# 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.

## Keywords

## 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

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