A fast difference scheme on a graded mesh for time-fractional and space distributed-order diffusion equation with nonsmooth data

The time-fractional and space distributed-order diffusion equation with a singularity at the initial time is investigated in this paper. For the solution of this equation, we propose a new difference scheme on a graded mesh. The theoretical underpinning for the suggested procedure is presented, which includes the stability and convergence of the difference scheme. Furthermore, the difference strategy proves to be unconditionally stable in this research. Additionally, we demonstrate that the difference technique is convergent and that the temporal convergence rate is faster than when a uniform mesh uses. Finally, a test example is provided to ensure that the theoretical analysis is accurate.