Parameter-Uniform Convergent Numerical Approach for Time-Fractional Singularly Perturbed Partial Differential Equations With Large Time Delay

Abstract

In this study, we consider a parameter-uniform convergent numerical approach for a class of time-fractional singularly perturbed partial differential equations (TF-SPDPDEs) with large delay in time that exhibits a regular exponential boundary layer on the right side of the spatial domain. An arbitrary very small parameter ε(0 < ε < <1) multiplies the highest-order derivative term of these singularly perturbed problems. The time-fractional derivative is considered in the Caputo sense with order α ∈ (0, 1). The numerical scheme comprises the L1 scheme and nonstandard finite difference method (FDM) for discretizing the time and space variables, respectively, on a uniform mesh. To show the parameter uniform convergence of the proposed method, the truncation error and stability analysis are discussed. The method is shown to be parameter-uniform convergent of order O((Δt)^2−α + Δx), where Δt and Δx are the step sizes in the time and space directions, respectively. In order to confirm the theoretical predictions, two numerical examples are presented, and the numerical results support the theoretical concepts discussed. Finally, to show the advantage of the proposed scheme, we made comparisons with the existing numerical methods in the literature, and the numerical results reveal that the present scheme is more accurate.