Singularly perturbed time-fractional convection–diffusion equations via exponential fitted operator scheme

Abstract

In this paper, we proposed an accurate $ϵ$-uniformly convergent numerical method to solve singularly perturbed time-fractional convection–diffusion equations via exponential fitted operator scheme. The time-fractional derivative is defined in the sense of Caputo with order $\eta \in (0,1)$. The time-fractional derivative is discretized by employing the Crank–Nicolson method on a uniform mesh, and an exponential fitted operator scheme along with the standard upwind method is used to mesh-grid the space domain. The truncation error and uniform stability of the discretized problems are examined in order to prove the parameter uniform convergence of the proposed scheme. It is demonstrated that the scheme is $ϵ$-uniformly convergent of order $O({\left(\Delta t\right)}^{2-\eta }+\Delta x),$ where $\Delta t$ and $\Delta x$ represent the step sizes of the time and space domains, respectively. Two numerical examples are provided in order to assess the accuracy of the suggested scheme and validate the theoretical concepts discussed. To demonstrate the efficiency of the numerical scheme presented, comparisons have been made with the numerical solution obtained by the finite difference method that exists in the literature. Consequently, it is observed that the results obtained by the present scheme are more accurate and have a better convergence rate.