Numerical methods for time-space fractional partial differential equations
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Fawang Liu
Abstract
In this chapter, numerical methods for time-space fractional partial differential equations are presented. Firstly, some preliminary knowledge for fractional operators is introduced. Secondly, finite difference methods for the one-dimensional time-space fractional advection-dispersion equation and time-space Caputo-Reisz fractional diffusion equation in two dimensions are considered, respectively. Thirdly, an unstructured mesh finite element method for the two-dimensional time-space Riesz fractional diffusion equation on an irregular convex domain is presented. Finally, a two-dimensional time-space fractional diffusion equation based on the fractional Laplacian operator is investigated.
Abstract
In this chapter, numerical methods for time-space fractional partial differential equations are presented. Firstly, some preliminary knowledge for fractional operators is introduced. Secondly, finite difference methods for the one-dimensional time-space fractional advection-dispersion equation and time-space Caputo-Reisz fractional diffusion equation in two dimensions are considered, respectively. Thirdly, an unstructured mesh finite element method for the two-dimensional time-space Riesz fractional diffusion equation on an irregular convex domain is presented. Finally, a two-dimensional time-space fractional diffusion equation based on the fractional Laplacian operator is investigated.
Chapters in this book
- Frontmatter I
- Preface V
- Contents IX
- Fundamental approaches for the numerical handling of fractional operators and time-fractional differential equations 1
- Time-fractional derivatives 23
- High-order finite difference methods for fractional partial differential equations 49
- Spectral methods for some kinds of fractional differential equations 101
- Spectral methods for fractional differential equations using generalized Jacobi functions 127
- Spectral and spectral element methods for fractional advection–diffusion–reaction equations 157
- Discontinuous Galerkin and finite element methods 185
- Numerical methods for time-space fractional partial differential equations 209
- Comparison of two radial basis collocation methods for Poisson problems with fractional Laplacian 249
- Particle tracking solutions of vector fractional differential equations: A review 275
- Singularities 287
- Fast numerical methods for space-fractional partial differential equations 307
- Fast methods for the computation of the Mittag-Leffler function 329
- Index 347
Chapters in this book
- Frontmatter I
- Preface V
- Contents IX
- Fundamental approaches for the numerical handling of fractional operators and time-fractional differential equations 1
- Time-fractional derivatives 23
- High-order finite difference methods for fractional partial differential equations 49
- Spectral methods for some kinds of fractional differential equations 101
- Spectral methods for fractional differential equations using generalized Jacobi functions 127
- Spectral and spectral element methods for fractional advection–diffusion–reaction equations 157
- Discontinuous Galerkin and finite element methods 185
- Numerical methods for time-space fractional partial differential equations 209
- Comparison of two radial basis collocation methods for Poisson problems with fractional Laplacian 249
- Particle tracking solutions of vector fractional differential equations: A review 275
- Singularities 287
- Fast numerical methods for space-fractional partial differential equations 307
- Fast methods for the computation of the Mittag-Leffler function 329
- Index 347
