Fast O(N) hybrid Laplace transform-finite difference method in solving 2D time fractional diffusion equation

It is time-memory consuming when numerically solving time fractional partial differential equations, as it requires O(N-2) computational cost and O(MN) memory complexity with finite difference methods, where, N and M are the total number of time steps and spatial grid points, respectively. To surmount this issue, we develop an efficient hybrid method with O(N) computational cost and O(M) memory complexity in solving two-dimensional time fractional diffusion equation. The presented method is based on the Laplace transform method and a finite difference scheme. The stability and convergence of the proposed method are analyzed rigorously by the means of the Fourier method. A comparative study drawn from numerical experiments shows that the hybrid method is accurate and reduces the computational cost, memory requirement as well as the CPU time effectively compared to a standard finite difference scheme.