Computational Fractional Dynamical Systems: Fractional Differential Equations and Applications

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