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E-raamat: Numerical Methods for Fractional Calculus

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(Shanghai University, China), (Shanghai University, China)
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Numerical Methods for Fractional CalculusSuurem pilt
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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.

Arvustused

"The book provides a survey of many different methods for the numerical computation of RiemannLiouville integrals of fractional order and of fractional derivatives of RiemannLiouville, Caputo, and Weyl type. Algorithms for the solution of associated ordinary differential equations and certain special classes of partial differential equations are presented as well." Zentralblatt MATH 1326

Foreword xi
Preface xiii
List of Figures
xv
List of Tables
xvii
1 Introduction to Fractional Calculus
1(28)
1.1 Fractional Integrals and Derivatives
1(9)
1.2 Some Other Properties of Fractional Derivatives
10(8)
1.2.1 Leibniz Rule for Fractional Derivatives
10(1)
1.2.2 Fractional Derivative of a Composite Function
11(1)
1.2.3 Behaviors Near and Far from the Lower Terminal
12(2)
1.2.4 Laplace Transforms of Fractional Derivatives
14(2)
1.2.5 Fourier Transforms of Fractional Derivatives
16(2)
1.3 Some Other Fractional Derivatives and Extensions
18(5)
1.3.1 Marchaud Fractional Derivative
18(1)
1.3.2 The Finite Parts of Integrals
19(1)
1.3.3 Directional Integrals and Derivatives in R2
20(1)
1.3.4 Partial Fractional Derivatives
21(2)
1.4 Physical Meanings
23(2)
1.5 Fractional Initial and Boundary Problems
25(4)
2 Numerical Methods for Fractional Integral and Derivatives
29(68)
2.1 Approximations to Fractional Integrals
29(11)
2.1.1 Numerical Methods Based on Polynomial Interpolation
30(4)
2.1.2 High-Order Methods Based on Gauss Interpolation
34(4)
2.1.3 Fractional Linear Multistep Methods
38(2)
2.2 Approximations to Riemann--Liouville Derivatives
40(8)
2.2.1 Grunwald--Letnikov Type Approximation
41(2)
2.2.2 L1, L2 and L2C Methods
43(5)
2.3 Approximations to Caputo Derivatives
48(7)
2.3.1 L1, L2 and L2C Methods
49(1)
2.3.2 Approximations Based on Polynomial Interpolation
49(3)
2.3.3 High-Order Methods
52(3)
2.4 Approximation to Riesz Derivatives
55(36)
2.4.1 High-Order Algorithms (I)
55(12)
2.4.2 High-Order Algorithms (II)
67(4)
2.4.3 High-Order Algorithms (III)
71(15)
2.4.4 Numerical Examples
86(5)
2.5 Matrix Approach
91(1)
2.6 Short Memory Principle
92(2)
2.7 Other Approaches
94(3)
3 Numerical Methods for Fractional Ordinary Differential Equations
97(28)
3.1 Introduction
97(1)
3.2 Direct Methods
98(2)
3.3 Integration Methods
100(10)
3.3.1 Numerical Examples
109(1)
3.4 Fractional Linear Multistep Methods
110(15)
4 Finite Difference Methods for Fractional Partial Differential Equations
125(94)
4.1 Introduction
125(1)
4.2 One-Dimensional Time-Fractional Equations
125(34)
4.2.1 Riemann--Liouville Type Subdiffusion Equations
127(1)
4.2.1.1 Explicit Euler Type Methods
127(3)
4.2.1.2 Implicit Euler Type Methods
130(6)
4.2.1.3 Crank--Nicolson Type Methods
136(6)
4.2.1.4 Integration Methods
142(2)
4.2.1.5 Numerical Examples
144(2)
4.2.2 Caputo Type Subdiffusion Equations
146(1)
4.2.2.1 Explicit Euler Type Methods
147(3)
4.2.2.2 Implicit Euler Type Methods
150(3)
4.2.2.3 FLMM Finite Difference Methods
153(4)
4.2.2.4 Numerical Examples
157(2)
4.3 One-Dimensional Space-Fractional Differential Equations
159(15)
4.3.1 One-Sided Space-Fractional Diffusion Equation
159(9)
4.3.2 Two-Sided Space-Fractional Diffusion Equation
168(2)
4.3.3 Riesz Space-Fractional Diffusion Equation
170(2)
4.3.4 Numerical Examples
172(2)
4.4 One-Dimensional Time-Space Fractional Differential Equations
174(9)
4.4.1 Time-Space Fractional Diffusion Equation with Caputo Derivative in Time
174(5)
4.4.2 Time-Space Fractional Diffusion Equation with Riemann-Liouville Derivative in Time
179(2)
4.4.3 Numerical Examples
181(2)
4.5 Fractional Differential Equations in Two Space Dimensions
183(36)
4.5.1 Time-Fractional Diffusion Equation with Riemann--Liouville Derivative in Time
185(13)
4.5.2 Time-Fractional Diffusion Equation with Caputo Derivative in Time
198(6)
4.5.3 Space-Fractional Diffusion Equation
204(4)
4.5.4 Time-Space Fractional Diffusion Equation with Caputo Derivative in Time
208(4)
4.5.5 Time-Space Fractional Diffusion Equation with Riemann-Liouville Derivative in Time
212(2)
4.5.6 Numerical Examples
214(5)
5 Galerkin Finite Element Methods for Fractional Partial Differential Equations
219(46)
5.1 Mathematical Preliminaries
219(5)
5.2 Galerkin FEM for Stationary Fractional Advection Dispersion Equation
224(6)
5.2.1 Notations and Polynomial Approximation
225(1)
5.2.2 Variational Formulation
226(3)
5.2.3 Finite Element Solution and Error Estimates
229(1)
5.3 Galerkin FEM for Space-Fractional Diffusion Equation
230(8)
5.3.1 Semi-Discrete Approximation
230(4)
5.3.2 Fully Discrete Approximation
234(4)
5.4 Galerkin FEM for Time-Fractional Differential Equations
238(13)
5.4.1 Semi-Discrete Schemes
238(3)
5.4.2 Fully Discrete Schemes
241(7)
5.4.3 Numerical Examples
248(3)
5.5 Galerkin FEM for Time-Space Fractional Differential Equations
251(14)
5.5.1 Semi-Discrete Approximations
253(3)
5.5.2 Fully Discrete Schemes
256(9)
Bibliography 265(14)
Index 279
Changpin Li is a full professor at Shanghai University. He earned his Ph.D. in computational mathematics from Shanghai University. Dr. Lis main research interests include numerical methods and computations for FPDEs and fractional dynamics. He was awarded the RiemannLiouville 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. Zengs research interests include numerical methods and computations for FPDEs.
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