- Research Article
- Open Access

# An Exponentially Fitted Method for Singularly Perturbed Delay Differential Equations

- Fevzi Erdogan
^{1}Email author

**2009**:781579

https://doi.org/10.1155/2009/781579

© Fevzi Erdogan. 2009

**Received:**4 November 2008**Accepted:**16 January 2009**Published:**26 January 2009

## 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

## 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 [1–4]. 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 is small. Therefore, it is important to develop suitable numerical methods to these problem, whose accuracy does not depend on the parameter value , that is, methods that are uniformly convergent with respect to the perturbation parameter [8–10]. 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 , 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

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

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, will denote a generic positive constant (possibly subscripted) that is independent of and of the mesh. Note that 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, denotes a continuous maximum norm on the corresponding interval.

Lemma 3.1.

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.

## 5. Analysis of the Method

where and are given by (4.10) and (4.13), respectively.

Lemma 5.1.

Proof.

The proof follows easily by induction in .

Lemma 5.2.

Proof.

It evidently follows from (5.2) by taking and .

Lemma 5.3.

Proof.

The same estimate is obtained for in the similar manner as above.

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

Theorem 5.4.

## 6. Numerical Results

The values of for which we solve the test problem are .

0.0033688 | 0.0016866 | 0.000843849 | 0.000422062 | 0.000211065 | |

0.998 | 0.999 | 0.999 | 0.999 | ||

0.00381473 | 0.00191236 | 0.000957428 | 0.000479026 | 0.000239591 | |

0.996 | 0.996 | 0.998 | 0.999 | ||

0.00386427 | 0.00194230 | 0.000973693 | 0.000487882 | 0.000243900 | |

0.992 | 0.996 | 0.998 | 0.999 | ||

0.00382489 | 0.00193278 | 0.000971476 | 0.00048701 | 0.000243823 | |

0.984 | 0.992 | 0.996 | 0998 | ||

0.00374366 | 0.00191245 | 0.000966391 | 0.000485738 | 0.000243505 | |

0.969 | 0.984 | 0.992 | 0.996 | ||

0.00358208 | 0.00187183 | 0.000956223 | 0.000433195 | 0.000242869 | |

0.936 | 0.969 | 0.984 | 0.992 | ||

0.00326581 | 0.00179104 | 0.000935915 | 0.000477811 | 0.000241598 | |

0.866 | 0.936 | 0.969 | 0.984 | ||

0.00268346 | 0.0016329 | 0.00895519 | 0.00467957 | 0.000239057 | |

0.716 | 0.866 | 0.936 | 0.969 |

0.00319858 | 0.00164347 | 0.000832995 | 0.000419339 | 0.000211065 | |

0.960 | 0.980 | 0.990 | 0.995 | ||

0.00600293 | 0.00300639 | 0.00150442 | 0.000752515 | 0.000376334 | |

0.997 | 0.999 | 0.999 | 1.00 | ||

0.00780800 | 0.00396966 | 0.00200100 | 0.00100461 | 0.000503328 | |

0.975 | 0.988 | 0.994 | 0.997 | ||

0.0185227 | 0.00951902 | 0.00482057 | 0.00242576 | 0.001216820 | |

0.960 | 0.981 | 0.990 | 0995 | ||

0.0388137 | 0.0202932 | 0.0103797 | 0.00525228 | 0.002641280 | |

0.935 | 0.967 | 0.9982 | 0.9916 | ||

0.0747962 | 0.0405973 | 0.0211784 | 0.0108201 | 0.005461600 | |

0.881 | 0.938 | 0.968 | 0.984 | ||

0.131822 | 0.0765885 | 0.0414891 | 0.0216210 | 0.011040200 | |

0.783 | 0.884 | 0.940 | 0.969 | ||

0.149561 | 0.133579 | 0.0774847 | 0.0419350 | 0.021842300 | |

0.163 | 0.785 | 0.885 | 0.941 |

## Authors’ Affiliations

## References

- Bellman R, Cooke KL:
*Differential-Difference Equations*. Academic Press, New York, NY, USA; 1963:xvi+462.MATHGoogle Scholar - Driver RD:
*Ordinary and Delay Differential Equations, Applied Mathematical Sciences*.*Volume 2*. Springer, New York, NY, USA; 1977:ix+501.View ArticleGoogle Scholar - McCartin BJ:
**Exponential fitting of the delayed recruitment/renewal equation.***Journal of Computational and Applied Mathematics*2001,**136**(1-2):343-356. 10.1016/S0377-0427(00)00625-7MATHMathSciNetView ArticleGoogle Scholar - Tian H:
**The exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag.***Journal of Mathematical Analysis and Applications*2002,**270**(1):143-149. 10.1016/S0022-247X(02)00056-2MATHMathSciNetView ArticleGoogle Scholar - Derstine MW, Gibbs HM, Hopf FA, Kaplan DL:
**Bifurcation gap in a hybrid optically bistable system.***Physical Review A*1982,**26**(6):3720-3722. 10.1103/PhysRevA.26.3720View ArticleGoogle Scholar - Longtin A, Milton JG:
**Complex oscillations in the human pupil light reflex with "mixed" and delayed feedback.***Mathematical Biosciences*1988,**90**(1-2):183-199. 10.1016/0025-5564(88)90064-8MathSciNetView ArticleGoogle Scholar - Mackey MC, Glass L:
**Oscillation and chaos in physiological control systems.***Science*1977,**197**(4300):287-289. 10.1126/science.267326View ArticleGoogle Scholar - Farrell PA, Hegarty AF, Miller JJH, O'Riordan E, Shishkin GI:
*Robust Computational Techniques for Boundary Layers, Applied Mathematics and Mathematical Computation*.*Volume 16*. Chapman & Hall/CRC, Boca Raton, Fla, USA; 2000:xvi+254.Google Scholar - Roos H-G, Stynes M, Tobiska L:
*Numerical Methods for Singularly Perturbed Differential Equations, Convection-Diffusion and Flow Problems, Springer Series in Computational Mathematics*.*Volume 24*. Springer, Berlin, Germany; 1996:xvi+348.View ArticleGoogle Scholar - Amiraliyev GM:
**Difference method for the solution of one problem of the theory dispersive waves.***Differentsial'nye Uravneniya*1990,**26:**2146-2154.Google Scholar - Amiraliyev GM, Erdogan F:
**Uniform numerical method for singularly perturbed delay differential equations.***Computers & Mathematics with Applications*2007,**53**(8):1251-1259. 10.1016/j.camwa.2006.07.009MATHMathSciNetView ArticleGoogle Scholar - Doolan EP, Miller JJH, Schilders WHA:
*Uniform Numerical Methods for Problems with Initial and Boundary Layers*. Boole Press, Dublin, Ireland; 1980:xv+324.MATHGoogle Scholar

## Copyright

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.