Singularities
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Martin Stynes
Abstract
First, it is demonstrated by simple examples that singularities are commonplace in the solutions of FD problems on bounded domains. Then for more general problems, bounds on derivatives of the solutions are presented to illustrate the exact nature of these singularities. It is shown that assuming more regularity than is generally true will restrict severely the class of problems under study. For numerical methods, singularities will often reduce the rate of convergence. Ways of addressing this deficiency are described - there are four main classes of methods designed specifically to handle singularities in the solutions of FD problems - and many references to the recent numerical research literature are given.
Abstract
First, it is demonstrated by simple examples that singularities are commonplace in the solutions of FD problems on bounded domains. Then for more general problems, bounds on derivatives of the solutions are presented to illustrate the exact nature of these singularities. It is shown that assuming more regularity than is generally true will restrict severely the class of problems under study. For numerical methods, singularities will often reduce the rate of convergence. Ways of addressing this deficiency are described - there are four main classes of methods designed specifically to handle singularities in the solutions of FD problems - and many references to the recent numerical research literature are given.
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
