A parameter-uniform numerical scheme for the parabolic singularly perturbed initial boundary value problems with large time delay

Numerical Methods Partial

2020

Self Cite

In this paper, a parameter-uniform numerical scheme for the solution of singularly perturbed parabolic convectiondiffusion problems with a delay in time defined on a rectangular domain is suggested. The presence of the small diffusion parameter leads to a parabolic right boundary layer. A collocation method consisting of cubic B-spline basis functions on an appropriate piecewise-uniform mesh is used to discretize the system of ordinary differential equations obtained by using Rothe's method on an equidistant mesh in the temporal direction. The parameter-uniform convergence of the method is shown by establishing the theoretical error bounds. The numerical results of the test problems validate the theoretical error bounds. KEYWORDS fitted-mesh, parameter-uniform convergence, singular perturbation, B-splines, time delay 1 INTRODUCTION Delay differential equations (DDEs) arise in the mathematical modeling of a large variety of practical phenomena, for instance, hydrodynamics of liquid helium [1], diffusion in polymers [2], thermo-elasticity [3], microscale heat transfer [4], physiological processes, diseases [5, 6], epidemiology, and the population dynamics [7, 8]. Singularly perturbed DDEs (SP-DDEs) are those in which in addition to the delay term the involved diffusion parameter, known as the perturbation parameter, is small. SP-DDEs arise in various practical phenomena, such as biological and chemical reactions, population growth [9], epidemiology [10]. A mathematical model [11] for a class of deterministic SP-PDE includes the following logistic equation (w (x, t)) t = (w (x, t)) xx + w (x, t)(1 − w (x, t −)), which arises in mathematical ecology for the evolution of a population with density w (x, t).

“…Now we give some basic results on the solution of the problem (2.1). The proof of the following maximum principle can be seen in [11].…”

Section: Problem Statementmentioning

confidence: 99%

{(w_{ϵ} (x, t))}_{t} = ϵ {(w_{ϵ} (x, t))}_{italicxx} + w_{ϵ} false(x, t false) false(1 - w_{ϵ} false(x, t - τ false) false),

which arises in mathematical ecology for the evolution of a population with density w ϵ ( x , t ).…”

Section: Introductionmentioning

confidence: 99%

A parameter‐uniform scheme for the parabolic singularly perturbed problem with a delay in time

Numerical Methods Partial

2020

Self Cite

“…Since the barycentric interpolation collocation method is not appropriate for the solution of SPPDEs for the delay, so Wang et al proposed a modified version of the barycentric interpolation collocation method. Recently, Kumar and Kumari proposed a parameter‐uniform numerical scheme comprising the Crank–Nicolson finite difference method consisting of a midpoint upwind scheme for the parabolic singularly perturbed IBVP for PDEs with large time delay. They have shown that the proposed difference scheme is second‐order accurate in time and almost first‐order accurate in the space.…”

Section: Problem Statement: Preliminariesmentioning

confidence: 99%

“…The global error is the measure of the contribution of the local error estimate at each time step and is given by E j = u ( x , t j ) − U ( x , t j ).Theorem The global error estimate at t j is given by

‖ E_{j} ‖ \leq C Δ t, j \leq T / Δ t .

Proof For the proof, the readers are referred to .…”

Section: Numerical Scheme: the Discretizationmentioning

confidence: 99%

A parameter‐uniform scheme for singularly perturbed partial differential equations with a time lag

Numerical Methods Partial

Kumari

2019

Self Cite

A numerical scheme for a class of singularly perturbed delay parabolic partial differential equations which has wide applications in the various branches of science and engineering is suggested. The solution of these problems exhibits a parabolic boundary layer on the lateral side of the rectangular domain which continuously depends on the perturbation parameter. For the small perturbation parameter, the standard numerical schemes for the solution of these problems fail to resolve the boundary layer(s) and the oscillations occur near the boundary layer. Thus, in this paper to resolve the boundary layer the extended cubic B-spline basis functions consisting of a free parameter are used on a fitted-mesh. The extended B-splines are the extension of classical B-splines. To find the best value of the optimization technique is adopted. The extended cubic B-splines are an advantage over the classical B-splines as for some optimized value of the solution obtained by the extended B-splines is better than the solution obtained by classical B-splines. The method is shown to be first-order accurate in t and almost the second-order accurate in x. It is also shown that this method is better than some existing methods. Several test problems are encountered to validate the theoretical results. KEYWORDS delay partial differential equations, extended B-splines, free parameter, parabolic boundary layers, parameter-uniform convergence, piecewise-uniform mesh, singular perturbation, time delay Numer Methods Partial Differential Eq. 2020;36:868-886. wileyonlinelibrary.com/journal/num

“…To solve SPPPDEs, Salama and AI-Amery (2017) discretized the time derivative using the Crank-Nicolson scheme and proposed a fourth-order compact scheme to solve the resulting system of ordinary differential equations. Kumar and Kumari (2019) proposed a stable numerical method comprising the Crank-Nicolson method and a midpoint upwind finite difference method on a piecewise uniform mesh. The method was shown unconditionally stable and parameter uniform convergent of order two in the temporal direction and of order one in spatial direction.…”

Section: Introductionmentioning

confidence: 99%

A new stable finite difference scheme and its convergence for time-delayed singularly perturbed parabolic PDEs

Chakravarthy

2020

Comp. Appl. Math.

In this study, we consider the time-delayed singularly perturbed parabolic PDEs (SPPPDEs). We know that the classical finite difference scheme will not produce good results for singular perturbation problems on a uniform mesh. Here, we propose a new stable finite difference (NSFD) scheme, which produces good results on a uniform mesh and also on an adaptive mesh. The NSFD scheme is constructed based on the stability of the analytical solution. Results are compared with the results available in the literature and observed that the proposed method is efficient over the existing methods for solving SPPPDEs.