Abstract
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(ταh2+τ4), where τ and h are time and space step sizes, respectively. We illustrate the accuracy and effectiveness of the method by providing several numerical examples.
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