# Uniform Second-Order Difference Method for a Singularly Perturbed Three-Point Boundary Value Problem

- Musa Çakır
^{1}Email author

**2010**:102484

https://doi.org/10.1155/2010/102484

© Musa Çakır. 2010

**Received: **21 June 2010

**Accepted: **15 October 2010

**Published: **20 October 2010

## Abstract

We consider a singularly perturbed one-dimensional convection-diffusion three-point boundary value problem with zeroth-order reduced equation. The monotone operator is combined with the piecewise uniform Shishkin-type meshes. We show that the scheme is second-order convergent, in the discrete maximum norm, independently of the perturbation parameter except for a logarithmic factor. Numerical examples support the theoretical results.

## 1. Introduction

where is the perturbation parameter, and, , and are given constants. The functions , and are sufficiently smooth. For the function has in general boundary layers at and .

Equations of this type arise in mathematical problems in many areas of mechanics and physics. Among these are the Navier-Stokes equations of fluid flow at high Reynolds number, mathematical models of liquid crystal materials and chemical reactions, shear in second-order fluids, control theory, electrical networks, and other physical models [1, 2].

Differential equations with a small parameter multiplying the highest order derivatives are called singularly perturbed differential equations. Typically, the solutions of such equations have steep gradients in narrow layer regions of the domain. Classical numerical methods are inappropriate for singularly perturbed problems. Therefore, it is important to develop suitable numerical methods to these problems, whose accuracy does not depend on the parameter value ; that is, methods that are convergence -uniformly [1–5]. One of the simplest ways to derive such methods consists of using a class of special piecewise uniform meshes (a Shishkin mesh), (see, e.g., [4–8] for motivation for this type of mesh), which are constructed a priori in function of sizes of parameter , the problem data, and the number of corresponding mesh points.

Three-point boundary value problems have been studied extensively in the literature. For a discussion of existence and uniqueness results and for applications of three-point problems, see [9–12] and the references cited in them. Some approaches to approximating this type of problem have also been considered [13, 14]. However, the algorithms developed in the papers cited above are mainly concerned with regular cases (i.e., when boundary layers are absent). Fitted difference scheme on an equidistant mesh for the numerical solution of the linear three-point reaction-diffusion problem have been studied in [15]. A uniform finite difference method, which is first-order convergent, on an S-mesh (Shishkin type mesh) for a singularly perturbed semilinear one-dimensional convection-diffusion three-point boundary value problem have also been studied in [16].

Computational methods for singularly perturbed problems with two small parameters have been studied in different ways [17–21]. In this paper, we propose the hybrid scheme for solving the nonlocal problem (1.1)-(1.2), which comprises three kinds of schemes, such as Samarskii's scheme [22], a finite difference scheme with uniform mesh, and finite difference scheme on piecewise uniform mesh. The considered algorithm is monotone.

We will prove that the method for the numerical solution of the three-point boundary value problem (1.1)-(1.2) is uniformly convergent of order on special piecewise equidistant mesh, in discrete maximum norm, independently of singular perturbation parameter. In Section 2, we present some analytical results of the three-point boundary value problem (1.1)-(1.2). In Section 3, we describe some monotone finite-difference discretization and introduce the piecewise uniform grid. In Section 4, we analyze the convergence properties of the scheme. Finally, numerical examples are presented in Section 5.

Notation 1.

Henceforth, denote the generic positive constants independent of and of the mesh parameter. Such a subscripted constant is also independent of and mesh parameter, but whose value is fixed.

Assumption.

In what follows, we will assume that , which is nonrestrictive in practice.

## 2. Properties of the Exact Solution

For constructing layer-adapted meshes correctly, we need to know the asymptotic behavior of the exact solution. This behavior will be used later in the analysis of the uniform convergence of the finite difference approximations defined in Section 3. For any continuous function , we use for the continuous maximum norm on the corresponding interval.

Lemma 2.1.

Proof.

## 3. Discretization and Piecewise Uniform Mesh

To approximate the solution of (1.1)-(1.2), we employ a finite difference scheme defined on a piecewise uniform Shishkin mesh. This mesh is defined as follows.

## 4. Uniform Error Estimates

Lemma 4.1.

holds.

Proof.

The proof is almost identical to that of [16, 23].

Lemma 4.2.

Proof.

The same estimate is obtained for and in a similar manner.

Combining Lemmas and gives us the following convergence result.

Theorem 4.3.

## 5. Algorithm and Numerical Results

- (a)

- (b)

2 | 0.0094302 | 0.0048322 | 0.0027402 | 0.0016792 | 0.0005534 |

1.78 | 1.87 | 1.95 | 1.98 | 2.02 | |

2 | 0.0095503 | 0.0056215 | 0.0033157 | 0.0017325 | 0.0005988 |

1.73 | 1.85 | 1.92 | 1.96 | 1.99 | |

2 | 0.0096054 | 0.0056215 | 0.0033157 | 0.0017325 | 0.0005988 |

1.76 | 1.85 | 1.92 | 1.96 | 1.99 | |

2 | 0.0095502 | 0.0056215 | 0.0033157 | 0.0017325 | 0.0005988 |

1.73 | 1.85 | 1.92 | 1.96 | 1.99 | |

2 | 0.0095502 | 0.0056215 | 0.0033157 | 0.0017325 | 0.0005988 |

1.73 | 1.85 | 1.92 | 1.96 | 1.99 | |

2 | 0.0095502 | 0.0056215 | 0.0033157 | 0.0017325 | 0.0005988 |

1.73 | 1.85 | 1.92 | 1.96 | 1.99 | |

2 | 0.0095502 | 0.0056215 | 0.0033157 | 0.0017325 | 0.0005988 |

1.73 | 1.85 | 1.92 | 1.96 | 1.99 | |

2 | 0.0095502 | 0.0056215 | 0.0033157 | 0.0017325 | 0.0005988 |

1.73 | 1.85 | 1.92 | 1.96 | 1.99 | |

0.0096054 | 0.0056215 | 0.0033157 | 0.0017325 | 0.0005988 | |

1.73 | 1.85 | 1.92 | 1.96 | 1.99 |

## Authors’ Affiliations

## References

- Nayfeh AH:
*Introduction to Perturbation Techniques*. John Wiley & Sons, New York, NY, USA; 1993.Google Scholar - O'Malley RE Jr.:
*Singular Perturbation Methods for Ordinary Differential Equations, Applied Mathematical Sciences*.*Volume 89*. Springer, New York, NY, USA; 1991:viii+225.View 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 - 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.Google Scholar - Roos H-G, Stynes M, Tobiska L:
*Robust Numerical Methods for Singularly Perturbed Differential Equations, Convection-Diffusion-Reaction and Flow Problems, Springer Series in Computational Mathematics*.*Volume 24*. 2nd edition. Springer, Berlin, Germany; 2008:xiv+604.Google Scholar - Linß T, Stynes M:
**A hybrid difference scheme on a Shishkin mesh for linear convection-diffusion problems.***Applied Numerical Mathematics*1999,**31**(3):255-270. 10.1016/S0168-9274(98)00136-6MATHMathSciNetView ArticleGoogle Scholar - Savin IA:
**On the rate of convergence, uniform with respect to a small parameter, of a difference scheme for an ordinary differential equation.***Computational Mathematics and Mathematical Physics*1995,**35**(11):1417-1422.MATHMathSciNetGoogle Scholar - Sun GF, Stynes M:
**A uniformly convergent method for a singularly perturbed semilinear reaction-diffusion problem with multiple solutions.***Mathematics of Computation*1996,**65**(215):1085-1109. 10.1090/S0025-5718-96-00753-3MATHMathSciNetView ArticleGoogle Scholar - Cziegis R:
**The numerical of singularly perturbed nonlocal problem.***Lietuvas Matematica Rink*1988,**28:**144-152.Google Scholar - Čiegis R:
**On the difference schemes for problems with nonlocal boundary conditions.***Informatica*1991,**2**(2):155-170.MATHMathSciNetGoogle Scholar - Nakhushev AM:
**Nonlocal boundary value problems with shift and their connection with loaded equations.***Differential Equations*1985,**21**(1):92-101.MathSciNetGoogle Scholar - Sapagovas MP, Chegis RYu:
**Numerical solution of some nonlocal problems.***Litovskiĭ Matematicheskiĭ Sbornik*1987,**27**(2):348-356.MATHMathSciNetGoogle Scholar - Liu B:
**Positive solutions of second-order three-point boundary value problems with change of sign.***Computers & Mathematics with Applications*2004,**47**(8-9):1351-1361. 10.1016/S0898-1221(04)90128-9MATHMathSciNetView ArticleGoogle Scholar - Ma R:
**Positive solutions for nonhomogeneous**-point boundary value problems.*Computers & Mathematics with Applications*2004,**47**(4-5):689-698. 10.1016/S0898-1221(04)90056-9MATHMathSciNetView ArticleGoogle Scholar - Amiraliyev GM, Çakir M:
**Numerical solution of the singularly perturbed problem with nonlocal boundary condition.***Applied Mathematics and Mechanics*2002,**23**(7):755-764. 10.1007/BF02456971MATHMathSciNetView ArticleGoogle Scholar - Cakir M, Amiraliyev GM:
**Numerical solution of a singularly perturbed three-point boundary value problem.***International Journal of Computer Mathematics*2007,**84**(10):1465-1481. 10.1080/00207160701296462MATHMathSciNetView ArticleGoogle Scholar - Gracia JL, O'Riordan E, Pickett ML:
**A parameter robust second order numerical method for a singularly perturbed two-parameter problem.***Applied Numerical Mathematics*2006,**56**(7):962-980. 10.1016/j.apnum.2005.08.002MATHMathSciNetView ArticleGoogle Scholar - Linß T, Roos H-G:
**Analysis of a finite-difference scheme for a singularly perturbed problem with two small parameters.***Journal of Mathematical Analysis and Applications*2004,**289**(2):355-366. 10.1016/j.jmaa.2003.08.017MATHMathSciNetView ArticleGoogle Scholar - Linß T:
**Layer-adapted meshes for convection-diffusion problems.***Computer Methods in Applied Mechanics and Engineering*2003,**192**(9-10):1061-1105. 10.1016/S0045-7825(02)00630-8MATHMathSciNetView ArticleGoogle Scholar - Clavero C, Gracia JL, Lisbona F:
**High order methods on Shishkin meshes for singular perturbation problems of convection-diffusion type.***Numerical Algorithms*1999,**22**(1):73-97. 10.1023/A:1019150606200MATHMathSciNetView ArticleGoogle Scholar - O'Malley RE Jr.:
**Two-parameter singular perturbation problems for second-order equations.***Journal of Mathematics and Mechanics*1967,**16:**1143-1164.MATHMathSciNetGoogle Scholar - Samarskii AA:
*Theory of Difference Schemes*. M. Nauka, Moscow, Russia; 1971.Google Scholar - Amiraliyev G, Çakir M:
**A uniformly convergent difference scheme for a singularly perturbed problem with convective term and zeroth order reduced equation.***International Journal of Applied Mathematics*2000,**2**(12):1407-1419.MATHMathSciNetGoogle 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.