A Numerical Method for a Singularly Perturbed Three-Point Boundary Value Problem

Abstract

The purpose of this paper is to present a uniform finite difference method for numerical solution of nonlinear singularly perturbed convection-diffusion problem with nonlocal and third type boundary conditions. The numerical method is constructed on piecewise uniform Shishkin type mesh. The method is shown to be convergent, uniformly in the diffusion parameter [straight epsilon] , of first order in the discrete maximum norm. Some numerical experiments illustrate in practice the result of convergence proved theoretically.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Musa Çakir 1 and Gabil M. Amiraliyev 2

Recommended by Michela Redivo-Zaglia

1, Department of Mathematics, Faculty of Sciences, 100. Y. University, 65080 Van, Turkey
2, Department of Mathematics, Faculty of Sciences, Sinop University, 57000 Sinop, Turkey

Received 30 October 2009; Accepted 13 April 2010

1. Introduction

This paper is concerned with [straight epsilon] -uniform numerical method for the singularly perturbed semilinear boundary-value problem (BVP): [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] where [straight epsilon] is a small positive parameter, the functions a(x)≥0, f(x,u) , and ψ(u), [straight phi](u) are sufficiently smooth on [0,[cursive l]], [0,[cursive l]]×... , and ... , respectively, and furthermore [figure omitted; refer to PDF] The solution u generally has boundary layers near x=0 and x=[cursive l] .

Singularly perturbed differential equations are characterized by the presence of a small parameter [straight epsilon] multiplying the highest-order derivatives. Such problems arise in many areas of applied mathematics. Among these are the Navier-Stokes equations of fluid flow at high Reynolds number, mathematical models of liquid crystal materials and chemical reactions, control theory, reaction-diffusion processes, quantum mechanics, and electrical networks. The solutions of singularly perturbed differential equations typically have steep gradients, in thin regions of the domain, whose magnitude depends inversely on some positive power of [straight epsilon] . Such regions are called either interior or boundary layers, depending on whether their location is in the interior or at the boundary of the domain. An overview of some existence and uniqueness results and applications of singularly perturbed equations can be found in [1-6].

It is known that these problems depend on a small positive parameter [straight epsilon] in such away that the solution exhibits a multiscale character; that is, there are thin transition layers where the solution varies rapidly, while away from layers it behaves regularly and varies slowly. The treatment of singularly perturbed problems presents severe difficulties that have to be addressed to ensure accurate numerical solutions. Therefore it is important to develop suitable numerical methods for solving these problems, whose accuracy does not depend on the value of parameter [straight epsilon] , that is, methods that are convergent [straight epsilon] -uniformly. These include fitted finite-difference methods, finite element methods using special elements such as exponential elements, and methods which use a priori refined or special piecewise uniform grids which condense in the boundary layers in a special manner. One of the simplest ways to derive parameter-uniform methods consists of using a class of special piecewise uniform meshes, such as Shishkin meshes (see [3, 6-15] for the motivation for this type of mesh), which are constructed a priori and depend on the parameter [straight epsilon] , the problem data, and the number of corresponding mesh points. The various approaches to numerical solution of differential equations with stepwise continuous solutions can be found in [2, 3, 6].

There is also an increasing interest in the application of Shishkin meshes to singularly perturbed convection-diffusion problems (see [16, 17] and references cited therein). However, much of the Shishkin mesh literature is concerned with linear or quasilinear singularly perturbed two-point problems with first-order reduced equations. In [18] has been obtained a result [straight epsilon] -uniform for the two-point boundary value problem of (1.1), by using a fitted operator method on uniform meshes.

In the present paper, we analyse a fitted finite-difference scheme on a Shishkin type mesh for the numerical solution of the semilinear nonlocal boundary value problem (1.1)-(1.3). The origin of the fitted finite difference method can be traced to [19]; for subsequent work on fitted operator method and its applications, see [2, 3]. Nonlocal boundary value problems have also been studied extensively in the literature. For a discussion of existence and uniqueness results and for applications of nonlocal problems see [20-26] and the references cited therein. Some approaches to approximating this type of problem have also been considered [20, 21, 26-28]. However, the algorithms developed in the papers cited above are mainly concerned with regular cases (i.e., when boundary layers are absent). In [27] has been studied the fitted difference schemes on an equidistant mesh for the numerical solution of the linear three-point reaction-diffusion problem.

The numerical method presented here comprises a fitted difference scheme on a piecewise uniform mesh. We have derived this approach on the basis of the method of integral identities using interpolating quadrature rules with the weight and remainder terms in integral form. This results in a local truncation error containing only first-order derivatives of exact solution and hence facilitates examination of the convergence. A summary of paper is as follows. Section 2 contains results concerning the exact solutions of problem (1.1)-(1.3). In Section 3, we describe the finite-difference discretization and construct a piecewise uniform mesh, which is fitted to the boundary layers. In Section 4, we present the error analysis for the approximate solution. Uniform convergence is proved in the discrete maximum norm. In the following section numerical results are presented, which are in agreement with the theoretical results. The approach to the construction of the discrete problem and the error analysis for the approximate solution are similar to those in [18, 27-29].

2. Continuous Solution

In this section, we give uniform bounds for the solution of the BVP (1.1)-(1.3), which will be used to analyze properties of appropriate difference problem.

Lemma 2.1.

Let a,f∈^C1 [0,[cursive l]] . Then the solution u(x) of problem (1.1)-(1.3) satisfies the inequalities [figure omitted; refer to PDF] where [figure omitted; refer to PDF] [figure omitted; refer to PDF] with [figure omitted; refer to PDF] providing that (∂f/∂x)(x,u) is bounded for x∈[0,[cursive l]] and |u|≤_C0 .

Proof.

We rewrite the problem (1.1)-(1.3) as [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Here we use the Maximum Principle: let L and _L0 be the differential operators in (2.5)-(2.6) and v∈^C2 [0,[cursive l]]. If _L0 v≥0, v([cursive l])≥0, and Lv≤0 for all, then v(x)≥0 for all x<0<[cursive l]. Then, from (2.5)-(2.6) we have the following inequality: [figure omitted; refer to PDF] Next, from boundary condition (2.7), we get [figure omitted; refer to PDF] By setting the value x=_{[cursive l]1} in the inequality (2.9), we obtain [figure omitted; refer to PDF] From (2.10) and (2.11), then we have [figure omitted; refer to PDF] which along with (2.9) leads to (2.1).

After establishing (2.1) the further part of the proof is almost identical to that of [28].

3. Discretization and Mesh

Let _ωN be any nonuniform mesh on [0,[cursive l]] : [figure omitted; refer to PDF] and _ω...N =_ωN ∪{_x0 =0,_xN =[cursive l]} . For each i≥1 we set the stepsize _hi =_xi -_xi-1 . Before describing our numerical method, we introduce some notation for the mesh functions. For any mesh function g(x) defined on _ω...N we use [figure omitted; refer to PDF]

The difference scheme we will construct follows from the identity [figure omitted; refer to PDF] with the basis functions ^{{_{[straight phi]i} (x)}i=1N-1} having the form [figure omitted; refer to PDF] where ^{[straight phi]i(1)} (x) and ^{[straight phi]i(2)} (x) , respectively, are the solutions of the following problems: [figure omitted; refer to PDF] The functions ^{[straight phi]i(1)} (x) and ^{[straight phi]i(2)} (x) can be explicitly expressed as follows: [figure omitted; refer to PDF] The coefficient _χi in (3.3) is given by [figure omitted; refer to PDF] Rearranging (3.3) gives [figure omitted; refer to PDF] with [figure omitted; refer to PDF] As consistent with [26, 27], we obtain the precise relation: [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

It then follows from (3.8) that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and after a simple calculation [figure omitted; refer to PDF]

To define an approximation for the boundary condition (1.2), we proceed our discretization process by [figure omitted; refer to PDF] with [figure omitted; refer to PDF] Here the function _{[straight phi]0} (x) is the solution of the following problem: [figure omitted; refer to PDF] In the analogous way, as in construction of (3.12), we obtain [figure omitted; refer to PDF] where [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF]

Now, it remains to define an approximation for the second boundary condition (1.3). Let _xN0 be the mesh point nearest to _{[cursive l]1} . Then [figure omitted; refer to PDF] where [figure omitted; refer to PDF] being ξ -intermediate point between u(_xN0 ) and u(_{[cursive l]1} ) .

Based on (3.12), (3.18), and (3.22), we propose the following difference scheme for approximating (1.1) and (1.3): [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] where θ, ^θ0(0) , and ^θ0(1) are defined by (3.14), (3.19), and (3.20), respectively.

The difference scheme (3.24)-(3.26) in order to be [straight epsilon] -uniform convergent, we will use the Shishkin mesh. For a divisible by 4 positive integer N , we divide each of the intervals [0,_σ1 ] and [[cursive l]-_σ2 ,[cursive l]] into N/4 equidistant subintervals and also [_σ1 ,[cursive l]-_σ2 ] into N/2 equidistant subintervals, where the transition points _σ1 and _σ2 , which separate the fine and coarse portions of the mesh, are obtained by taking [figure omitted; refer to PDF] where _μ1 and _μ2 are given in Lemma 2.1. In practice one usually has _σi <<[cursive l] (i=1,2) ; so the mesh is fine on [0,_σ1 ], [[cursive l]-_σ2 ,[cursive l]] , and coarse on [_σ1 ,[cursive l]-_σ2 ] . Hence, if we denote by ^h(1) , ^h(2) , and ^h(3) the step-size in [0,_σ1 ], [_σ1 ,[cursive l]-_σ2 ] , and [[cursive l]-_σ2 ,[cursive l]] , respectively, we have [figure omitted; refer to PDF] and so [figure omitted; refer to PDF] In the rest of the paper we only consider this mesh.

We note that on this mesh the coefficient _θi which is defined by (3.14) simplifies to [figure omitted; refer to PDF] For the evaluation of the rest values _θN/4 and _θ3N/4 we will use the form (3.14).

4. Error Analysis

Let z=y-u, x∈_ω...N . Then for the error of the difference scheme (3.24)-(3.26) we get [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] where the truncation errors _Ri , ^r(0) , and ^r(1) are defined by (3.9), (3.21), and (3.23), respectively.

Lemma 4.1.

The solution _zi of problem (4.1)-(4.3) satisfies [figure omitted; refer to PDF]

Proof.

The problem (4.1)-(4.3) can be rewritten as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] _y...0 , _y...N0 , _y...i -intermediate points called for by the mean value theorem.

Since the discrete maximum principle is valid here, we have the proof of (4.4) by analogy with the proof of Lemma 2.1.

Lemma 4.2.

Under the above assumptions of Section 1 and Lemma 2.1, for the error functions _Ri , ^r(0) , and ^r(1) , the following estimates hold: [figure omitted; refer to PDF]

Proof.

From explicit expression (3.9) for _Ri , on an arbitrary mesh we have [figure omitted; refer to PDF] This inequality together with (2.3) enables us to write [figure omitted; refer to PDF] in which [figure omitted; refer to PDF]

We consider first the case _σ1 =_σ2 =[cursive l]/4 , and so [cursive l]/4<^μk-1 [straight epsilon]ln N, k=1,2 , and ^h(1) =^h(2) =^h(3) =h=[cursive l]^N-1 . Hereby, from (4.9) we can write [figure omitted; refer to PDF] Hence [figure omitted; refer to PDF]

We now consider the case _σ1 =^μ1-1 [straight epsilon]ln N and _σ2 =^μ2-1 [straight epsilon]ln N , and so ^μk-1 [straight epsilon]ln N<[cursive l]/4, k=1,2 and estimate _Ri on [0,_σ1 ], [_σ1 ,[cursive l]-_σ2 ] , and [[cursive l]-_σ2 ,[cursive l]] separately. In the layer region [0,_σ1 ] , the inequality (4.9) reduces to [figure omitted; refer to PDF] Hence [figure omitted; refer to PDF]

The same estimate is obtained in the layer region [[cursive l]-_σ2 ,[cursive l]] in the similar manner.

It remains to estimate _Ri for N/4+1≤i≤3N/4-1 . In this case we are able to write (4.9) as [figure omitted; refer to PDF] Since _xi =^μ1-1 [straight epsilon]ln N+(i-N/4)^h(2) , it follows that [figure omitted; refer to PDF] Also, if we rewrite the mesh points in the form _xi =[cursive l]-_σ2 -(3N/4-i)^h(2) , evidently [figure omitted; refer to PDF] The last two inequalities together with (4.15) give the bound [figure omitted; refer to PDF]

It remains to estimate _Ri for the mesh points _xN/4 and _x3N/4 . For the mesh point _xN/4 , inequality (4.9) reduces to [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] it then follows that [figure omitted; refer to PDF]

The same estimate is obtained for i=3N/4 in the similar manner.

The same estimate is valid when only one of the values _σ1 and _σ2 is equal to [cursive l]/4 .

Next, we estimate the remainder term ^r(0) . From the explicit expression (3.21), taking into consideration that ^{(δ+βθ0(1) )-1} ≤^δ-1 and |_{[straight phi]0} (x)|≤1 , we obtain [figure omitted; refer to PDF] Hence [figure omitted; refer to PDF]

Finally, we estimate the remainder term ^r(1) . From the expression (3.23) we obtain [figure omitted; refer to PDF] where we have assumed that _xN0 is on the left-hand side of _{[cursive l]1} (if _xN0 is on right side of _{[cursive l]1} , the integral will be over (_{xN0 -1} ,_xN0 )). In the same manner as above we therefore obtain from here that [figure omitted; refer to PDF]

Thus Lemma 4.2 is proved.

Combining the two previous lemmas gives us the following convergence result.

Theorem 4.3.

Let u(x) be the solution of (1.1)-(1.3) and y the solution (3.24)-(3.26). Then [figure omitted; refer to PDF]

5. Numerical Results

In this section, we present some numerical results which illustrate the present method.

Example 5.1.

Consider the test problem: [figure omitted; refer to PDF]

The exact solution of our test problem is unknown. Therefore, we use a double-mesh method [2] to estimate the errors and compute the experimental rates of convergence in our computed solutions; that is, we compare the computed solution with the solution on a mesh that is twice as fine (for details see [13, 28]). The error estimates obtained in this way are denoted by [figure omitted; refer to PDF] where ^{y...i[straight epsilon],2N} is the approximate solution of the respective method on the mesh [figure omitted; refer to PDF] with [figure omitted; refer to PDF] The convergence rates are [figure omitted; refer to PDF] Approximations to the [straight epsilon] -uniform rates of convergence are estimated from [figure omitted; refer to PDF] The corresponding [straight epsilon] -uniform convergence rates are computed using the formula [figure omitted; refer to PDF]

To solve the nonlinear problem (3.24)-(3.26) we use the following iteration technique: [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF]

To solve (5.8)-(5.10), we take the initial approximation as ^yi(0) =^xi2 and the stopping criterion is [figure omitted; refer to PDF]

The computed maximum pointwise errors ^{e[straight epsilon]N} and ^{e[straight epsilon]2N} , and the orders of uniform convergence ^{p[straight epsilon]N} for different values of [straight epsilon] and N , based on the double-mesh principle are presented in Tables 1 and 2. The results established here are that the discrete solution is uniformly convergent with respect to the perturbation parameter ^pN and the errors are uniformly convergent with rates of almost unity as predicted by our theoretical analysis.

Table 1: For the case of a(x)≠0 , approximate errors ^{e[straight epsilon]N} and ^eN and the computed orders of convergence ^{p[straight epsilon]N} on the piecewise uniform mesh _ωN for various values of [straight epsilon] and N .

[straight epsilon]	N=8	N=16	N=32	N=64	N=128	N=256
10-2	0.02233220	0.01350608	0.00954256	0.00428337	0.00310038	0.00176533
	0.77	0.82	0.86	0.93	0.98	0.99
10-4	0.02233215	0.01376835	0.00850615	0.00538213	0.00315446	0.00160520
	0.75	0.80	0.85	0.92	0.97	0.99
10-6	0.02233210	0.01376065	0.00851454	0.00540203	0.00314356	0.00160544
	0.75	0.80	0.85	0.92	0.97	0.99
10-8	0.02233205	0.01376074	0.00851405	0.00540203	0.00314356	0.00160522
	0.75	0.80	0.85	0.92	0.97	0.99
10-10	0.02233209	0.01376065	0.00851405	0.00540203	0.00314356	0.00160522
	0.75	0.80	0.85	0.92	0.97	0.99
10-12	0.02233209	0.01376065	0.00851405	0.00540203	0.00314356	0.00160522
	0.75	0.80	0.85	0.92	0.97	0.99
10-14	0.02233209	0.01376065	0.00851405	0.00540203	0.00314356	0.00160522
	0.75	0.80	0.85	0.92	0.97	0.99
10-16	0.02233213	0.01376074	0.00851405	0.00540203	0.00314356	0.00160522
	0.75	0.80	0.85	0.92	0.97	0.99
10-18	0.02233213	0.01376074	0.00851405	0.00540203	0.00314356	0.00160522
	0.75	0.80	0.85	0.92	0.97	0.99
10-20	0.02233213	0.01376074	0.00851405	0.00540203	0.00314356	0.00160522
	0.75	0.80	0.85	0.92	0.97	0.99
eN	0.02233220	0.01376835	0.00954256	0.00540203	0.00315446	0.00176533
pN	0.75	0.80	0.85	0.92	0.97	0.99

Table 2: For the case a(0)=0 of approximate errors ^{e[straight epsilon]N} and ^eN and the computed orders of convergence ^{p[straight epsilon]N} on the piecewise uniform mesh _ωN for various values of [straight epsilon] and N .

[straight epsilon]	N=8	N=16	N=32	N=64	N=128	N=256
10-2	0.02453225	0.01721110	0.01123085	0.00652485	0.00439674	0.00189543
	0.68	0.79	0.83	0.88	0.96	0.99
10-4	0.02453220	0.01716585	0.01120973	0.00651194	0.00439686	0.00188515
	0.67	0.76	0.80	0.85	0.94	0.99
10-6	0.02453221	0.01706582	0.01120403	0.00651089	0.0043 9225	0.00188523
	0.67	0.76	0.80	0.85	0.94	0.99
10-8	0.02453215	0.01706553	0.01120282	0.00651046	0.00439214	0.00188517
	0.67	0.76	0.80	0.85	0.94	0.99
10-10	0.02453215	0.01706593	0.01120275	0.00651093	0.00439214	0.00188517
	0.67	0.76	0.80	0.85	0.94	0.99
10-12	0.02453216	0.01706452	0.01120270	0.00651025	0.00439214	0.00188517
	0.67	0.76	0.80	0.85	0.94	0.99
10-14	0.02453209	0.01706425	0.01120256	0.00651025	0.00439214	0.00188517
	0.67	0.76	0.80	0.85	0.94	0.99
10-16	0.02453209	0.01706425	0.01120256	0.00651025	0.00439214	0.00188517
	0.67	0.76	0.80	0.85	0.94	0.99
10-18	0.02453209	0.01706425	0.01120256	0.00651025	0.00439214	0.00188517
	0.67	0.76	0.80	0.85	0.94	0.99
10-20	0.02453209	0.01706425	0.01120256	0.00651025	0.00439214	0.00188517
	0.67	0.76	0.80	0.85	0.94	0.99
eN	0.02453225	0.01716585	0.01123085	0.00652485	0.00439686	0.00189543
pN	0.68	0.79	0.83	0.88	0.96	0.99

Example 5.2.

Consider the test problem: [figure omitted; refer to PDF] The exact solution of our test problem is unknown. In the same manner as above we solve this problem.

Acknowledgment

The authors wish to thank the referees for their suggestions and comments which helped improve the quality of manuscript.

References

[1] E. M. Dejager, J. Furu The Theory of Singular Perturbations , Elsevier Science, 1996.

[2] E. P. Doolan, J. J. H. Miller, W. H. A. Schilders Uniform Numerical Methods for Problems with Initial and Boundary Layers , pp. xv+324, Boole Press, Dublin, Ireland, 1980.

[3] P. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O'Riordan, G. I. Shishkin Robust Computational Techniques for Boundary Layers , vol. 16, of Applied Mathematics, pp. xvi+254, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2000.

[4] A. H. Nayfeh Introduction to Perturbation Techniques , pp. xiv+519, John Wiley & Sons, New York, NY, USA, 1993.

[5] R. E. O'Malley Jr. Singular Perturbation Methods for Ordinary Differential Equations , vol. 89, of Applied Mathematical Sciences, pp. viii+225, Springer, New York, NY, USA, 1991.

[6] H.-G. Roos, M. Stynes, L. Tobiska Robust Numerical Methods for Singularly Perturbed Differential Equations , vol. 24, of Springer Series in Computational Mathematics, pp. xiv+604, Springer, Berlin, Germany, 2008., 2nd.

[7] P. W. Hemker A numerical study of stiff two-point boundary value problems , Ph.D. thesis, Mathematical Center MCT 80, Amsterdam, The Netherlands, 1977.

[8] C. Clavero, J. L. Gracia, F. Lisbona, "High order methods on Shishkin meshes for singular perturbation problems of convection-diffusion type," Numerical Algorithms , vol. 22, no. 1, pp. 73-97, 1999.

[9] P. A. Farrell, E. O'Riordan, G. I. Shishkin, "A class of singularly perturbed semilinear differential equations with interior layers," Mathematics of Computation , vol. 74, no. 252, pp. 1759-1776, 2005.

[10] J. L. Gracia, E. O'Riordan, M. L. Pickett, "A parameter robust second order numerical method for a singularly perturbed two-parameter problem," Applied Numerical Mathematics , vol. 56, no. 7, pp. 962-980, 2006.

[11] E. O'Riordan, M. L. Pickett, G. I. Shishkin, "Parameter-uniform finite difference schemes for singularly perturbed parabolic diffusion-convection-reaction problems," Mathematics of Computation , vol. 75, no. 255, pp. 1135-1154, 2006.

[12] H.-G. Roos, "Ten ways to generate the Il'in and related schemes," Journal of Computational and Applied Mathematics , vol. 53, no. 1, pp. 43-59, 1994.

[13] T. Linß, M. Stynes, "A hybrid difference scheme on a Shishkin mesh for linear convection-diffusion problems," Applied Numerical Mathematics , vol. 31, no. 3, pp. 255-270, 1999.

[14] I. A. Savin, "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 , vol. 35, no. 11, pp. 1417-1422, 1995.

[15] G. F. Sun, M. Stynes, "A uniformly convergent method for a singularly perturbed semilinear reaction-diffusion problem with multiple solutions," Mathematics of Computation , vol. 65, no. 215, pp. 1085-1109, 1996.

[16] T. Linß, "Layer-adapted meshes for convection-diffusion problems," Computer Methods in Applied Mechanics and Engineering , vol. 192, no. 9-10, pp. 1061-1105, 2003.

[17] T. Linß, H.-G. Roos, "Analysis of a finite-difference scheme for a singularly perturbed problem with two small parameters," Journal of Mathematical Analysis and Applications , vol. 289, no. 2, pp. 355-366, 2004.

[18] G. Amiraliyev, M. Çakir, "A uniformly convergent difference scheme for a singularly perturbed problem with convective term and zeroth order reduced equation," International Journal of Applied Mathematics , vol. 2, no. 12, pp. 1407-1419, 2000.

[19] A. M. Il'in, "A difference scheme for a differential equation with a small parameter multiplying the highest derivative," Mathematical Notes of the Academy of Science of the USSR , vol. 6, pp. 237-248, 1969.

[20] R. Ciegis, "The numerical of singularly perturbed non local problem," Lietuvas Matematica Rink , vol. 28, pp. 144-152, 1988.

[21] R. Ciegis, "On the difference schemes for problems with nonlocal boundary conditions," Informatica (Lietuva) , vol. 2, no. 2, pp. 155-170, 1991.

[22] C. P. Gupta, S. I. Trofimchuk, "A sharper condition for the solvability of a three-point second order boundary value problem," Journal of Mathematical Analysis and Applications , vol. 205, no. 2, pp. 586-597, 1997.

[23] B. Liu, "Positive solutions of second-order three-point boundary value problems with change of sign," Computers & Mathematics with Applications , vol. 47, no. 8-9, pp. 1351-1361, 2004.

[24] R. Ma, "Positive solutions for nonhomogeneous m-point boundary value problems," Computers & Mathematics with Applications , vol. 47, no. 4-5, pp. 689-698, 2004.

[25] A. M. Nahushev, "On non local boundary value problems," Differential Equations , vol. 21, pp. 92-101, 1985.

[26] M. P. Sapagovas, R. Yu. Ciegis, "Numerical solution of some nonlocal problems," Lietuvas Matematica Rink , vol. 27, no. 2, pp. 348-356, 1987.

[27] G. M. Amiraliyev, M. Çakir, "Numerical solution of the singularly perturbed problem with nonlocal boundary condition," Applied Mathematics and Mechanics , vol. 23, no. 7, pp. 755-764, 2002.

[28] M. Çakir, G. M. Amiraliyev, "Numerical solution of a singularly perturbed three-point boundary value problem," International Journal of Computer Mathematics , vol. 84, no. 10, pp. 1465-1481, 2007.

[29] G. M. Amiraliyev, H. Duru, "A uniformly convergent difference method for the periodical boundary value problem," Computers & Mathematics with Applications , vol. 46, no. 5-6, pp. 695-703, 2003.