A domain decomposition method for solving singularly perturbed parabolic reaction‐diffusion problems with time delay

Abstract

We design and analyse a domain decomposition method for solving singularly perturbed parabolic reaction-diffusion problems with time delay. Using the asymptotic behavior of the solution, we decompose the original domain of the problem into three overlapping subdomains, two of which are boundary layer subdomains and one is a regular subdomain. On each subdomain, we discretize the problem by the backward Euler scheme in the time direction and the central difference scheme in the spatial direction. The proposed method is shown to be uniformly convergent, having almost second order in space and first order in time. In addition, we prove that the proposed method converges much faster for small values of perturbation parameter ε. At the end, some numerical results are given in support of theoretical findings.

A Uniformly Convergent Collocation Method for Singularly Perturbed Delay Parabolic Reaction-Diffusion Problem

“…It is clear from Table 4 that extended B-spline performs good result than classical cubic B-spline. Figure 1 depicts the numerical simulation of solution profile at N = 2 7 , Δt = 0:1/2 4 , ε = 2 −12 , and λ = −0:55, which indicates parabolic boundary layers at x = 0 and x = 1.…”

“…The numerical solution of delay partial differential equation depends not only on the solution at a present stage but also at some past stages. Many researchers have proposed different numerical methods to solve singularly perturbed time delay parabolic reaction-diffusion equations; for instance, see [3][4][5][6][7][8]. Singularly perturbed two-parameter time delay parabolic problems are studied in [9] using hybrid method based on a uniform mesh in time and a layer-adapted Shishkin mesh in space.…”

Robust numerical schemes for singularly perturbed delay parabolic convection-diffusion problems with degenerate coefficient

“…Sunil and Mukesh [8] constructed a hybrid scheme consistng of HODIE type on generalized Shishkin mesh in spatial direction and implicit Euler scheme on uniform mesh in time direction for the numerical approximation of singularly perturbed parabolic delay differential reaction diffusion problems. Joginder et al [12] designed and analyzed a domain decomposition method for the numerical solution of SPPDDEs.…”

Computational study for a class of time-dependent singularly perturbed parabolic partial differential equation through tension spline

“…Singularly perturbed delay differential equations are in to the picture in different fields of science and engineering, for instance, control theory (Van Harten and Schumacher 1978), chemical processes (Adomian and Rach 1983), Problems with water quality in river networks (Samarskii and Vabishchevich 1995;Tikhonov and Samarskii 1972), semiconductor drift-diffusion model (McCartin 1985), chemical kinetics (Epstein 1992), etc. In recent years, several computational techniques have been constructed for singularly perturbed time delayed parabolic partial differential equation (PDE), for instance, fitted mesh finite difference method (Ansari et al 2007;Selvi and Ramanujam 2017), fitted operator finite difference method (Bashier and Patidar 2011), high-order parameter-uniform scheme (Kumar and Kumar 2014), upwind finite difference scheme with adaptive mesh (Gowrisankar and Natesan 2014), fitted Numerov method (Rao and Chakravarthy 2014), an overlapping Schwarz domain decomposition (Singh et al 2018).…”