A fourth-order compact solution of the two-dimensional modified anomalous fractional sub-diffusion equation with a nonlinear source term

This work is concerned to the study of high order difference scheme for the solution of a two-dimensional modified anomalous sub-diffusion equation with a nonlinear source term which describes processes that become less anomalous as time progresses. The space fractional derivatives are described in the Riemann–Liouville sense. In the proposed scheme we discretize the space derivatives with a fourth-order compact scheme and use the Grünwald–Letnikov discretization of the Riemann–Liouville derivatives to obtain a fully discrete implicit scheme. We prove the stability and convergence of proposed scheme using the Fourier analysis. The convergence order of the proposed method is O(τ+hx4+hy4). Comparison of numerical results with analytical solutions demonstrates the unconditional stability and high accuracy of proposed scheme.