Research

# On a difference scheme of second order of accuracy for the Bitsadze-Samarskii type nonlocal boundary-value problem

Allaberen Ashyralyev1 and Elif Ozturk2*

Author Affiliations

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

2 Department of Econometrics, Canakkale Onsekiz Mart University, Canakkale, 17200, Turkey

For all author emails, please log on.

Boundary Value Problems 2014, 2014:14  doi:10.1186/1687-2770-2014-14

 Received: 11 October 2013 Accepted: 16 December 2013 Published: 13 January 2014

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this study, the Bitsadze-Samarskii type nonlocal boundary-value problem with integral condition for an elliptic differential equation in a Hilbert space H with self-adjoint positive definite operator A is considered. The second order of the accuracy difference scheme for the approximate solutions of this nonlocal boundary-value problem is presented. The well-posedness of this difference scheme in Hölder spaces with a weight is proved. The theoretical statements for the solution of this difference scheme are supported by the results of numerical example.

##### Keywords:
well-posedness; difference scheme; elliptic equation

### 1 Introduction

In 1969 Bitsadze and Samarskii [1] stated and studied a new problem in which a nonlocal condition is related to the values of the solution on parts of the boundary and on an interior curve for a uniformly elliptic equation. Furthermore, in [2-16] the Bitsadze-Samarskii type nonlocal boundary-value problems were investigated for the various differential and difference equations of elliptic type. The role played by coercive inequalities in the study of local boundary-value problems for elliptic differential equations is well known [17]. Methods of solutions of elliptic differential and difference equations have been studied extensively by many researchers (see [18-27] and the references therein). In the present paper we consider the Bitsadze-Samarskii type nonlocal boundary-value problem with integral condition,

(1)

for the differential equation of elliptic type in a Hilbert space H with the self-adjoint positive definite operator A with a closed domain . Here, let be a given abstract continuous function defined on with values in H, φ, and ψ are elements of and is a scalar function. A function is called a solution of problem (1) if the following conditions are satisfied:

i. is a twice continuously differentiable on the segment .

ii. The element belongs to for all , and the function is continuous on the segment .

iii. satisfies the equation and nonlocal boundary conditions (1).

The paper is organized as follows. In Section 2 the second order of the accuracy difference scheme for the approximate solution (1) is presented. The stability, the almost coercive stability, and the coercive stability estimates for the solution of the difference scheme for an approximate solution of the nonlocal boundary-value problem with integral condition for elliptic equations are obtained. Section 3 contains the applications of Section 2. The final section is devoted to the numerical result. Theoretical statements for the solution of the second order of the accuracy difference scheme is supported by a numerical experiment.

### 2 The second order of the accuracy difference scheme

Let us associate the nonlocal boundary-value problem (1) with the corresponding difference problem,

(2)

We will study the problem (2) under the following assumption:

(3)

It is well known [28] that for a self-adjoint positive definite operator A it follows that is self-adjoint positive definite and , which is defined on the whole space H is a bounded operator. Here, I is the unit operator. Furthermore, we have

(4)

In this paper, positive constants, which can differ in time (hence they are not a subject of precision considerations) will be indicated with M. On the other hand is used to focus on the fact that the constant depends only on  .

Lemma 1The operator

has an inverse

and the following estimate is satisfied:

(5)

whereMdoes not depend onτ.

The proof of the estimate (5) is based on the estimate

(6)

Here

The estimate (6) follows from the spectral representation of A and the Cauchy inequality.

Theorem 2For any, , the solution of the problem (2) exists and the following formula holds:

(7)

for,

for.

Proof

(8)

has a solution and the following formula holds [29]:

(9)

Applying formula (9) and the nonlocal boundary condition

we obtain

Since the operator

has an inverse , it follows that

Theorem 2 is proved. □

Let be the linear space of the mesh functions with values in the Hilbert space H. We denote by and , , Banach spaces with the norms

The nonlocal boundary-value problem (2) is said to be stable in if we have the inequality

Theorem 3The solutions of the difference scheme (2) under the assumption (3) satisfy the stability estimate

(10)

Proof By [29],

(11)

is proved for the solution of difference scheme (8). Then the proof of (10) is based on (11) and on the estimate

Using the formula (7) and the estimates (4), (5), we get

Theorem 3 is proved. □

Theorem 4The solutions of the difference problem (2) inunder the assumption (3) obey the almost coercive inequality

Proof By [29],

is proved for the solution of the boundary-value problem (8). Using the estimates (4), (5) and the formula (7), we obtain

(12)

for the solution of difference scheme (2). Applying formula (7) and , we get

where

(13)

(14)

To this end it suffices to show that

(15)

and

(16)

The estimate (15) follows from formula (13) and the estimates (4), (5). Using formula (14) and the estimates (4), (5), we obtain

From the last estimate and the estimate (15) follows the estimate (12). Theorem 4 is proved. □

Theorem 5The difference problem (2) is well posed in the Hölder spacesunder the assumption (3) and the following coercivity inequality holds:

(17)

Proof By [29],

(18)

is proved for the solution of difference scheme (8). Then the proof of (17) is based on (18) and on the estimate

Applying the triangle inequality, formula (7) and the estimate (15), we get

To this end it suffices to show that

(19)

Applying formula (14), we get

(20)

where

Second, let us estimate for any separately. We start with . Using estimates (4), (5) and the definition of the norm of the space , we get

From (3) it follows that

Now, let us estimate . Using estimates (4), (5) and the definition of the norm of the space , we obtain

The sum

is the lower Darboux integral sum for the integral

It follows that

By the lower Darboux integral sum for the integral it concludes that

For , applying (4), (5) and the definition of the norm of the space , we get

The sum

is the lower Darboux integral sum for the integral

Since

it follows that

By the lower Darboux integral sum for the integral it follows that

Combining and , we get

From (3) it follows that

Next, let us estimate . Using the estimates (4), (5), and the definition of the norm space , we obtain

The sum

is the lower Darboux integral sum for the integral

Since

it follows that

By the lower Darboux integral sum for the integral it follows that

Finally, let us estimate . Using the estimates (4), (5), and the definition of the norm space , we get

The sum

is the lower Darboux integral sum for the integral

Thus, we show that

By the lower Darboux integral sum for the integral it follows that

Combining and , we get

From (3) it follows that

Combining estimates for , we get the estimate (19). Theorem 5 is proved. □

### 3 Applications

Now, the application of Theorems 3-5 will be given. First, we consider the mixed boundary-value problem for elliptic equation

(21)

where , , and are given sufficiently smooth functions and , , . The discretization of problem (21) is carried out in two steps. In the first step, let us define the grid space

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

To the differential operator A generated by the problem (21) we assign the difference operator by the formula

(22)

acting in the space of the grid functions satisfying the conditions , . It is know that is a self-adjoint positive definite operator in . With the help of , we arrive at the nonlocal boundary-value problem

(23)

for an infinite system of ordinary differential equations. Therefore, in the second step, equation (23) is replaced by the difference scheme (2), and we get the following difference scheme:

(24)

for the numerical solution of (21).

Theorem 6Letτandhbe sufficiently small positive numbers. Then under the assumption (3), the solution of the difference scheme (24) satisfies the following stability and almost coercivity estimates:

The proof of Theorem 6 is based on Theorems 3 and 4, on the estimate

(25)

and on the symmetry properties of the difference operator defined by the formula (22) in .

Theorem 7Letτandbe sufficiently small positive numbers. Then under the assumption (3), the solution of the difference scheme (24) satisfies the following coercivity estimate:

The proof of Theorem 7 is based on Theorem 5 and the symmetry properties of the difference operator defined by formula (22).

Second, let Ω be the unit open cube in with boundary S, . In , the Dirichlet-Bitsadze-Samarskii type mixed boundary-value problem for the multidimensional elliptic equation

(26)

is considered. We will study the problem (26) under the assumption (3). Here, (), , () and (, ) are smooth functions and . The discretization of problem (26) is carried out in two steps. In the first step let us define the grid sets

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

To the differential operator A generated by the problem (26), we assign the difference operator by the formula

(27)

acting in the space of the grid functions , satisfying the conditions for all . 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

(28)

In the second step, (28) is replaced by the difference scheme (2), and we get the following difference scheme:

(29)

for the numerical solution of (26).

Theorem 8Letτandbe sufficiently small positive numbers. Then under the assumption (3) the solution of the difference scheme (29) satisfies the following stability estimates:

The proof of Theorem 8 is based on Theorem 3 and the symmetry properties of the difference operator defined by (27) in .

Theorem 9Letτandbe sufficiently small positive numbers. Then under the assumption (3) the solution of the difference scheme (29) satisfies the following almost coercivity estimates:

The proof of Theorem 9 is based on Theorem 4, on the estimate (25), on the symmetry properties of the difference operator defined by (27) in , and on the following theorem on the coercivity inequality for the solution of the elliptic difference problem in .

Theorem 10For the solutions of the elliptic difference problem

the following coercivity inequality holds[30]:

Theorem 11Letτandbe sufficiently small positive numbers. Then under the assumption (3) the solution of the difference scheme (29) satisfies the following coercivity stability estimate:

The proof of Theorem 11 is based on Theorem 5, on the symmetry properties of the difference operator defined by the formula (27), and on Theorem 10 on the coercivity inequality for the solution of the elliptic difference equation in .

### 4 Numerical results

We consider the Bitsadze -Samarskii type nonlocal boundary problem for the elliptic equation

(30)

The exact solution of this problem is

In the present part for the approximate solutions of the Bitsadze-Samarskii type nonlocal boundary-value problem (30), we will use the first and second orders of the accuracy difference schemes with grid intervals , for t and x, respectively. For the approximate solution of the nonlocal boundary Bitsadze-Samarskii type problem (30), we consider the set of a family of grid points depending on the small parameters τ and h,

Applying the first order of the accuracy difference scheme from [31] and the second order of the accuracy difference scheme (2) for the approximate solutions of the problem, we have the second-order difference equations with respect to n with matrix coefficients. To solve these difference equations, we have applied the procedure of a modified Gauss elimination method for the difference equations with respect to n with matrix coefficients. To obtain the solution of (2), we use MATLAB programming. The errors are computed by

of numerical solutions for different values of M and N, where represents the exact solution and represents the numerical solution at . The results are shown in Table 1, respectively.

Table 1. The errors for first- and second-order difference scheme

### 5 Conclusion

In this paper, the second order of the accuracy difference scheme for the approximate solution of the Bitsadze-Samarskii type nonlocal boundary-value problem with the integral condition for elliptic equations is presented. Theorems on the stability estimates, almost coercive stability estimates, and coercive stability estimates for the solution of difference scheme for elliptic equations are proved. The theoretical statements for the solution of this difference scheme are supported by the result of a numerical example. As can be seen from Table 1, the second order of the accuracy difference scheme is more accurate than the first order of the accuracy difference scheme.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

EO carried out the studies, participated in the sequence alignment and drafted the manuscript and AA carried out the studies, participated in the sequence alignment. All authors read and approved the final manuscript.

### Acknowledgements

The authors would like to thank Prof. Dr. PE Sobolevskii for his helpful suggestions on the improvement of this paper.

### References

1. Bitsadze, AV, Samarskii, AA: On some simplest generalizations of linear elliptic problems. Dokl. Akad. Nauk SSSR. 185(4), 739–740 (1969)

2. Samarskii, AA: Some problems in differential equation theory. Differ. Uravn.. 16(11), 1925–1935 (1980)

3. Sapagovas, MP: Difference method of increased order of accuracy for the Poisson equation with nonlocal conditions. Differ. Equ.. 44(7), 1018–1028 (2008). Publisher Full Text

4. Ashyralyev, A: A note on the Bitsadze-Samarskii type nonlocal boundary value problem in a Banach space. J. Math. Anal. Appl.. 344(1), 557–573 (2008). Publisher Full Text

5. Ashyralyev, A, Ozturk, E: The numerical solution of Bitsadze-Samarskii nonlocal boundary value problems with the Dirichlet-Neumann condition. Abstr. Appl. Anal.. 2012, Article ID 730804 (2012)

6. Ashyralyev, A, Ozturk, E: On a difference scheme of fourth-order of accuracy for the Bitsadze-Samarskii type nonlocal boundary value problem. Math. Methods Appl. Sci.. 36(8), 936–955 (2013). Publisher Full Text

7. Ashyralyev, A, Ozturk, E: On Bitsadze-Samarskii type nonlocal boundary value problems for elliptic differential and difference equations: well-posedness. Appl. Math. Comput.. 219(3), 1093–1107 (2012). Publisher Full Text

8. Volkova, EA, Dosiyev, AA, Buranay, SC: On the solution of a nonlocal problem. Comput. Math. Appl.. 66(3), 330–338 (2013). Publisher Full Text

9. Ashyralyev, A, Tetikoğlu, FS: A note on Bitsadze-Samarskii type nonlocal boundary value problems: well-posedness. Numer. Funct. Anal. Optim.. 34(9), 939–975 (2013). Publisher Full Text

10. Berikelashvili, G: On a nonlocal boundary value problem for a two-dimensional elliptic equation. Comput. Methods Appl. Math.. 3(1), 35–44 (2003)

11. Gordeziani, DG: On a method of resolution of Bitsadze-Samarskii boundary value problem. Abst. Rep. Inst. Appl. Math. Tbilisi State Univ.. 2, 38–40 (1970)

12. Kapanadze, DV: On the Bitsadze-Samarskii nonlocal boundary value problem. Differ. Equ.. 23(3), 543–545 (1987)

13. Il’in, VA, Moiseev, EI: Two-dimensional nonlocal boundary value problems for Poisson’s operator in differential and difference variants. Mat. Model.. 1(2), 139–159 (1990)

14. Berikelashvili, GK: On the convergence rate of the finite-difference solution of a nonlocal boundary value problem for a second-order elliptic equation. Differ. Equ.. 39(7), 945–953 (2003)

15. Skubaczewski, AL: Solvability of elliptic problems with Bitsadze-Samarskii boundary conditions. Differ. Uravn.. 21(4), 701–706 (1985)

16. Ashyralyev, A, Tetikoğlu, FS: FDM for elliptic equations with Bitsadze-Samarskii-Dirichlet conditions. Abstr. Appl. Anal.. 2012, Article ID 454831 (2012)

17. Ladyzhenskaya, OA, Ural’tseva, NN: Linear and Quasilinear Equations of Elliptic Type, Nauka, Moscow (1973) (Russian)

18. Gorbachuk, VL, Gorbachuk, ML: Boundary Value Problems for Differential-Operator Equations, Naukova Dumka, Kiev (1984) (Russian)

19. Grisvard, P: Elliptic Problems in Nonsmooth Domains, Pitman, London (1985)

20. Agmon, S: Lectures on Elliptic Boundary Value Problems, Van Nostrand, Princeton (1965)

21. Krein, SG: Linear Differential Equations in Banach Space, Nauka, Moscow (1966) (Russian)

22. Skubachevskii, AL: Elliptic Functional Differential Equations and Applications, Birkhäuser, Basel (1997)

23. Agarwal, R, Bohner, M, Shakhmurov, VB: Maximal regular boundary value problems in Banach-valued weighted spaces. Bound. Value Probl.. 18, 9–42 (2005)

24. Sobolevskii, PE: Well-posedness of difference elliptic equations. Discrete Dyn. Nat. Soc.. 1(4), 219–231 (1997)

25. Ashyralyev, A: Well-posed solvability of the boundary value problem for difference equations of elliptic type. Nonlinear Anal., Theory Methods Appl.. 24(2), 251–256 (1995). Publisher Full Text

26. Ashyralyev, A, Altay, N: A note on the well-posedness of the nonlocal boundary value problem for elliptic difference equations. Appl. Math. Comput.. 175(1), 49–60 (2006). Publisher Full Text

27. Agmon, S, Douglis, SA, Nirenberg, L: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Commun. Pure Appl. Math.. 17, 35–92 (1964). Publisher Full Text

28. Sobolevskii, PE: On elliptic equations in a Banach space. Differ. Uravn.. 4(7), 1346–1348 (Russian) (1969)

29. Ashyralyev, A, Sobolevskii, PE: New Difference Schemes for Partial Differential Equations, Birkhäuser, Basel (2004)

30. Sobolevskii, PE: On Difference Method for Approximate Solution of Differential Equations, Izdat. Voronezh. Gosud. Univ., Voronezh (1975)

31. Ozturk, E: Nonlocal boundary value problems for elliptic differential and difference equations. PhD thesis, Mathematics Department, Uludag University (2013) (Turkish)