Error analysis of a finite difference scheme on a modified graded mesh for a nonlocal-in-time parabolic equation

In this paper, a novel numerical method is developed to solve a parabolic diffusion equation with a nonlocal-in-time operator. First, drawing upon the mesh equidistribution principle, a modified graded mesh along with its properties is introduced. Additionally, the spatial derivative is discretized using the standard central difference scheme on a uniform mesh, whereas the time nonlocal operator is approximated by employing a finite difference scheme on the aforementioned modified graded mesh. Furthermore, the stability and convergence of our proposed scheme are demonstrated. Finally, the theoretical findings are corroborated through a series of numerical experiments.