdpi.com/2504-3110/1/1/1 Fractal and Fractional are two words referring to some characteristics and fundamental problems which arise in all fields of science and technology. [...] Fractal Fract, Vol. 1, Pages 1: Fractal and Fractional

Fractal and Fractional doi: 10.3390/fractalfract1010001

Authors: Carlo Cattani

Fractal and Fractional are two words referring to some characteristics and fundamental problems which arise in all fields of science and technology. [...]

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Fractal and Fractional Carlo Cattani doi: 10.3390/fractalfract1010001 Fractal and Fractional 2017-03-26 Fractal and Fractional 2017-03-26 1 1 Editorial 1 10.3390/fractalfract1010001 http://www.mdpi.com/2504-3110/1/1/1