Introduction to Fractional Differential Equations (Nonlinear Systems and Complexity)

✍ Scribed by Constantin Milici, Gheorghe Drăgănescu, J. Tenreiro Machado

• Publisher: Springer
• Year: 2018
• Tongue: English
• Leaves: 199
• Series: Nonlinear Systems and Complexity (Book 25)
• Edition: 1st ed. 2019
• Category: Library

✦ Synopsis

This book introduces a series of problems and methods insufficiently discussed in the field of Fractional Calculus – a major, emerging tool relevant to all areas of scientific inquiry. The authors present examples based on symbolic computation, written in Maple and Mathematica, and address both mathematical and computational areas in the context of mathematical modeling and the generalization of classical integer-order methods. Distinct from most books, the present volume fills the gap between mathematics and computer fields, and the transition from integer- to fractional-order methods.

✦ Table of Contents

Preface
References
Contents
Acronyms
List of Symbols
1 Special Functions
1.1 Euler's Function
1.1.1 Gamma Function
1.1.2 Beta Function
1.2 Integral Functions
1.3 Mittag-Leffler Function
1.4 Function E(t,α,a)
References
2 Fractional Derivative and Fractional Integral
2.1 Fractional Integral and Derivative
References
3 The Laplace Transform
3.1 Calculus of the Images
3.2 Calculus of the Original Function
3.2.1 Calculus of Original Using Residues
3.2.2 Calculus of Original with Post's Inversion Formula
3.3 The Properties of the Laplace Transform
3.3.1 The Property of Linearity
3.3.2 Similarity Theorem
3.3.3 The Differentiation and Integration Theorems
3.3.4 Delay Theorem
3.3.5 Displacement Theorem
3.3.6 Multiplication Theorem
3.3.7 Properties of the Inverse Laplace Transform
3.4 Laplace Transform of the Fractional Integrals and Derivatives
3.4.1 Fractional Integrals
3.4.2 Fractional Derivatives
References
4 Fractional Differential Equations
4.1 The Existence and Uniqueness Theorem for Initial Value Problems
4.2 Linear Fractional Differential Equations
4.3 Nonlinear Equations
4.3.1 The Adomian Decomposition Method
4.3.2 Decomposition of Nonlinear Equations
4.3.3 Perturbation Method
4.4 Fractional Systems of Differential Equations
4.4.1 Linear Systems
4.4.2 Nonlinear Systems
(A) Method of Successive Approximations
(B) Method of Laplace's Transform
References
5 Generalized Systems
5.1 Cornu Fractional System
5.1.1 Cos and Sin Fractional of Type Fresnel
5.1.2 Cornu Fractional System and Curve
5.1.3 Cornu Generalized Curve/System
5.1.4 Cornu Fractional System in a Plane
5.1.5 Fractional Cornu Spiral on the Sphere
5.1.6 Fractional Cornu Spiral on the Cone
5.2 Power Series
5.2.1 The Müntz Theorem
5.2.2 Lane-Emden Equation
5.2.3 The Taylor Series Method
5.2.4 The Generalized Hermite Equation
5.2.5 The Generalized Legendre Equation
5.2.6 The Generalized Bessel Equation
5.2.7 Nonlinear Systems
Lotka System
Lorenz Fractional Attractor
References
6 Numerical Methods
6.1 Variational Iteration Method for Fractional Differential Equations
6.2 The Least Squares Method
6.3 The Galerkin Method for Fractional Differential Equations
6.4 Euler's Method
6.5 Runge–Kutta Methods for Fractional Differential Equation
6.5.1 The Second Order Runge–Kutta Method
6.5.2 The Fourth Order Runge–Kutta Method
6.5.3 A More General System
6.5.4 A Vectorial Runge–Kutta Algorithm
References
Index

✦ Subjects

Mathematics;Calculus; Differential equations