Numerical methods for fractional calculus

$Cover image for Numerical methods for fractional calculus$

Başlık:

Yazar:

Li, Changpin.

ISBN:

9781482253801

Ek Yazar:

Li, Changpin.

Fiziksel Tanım:

xviii, 281 pages : illustrations ; 25 cm.

Series:

Chapman & Hall/CRC numerical analysis and scientific computing

Chapman & Hall/CRC numerical analysis and scientific computing.

General Note:

"A Chapman & Hall book."

Contents:

1. Introduction to fractional calculus -- 2. Numerical methods for fractional integral and derivatives -- 3. Numerical methods for fractional ordinary differential equations -- 4. Finite difference methods for fractional partial differential equations -- 5. Galerkin finite element methods for fractional partial differential equations.

Konu Terimleri:

Kesirli kalkulus.

Sayısal analiz.

Fractional integrals.

Fractional calculus.

Numerical analysis.

Added Author:

Zeng, Fanhai.

Mevcut:*

Library	Materyal Türü	Barkod	Yer Numarası	Durum
Searching... Pamukkale Merkez Kütüphanesi	Kitap	0088648	QA297 L69311 2015	Searching... Unknown

Bound With These Titles

On Order

Özet

Numerical Methods for Fractional Calculus presents numerical methods for fractional integrals and fractional derivatives, finite difference methods for fractional ordinary differential equations (FODEs) and fractional partial differential equations (FPDEs), and finite element methods for FPDEs.

The book introduces the basic definitions and properties of fractional integrals and derivatives before covering numerical methods for fractional integrals and derivatives. It then discusses finite difference methods for both FODEs and FPDEs, including the Euler and linear multistep methods. The final chapter shows how to solve FPDEs by using the finite element method.

This book provides efficient and reliable numerical methods for solving fractional calculus problems. It offers a primer for readers to further develop cutting-edge research in numerical fractional calculus. MATLAB® functions are available on the book's CRC Press web page.

Author Notes

Changpin Li is a full professor at Shanghai University. He earned his Ph.D. in computational mathematics from Shanghai University. Dr. Li's main research interests include numerical methods and computations for FPDEs and fractional dynamics. He was awarded the Riemann-Liouville Award for Best FDA Paper (theory) in 2012. He is on the editorial board of several journals, including Fractional Calculus and Applied Analysis , International Journal of Bifurcation and Chaos , and International Journal of Computer Mathematics .

Fanhai Zeng is visiting Brown University as a postdoc fellow. He earned his Ph.D. in computational mathematics from Shanghai University. Dr. Zeng's research interests include numerical methods and computations for FPDEs.

Foreword	p. xi
Preface	p. xiii
List of Figures	p. xv
List of Tables	p. xvii
1 Introduction to Fractional Calculus	p. 1
1.1 Fractional Integrals and Derivatives	p. 1
1.2 Some Other Properties of Fractional Derivatives	p. 10
1.2.1 Leibniz Rule for Fractional Derivatives	p. 10
1.2.2 Fractional Derivative of a Composite Function	p. 11
1.2.3 Behaviors Near and Far from the Lower Terminal	p. 12
1.2.4 Laplace Transforms of Fractional Derivatives	p. 14
1.2.5 Fourier Transforms of Fractional Derivatives	p. 16
1.3 Some Other Fractional Derivatives and Extensions	p. 18
1.3.1 Marchaud Fractional Derivative	p. 18
1.3.2 The Finite Parts of Integrals	p. 19
1.3.3 Directional Integrals and Derivatives in R 2	p. 20
1.3.4 Partial Fractional Derivatives	p. 21
1.4 Physical Meanings	p. 23
1.5 Fractional Initial and Boundary Problems	p. 25
2 Numerical Methods for Fractional Integral and Derivatives	p. 29
2.1 Approximations to Fractional Integrals	p. 29
2.1.1 Numerical Methods Based on Polynomial Interpolation	p. 30
2.1.2 High-Order Methods Based on Gauss Interpolation	p. 34
2.1.3 Fractional Linear Multistep Methods	p. 38
2.2 Approximations to Riemann-Liouville Derivatives	p. 40
2.2.1 Grunwald-Letnikov Type Approximation	p. 41
2.2.2 L1, L2 and L2C Methods	p. 43
2.3 Approximations to Caputo Derivatives	p. 48
2.3.1 L1, L2 and L2C Methods	p. 49
2.3.2 Approximations Based on Polynomial Interpolation	p. 49
2.3.3 High-Order Methods	p. 52
2.4 Approximation to Riesz Derivatives	p. 55
2.4.1 High-Order Algorithms (I)	p. 55
2.4.2 High-Order Algorithms (II)	p. 67
2.4.3 High-Order Algorithms (III)	p. 71
2.4.4 Numerical Examples	p. 86
2.5 Matrix Approach	p. 91
2.6 Short Memory Principle	p. 92
2.7 Other Approaches	p. 94
3 Numerical Methods for Fractional Ordinary Differential Equations	p. 97
3.1 Introduction	p. 97
3.2 Direct Methods	p. 98
3.3 Integration Methods	p. 100
3.3.1 Numerical Examples	p. 109
3.4 Fractional Linear Multistep Methods	p. 110
4 Finite Difference Methods for Fractional Partial Differential Equations	p. 125
4.1 Introduction	p. 125
4.2 One-Dimensional Time-Fractional Equations	p. 125
4.2.1 Riemann-Liouville Type Subdiffusion Equations	p. 127
4.2.1.1 Explicit Euler Type Methods	p. 127
4.2.1.2 Implicit Euler Type Methods	p. 130
4.24.3 Crank-Nicolson Type Methods	p. 136
4.24.4 Integration Methods	p. 142
4.2.1.5 Numerical Examples	p. 144
4.2.2 Caputo Type Subdiffusion Equations	p. 146
4.2.2.1 Explicit Euler Type Methods	p. 147
4.2.2.2 Implicit Euler Type Methods	p. 150
4.2.2.3 FLMM Finite Difference Methods	p. 153
4.2.2.4 Numerical Examples	p. 157
4.3 One-Dimensional Space-Fractional Differential Equations	p. 159
4.3.1 One-Sided Space-Fractional Diffusion Equation	p. 159
4.3.2 Two-Sided Space-Fractional Diffusion Equation	p. 168
4.3.3 Riesz Space-Fractional Diffusion Equation	p. 170
4.3.4 Numerical Examples	p. 172
4.4 One-Dimensional Time-Space Fractional Differential Equations	p. 174
4.4.1 Time-Space Fractional Diffusion Equation with Caputo Derivative in Time	p. 174
4.4.2 Time-Space Fractional Diffusion Equation with Riemann-Liouville Derivative in Time	p. 179
4.4.3 Numerical Examples	p. 181
4.5 Fractional Differential Equations in Two Space Dimensions	p. 183
4.5.1 Time-Fractional Diffusion Equation with Riemann-Liouville Derivative in Time	p. 185
4.5.2 Time-Fractional Diffusion Equation with Caputo Derivative in Time	p. 198
4.5.3 Space-Fractional Diffusion Equation	p. 204
4.5.4 Time-Space Fractional Diffusion Equation with Caputo Derivative in Time	p. 208
4.5.5 Time-Space Fractional Diffusion Equation with Riemann-Liouville Derivative in Time	p. 212
4.5.6 Numerical Examples	p. 214
5 Galerkin Finite Element Methods for Fractional Partial Differential Equations	p. 219
5.1 Mathematical Preliminaries	p. 219
5.2 Galerkin FEM for Stationary Fractional Advection Dispersion Equation	p. 224
5.2.1 Notations and Polynomial Approximation	p. 225
5.2.2 Variational Formulation	p. 226
5.2.3 Finite Element Solution and Error Estimates	p. 229
5.3 Galerkin FEM for Space-Fractional Diffusion Equation	p. 230
5.3.1 Semi-Discrete Approximation	p. 230
5.3.2 Fully Discrete Approximation	p. 234
5.4 Galerkin FEM for Time-Fractional Differential Equations	p. 238
5.4.1 Semi-Discrete Schemes	p. 238
5.4.2 Fully Discrete Schemes	p. 241
5.4.3 Numerical Examples	p. 248
5.5 Galerkin FEM for Time-Space Fractional Differential Equations	p. 251
5.5.1 Semi-Discrete Approximations	p. 253
5.5.2 Fully Discrete Schemes	p. 256
Bibliography	p. 265
Index	p. 279

Mevcut:*

Bound With These Titles

On Order

Özet

Özet

Author Notes

Table of Contents