 Research Article
 Open Access
Uniform SecondOrder Difference Method for a Singularly Perturbed ThreePoint Boundary Value Problem
 Musa Çakır^{1}Email author
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 onedimensional convectiondiffusion threepoint boundary value problem with zerothorder reduced equation. The monotone operator is combined with the piecewise uniform Shishkintype meshes. We show that the scheme is secondorder convergent, in the discrete maximum norm, independently of the perturbation parameter except for a logarithmic factor. Numerical examples support the theoretical results.
Keywords
 Finite Difference Scheme
 Mesh Point
 Mesh Function
 Liquid Crystal Material
 Shishkin Mesh
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 NavierStokes equations of fluid flow at high Reynolds number, mathematical models of liquid crystal materials and chemical reactions, shear in secondorder 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.
Threepoint boundary value problems have been studied extensively in the literature. For a discussion of existence and uniqueness results and for applications of threepoint 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 threepoint reactiondiffusion problem have been studied in [15]. A uniform finite difference method, which is firstorder convergent, on an Smesh (Shishkin type mesh) for a singularly perturbed semilinear onedimensional convectiondiffusion threepoint 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 threepoint 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 threepoint boundary value problem (1.1)(1.2). In Section 3, we describe some monotone finitedifference 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 layeradapted 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
and is defined by (3.14).
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)
Approximate errors and and the computed orders of convergence on the piecewise uniform mesh for various values of and .







2^{2}  0.0094302  0.0048322  0.0027402  0.0016792  0.0005534 
1.78  1.87  1.95  1.98  2.02  
2^{4}  0.0095503  0.0056215  0.0033157  0.0017325  0.0005988 
1.73  1.85  1.92  1.96  1.99  
2^{6}  0.0096054  0.0056215  0.0033157  0.0017325  0.0005988 
1.76  1.85  1.92  1.96  1.99  
2^{8}  0.0095502  0.0056215  0.0033157  0.0017325  0.0005988 
1.73  1.85  1.92  1.96  1.99  
2^{10}  0.0095502  0.0056215  0.0033157  0.0017325  0.0005988 
1.73  1.85  1.92  1.96  1.99  
2^{12}  0.0095502  0.0056215  0.0033157  0.0017325  0.0005988 
1.73  1.85  1.92  1.96  1.99  
2^{14}  0.0095502  0.0056215  0.0033157  0.0017325  0.0005988 
1.73  1.85  1.92  1.96  1.99  
2^{16}  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 HG, Stynes M, Tobiska L: Robust Numerical Methods for Singularly Perturbed Differential Equations, ConvectionDiffusionReaction 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 convectiondiffusion problems. Applied Numerical Mathematics 1999,31(3):255270. 10.1016/S01689274(98)001366MATHMathSciNetView 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):14171422.MATHMathSciNetGoogle Scholar
 Sun GF, Stynes M: A uniformly convergent method for a singularly perturbed semilinear reactiondiffusion problem with multiple solutions. Mathematics of Computation 1996,65(215):10851109. 10.1090/S0025571896007533MATHMathSciNetView ArticleGoogle Scholar
 Cziegis R: The numerical of singularly perturbed nonlocal problem. Lietuvas Matematica Rink 1988, 28: 144152.Google Scholar
 Čiegis R: On the difference schemes for problems with nonlocal boundary conditions. Informatica 1991,2(2):155170.MATHMathSciNetGoogle Scholar
 Nakhushev AM: Nonlocal boundary value problems with shift and their connection with loaded equations. Differential Equations 1985,21(1):92101.MathSciNetGoogle Scholar
 Sapagovas MP, Chegis RYu: Numerical solution of some nonlocal problems. Litovskiĭ Matematicheskiĭ Sbornik 1987,27(2):348356.MATHMathSciNetGoogle Scholar
 Liu B: Positive solutions of secondorder threepoint boundary value problems with change of sign. Computers & Mathematics with Applications 2004,47(89):13511361. 10.1016/S08981221(04)901289MATHMathSciNetView ArticleGoogle Scholar
 Ma R:Positive solutions for nonhomogeneous point boundary value problems. Computers & Mathematics with Applications 2004,47(45):689698. 10.1016/S08981221(04)900569MATHMathSciNetView ArticleGoogle Scholar
 Amiraliyev GM, Çakir M: Numerical solution of the singularly perturbed problem with nonlocal boundary condition. Applied Mathematics and Mechanics 2002,23(7):755764. 10.1007/BF02456971MATHMathSciNetView ArticleGoogle Scholar
 Cakir M, Amiraliyev GM: Numerical solution of a singularly perturbed threepoint boundary value problem. International Journal of Computer Mathematics 2007,84(10):14651481. 10.1080/00207160701296462MATHMathSciNetView ArticleGoogle Scholar
 Gracia JL, O'Riordan E, Pickett ML: A parameter robust second order numerical method for a singularly perturbed twoparameter problem. Applied Numerical Mathematics 2006,56(7):962980. 10.1016/j.apnum.2005.08.002MATHMathSciNetView ArticleGoogle Scholar
 Linß T, Roos HG: Analysis of a finitedifference scheme for a singularly perturbed problem with two small parameters. Journal of Mathematical Analysis and Applications 2004,289(2):355366. 10.1016/j.jmaa.2003.08.017MATHMathSciNetView ArticleGoogle Scholar
 Linß T: Layeradapted meshes for convectiondiffusion problems. Computer Methods in Applied Mechanics and Engineering 2003,192(910):10611105. 10.1016/S00457825(02)006308MATHMathSciNetView ArticleGoogle Scholar
 Clavero C, Gracia JL, Lisbona F: High order methods on Shishkin meshes for singular perturbation problems of convectiondiffusion type. Numerical Algorithms 1999,22(1):7397. 10.1023/A:1019150606200MATHMathSciNetView ArticleGoogle Scholar
 O'Malley RE Jr.: Twoparameter singular perturbation problems for secondorder equations. Journal of Mathematics and Mechanics 1967, 16: 11431164.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):14071419.MATHMathSciNetGoogle Scholar
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