A numerical algorithm with parallel implementation for variable-order fractional mobile/immobile equation

The numerical solution of the variable-order time fractional mobile/immobile (VO-TFMIE) equation is computationally challenging due to the nonlocal property of fractional operators. In this paper, an efficient numerical scheme, that is explicit decoupled group (EDG) method is proposed for solving the VO-TFMIE in two space dimensions. In this method, the variable-order temporal derivative is approximated by means of the L1 scheme, while the spatial derivatives are discretized by means of a skewed finite difference scheme. A Crank-Nicolson (C-N) difference scheme is also developed for comparison purposes. Stability and convergence results of the suggested methods are discussed. Several numerical simulations are conducted to solve the problem under consideration. The results validate the accuracy of the suggested methods, and the computational superiority of the EDG method over the C-N difference scheme. To enhance the computational efficiency furthermore, parallel implementation of the EDG method is also considered in this work. Lastly, the effectiveness and efficiency of the parallel EDG (PEDG) method in solving the considered problem is confirmed through numerical experiments. The obtained numerical results show the pivotal role of parallel computing in diminishing the computational limitations associated with the simulation of multi-dimensional variable-order fractional differential equations.