A fully discrete scheme based on cubic splines and its analysis for time-fractional reaction–diffusion equations exhibiting weak initial singularity

Abstract

The aim of this paper is to design and analyze a robust fully discrete scheme based on cubic splines for numerically solving a time-fractional reaction–diffusion equation (TFRDE) with smooth and non-smooth solutions. The solution of this problem exhibits a weak singularity near time t=0. The method combines the L1 scheme on a uniform or graded mesh to discretize in time and a cubic spline difference scheme on a uniform mesh to discretize in space. Further, the stability and convergence for both the smooth and non-smooth solutions are analyzed separately. Moreover, due to the use of a cubic spline difference scheme to discretize the space variable, the analysis of the fully discrete scheme requires some innovative ideas. In the end, two test problems are considered to demonstrate the validity and authenticity of the proposed method. The first example deals with the smooth solution, whereas in the second example, we consider a non-smooth solution.