On a difference scheme of second order of accuracy for the Bitsadze-Samarskii type nonlocal boundary-value problem
© Ashyralyev and Ozturk; licensee Springer. 2014
Received: 11 October 2013
Accepted: 16 December 2013
Published: 13 January 2014
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.
Keywordswell-posedness difference scheme elliptic equation
is a twice continuously differentiable on the segment .
The element belongs to for all , and the function is continuous on the segment .
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
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 .
where M does not depend on τ.
The estimate (6) follows from the spectral representation of A and the Cauchy inequality.
Theorem 2 is proved. □
Theorem 3 is proved. □
From the last estimate and the estimate (15) follows the estimate (12). Theorem 4 is proved. □
Combining estimates for , we get the estimate (19). Theorem 5 is proved. □
for the numerical solution of (21).
and on the symmetry properties of the difference operator defined by the formula (22) in .
The proof of Theorem 7 is based on Theorem 5 and the symmetry properties of the difference operator defined by formula (22).
for the numerical solution of (26).
The proof of Theorem 8 is based on Theorem 3 and the symmetry properties of the difference operator defined by (27) in .
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 .
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
The errors for first- and second-order difference scheme
N = M = 20
N = M = 40
N = M = 80
First-order difference scheme
Second-order difference scheme
6.245e − 004
1.562e − 004
3.906e − 005
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.
The authors would like to thank Prof. Dr. PE Sobolevskii for his helpful suggestions on the improvement of this paper.
- Bitsadze AV, Samarskii AA: On some simplest generalizations of linear elliptic problems. Dokl. Akad. Nauk SSSR 1969, 185(4):739-740.MathSciNetGoogle Scholar
- Samarskii AA: Some problems in differential equation theory. Differ. Uravn. 1980, 16(11):1925-1935.MathSciNetGoogle Scholar
- Sapagovas MP: Difference method of increased order of accuracy for the Poisson equation with nonlocal conditions. Differ. Equ. 2008, 44(7):1018-1028. 10.1134/S0012266108070148MATHMathSciNetView ArticleGoogle Scholar
- Ashyralyev A: A note on the Bitsadze-Samarskii type nonlocal boundary value problem in a Banach space. J. Math. Anal. Appl. 2008, 344(1):557-573. 10.1016/j.jmaa.2008.03.008MATHMathSciNetView ArticleGoogle Scholar
- Ashyralyev A, Ozturk E: The numerical solution of Bitsadze-Samarskii nonlocal boundary value problems with the Dirichlet-Neumann condition. Abstr. Appl. Anal. 2012., 2012: Article ID 730804Google Scholar
- 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. 2013, 36(8):936-955. 10.1002/mma.2650MATHMathSciNetView ArticleGoogle Scholar
- Ashyralyev A, Ozturk E: On Bitsadze-Samarskii type nonlocal boundary value problems for elliptic differential and difference equations: well-posedness. Appl. Math. Comput. 2012, 219(3):1093-1107. 10.1016/j.amc.2012.07.016MATHMathSciNetView ArticleGoogle Scholar
- Volkova EA, Dosiyev AA, Buranay SC: On the solution of a nonlocal problem. Comput. Math. Appl. 2013, 66(3):330-338. 10.1016/j.camwa.2013.05.010MathSciNetView ArticleGoogle Scholar
- Ashyralyev A, Tetikoğlu FS: A note on Bitsadze-Samarskii type nonlocal boundary value problems: well-posedness. Numer. Funct. Anal. Optim. 2013, 34(9):939-975. 10.1080/01630563.2012.738458MATHMathSciNetView ArticleGoogle Scholar
- Berikelashvili G: On a nonlocal boundary value problem for a two-dimensional elliptic equation. Comput. Methods Appl. Math. 2003, 3(1):35-44.MATHMathSciNetGoogle Scholar
- Gordeziani DG: On a method of resolution of Bitsadze-Samarskii boundary value problem. Abst. Rep. Inst. Appl. Math. Tbilisi State Univ. 1970, 2: 38-40.Google Scholar
- Kapanadze DV: On the Bitsadze-Samarskii nonlocal boundary value problem. Differ. Equ. 1987, 23(3):543-545.MATHMathSciNetGoogle Scholar
- Il’in VA, Moiseev EI: Two-dimensional nonlocal boundary value problems for Poisson’s operator in differential and difference variants. Mat. Model. 1990, 1(2):139-159.MathSciNetGoogle Scholar
- 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. 2003, 39(7):945-953.MATHMathSciNetView ArticleGoogle Scholar
- Skubaczewski AL: Solvability of elliptic problems with Bitsadze-Samarskii boundary conditions. Differ. Uravn. 1985, 21(4):701-706.Google Scholar
- Ashyralyev A, Tetikoğlu FS: FDM for elliptic equations with Bitsadze-Samarskii-Dirichlet conditions. Abstr. Appl. Anal. 2012., 2012: Article ID 454831Google Scholar
- Ladyzhenskaya OA, Ural’tseva NN: Linear and Quasilinear Equations of Elliptic Type. Nauka, Moscow; 1973. (Russian)MATHGoogle Scholar
- Gorbachuk VL, Gorbachuk ML: Boundary Value Problems for Differential-Operator Equations. Naukova Dumka, Kiev; 1984. (Russian)Google Scholar
- Grisvard P: Elliptic Problems in Nonsmooth Domains. Pitman, London; 1985.MATHGoogle Scholar
- Agmon S: Lectures on Elliptic Boundary Value Problems. Van Nostrand, Princeton; 1965.MATHGoogle Scholar
- Krein SG: Linear Differential Equations in Banach Space. Nauka, Moscow; 1966. (Russian)Google Scholar
- Skubachevskii AL Operator Theory - Advances and Applications. In Elliptic Functional Differential Equations and Applications. Birkhäuser, Basel; 1997.Google Scholar
- Agarwal R, Bohner M, Shakhmurov VB: Maximal regular boundary value problems in Banach-valued weighted spaces. Bound. Value Probl. 2005, 18: 9-42.MathSciNetGoogle Scholar
- Sobolevskii PE: Well-posedness of difference elliptic equations. Discrete Dyn. Nat. Soc. 1997, 1(4):219-231.MATHView ArticleGoogle Scholar
- Ashyralyev A: Well-posed solvability of the boundary value problem for difference equations of elliptic type. Nonlinear Anal., Theory Methods Appl. 1995, 24(2):251-256. 10.1016/0362-546X(94)E0003-YMATHMathSciNetView ArticleGoogle Scholar
- Ashyralyev A, Altay N: A note on the well-posedness of the nonlocal boundary value problem for elliptic difference equations. Appl. Math. Comput. 2006, 175(1):49-60. 10.1016/j.amc.2005.07.013MATHMathSciNetView ArticleGoogle Scholar
- 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. 1964, 17: 35-92. 10.1002/cpa.3160170104MATHMathSciNetView ArticleGoogle Scholar
- Sobolevskii PE: On elliptic equations in a Banach space. Differ. Uravn. 1969, 4(7):1346-1348. (Russian)MathSciNetGoogle Scholar
- Ashyralyev A, Sobolevskii PE: New Difference Schemes for Partial Differential Equations. Birkhäuser, Basel; 2004.MATHView ArticleGoogle Scholar
- Sobolevskii PE: On Difference Method for Approximate Solution of Differential Equations. Izdat. Voronezh. Gosud. Univ., Voronezh; 1975.Google Scholar
- Ozturk, E: Nonlocal boundary value problems for elliptic differential and difference equations. PhD thesis, Mathematics Department, Uludag University (2013) (Turkish)Google Scholar
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