The implicit midpoint method for the modified anomalous sub-diffusion equation with a nonlinear source term

In this paper, the implicit midpoint method is used to solve the semi-discrete modified anomalous sub-diffusion equation with a nonlinear source term, and the weighted and shifted Grünwald-Letnikov difference operator and the compact difference operator are applied to approximate the Riemann-Liouville fractional derivative and space partial derivative respectively, then the new numerical scheme is constructed. The stability and the convergence of this method are analyzed. Numerical experiment demonstrates the high accuracy of this method and confirm our theoretical results.