A Numerical Method on Graded Meshes for Caputo Time-Fractional Subdiffusion Equation with Singular Source Term

In this paper, we propose an efficient numerical method to solve initial-boundary value problems of Caputo fractional subdiffusion equations with a class of source terms weakly singular in time. To do this, we firstly derive fractional Volterra integro-differential equations equivalent to the given problems and establish some regularity results for solutions to them with respect to first or higher order time derivatives. These results are often called a weak regularity at the initial time of the solution. To effectively approximate the solution with the initial weak regularity, employing a temporal graded mesh, we propose a new numerical approach combined with some fractional quadrature formulas and Galerkin finite element method. Also, for our semi-discretization and fully discrete schemes, the stability and convergence analysis are discussed in details. Finally, we present a numerical example on a uniform and several graded temporal meshes to illustrate our main results.