Abstract and Applied Analysis

Volume 2012 (2012), Article ID 730804, 13 pages

doi:10.1155/2012/730804

Research Article

The Numerical Solution of the Bitsadze-Samarskii Nonlocal Boundary Value Problems with the Dirichlet-Neumann Condition

Allaberen Ashyralyev1,2 and Elif Ozturk³

¹Department of Mathematics, Fatih University, 34500 Istanbul, Turkey

²Department of Mathematics, International Turkmen-Turkish University, 74400 Ashgabat, Turkmenistan

³Department of Mathematics, Uludag University, 16059 Bursa, Turkey

Received 19 March 2012; Accepted 1 May 2012

Academic Editor: Valery  Covachev

Copyright © 2012 Allaberen Ashyralyev and Elif Ozturk. 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.

Abstract.

We are interested in studying the stable difference schemes for the numerical solution of the nonlocal boundary value problem with the Dirichlet-Neumann condition for the multidimensional elliptic equation. The first and second orders of accuracy difference schemes are presented. A procedure of modified Gauss elimination method is used for solving these difference schemes for the two-dimensional elliptic differential equation. The method is illustrated by numerical examples.

1. Introduction

Methods of solution of the Bitsadze-Samarskii nonlocal boundary value problems for elliptic differential equations have been studied extensively by many researchers (see [1–22] and the references given therein).

Let   be the unit open cube in with boundary . In , the Bitsadze-Samarskii-type nonlocal boundary value problem for the multidimensional elliptic equation

is considered. Here , and are given smooth functions, , is a positive number, and is the normal vector to . We are interested in studying the stable difference schemes for the numerical solution of the nonlocal boundary value problem (1). The first and second orders of accuracy difference schemes are presented. The stability and almost coercive stability of these difference schemes are established. A procedure of modified Gauss elimination method is used for solving these difference schemes in the case of two-dimensional elliptic partial differential equations.

2. Difference Schemes: The Stability and Coercive Stability Estimates

The discretization of problem (1) is carried out in two steps. In the first step, let us define the grid sets

We introduce the Hilbert space and of the grid functions defined on , equipped with the norms

To the differential operator generated by problem (1), we assign the difference operator by the formula

acting in the space of grid functions , satisfying the conditions for all and for all . Here, is an approximation to . It is known that is a self-adjoint positive definite operator in . With the help of , we arrive at the nonlocal boundary value problem

for an infinite system of ordinary differential equations. In the second step, we replace problem (2.4) by the first and second orders of accuracy difference schemes

To formulate our result on well-posedness, we will give definition of and . Let be the linear space of mesh functions with values in the Hilbert space . We denote normed space with the norm

and normed space with the norm

Theorem 2.1. Let and be sufficiently small positive numbers. Then, the solutions of difference schemes (2.5) and (2.6) satisfy the following stability and almost coercive stability estimates

Here, and do not depend on,,, and ,.

Theorem 2.2. Let and be sufficiently small positive numbers. Then, the solution of difference schemes (2.5) and (2.6) satisfies the following coercive stability estimate:

is independent of ,, and .

Proofs of Theorems 2.1 and 2.2 are based on the symmetry properties of operator defined by formula (2.3) and on the following formulas:

for difference scheme (2.5), and

for difference scheme (2.6). Here,

and on the following theorem on the coercivity inequality for the solution of the elliptic difference problem in .

Theorem 2.3 (see [22]). For the solution of the elliptic difference problem

the following coercivity inequality holds:

where does not depend on and .

Note that we have not been able to obtain sharp estimate for the constants figuring in the stability estimates. Hence, in the following section, we study difference schemes (2.5) and (2.6) by numerical experiments.

3. Numerical Results

For the numerical result, we consider the nonlocal boundary value problem

for the elliptic equation. The exact solution of (3.1) is

For the approximate solution of the nonlocal boundary Bitsadze-Samarskii problem (3.1), we consider the set of a family of grid points depending on the small parameters and

Firstly, applying difference scheme (2.5), we present the first order of accuracy difference scheme for the approximate solution of problem (3.1) is

Then, we have an system of linear equations and we will write them in the matrix form

where

and is an identity matrix and

where ,

Here,

So, we have a second-order difference equation with respect to matrix coefficients. To solve this difference equation, we have applied a procedure of modified Gauss elimination method for difference equation with respect to matrix coefficients. Hence, we seek a solution of the matrix equation in the following form:

where are square matrix and are column matrix and is the zero matrix and is the zero matrix. Secondly, applying difference scheme (2.6), we present the following second order of accuracy difference scheme for the approximate solutions of problem (3.1):

So, we have again an system of linear equations and we will write in the matrix form

where

where and .

Here,

Thus, we have a second-order difference equation with respect to matrix coefficients. To solve this difference equation, we have applied the same procedure of modified Gauss elimination method (3.10) for difference equation with respect to matrix coefficients with

Now, we will give the results of the numerical analysis. The errors computed by

of the numerical solutions for different values of and , where represents the exact solution and represents the numerical solution at . Table 1 gives the error analysis between the exact solution and solutions derived by difference schemes for , 40, and 60, respectively.

Table 1: Error analysis.

Difference schemes

Difference scheme (2.5)
0.0049
0.0025
0.0012

Difference scheme (2.6)

4. Conclusion

In this work, the first and second orders of accuracy difference schemes for the approximate solution of the Bitsadze-Samarskii nonlocal boundary value problem for elliptic equations are presented. Theorems on the stability estimates, almost coercive stability estimates, and coercive stability estimates for the solution of difference schemes for elliptic equations are proved. The theoretical statements for the solution of these difference schemes are supported by the results of numerical examples. The second order of accuracy difference scheme is more accurate comparing with the first order of accuracy difference scheme. As a future work, high orders of accuracy difference schemes for the approximate solutions of this problem could be established. Theorems on the stability estimates, almost coercive stability estimates, and coercive stability estimates for the solution of difference schemes for elliptic equations could be proved.

References

S. Agmon, Lectures on Elliptic Boundary Value Problems, D. Van Nostrand, London, UK, 1965.

S. G. Krein, Linear Differential Equations in Banach Space, Nauka, Moscow, Russia, 1966.

A. L. Skubachevskii, Elliptic Functional-Differential Equations and Applications, vol. 91 of Operator Theory: Advances and Applications, Birkhäuser, Basel, Switzerland, 1997.

A. V. Bitsadze and A. A. Samarskiĭ, “Some elementary generalizations of linear elliptic boundary value problems,” Doklady Akademii Nauk SSSR, vol. 185, pp. 739–740, 1969.

P. E. Sobolevskii, “Well-posedness of difference elliptic equation,” Rendiconti dell'Istituto di Matematica dell'Università di Trieste, vol. 28, no. suppl., pp. 337–382, 1996.

A. Ashyralyev, “Well-posed solvability of the boundary value problem for difference equations of elliptic type,” Nonlinear Analysis. Theory, Methods & Applications, vol. 24, no. 2, pp. 251–256, 1995.

A. Ashyralyev and P. E. Sobolevskii, New Difference Schemes for Partial Differential Equations, vol. 148 of Operator Theory: Advances and Applications, Birkhäuser, Basel, Switzerland, 2004.

S. Agmon, A. Douglis, and L. Nirenberg, “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions—II,” Communications on Pure and Applied Mathematics, vol. 17, pp. 35–92, 1964.

G. Berikelashvili, “On a nonlocal boundary-value problem for two-dimensional elliptic equation,” Computational Methods in Applied Mathematics, vol. 3, no. 1, pp. 35–44, 2003.

D. G. Gordeziani, “On a method of resolution of Bitsadze-Samarskii boundary value problem,” Abstracts of Reports of Institute of Applied Mathematics, Tbilisi State University, vol. 2, pp. 38–40, 1970.

D. V. Kapanadze, “On a nonlocal Bitsadze-Samarskiĭ boundary value problem,” Differentsial'nye Uravneniya, vol. 23, no. 3, pp. 543–545, 1987.

V. A. Il'in and E. I. Moiseev, “A two-dimensional nonlocal boundary value problem for the Poisson operator in the differential and the difference interpretation,” Matematicheskoe Modelirovanie, vol. 2, no. 8, pp. 139–156, 1990.

A. Ashyralyev, “A note on the Bitsadze-Samarskii type nonlocal boundary value problem in a Banach space,” Journal of Mathematical Analysis and Applications, vol. 344, no. 1, pp. 557–573, 2008.

A. P. Soldatov, “A problem of Bitsadze-Samarskiĭ type for second-order elliptic systems on the plane,” Rossiĭskaya Akademiya Nauk. Doklady Akademii Nauk, vol. 410, no. 5, pp. 607–611, 2006.

R. P. Agarwal, D. O'Regan, and V. B. Shakhmurov, “-separable boundary value problems in Banach-valued function spaces,” Applied Mathematics and Computation, vol. 210, no. 1, pp. 48–63, 2009.

V. Shakhmurov, “Linear and nonlinear abstract equations with parameters,” Nonlinear Analysis. Theory, Methods & Applications, vol. 73, no. 8, pp. 2383–2397, 2010.

A. A. Dosiev and B. S. Ashirov, “On a numerical solution of a multipoint problem for a second-order ordinary differential equation with singular coefficients,” in Approximate Methods for Solving Operator Equations, pp. 41–46, Bakinskii Gosudarstvennogo Universiteta, Baku, Azerbaijan, 1991.

A. Ashyralyev, “Nonlocal boundary-value problems for abstract elliptic equations: well-posedness in Bochner spaces,” AIP Conference Proceedings, vol. 1309, pp. 66–85, 2010.

M. P. Sapagovas, “A difference method of increased order of accuracy for the Poisson equation with nonlocal conditions,” Differentsial'nye Uravneniya, vol. 44, no. 7, pp. 988–998, 2008.

A. Ashyralyev, C. Cuevas, and S. Piskarev, “On well-posedness of difference schemes for abstract elliptic problems in spaces,” Numerical Functional Analysis and Optimization, vol. 29, no. 1-2, pp. 43–65, 2008.

A. Ashyralyev, “Well-posedness of the difference schemes for elliptic equations in spaces,” Applied Mathematics Letters, vol. 22, no. 3, pp. 390–395, 2009.

P. E. Sobolevskii, Difference Methods for the Approximate Solution of Differential Equations, Voronezh State University Press, Voronezh, Russia, 1975.