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

Abstract

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).

Parameter-uniformly convergent numerical scheme for singularly perturbed delay parabolic differential equation via extended B-spline collocation

“…A fixed delay of length τ is induced by the finite speed of the controller. Another typical example of SPDDEs is the following logistic equation [8]:…”

“…In the last few years, several numerical approaches have been developed for the solution of time dependent singularly perturbed differential equations [8]. However, in the case of these equations with time delay, few numerical methods have been developed, and studies are still at an early stage [8].…”

“…In the last few years, several numerical approaches have been developed for the solution of time dependent singularly perturbed differential equations [8]. However, in the case of these equations with time delay, few numerical methods have been developed, and studies are still at an early stage [8]. Some of them, which treat time dependent singularly perturbed delay parabolic convectiondiffusion initial boundary value problems are listed here.…”

An efficient parameter uniform spline-based technique for singularly perturbed weakly coupled reaction-diffusion systems

“…This phenomenon determines the evolution of parameter-uniform numerical methods i.e., the methods in which the error constant is independent of ε and of the mesh parameter. Various ε-uniform numerical schemes such as the variational method, the finite difference methods (FDMs), the rational spectral collocation methods, the finite element methods (FEMs), the adaptive mesh methods, and the layer-adapted mesh methods have been developed in the literature for singularly perturbed boundary value problems (SPBVPs) (readers are referred to [1,[11][12][13][18][19][20]32] and the references therein). Although the Shishkin mesh is one of the simplest non-uniform meshes; it has a drawback, that is before one attempts to solve the differential equation, significant information about the exact solution must be known.…”

A Nonstandard Fitted Operator Method for Singularly Perturbed Parabolic Reaction-Diffusion Problems with a Large Time Delay

“…Kumar and Kumari [14] introduced a cubic B-spline method over tted mesh for a class of singularly perturbed delay parabolic partial di erential equations rst-order in t and second-order accurate in x. Dagnachew et al [15] designed a tted operator method for singularly perturbed parabolic convection di usion equation with small space delays, and the method have second order in both spatial and temporary variables. Kumar [16] developed a collocation method consisting of cubic B-spline basis functions over piecewise-uniform mesh for singularly perturbed parabolic convection diffusion problems with a time delay. Govindarao et al [17] presented for the solution of singularly perturbed delay parabolic reaction-diffusion initial-boundary-value problem using Shishkin mesh strategy.…”