Time-fractional derivatives
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Xuan Zhao
Abstract
The definitions of the fractional integral, Grünwald-Letnikov fractional derivative, Riemann-Liouville fractional derivative, and Caputo fractional derivative are presented. The numerical approximations for the Riemann-Liouville fractional derivative based on the shifted Grünwald-Letnikov formula are provided. The L1 interpolation approximation and L2-1σ interpolation approximation for the Caputo fractional derivative are given. The finite difference methods based on Grünwald-Letnikov formula, L1 formula and L2-1σ formula for the fractional ordinary equation are derived. Four finite difference schemes based on the first-order Grünwald-Letnikov formula, the second-order shifted Grünwald-Letnikov formula, the L1 formula and the L2-1σ formula are constructed for the time-fractional subdiffusion equations. Two difference schemes by using the L1 formula and the L2-1σ formula are developed for the time-fractional diffusion-wave equations. For each scheme, the convergence result is given.
Abstract
The definitions of the fractional integral, Grünwald-Letnikov fractional derivative, Riemann-Liouville fractional derivative, and Caputo fractional derivative are presented. The numerical approximations for the Riemann-Liouville fractional derivative based on the shifted Grünwald-Letnikov formula are provided. The L1 interpolation approximation and L2-1σ interpolation approximation for the Caputo fractional derivative are given. The finite difference methods based on Grünwald-Letnikov formula, L1 formula and L2-1σ formula for the fractional ordinary equation are derived. Four finite difference schemes based on the first-order Grünwald-Letnikov formula, the second-order shifted Grünwald-Letnikov formula, the L1 formula and the L2-1σ formula are constructed for the time-fractional subdiffusion equations. Two difference schemes by using the L1 formula and the L2-1σ formula are developed for the time-fractional diffusion-wave equations. For each scheme, the convergence result is given.
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
