Computational Fractional Dynamical Systems: Fractional Differential Equations and Applications

Description

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. Cover Title Page Copyright Page Contents Preface Acknowledgments About the Authors Chapter 1 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 (R-L) 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 Chapter 2 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 Chapter 3 Adomian Decomposition Method 3.1 Introduction 3.2 Basic Idea of ADM 3.3 Numerical Examples References Chapter 4 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 Chapter 5 Homotopy Perturbation Method 5.1 Introduction 5.2 Procedure for HPM 5.3 Numerical Examples References Chapter 6 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 Chapter 7 Fractional Differential Transform Method 7.1 Introduction 7.2 Fractional Differential Transform Method 7.3 Illustrative Examples References Chapter 8 Fractional Reduced Differential Transform Method 8.1 Introduction 8.2 Description of FRDTM 8.3 Numerical Examples References Chapter 9 Variational Iterative Method 9.1 Introduction 9.2 Procedure for VIM 9.3 Examples References Chapter 10 Weighted Residual Methods 10.1 Introduction 10.2 Collocation Method 10.3 Least-Square Method 10.4 Galerkin Method 10.5 Numerical Examples References Chapter 11 Boundary Characteristic 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 Chapter 12 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 Chapter 13 Homotopy Analysis Method 13.1 Introduction 13.2 Theory of Homotopy Analysis Method 13.3 Convergence Theorem of HAM 13.4 Test Examples References Chapter 14 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 Chapter 15 q-Homotopy Analysis Method 15.1 Introduction 15.2 Theory of q-HAM 15.3 Illustrative Examples References Chapter 16 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 Chapter 17 (G/G)-Expansion Method 17.1 Introduction 17.2 Description of the (G/G)-Expansion Method 17.3 Application Problems References Chapter 18 (G/G2)-Expansion Method 18.1 Introduction 18.2 Description of the (G/G2)-Expansion Method 18.3 Numerical Examples References Chapter 19 (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 Chapter 20 The Modified Simple Equation Method 20.1 Introduction 20.2 Procedure of the Modified Simple Equation Method 20.3 Application Problems References Chapter 21 Sine-Cosine Method 21.1 Introduction 21.2 Details of the Sine-Cosine Method 21.3 Numerical Examples References Chapter 22 Tanh Method 22.1 Introduction 22.2 Description of the Tanh Method 22.3 Numerical Examples References Chapter 23 Fractional Subequation Method 23.1 Introduction 23.2 Implementation of the Fractional Subequation Method 23.3 Numerical Examples References Chapter 24 Exp-Function Method 24.1 Introduction 24.2 Procedure of the Exp-Function Method 24.3 Numerical Examples References Chapter 25 Exp(−φ(ξ))-Expansion Method 25.1 Introduction 25.2 Methodology of the Exp(−φ(ξ))-Expansion Method 25.3 Numerical Examples References Index EULA