Crank–Nicolson method for solving time-fractional singularly perturbed delay partial differential equations

Abstract

In this paper, we propose a uniformly convergent numerical scheme for a class of time-fractional singularly perturbed delay partial differential equations exhibiting a right regular boundary layer. The time-fractional derivative is considered in the Caputo sense with order β∈(0,1). The domain is discretized with a uniform mesh in both time and space directions. The numerical scheme comprises the discretization technique given by the Crank–Nicolson method in the temporal direction and the non-standard finite difference method in the spatial direction. To show the parameter uniform convergence of the proposed method, the boundedness of truncation error and stability analysis are performed. It is shown that the proposed scheme is second-order convergent in the temporal direction and first-order convergent in the spatial direction. Numerical examples are presented, and their numerical simulations confirm the theoretical findings.