# An Exponentially Fitted Method for Singularly Perturbed Delay Differential Equations

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## 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, 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

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.

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.

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

## 5. Analysis of the Method

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

Lemma 5.1.

Proof.

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

Lemma 5.2.

Proof.

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

Lemma 5.3.

Proof.

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.

## 6. Numerical Results

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

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

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 |

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

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 |

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