Solving a generalized fractional diffusion equation with variable fractional order and moving boundary by two numerical methods: FDM vs FVM

Abstract

In this paper, both the finite difference method (FDM) and the finite volume method (FVM) are employed to solve the fractional partial differential diffusion equation with temporal dimension and one spatial dimension. In this case, the boundary on the right of the domain is moving with time, while the variable fractional order is depicted as a function of both time and space. Special technique has been proposed to deal with the moving boundary which not only involves the computational difficulty and also accumulates the error. The accuracy and computational resource consumption of the two methods are compared in four designed cases with different functions of moving boundaries and fractional orders. The results show that the computation cost of FDM and FVM is almost the same in problems with one-dimensional space, but the accuracy of the FDM is higher than that of the FVM. Besides, compared with linear cases, the computational accuracy of both methods decreases significantly with nonlinear functions of fractional derivative and moving boundary.