Computational Fractional Dynamical Systems: Fractional Differential Equations and Applications

Computational Fractional Dynamical Systems A rigorous presentation of different expansion and semi-analytical methods for fractional differential equations Fractional differential equations, differential and integral operators with non-integral powers, are used in various science and engineering applications. Over the past several decades, the popularity of the fractional derivative has increased significantly in diverse areas such as electromagnetics, financial mathematics, image processing, and materials science. Obtaining analytical and numerical solutions of nonlinear partial differential equations of fractional order can be challenging and involve the development and use of different methods of solution. Computational Fractional Dynamical Systems: Fractional Differential Equations and Applications presents a variety of computationally efficient semi-analytical and expansion methods to solve different types of fractional models. Rather than focusing on a single computational method, this comprehensive volume brings together more than 25 methods for solving an array of fractional-order models. The authors employ a rigorous and systematic approach for addressing various physical problems in science and engineering. Covers various aspects of efficient methods regarding fractional-order systems Presents different numerical methods with detailed steps to handle basic and advanced equations in science and engineering Provides a systematic approach for handling fractional-order models arising in science and engineering Incorporates a wide range of methods with corresponding results and validation Computational Fractional Dynamical Systems: Fractional Differential Equations and Applications is an invaluable resource for advanced undergraduate students, graduate students, postdoctoral researchers, university faculty, and other researchers and practitioners working with fractional and integer order differential equations.

Table of Contents:
Preface Acknowledgments About the Authors                 Introduction to Fractional Calculus 1.1.          Introduction 1.2.          Birth of fractional calculus 1.3.          Useful mathematical functions       1.3.1.       The gamma function       1.3.2.       The beta function       1.3.3.       The Mittag-Leffler function            1.3.4.       The Mellin-Ross function       1.3.5.       The Wright function       1.3.6.       The error function       1.3.7.       The hypergeometric function 1.3.8.       The H-function 1.4.          Riemann–Liouville fractional integral and derivative 1.5.          Caputo fractional derivative 1.6.          Grünwald-Letnikov fractional derivative and integral 1.7.          Riesz fractional derivative and integral 1.8.          Modified Riemann-Liouville derivative       1.9.          Local fractional derivative 1.9.1.       Local fractional continuity of a function 1.9.2.       Local fractional derivative                 References                   Recent Trends in Fractional Dynamical Models and Mathematical Methods 2.1.          Introduction 2.2.          Fractional calculus: A generalization of integer-order calculus 2.3.          Fractional derivatives of some functions and their graphical illustrations 2.4.          Applications of fractional calculus 2.4.1.       N.H. Abel and Tautochronous problem 2.4.2.       Ultrasonic wave propagation in human cancellous bone 2.4.3.       Modeling of speech signals using fractional calculus 2.4.4.       Modeling the cardiac tissue electrode interface using fractional calculus 2.4.5.     Application of fractional calculus to the sound waves propagation in rigid porous                      Materials 2.4.6.        Fractional calculus for lateral and longitudinal control of autonomous vehicles 2.4.7.        Application of fractional calculus in the theory of viscoelasticity 2.4.8.        Fractional differentiation for edge detection 2.4.9.        Wave propagation in viscoelastic horns using a fractional calculus rheology model 2.4.10.      Application of fractional calculus to fluid mechanics 2.4.11.      Radioactivity, exponential decay and population growth 2.4.12.      The Harmonic oscillator 2.5.           Overview of some analytical/numerical methods 2.5.1.        Fractional Adams–Bashforth/Moulton methods 2.5.2.        Fractional Euler method 2.5.3.          Finite difference method 2.5.4.          Finite element method 2.5.5.        Finite volume method 2.5.6.        Meshless method 2.5.7.        Reproducing kernel Hilbert space method 2.5.8.        Wavelet method 2.5.9.        The Sine-Gordon expansion method 2.5.10.      The Jacobi elliptic equation method 2.5.11.      The generalized Kudryashov method                  References                   Adomian Decomposition Method (ADM) 3.1.           Introduction 3.2.           Basic Idea of  ADM 3.3.           Numerical Examples                  References                   Adomian Decomposition Transform Method 4.1.            Introduction 4.2.            Transform methods for the Caputo sense derivatives 4.3.            Adomian decomposition Laplace transform method (ADLTM) 4.4.            Adomian decomposition Sumudu transform method (ADSTM) 4.5.            Adomian decomposition Elzaki transform method (ADETM) 4.6.            Adomian decomposition Aboodh transform method (ADATM) 4.7.            Numerical Examples 4.7.1.         Implementation of ADLTM 4.7.2.         Implementation of ADSTM 4.7.3.         Implementation of ADETM 4.7.4.         Implementation of ADATM                        References                   Homotopy Perturbation Method (HPM) 5.1.            Introduction 5.2.            Procedure of HPM 5.3.            Numerical examples                   References                   Homotopy Perturbation Transform Method 6.1.            Introduction 6.2.            Transform methods for the Caputo sense derivatives 6.3.            Homotopy perturbation Laplace transform method (HPLTM) 6.4.            Homotopy perturbation Sumudu transform method (HPSTM) 6.5.            Homotopy perturbation Elzaki transform method (HPETM) 6.6.            Homotopy perturbation Aboodh transform method (HPATM) 6.7.            Numerical Examples 6.7.1.         Implementation of HPLTM 6.7.2.         Implementation of HPSTM 6.7.3.         Implementation of HPETM 6.7.4.         Implementation of HPATM                   References                   Fractional Differential Transform Method 7.1.            Introduction 7.2.            Fractional differential transform method 7.3.            Illustrative Examples                   References                   Fractional Reduced Differential Transform Method 8.1.            Introduction 8.2.            Description of FRDTM 8.3.            Numerical Examples                   References                   Variational Iterative Method 9.1.            Introduction 9.2.            Procedure for VIM 9.3.            Examples                   References                        Method of Weighted Residuals  10.1.         Introduction        10.2.         Collocation method        10.3.         Least-square method        10.4.         Galerkin method        10.5.         Numerical Examples                   References                    Boundary Characteristics Orthogonal Polynomials  11.1.         Introduction  11.2.         Gram–Schmidt orthogonalization procedure  11.3.         Generation of BCOPs  11.4.         Galerkin method with BCOPs  11.5.         Least-Square method with BCOPs  11.6.         Application Problems                   References                    Residual Power Series Method 12.1.           Introduction 12.2.           Theorems and lemma related to RPSM 12.3.           Basic idea of RPSM 12.4.           Convergence Analysis 12.5.           Examples                    References                   Homotopy Analysis Method 13.1.           Introduction 13.2.           Theory of homotopy analysis method 13.3.           Convergence theorem of HAM 13.4.           Test Examples                    References                   Homotopy Analysis Transform Method 14.1.           Introduction       14.2.           Transform methods for the Caputo sense derivative       14.3.           Homotopy analysis Laplace transform method (HALTM)       14.4.           Homotopy analysis Sumudu transform method (HASTM)       14.5.           Homotopy analysis Elzaki transform method (HAETM)       14.6.           Homotopy analysis Aboodh transform method (HAATM)       14.7.           Numerical Examples       14.7.1.         Implementation of HALTM       14.7.2.         Implementation of HASTM       14.7.3.         Implementation of HAETM       14.7.4.         Implementation of HAATM                          References                    q-Homotopy Analysis Method  15.1.         Introduction  15.2.         Theory of q-HAM  15.3.         Illustrative Examples                   References                     q-Homotopy Analysis transform Method   16.1.         Introduction   16.2.         Transform methods for the Caputo sense derivative         16.3.         q-homotopy analysis Laplace transform method (q-HALTM)         16.4.         q-homotopy analysis Sumudu transform method (q-HASTM)         16.5.         q-homotopy analysis Elzaki transform method (q-HAETM)         16.6.         q-homotopy analysis Aboodh transform method (q-HAATM)         16.7.         Test Problems         16.7.1.        Implementation of q-HALTM         16.7.2.        Implementation of q-HASTM         16.7.3.        Implementation of q-HAETM         16.7.4.        Implementation of q-HAATM                           References                     (G'/G)-Expansion Method    17.1.          Introduction    17.2.          Description of the (G'/G)-expansion method    17.3.          Application Problems                      References                     (G’/G^2)-Expansion Method    18.1.          Introduction  18.2.            Description of the (G’/G^2)-expansion method  18.3.            Numerical Examples                      References                     (G’/G,1/G)-Expansion Method   19.1.           Introduction   19.2.           Algorithm of the (G’/G,1/G)-expansion method   19.3.           Illustrative Examples                      References                    The modified simple equation method  20.1.           Introduction  20.2.           Procedure of the modified simple equation method  20.3.           Application Problems                     References                    Sine-Cosine Method  21.1.           Introduction  21.2.           Details of Sine-Cosine method  21.3.           Numerical Examples                     References                    Tanh Method  22.1.            Introduction  22.2.            Description of the Tanh method  22.3.            Numerical Examples                      References                    Fractional sub-equation method  23.1.            Introduction  23.2.            Implementation of the fractional sub-equation method  23.3.            Numerical Examples                      References                    Exp-function Method  24.1.           Introduction  24.2.           Procedure of the Exp-function method  24.3.           Numerical Examples                     References                    Exp(-φ(ξ))-expansion method  25.1.          Introduction  25.2.          Methodology of the exp(-φ(ξ))-expansion method  25.3.          Numerical Examples                    References Index