An accurate numerical method for solving the variable-order fractional diffusion problem

Authors
Farnaz Kheirkhah
Mojtaba Hajipour
Sahand University of Technology
Abstract
In this paper, a high-order numerical method is designed and implemented to solve a boundary value problem governed by the variable-order fractional diffusion equation. This equation contains a variable-order fractional time-derivative and a second-order spatial-derivative. To develop this novel method, a compact finite difference formula and a weighted shifted Grunwald-Letnikov operator are used for spatial and temporal discretization, respectively. It is shown that this method is of fourth- and second-order of convergence accuracy in spatial and time directions, respectively. Also, the solvability, stability and convergence of the peresent method are investigated. To verify the efficiency and high accuracy of this method, some numerical examples and comparative results are presented.
Keywords
Variable-order fractional derivative
fractional diffusion equation
Numerical method
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