Comparison of two radial basis collocation methods for Poisson problems with fractional Laplacian
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Guofei Pang
Abstract
Research on meshless methods for partial differential equations with fractional Laplacian, especially in high spatial dimensions, is rare. This chapter compares two existing radial basis collocation methods on 1D, 2D, and 3D fractional Poisson problems with zero non-local boundary condition. Benchmark solutions over unit balls in 1D, 2D, and 3D spaces are employed to validate the methods. The two methods are compared in terms of solution accuracy, computational cost, and flexibility. Numerical results show that the two methods have comparable solution accuracy and same-order time complexity, but varied flexibilities. Additionally, the influences of fractional order on the solution accuracy differ from each other.
Abstract
Research on meshless methods for partial differential equations with fractional Laplacian, especially in high spatial dimensions, is rare. This chapter compares two existing radial basis collocation methods on 1D, 2D, and 3D fractional Poisson problems with zero non-local boundary condition. Benchmark solutions over unit balls in 1D, 2D, and 3D spaces are employed to validate the methods. The two methods are compared in terms of solution accuracy, computational cost, and flexibility. Numerical results show that the two methods have comparable solution accuracy and same-order time complexity, but varied flexibilities. Additionally, the influences of fractional order on the solution accuracy differ from each other.
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
