An α-Robust Galerkin Spectral Method for the Nonlinear Distributed-Order Time-Fractional Diffusion Equations with Initial Singularity

In this paper, we numerically solve the nonlinear time-fractional diffusion equation of distributed order on an unbounded domain with a weak singularity. A fully discrete implicit scheme is developed based on the L1 formula on graded meshes in time and the Galerkin spectral method using the Laguerre function in space. We obtained an α-robust discrete Gronwall inequality and the a priori error estimation of the numerical solution. Then, the existence and uniqueness of the numerical solution are discussed. Next, we present the α-robust stability and convergence of the fully discrete scheme, where the convergence was obtained based on the regularity conditions of the exact solution. A numerical example demonstrates the validity of the theoretical results.