High-order finite difference methods for fractional partial differential equations
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Hengfei Ding
Abstract
More than six kinds of fractional derivatives have been described. Normally, the time-fractional derivatives appear in the Caputo or Riemann-Liouville sense. As for the space-fractional derivative, it is commonly defined as an operator inverse to the Riesz potential and referred to as the Riesz derivative. Here, we mainly focus on introducing numerical approximations for these fractional derivatives. Meanwhile, we give applications of these numerical approximation formulas in fractional partial differential equations.
Abstract
More than six kinds of fractional derivatives have been described. Normally, the time-fractional derivatives appear in the Caputo or Riemann-Liouville sense. As for the space-fractional derivative, it is commonly defined as an operator inverse to the Riesz potential and referred to as the Riesz derivative. Here, we mainly focus on introducing numerical approximations for these fractional derivatives. Meanwhile, we give applications of these numerical approximation formulas in fractional partial differential equations.
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
