An efficient computational approach and its analysis for the Caputo time‐fractional convection reaction–diffusion equation in two‐dimensional space

This work concerns with a novel computational approach to solve a two‐dimensional Caputo time‐fractional convection reaction–diffusion (CTFCRD) equation with the weak singularity at the initial time. In this method, the ‐scheme on graded mesh is used to approximate the Caputo temporal derivative, while the spatial derivatives are approximated by using a compact finite difference method (CFDM) coupled with alternating direction implicit (ADI) scheme. We present a methodology to examine the optimal error estimates of the proposed scheme, in terms of ‐norm. Convergence and stability results are proved. It is shown that the suggested method yields an optimal order, that is, min(2‐ , , 2 +1) in time direction, where is the order of time fractional derivative and is the grading parameter and has a fourth‐order of convergence in space direction. Finally, two numerical examples are provided to verify the theoretical estimates and to illustrate the accuracy and efficiency of the method.