Spectral and spectral element methods for fractional advection–diffusion–reaction equations
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Anna Lischke
Abstract
We review recent advances in spectral and spectral element methods for a class of fractional partial differential equations. We focus on linear advection- diffusion-reaction equations in one and two dimensions. In particular, we discuss how to deal with boundary and interior singularity of solutions on finite intervals, on the half line, and on two-dimensional domains. Regularity theory for equations with fractional Laplacian is also presented. Finally, we present two approaches to discretize equations with space-time fractional derivatives. Numerical results are presented for both one- and two-dimensional problems.
Abstract
We review recent advances in spectral and spectral element methods for a class of fractional partial differential equations. We focus on linear advection- diffusion-reaction equations in one and two dimensions. In particular, we discuss how to deal with boundary and interior singularity of solutions on finite intervals, on the half line, and on two-dimensional domains. Regularity theory for equations with fractional Laplacian is also presented. Finally, we present two approaches to discretize equations with space-time fractional derivatives. Numerical results are presented for both one- and two-dimensional problems.
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
