A high-order and unconditionally stable scheme for the modified anomalous fractional sub-diffusion equation with a nonlinear source term

The aim of this paper is to study the high order difference scheme for the solution of modified anomalous fractional sub-diffusion equation. The time fractional derivative is described in the Riemann–Liouville sense. In the proposed scheme we discretize the space derivative with a fourth-order compact scheme and use the Grünwald–Letnikov discretization of the Riemann–Liouville derivative to obtain a fully discrete implicit scheme. We analyze the solvability, stability and convergence of the proposed scheme using the Fourier method. The convergence order of method is O ( τ + h 4 ) . Numerical examples demonstrate the theoretical results and high accuracy of the proposed scheme.