Jena P.J., Jena S.K., Chakraverty S. Computational Fractional Dynamical Systems: Fractional Differential Equations and Applications

Hoboken: Wiley, 2022. — 268 p.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.About the Authors
Introduction to Fractional CalculusBirth of Fractional Calculus
Useful Mathematical Functions
The Gamma Function
The Beta Function
The Mittag-Leffler Function
The Mellin-Ross Function
The Wright Function
The Error Function
The Hypergeometric Function
The H-Function
Riemann–Liouville (R-L) Fractional Integral and Derivative
Caputo Fractional Derivative
Grünwald–Letnikov Fractional Derivative and Integral
Riesz Fractional Derivative and Integral
Modified Riemann–Liouville Derivative
Local Fractional Derivative
Local Fractional Continuity of a Function
Local Fractional DerivativeRecent Trends in Fractional Dynamical Models and Mathematical MethodsFractional Calculus: A Generalization of Integer-Order Calculus
Fractional Derivatives of Some Functions and Their Graphical Illustrations
Applications of Fractional Calculus
NH Abel and Tautochronous problem
Ultrasonic Wave Propagation in Human Cancellous Bone
Modeling of Speech Signals Using Fractional Calculus
Modeling the Cardiac Tissue Electrode Interface Using Fractional Calculus
Application of Fractional Calculus to the Sound Waves Propagation in Rigid Porous Materials
Fractional Calculus for Lateral and Longitudinal Control of Autonomous Vehicles
Application of Fractional Calculus in the Theory of Viscoelasticity
Fractional Differentiation for Edge Detection
Wave Propagation in Viscoelastic Horns Using a Fractional Calculus Rheology Model
Application of Fractional Calculus to Fluid Mechanics
Radioactivity, Exponential Decay, and Population Growth
The Harmonic Oscillator
Overview of Some Analytical/Numerical Methods
Fractional Adams–Bashforth/Moulton Methods
Fractional Euler Method
Finite Difference Method
Finite Element Method
Finite Volume Method
Meshless Method
Reproducing Kernel Hilbert Space Method
Wavelet Method
The Sine-Gordon Expansion Method
The Jacobi Elliptic Equation Method
The Generalized Kudryashov MethodAdomian Decomposition MethodBasic Idea of ADM
Numerical ExamplesAdomian Decomposition Transform MethodTransform Methods for the Caputo Sense Derivatives
Adomian Decomposition Laplace Transform Method (ADLTM)
Adomian Decomposition Sumudu Transform Method (ADSTM)
Adomian Decomposition Elzaki Transform Method (ADETM)
Adomian Decomposition Aboodh Transform Method (ADATM)
Numerical Examples
Implementation of ADLTM
Implementation of ADSTM
Implementation of ADETM
Implementation of ADATMHomotopy Perturbation MethodProcedure for HPM
Numerical ExamplesHomotopy Perturbation Transform MethodTransform Methods for the Caputo Sense Derivatives
Homotopy Perturbation Laplace Transform Method (HPLTM)
Homotopy Perturbation Sumudu Transform Method (HPSTM)
Homotopy Perturbation Elzaki Transform Method (HPETM)
Homotopy Perturbation Aboodh Transform Method (HPATM)
Numerical Examples
Implementation of HPLTM
Implementation of HPSTM
Implementation of HPETM
Implementation of HPATMFractional Differential Transform MethodFractional Differential Transform Method
Illustrative ExamplesFractional Reduced Differential Transform MethodDescription of FRDTM
Numerical ExamplesVariational Iterative MethodProcedure for VIM
ExamplesWeighted Residual MethodsCollocation Method
Least-Square Method
Galerkin Method
Numerical ExamplesBoundary Characteristic Orthogonal PolynomialsGram–Schmidt Orthogonalization Procedure
Generation of BCOPs
Galerkin Method with BCOPs
Least-Square Method with BCOPs
Application ProblemsResidual Power Series MethodTheorems and Lemma Related to RPSM
Basic Idea of RPSM
Convergence Analysis
ExamplesHomotopy Analysis MethodTheory of Homotopy Analysis Method
Convergence Theorem of HAM
Test ExamplesHomotopy Analysis Transform MethodTransform Methods for the Caputo Sense Derivative
Homotopy Analysis Laplace Transform Method (HALTM)
Homotopy Analysis Sumudu Transform Method (HASTM)
Homotopy Analysis Elzaki Transform Method (HAETM)
Homotopy Analysis Aboodh Transform Method (HAATM)
Numerical Examples
Implementation of HALTM
Implementation of HASTM
Implementation of HAETM
Implementation of HAATMq-Homotopy Analysis MethodTheory of q-HAM
Illustrative Examplesq-Homotopy Analysis Transform MethodTransform Methods for the Caputo Sense Derivative
q-Homotopy Analysis Laplace Transform Method (q-HALTM)
q-Homotopy Analysis Sumudu Transform Method (q-HASTM)
q-Homotopy Analysis Elzaki Transform Method (q-HAETM)
q-Homotopy Analysis Aboodh Transform Method (q-HAATM)
Test Problems
Implementation of q-HALTM
Implementation of q-HASTM
Implementation of q-HAETM
Implementation of q-HAATM(G/G)-Expansion MethodDescription of the (G/G)-Expansion Method
Application Problems(G/G2)-Expansion MethodDescription of the (G/G2)-Expansion Method
Numerical Examples(G/G, 1/G)-Expansion MethodAlgorithm of the (G/G,1/G)-Expansion Method
Illustrative ExamplesThe Modified Simple Equation MethodProcedure of the Modified Simple Equation Method
Application ProblemsSine-Cosine MethodDetails of the Sine-Cosine Method
Numerical ExamplesTanh MethodDescription of the Tanh Method
Numerical ExamplesFractional Subequation MethodImplementation of the Fractional Subequation Method
Numerical ExamplesExp-Function MethodProcedure of the Exp-Function Method
Numerical ExamplesExp(−φ(ξ))-Expansion MethodMethodology of the Exp(−φ(ξ))-Expansion Method
Numerical Examples