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Advances in Difference Equations

, 2009:781579 | Cite as

An Exponentially Fitted Method for Singularly Perturbed Delay Differential Equations

  • Fevzi Erdogan
Open Access
Research Article

Abstract

This paper deals with singularly perturbed initial value problem for linear first-order delay differential equation. An exponentially fitted difference scheme is constructed in an equidistant mesh, which gives first-order uniform convergence in the discrete maximum norm. The difference scheme is shown to be uniformly convergent to the continuous solution with respect to the perturbation parameter. A numerical example is solved using the presented method and compared the computed result with exact solution of the problem.

Keywords

Perturbation Parameter Uniform Mesh Singular Perturbation Problem Shishkin Mesh Special Mesh 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

Delay differential equations play an important role in the mathematical modelling of various practical phenomena in the biosciences and control theory. Any system involving a feedback control will almost always involve time delays. These arise because a finite time is required to sense information and then react to it. 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 [1, 2, 3, 4]. Such problems arise frequently in the mathematical modelling of various practical phenomena, for example, in the modelling of several physical and biological phenomena like the optically bistable devices [5], description of the human pupil-light reflex [6], a variety of models for physiological processes or diseases and variational problems in control theory where they provide the best, and in many cases the only realistic simulation of the observed phenomena [7].

It is well known that standard discretization methods for solving singular perturbation problems are unstable and fail to give accurate results when the perturbation parameter Open image in new window is small. Therefore, it is important to develop suitable numerical methods to these problem, whose accuracy does not depend on the parameter value Open image in new window , that is, methods that are uniformly convergent with respect to the perturbation parameter [8, 9, 10]. One of the simplest ways to derive such methods consists of using an exponentially fitted difference scheme (see, e.g., [10] for motivation for this type of mesh), which are constructed a priori and depend of the parameter Open image in new window , the problem data and the number of corresponding mesh points. In the direction of numerical treatment for first-order singularly perturbed delay differential equations, several can be seen in [4, 7, 11].

In order to construct parameter-uniform numerical methods, two different techniques are applied. Firstly, the numerical methods of exponential fitting type (fitting operators) (see [9]), which have coefficients of exponential type adapted to the singular perturbation problems. Secondly, the special mesh approach (see [11, 12]), which constructs meshes adapted to the solution of the problem.

In the works of Amiraliyev and Erdogan [11], special meshes (Shishkin mesh) have been used. The method that we propose in this paper uses exponential fitting schemes, which have coefficients of exponential type.

2. Statement of the Problem

Consider a model problem for the initial value problems for singularly perturbed delay differential equations with delay in the interval Open image in new window = [0,T]:

where Open image in new window , Open image in new window , Open image in new window and Open image in new window , for Open image in new window and Open image in new window (for simplicity we suppose that Open image in new window is integer). Open image in new window is the perturbation parameter, Open image in new window , Open image in new window , Open image in new window , and Open image in new window are given sufficiently smooth functions satisfying certain regularity conditions to be specified and r is a constant delay. The solution Open image in new window displays in general boundary layers at the right side of each points Open image in new window for small values of Open image in new window .

In this paper, we present the completely exponentially fitted difference scheme on the uniform mesh. The difference scheme is constructed by the method of integral identities with the use of exponentially basis functions and interpolating quadrature rules with weight and remainder terms integral form [10]. This method of approximation has the advantage that the schemes can also be effective in the case when the continuous problem is considered under certain restrictions.

In the present paper, we analyze a fitted difference scheme on a uniform mesh for the numerical solution of the problem (2.1). In Section 2, we describe the problem. In Section 3, we state some important properties of the exact solution. In Section 4, we construct a numerical scheme for solving the initial value problem (2.1) based on an exponentially fitted difference scheme on a uniform mesh. In Section 5, we present the error analysis for approximate solution. Uniform convergence is proved in the discrete maximum norm. A numerical example in comparison with their exact solution is being presented in Section 6. The approach to construct discrete problem and error analysis for approximate solution is similar to those ones from [10, 11].

Notation

Throughout the paper, Open image in new window will denote a generic positive constant (possibly subscripted) that is independent of Open image in new window and of the mesh. Note that Open image in new window is not necessarily the same at each occurrence.

3. The Continuous Problem

Here, we show some properties of the solution of (2.1), which are needed in later sections for the analysis of appropriate numerical solution. Let, for any continuous function g, Open image in new window denotes a continuous maximum norm on the corresponding interval.

Lemma 3.1.

Let Open image in new window , Open image in new window . Then, for the solution Open image in new window of the problem (2.1) the following estimates hold

Proof.

see [11].

4. Discretization and Mesh

In this section, we construct a numerical scheme for solving the initial value problem (2.1) based upon an exponential fitting on a uniform mesh.

which contains Open image in new window mesh points at each subinterval Open image in new window :
and consequently
To simplify the notation, we set Open image in new window for any function Open image in new window , and moreover Open image in new window denotes an approximation of Open image in new window at Open image in new window . For any mesh function Open image in new window defined on Open image in new window we use
The approach of generating difference methods through integral identity
with the exponential basis functions
We note that function Open image in new window is the solution of the problem
The relation (4.5) is rewritten as
with the remainder term
Taking into account (4.5) and using interpolating rules with the weight (see [10]), we obtain the following relations:
and a simple calculation gives us
As a consequence of the (4.11), we propose the following difference scheme for approximation (2.1):

where Open image in new window is defined by (4.13).

5. Analysis of the Method

To investigate the convergence of the method, note that the error function Open image in new window , Open image in new window , is the solution of the discrete problem

where Open image in new window and Open image in new window are given by (4.10) and (4.13), respectively.

Lemma 5.1.

Let Open image in new window be approximate solution of (2.1). Then the following estimate holds

Proof.

The proof follows easily by induction in Open image in new window .

Lemma 5.2.

Let Open image in new window be solution of (5.1). Then following estimate holds

Proof.

It evidently follows from (5.2) by taking Open image in new window and Open image in new window .

Lemma 5.3.

Under the above assumptions of Section 2 and Lemma 3.1, for the error function Open image in new window , the following estimate holds

Proof.

To this end, it suffices to establish that the functions Open image in new window , involved in the expression for Open image in new window , admit the estimate
Using the mean value theorem, we get
and taking also into account that Open image in new window and using Lemma 3.1, we have
For Open image in new window , in view of Open image in new window and using Lemma 3.1, we obtain
and after replacing Open image in new window this reduces to
which yields

The same estimate is obtained for Open image in new window in the similar manner as above.

Combining the previous lemmas we get the following final estimate, that is, uniformly convergent estimate.

Theorem 5.4.

Let Open image in new window be the solution of (2.1) and Open image in new window be the solution of (4.14). Then the following estimate holds

6. Numerical Results

We begin with an example from Driver [2] for which we possess the exact solution.
The exact solution for Open image in new window is given by
We define the computed parameter-uniform maximum error Open image in new window as follows:
where Open image in new window is the numerical approximation to Open image in new window for various values of Open image in new window . We also define the computed parameter-uniform convergence rates for each Open image in new window :

The values of Open image in new window for which we solve the test problem are Open image in new window .

These convergence rates are increasing as Open image in new window increases for any fixed Open image in new window . Tables 1 and 2 thus verify the Open image in new window -uniform convergence of the numerical solutions and the computed rates are in agreement with our theoretical analysis.
Table 1

Maximum errors Open image in new window and convergence rates Open image in new window on Open image in new window

Open image in new window

Open image in new window

Open image in new window

Open image in new window

Open image in new window

Open image in new window

Open image in new window

0.0033688

0.0016866

0.000843849

0.000422062

0.000211065

 

0.998

0.999

0.999

0.999

 

Open image in new window

0.00381473

0.00191236

0.000957428

0.000479026

0.000239591

 

0.996

0.996

0.998

0.999

 

Open image in new window

0.00386427

0.00194230

0.000973693

0.000487882

0.000243900

 

0.992

0.996

0.998

0.999

 

Open image in new window

0.00382489

0.00193278

0.000971476

0.00048701

0.000243823

 

0.984

0.992

0.996

0998

 

Open image in new window

0.00374366

0.00191245

0.000966391

0.000485738

0.000243505

 

0.969

0.984

0.992

0.996

 

Open image in new window

0.00358208

0.00187183

0.000956223

0.000433195

0.000242869

 

0.936

0.969

0.984

0.992

 

Open image in new window

0.00326581

0.00179104

0.000935915

0.000477811

0.000241598

 

0.866

0.936

0.969

0.984

 

Open image in new window

0.00268346

0.0016329

0.00895519

0.00467957

0.000239057

 

0.716

0.866

0.936

0.969

 
Table 2

Maximum errors Open image in new window and convergence rates Open image in new window on Open image in new window

Open image in new window

Open image in new window

Open image in new window

Open image in new window

Open image in new window

Open image in new window

Open image in new window

0.00319858

0.00164347

0.000832995

0.000419339

0.000211065

 

0.960

0.980

0.990

0.995

 

Open image in new window

0.00600293

0.00300639

0.00150442

0.000752515

0.000376334

 

0.997

0.999

0.999

1.00

 

Open image in new window

0.00780800

0.00396966

0.00200100

0.00100461

0.000503328

 

0.975

0.988

0.994

0.997

 

Open image in new window

0.0185227

0.00951902

0.00482057

0.00242576

0.001216820

 

0.960

0.981

0.990

0995

 

Open image in new window

0.0388137

0.0202932

0.0103797

0.00525228

0.002641280

 

0.935

0.967

0.9982

0.9916

 

Open image in new window

0.0747962

0.0405973

0.0211784

0.0108201

0.005461600

 

0.881

0.938

0.968

0.984

 

Open image in new window

0.131822

0.0765885

0.0414891

0.0216210

0.011040200

 

0.783

0.884

0.940

0.969

 

Open image in new window

0.149561

0.133579

0.0774847

0.0419350

0.021842300

 

0.163

0.785

0.885

0.941

 

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Copyright information

© Fevzi Erdogan. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Faculty of Sciences, Department of MathematicsYuzuncu Yil UniversityVanTurkey

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