A high-order scheme for time-space fractional diffusion equations with Caputo-Riesz derivatives. (English) Zbl 1524.65392
Summary: In this paper, we present a high-order approach for solving one- and two-dimensional time-space fractional diffusion equations (FDEs) with Caputo-Riesz derivatives. To design the scheme, the Caputo temporal derivative is approximated using a high-order method, and the spatial Riesz derivative is discretized by the second-order weighted and shifted Grünwald difference (WSGD) method. It is proved that the scheme is unconditionally stable and convergent with the order of \(O(\tau^\alpha h^2+\tau^4)\), where \(\tau\) and \(h\) are time and space step sizes, respectively. We illustrate the accuracy and effectiveness of the method by providing several numerical examples.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 35R11 | Fractional partial differential equations |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 26A33 | Fractional derivatives and integrals |
| 65N06 | Finite difference methods for boundary value problems involving PDEs |
Keywords:
Caputo derivative; fractional diffusion equation; Riesz derivative; fractional kinetic equation; weighted and shifted Grünwald difference methodReferences:
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