A fast modified $$\overline{L1}$$ finite difference method for time fractional diffusion equations with weakly singular solution

In this paper, we establish a fast modified \(\overline{L1}\) finite difference method for the time fractional diffusion equation with weakly singular solution at the initial moment. First, the time fractional derivative is approximated by the modified \(\overline{L1}\) formula on graded meshes, and the spatial derivative is approximated by the standard central difference formula on uniform meshes. Therefore, a numerical scheme for the time fractional diffusion equation is obtained. Then, the Von-Neumann stability analysis method is used to analyze the stability of the scheme, and the truncation error estimate is given. On the other hand, the time fractional derivative is nonlocal, which has historical dependency, thus, the cost of computation and memory consumption are expensive. Based on the sum-of exponentials approximation (SOE) technique, we optimize the numerical format, reduce the complex amount from \(O(M\hat{N})\) to \(O(M N_{exp})\), and the amount of computation from \(O(M\hat{N}^2)\) to \(O(M\hat{N}N_{exp})\), where M, \(\hat{N}\) and \(N_{exp}\) represent the number of spatial points, the number of temporal points, and the exponential amount, respectively. Finally, numerical examples verify the effectiveness of the scheme and theoretical analysis.