A numerical method based on finite difference for boundary value problems for singularly perturbed delay differential equations

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Abstract

A boundary value problem for second order singularly perturbed delay differential equation is considered. When the delay argument is sufficiently small, to tackle the delay term, the researchers [M.K. Kadalbajoo, K.K. Sharma, Numerical analysis of singularly perturbed delay differential equations with layer behavior, Appl. Math. Comput. 157 (2004) 11–28, R.E. O’Malley, Jr., Singular Perturbation Methods for Ordinary Differential Equations, Springer-Verlag, New York, 1991] used Taylor’s series expansion and presented an asymptotic as well as numerical approach to solve such type boundary value problem. But the existing methods in the literature fail in the case when the delay argument is bigger one because in this case, the use of Taylor’s series expansion for the term containing delay may lead to a bad approximation.

In this paper to short out this problem, we present a numerical scheme for solving such type of boundary value problems, which works nicely in both the cases, i.e., when delay argument is bigger one as well as smaller one. To handle the delay argument, we construct a special type of mesh so that the term containing delay lies on nodal points after discretization. The proposed method is analyzed for stability and convergence. To demonstrate the efficiency of the method and how the size of the delay argument and the coefficient of the delay term affects the layer behavior of the solution several test examples are considered.

Introduction

In this paper, we extend the numerical study of boundary value problems for singularly perturbed delay differential equations of the convection–diffusion type with delay in the convection term which was initiated in [4]. A singularly perturbed delay differential equation is an ordinary differential equation in which the highest derivative is multiplied by a small parameter and involving at least one delay term. In the past, less attention had been paid for the numerical solution of singularly perturbed delay differential equations. But in recent years, there has been a growing interest in the numerical treatment of such differential equations. This is due to the versatility of such types of differential equations in the mathematical modeling of processes in various application fields, for e.g., the first exit time problem in the modeling of the activation of neuronal variability [5], in the study of bistable devices [1], evolutionary biology [10], in a variety of models for physiological processes or diseases [8], [10], to describe the human pupil-light reflex [7] and variational problems in control theory [3] where they provide the best and in many cases the only realistic simulation of the observed phenomena. For the numerical treatment for first order singularly perturbed delay differential equations, one can see the thesis by Tian [9]. Lange and Miura [5] gave an asymptotic approach to solve boundary value problems for the second order singularly perturbed differential difference equation with small shifts [5], [6].

To approximate the solution of such boundary value problems, we consider the two cases on the basis of size of the delay (i) when the delay is of small order of the singular perturbation parameter and (ii) when the delay is of capital order of the singular perturbation parameter. In the first case, the numerical schemes proposed in [4] work nicely, but they fail in the second case, i.e., when the delay is of capital order of the singular perturbation parameter. This happens because there, we use Taylor’s series to approximate the term containing the delay which is valid provided the delay is of o(ε) but may lead to a bad approximation in the case when the delay is of O(ε). Here, we propose a generic numerical approach to solve the boundary value problem for singularly perturbed delay differential equations which works nicely in both the cases, i.e., whether the delay is of O(ε) or of o(ε). The numerical scheme comprises a standard upwind finite difference scheme on a special type of mesh to tackle the delay argument. In this paper, we also relax the restriction on the coefficient of the reaction term which was imposed in [4]. The stability and error analysis for the proposed scheme is given in both the cases, when the sign of the coefficient of the reaction term is negative or positive. An extensive amount of computation work has been carried out to demonstrate the method and to show the effect of delay on the boundary layer behavior (which is exhibited due to the presence of the singular perturbation parameter) of the solution of the problem.

Section snippets

Description of the problem

Consider a model problem for the boundary value problems for singularly perturbed delay differential equations with delay in the convection term [5], [6]εy(x)+a(x)y(x-δ)+b(x)y(x)=f(x)on 0<x<1, 0<ε1, subject to the interval and boundary conditionsy(x)=ϕ(x),-δx0,y(1)=γ,where a(x), b(x), f(x) and ϕ(x) are smooth functions, γ is a constant and δ is the delay. For a function y(x) be a smooth solution to the problem (2.1), (2.2a), (2.2b), it must satisfy the boundary value problem (2.1), (2.2a),

Layer on the left side

Here, we consider the case when the solution of the boundary value problem (2.1), (2.2a), (2.2b) exhibits boundary layer behavior on the left side of the interval [0,1], i.e., we consider the case when a(x)M>0 x[0,1], M being a positive constant.

Layer on the right side

Here, we consider the case when the solution of the boundary value problem (2.1), (2.2a), (2.2b) exhibits layer behavior on the right side of the interval [0,1], i.e., we consider the case when a(x)-M<0, M being a positive constant and construct a numerical scheme to solve the problem (2.1), (2.2a), (2.2b).

Numerical results

For δ=0, the solution to the boundary value problem (2.1), (2.2a), (2.2b) exhibits layer behavior at one of the boundary ends. To demonstrate the effect of delay on the layer behavior of the solution and the efficiency of the method, we consider the examples given below and solve them using the present method. We have plotted the graphs of the solution of the problem for ε=0.01 with different values of δ to show the effect of delay on the boundary layer solution.

The maximum absolute error for

Discussion

A boundary value problem for linear second order singularly perturbed delay differential equations with delay in the convection term is considered. To obtain an approximate solution for such types of boundary value problems, a generic numerical approach based on finite difference is presented. A numerical study of boundary value problems for second order singularly perturbed differential difference equations of the convection–diffusion type with delay initiated in paper [4] wherein the authors

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