L1 Method on Nonuniform Meshes for Linear Time-Fractional Diffusion Equations with Constant Time Delay

This paper is concerned with numerical solution of the linear time-fractional diffusion equations with constant time delay. First we investigate the existence, uniqueness and regularity of the exact solution of such equations. Focusing on the time derivative discontinuities behavior of the solutions of the equations at multiple points generated by the time delay and the Caputo fractional derivative, we propose a numerical method based on the L1 method on time nonuniform meshes and the standard finite element method in space. Moreover, the results of stability and error estimate for the method are obtained under the regularity condition. The validity of the proposed method is verified by numerical experiments.