# A Parameter Robust Method for Singularly Perturbed Delay Differential Equations

- Fevzi Erdogan
^{1}Email author

**2010**:325654

https://doi.org/10.1155/2010/325654

© Fevzi Erdogan. 2010

**Received: **29 April 2010

**Accepted: **17 July 2010

**Published: **1 August 2010

## Abstract

Uniform finite difference methods are constructed via nonstandard finite difference methods for the numerical solution of singularly perturbed quasilinear initial value problem for delay differential equations. A numerical method is constructed for this problem which involves the appropriate Bakhvalov meshes on each time subinterval. The method is shown to be uniformly convergent with respect to the perturbation parameter. A numerical example is solved using the presented method, and the computed result is compared with exact solution of the problem.

## 1. Introduction

Delay differential equations are used to model a large variety of practical phenomena in the biosciences, engineering and control theory, and in many other areas of science and technology, in which the time evolution depends not only on present states but also on states at or near a given time in the past (see, e.g., [1–4]). If we restrict the class of delay differential equations to a class in which the highest derivative is multiplied by a small parameter, then it is said to be a singularly perturbed delay differential equation. Such problems arise in the mathematical modeling of various practical phenomena, for example, in population dynamics [4], the study of bistable devices [5], description of the human pupil-light reflex [6], and variational problems in control theory [7]. In the direction of numerical study of singularly perturbed delay differential equation, much can be seen in [8–16].

The numerical analysis of singular perturbation cases has always been far from trivial because of the boundary layer behavior of the solution. Such problems undergo rapid changes within very thin layers near the boundary or inside the problem domain. It is well known that standard numerical methods for solving singular perturbation problems do not give a satisfactory result when the perturbation parameter is sufficiently small. Therefore, it is important to develop suitable numerical methods for these problems, whose accuracy does not depend on the perturbation parameter, that is, methods that are uniformly convergent with respect to the perturbation parameter [17–20].

In order to construct parameter-uniform numerical methods for singularly perturbed differential equations, two different techniques are applied. Firstly, the fitted operator approach [20] which has coefficients of exponential type adapted to the singular perturbation problems. Secondly, the special mesh approach [19], which constructs meshes adapted to the solution of the problem.

The work contained in this paper falls under the second category. We use the nonstandard finite difference methods originally developed by Bakhvalov for some other problems. One of the simplest ways to derive such methods consists of using a class of special meshes (such as Bakhvalov meshes; see, e.g., [18–24]), which is constructed a priori and depend on the perturbation parameter, the problem data, and the number of corresponding mesh points.

The solution, , displays in general boundary layers on the right side of each point for small values of .

In the present paper we discretize (1.1)-(1.2) using a numerical method which is composed of an implicit finite difference scheme on special Bakhvalov meshes for the numerical solution on each timesubinterval. In Section 2, we state some important properties of the exact solution. In Section 3, we describe the finite difference discretization and introduce Bakhvalov-Shishkin mesh and Bakhvalov mesh. In Section 4, we present the error analysis for the approximate solution. Uniform convergence is proved in the discrete maximum norm. In Section 5, a test example is considered and a comparison of the numerical and exact solutions is presented.

In the works of Amiraliyev and Erdogan [9], special meshes (Shishkin mesh) have been used. The method that we propose in this paper uses Bakhvalov-type meshes.

Throughout the paper, denotes a generic positive constant independent of and the mesh parameter. Some specific, fixed constants of this kind are indicated by subscripting .

## 2. The Continuous Problem

Before defining the mesh and the finite difference scheme, we show some results about the behavior with respect to the perturbation parameter of the exact solution of problem (1.1)-(1.2) and its derivatives, which we will use in later section for the analysis of an appropriate numerical solution. For any continuous function , denotes a continuous maximum norm on the corresponding closed interval ; in particular we will use .

Lemma 2.1.

Proof.

which proves (2.1).

which proves (2.3).

## 3. Discretization and Mesh

We consider two special discretization meshes, both dense in the boundary layer. We illustrate that the essential idea of Bakhvalov [21] by constructing special nonuniform meshes and has been combined with various difference schemes in numerous papers [22, 23].

### 3.1. Bakhvalov-Shishkin Mesh

where and ( are some constants. We will assume throughout the paper that , as is generally the case in practice.

### 3.2. Bakhvalov Mesh

## 4. Stability and Convergence Analysis

where the truncation error is given by (3.7).

Lemma 4.1.

Proof.

The proof follows easily by induction in , by analogy with differential case.

Lemma 4.2.

Proof.

It evidently follows from (4.2) by taking and .

Lemma 4.3.

Proof.

Thus, the proof is completed.

Combining the previous lemmas gives us the following convergence result.

Theorem 4.4.

## 5. Numerical Results

0.00978429 | 0.00493577 | 0.00247899 | 0.00124229 | 0.00062184 | |

0.987 | 0.993 | 0.996 | 0.998 | ||

0.016348 | 0.00831665 | 0.0041954 | 0.00210714 | 0.00105595 | |

0.975 | 0.987 | 0.993 | 0.996 | ||

0.0230541 | 0.0118195 | 0.00598914 | 0.00301454 | 0.00151234 | |

0.963 | 0.980 | 0.990 | 0.995 | ||

0.0298948 | 0.0154465 | 0.00785801 | 0.00396404 | 0.00199094 | |

0.952 | 0.975 | 0.987 | 0.993 | ||

0.0366571 | 0.0190685 | 0.0097511 | 0.00492979 | 0.00247866 | |

0.942 | 0.967 | 0.984 | 0.991 | ||

0.0432959 | 0.022705 | 0.0116405 | 0.00589844 | 0.00296889 | |

0.931 | 0.963 | 0.980 | 0.990 | ||

0.0493475 | 0.0262615 | 0.0135164 | 0.00686448 | 0.00345923 | |

0.911 | 0.958 | 0.977 | 0.988 | ||

0.0560001 | 0.0297789 | 0.0153866 | 0.00782756 | 0.00394867 | |

0.911 | 0.52 | 0.975 | 0.987 |

0.0120441 | 0.00609088 | 0.00306261 | 0.00153567 | 0.00076893 | |

0.983 | 0.991 | 0.995 | 0.997 | ||

0.0204344 | 0.0106567 | 0.00542574 | 0.0027399 | 0.00137664 | |

0.939 | 0.973 | 0.985 | 0.992 | ||

0.0206243 | 0.0123374 | 0.00663473 | 0.00346693 | 0.00178218 | |

0.741 | 0.894 | 0.936 | 0.960 | ||

0.0251094 | 0.0129806 | 0.00660313 | 0.00346667 | 0.00192158 | |

0.951 | 0.975 | 0.929 | 0.951 | ||

0.0308922 | 0.0160402 | 0.00819434 | 0.00414173 | 0.00208219 | |

0.945 | 0.968 | 0.992 | 0.996 | ||

0.0358208 | 0.0190729 | 0.00978373 | 0.00495569 | 0.00249403 | |

0.909 | 0.963 | 0.981 | 0.990 | ||

0.0418982 | 0.0220722 | 0.0113657 | 0.00576776 | 0.00290598 | |

0.924 | 0.957 | 0.978 | 0.988 | ||

0.0471824 | 0.0250121 | 0.0129303 | 0.00657754 | 0.0033174 | |

0.915 | 0.951 | 0.975 | 0.987 |

0.0140074 | 0.00709303 | 0.00356936 | 0.00179046 | 0.000896684 | |

0.987 | 0.993 | 0.996 | 0.998 | ||

0.0241181 | 0.0143603 | 0.00831665 | 0.00471467 | 0.00263101 | |

0.975 | 0.987 | 0.993 | 0.996 | ||

0.0230541 | 0.0137267 | 0.00794974 | 0.00450667 | 0.00251493 | |

0.963 | 0.980 | 0.990 | 0.995 | ||

0.0227881 | 0.0135684 | 0.00785801 | 0.00445467 | 0.00248591 | |

0.952 | 0.975 | 0.987 | 0.993 | ||

0.0227216 | 0.0135288 | 0.00783508 | 0.00444167 | 0.00247866 | |

0.942 | 0.967 | 0.984 | 0.991 | ||

0.0227050 | 0.0135189 | 0.00782935 | 0.00443842 | 0.00247684 | |

0.931 | 0.963 | 0.980 | 0.990 | ||

0.0227008 | 0.0135164 | 0.0782791 | 0.00443761 | 0.00247639 | |

0.911 | 0.958 | 0.977 | 0.988 | ||

0.0226998 | 0.0135158 | 0.00782756 | 0.0044374 | 0.00247628 | |

0.911 | 0.52 | 0.975 | 0.987 |

0.0121386 | 0.00613925 | 0.00308755 | 0.00154829 | 0.000775281 | |

0.983 | 0.991 | 0.995 | 0.997 | ||

0.0202600 | 0.0120754 | 0.00698853 | 0.00396095 | 0.00221017 | |

0.939 | 0.973 | 0.985 | 0.992 | ||

0.0206243 | 0.0115426 | 0.00668021 | 0.0037862 | 0.00211266 | |

0.741 | 0.894 | 0.936 | 0.960 | ||

0.0191427 | 0.0114094 | 0.00660313 | 0.00374251 | 0.00208828 | |

0.951 | 0.975 | 0.929 | 0.951 | ||

0.0190868 | 0.0113761 | 0.00658386 | 0.00373159 | 0.00208219 | |

0.945 | 0.968 | 0.992 | 0.996 | ||

0.0190729 | 0.0113678 | 0.00657904 | 0.00372886 | 0.00208066 | |

0.909 | 0.963 | 0.981 | 0.990 | ||

0.0190694 | 0.0113657 | 0.0657784 | 0.00372817 | 0.00208028 | |

0.924 | 0.957 | 0.978 | 0.988 | ||

0.0190685 | 0.0113652 | 0.00657754 | 0.003728 | 0.00208019 | |

0.915 | 0.951 | 0.975 | 0.987 |

## Authors’ Affiliations

## References

- Bellman R, Cooke KL:
*Differential-Difference Equations*. Academic Press, New York, NY, USA; 1963:xvi+462.Google Scholar - Driver RD:
*Ordinary and Delay Differential Equations, Applied Mathematical Sciences*.*Volume 2*. Springer, New York, NY, USA; 1977:ix+501.View ArticleGoogle Scholar - Bellen A, Zennaro M:
*Numerical Methods for Delay Differential Equations, Numerical Mathematics and Scientific Computation*. Oxford University Press, Oxford, UK; 2003:xiv+395.View ArticleMATHGoogle Scholar - Kuang Y:
*Delay Differential Equations with Applications in Population Dynamic, Mathematics in Science and Engineering*.*Volume 191*. Academic Press, Boston, Mass, USA; 1993:xii+398.MATHGoogle Scholar - Chow S-N, Mallet-Paret J: Singularly perturbed delay-differential equations. In
*Coupled Nonlinear Oscillators*. Edited by: Chandra J, Scott AC. North-Holland, Amsterdam, The Netherlands; 1983:7–12.Google 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 - 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.2006.07.009MathSciNetView ArticleMATHGoogle Scholar - Amiraliyev GM, Erdogan F: A finite difference scheme for a class of singularly perturbed initial value problems for delay differential equations.
*Numerical Algorithms*2009, 52(4):663–675. 10.1007/s11075-009-9306-zMathSciNetView ArticleMATHGoogle Scholar - Mallet-Paret J, Nussbaum RD: A differential-delay equation arising in optics and physiology.
*SIAM Journal on Mathematical Analysis*1989, 20(2):249–292. 10.1137/0520019MathSciNetView ArticleMATHGoogle Scholar - Mallet-Paret J, Nussbaum RD: Multiple transition layers in a singularly perturbed differential-delay equation.
*Proceedings of the Royal Society of Edinburgh A*1993, 123(6):1119–1134. 10.1017/S0308210500029772MathSciNetView ArticleMATHGoogle Scholar - Maset S: Numerical solution of retarded functional differential equations as abstract Cauchy problems.
*Journal of Computational and Applied Mathematics*2003, 161(2):259–282. 10.1016/j.cam.2003.03.001MathSciNetView ArticleMATHGoogle 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)00625-7MathSciNetView ArticleMATHGoogle Scholar - Kadalbajoo MK, Sharma KK:
-uniform fitted mesh method for singularly perturbed differential-difference equations: mixed type of shifts wtih layer behavior.
*International Journal of Computer Mathematics*2004, 81(1):49–62. 10.1080/00207160310001606052MathSciNetView ArticleMATHGoogle Scholar - Tian H:
*Numerical treatment of singularly perturbed delay differential equations, Ph.D. thesis*. Department of Mathematics, University of Manchester; 2000.Google 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)00056-2MathSciNetView ArticleMATHGoogle 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 - Farrell PA, Hegarty AF, Miller JJH, O'Riordan E, Shishkin GI:
*Robust Computational Techniques for Boundary Layers, Applied Mathematics*.*Volume 16*. Chapman & Hall/CRC, Boca Raton, Fla, USA; 2000:xvi+254.MATHGoogle Scholar - Miller JJ, ORiordan E, Shishkin GI:
*Fitted Numerical Methods for Singular Perturbation Problems, Error Estimates in the Maximum Error for Linear Problems in One and Two Dimensions*. World Scientific, Singapore; 1996.View ArticleGoogle 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.MATHGoogle Scholar - Bakhvalov NS: Towards optimization of methods for solving boundary value problems in the presence of boundary layers.
*Zhurnal Vychislitelnoi Matematiki I Matematicheskoi Fiziki*1969, 9: 841–859.Google Scholar - Kopteva N: Uniform pointwise convergence of difference schemes for convection-diffusion problems on layer-adapted meshes.
*Computing*2001, 66(2):179–197. 10.1007/s006070170034MathSciNetView ArticleMATHGoogle Scholar - Linss T: Analysis of a Galerkin finite element method on a Bakhvalov-Shishkin mesh for a linear convection-diffusion problem.
*IMA Journal of Numerical Analysis*2000, 20(4):621–632. 10.1093/imanum/20.4.621MathSciNetView ArticleMATHGoogle Scholar - Vulanović R: A priori meshes for singularly perturbed quasilinear two-point boundary value problems.
*IMA Journal of Numerical Analysis*2001, 21(1):349–366. 10.1093/imanum/21.1.349MathSciNetView ArticleMATHGoogle Scholar

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