A Second-Order Crank-Nicolson-Type Scheme for Nonlinear Space–Time Reaction–Diffusion Equations on Time-Graded Meshes

Abstract

A weak singularity in the solution of time-fractional differential equations can degrade the accuracy of numerical methods when employing a uniform mesh, especially with schemes involving the Caputo derivative (order α,), where time accuracy is of the order (2−α) or (1+α). To deal with this problem, we present a second-order numerical scheme for nonlinear time–space fractional reaction–diffusion equations. For spatial resolution, we employ a matrix transfer technique. Using graded meshes in time, we improve the convergence rate of the algorithm. Furthermore, some sharp error estimates that give an optimal second-order rate of convergence are presented and proven. We discuss the stability properties of the numerical scheme and elaborate on several empirical examples that corroborate our theoretical observations.