Fast numerical methods for space-fractional partial differential equations
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Hong Wang
Abstract
Because of the non-local nature of fractional differential operators, numerical methods for space-fractional partial differential equations (sFPDEs) often generate dense or full stiffness matrices with complex structures. The scenario is complicated further by the fact that linear elliptic and parabolic FPDEs with smooth data defined in smooth domains may generate solutions with boundary layers. Consequently, the numerical simulations of sFPDEs have significantly increased computational complexity and memory requirements, compared to their integer-order analogs. In this chapter we address the computational issues of sFPDEs, outline some of the recent developments of fast and accurate numerical methods for sFPDEs, and briefly discuss possible future directions in the field.
Abstract
Because of the non-local nature of fractional differential operators, numerical methods for space-fractional partial differential equations (sFPDEs) often generate dense or full stiffness matrices with complex structures. The scenario is complicated further by the fact that linear elliptic and parabolic FPDEs with smooth data defined in smooth domains may generate solutions with boundary layers. Consequently, the numerical simulations of sFPDEs have significantly increased computational complexity and memory requirements, compared to their integer-order analogs. In this chapter we address the computational issues of sFPDEs, outline some of the recent developments of fast and accurate numerical methods for sFPDEs, and briefly discuss possible future directions in the field.
Kapitel in diesem Buch
- 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
Kapitel in diesem Buch
- 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
