An efficient computational technique for solving a time-fractional reaction-subdiffusion model in 2D space

The solution of time fractional reaction-subdiffusion (TFRS) equation with the Caputo time fractional derivative of order alpha is an element of(0, 1), in general, exhibits a mild singularity at the initial time. In this article, we propose an efficient numerical technique based on a graded mesh in time to overcome the singular behavior of the solution at t = 0. In this technique, the temporal fractional derivative is approximated by means of the L-1 scheme on nonuniform mesh, while the spatial derivative is approximated by means of a high-order alternating direction implicit (ADI) compact finite difference scheme. Convergence and stability analysis of the suggested method are investigated. To demonstrate the applicability and accuracy of the method, we employ the resulting fully-discrete numerical scheme to solve some numerical examples. The results confirm that the rate of convergence of the proposed method is min(2 - alpha, nu alpha, 2 alpha + 1) in time, where nu is a grading exponent used for generating graded mesh points, while it is four in space. The theoretical result is supported by the numerical one. The results obtained by the proposed technique on graded time steps are compared with those obtained by the method on uniform time steps.