About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2010 (2010), Article ID 705172, 17 pages
http://dx.doi.org/10.1155/2010/705172
Research Article

On Second Order of Accuracy Difference Scheme of the Approximate Solution of Nonlocal Elliptic-Parabolic Problems

1Department of Mathematics, Fatih University, Buyukcekmece, Istanbul 34500, Turkey
2Department of Mathematics, ITTU, Ashgabat 744012, Turkmenistan
3Vocational School, Fatih University, Buyukcekmece, Istanbul 34500, Turkey
4Department of Mathematics, Yildiz Technical University, 34210 Istanbul, Turkey

Received 16 February 2010; Accepted 15 May 2010

Academic Editor: Ağacik Zafer

Copyright © 2010 Allaberen Ashyralyev and Okan Gercek. 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

A second order of accuracy difference scheme for the approximate solution of the abstract nonlocal boundary value problem , , , , for differential equations in a Hilbert space with a self-adjoint positive definite operator is considered. The well posedness of this difference scheme in Hölder spaces is established. In applications, coercivity inequalities for the solution of a difference scheme for elliptic-parabolic equations are obtained and a numerical example is presented.

1. Introduction

The role played by coercive inequalities in the study of boundary value problems for elliptic and parabolic partial differential equations is well known (see [14]).

Nonlocal problems are widely used for mathematical modeling of various processes of physics, biology, chemistry, ecology, engineering, and industry when it is impossible to determine the boundary or initial values of the unknown function. Theory and numerical methods of solutions of the nonlocal boundary value problems for partial differential equations of variable type have been studied extensively by many researchers (see, e.g., [534] and the references therein).

In paper [35], the nonlocal boundary value problem for the differential equation in a Hilbert space with the self-adjoint positive definite operator was considered. The well posedness of problem (1.1) in Hölder spaces was established. The first order of accuracy difference scheme for approximate solutions of nonlocal boundary value problem (1.1) was presented. In applications, the coercivity inequalities for solutions of difference schemes for elliptic-parabolic equations were obtained.

In the present paper, the second order of accuracy difference scheme generated by Crank-Nicholson difference scheme for the approximate solution of problem (1.1) is presented. The well posedness of difference scheme (1.2) in Hölder spaces is established. As an application, coercivity inequalities for solutions of difference schemes for elliptic-parabolic equations are obtained. A numerical example is given.

2. The Formula for the Solution of Problem (1.2)

The following operators: exist and are bounded for a self-adjoint positive operator . Here

Theorem 2.1. For any , and the solution of problem (1.2) exists and the following formula holds:

Proof. For any and the solution of the auxiliary inverse Cauchy difference problem exists and the following formula holds [36]
Putting we get (2.4).
Now, we consider the following auxiliary difference problem
It is well known that for the solution of (2.8) the following formula holds [37, 38]:
Applying (2.7) and putting in (2.9), we get (2.3).
For using (2.3), (2.4), and the condition we obtain the operator equation
The operator has an inverse
Hence, we obtain that
This concludes the proof of Theorem 2.1.

3. Main Theorems

Here, we study well posedness of problem (1.2). First, we give some necessary estimates for , and For a self-adjoint positive operator , the following estimates are satisfied [36, 38, 39]: where is independent of From these estimates, it follows that

Let be the linear space of mesh functions defined on with values in the Hilbert space Next on we denote and Banach spaces with the norms

Nonlocal boundary value problem (1.2) is said to be stable in if we have the inequality where is independent of not only but also .

Theorem 3.1. Nonlocal boundary value problem (1.2) is stable in norm.

Proof. By [38], we have for the solution of boundary value problem (2.8).
By [36], we get for the solution of an inverse Cauchy difference problem (2.6). Then, the proof of Theorem 3.1 is based on the stability inequalities (3.6), (3.7), and on the estimates for the solution of the boundary value problem (1.2). Estimates (3.8) and (3.9) follow from formula (2.5) and estimates (3.1), (3.2), and (3.3) which conclude the proof of Theorem 3.1.

Theorem 3.2. Assume that and Then, for the solution of difference problem (1.2), we have the following almost coercivity inequality: where does not dependent on not only but also .

Proof. By [40], we have for the solution of an inverse Cauchy difference problem (2.6).
By [38], we get for the solution of boundary value problem (2.8).
Then, the proof of Theorem 3.2 is based on almost coercivity inequalities (3.11), (3.12), and on the estimates for the solution of boundary value problem (1.2). The proof of these estimates follows the scheme of the papers [38, 40] and relies on both the formula (2.5) and the estimates (3.1), (3.2), and (3.3).
This concludes the proof of Theorem 3.2.

Let be the Banach spaces with the norms

Theorem 3.3. Let the assumptions of Theorem 3.2 be satisfied. Then, boundary value problem (1.2) is well posed in Hölder spaces and , and the following coercivity inequalities hold: where is independent of not only and but also and .

Proof. By [39, 40], for the solution of an inverse Cauchy difference problem (2.6) can be written.
By [37, 38], we get for the solution of boundary value problem (2.8).
Then, the proof of Theorem 3.3 is based on coercivity inequalities (3.16)–(3.18), and the estimates for the solution of boundary value problem (1.2).
Estimates (3.19) and (3.20) follow from the formulas for the solution of problem (1.2) and estimates (3.1), (3.2), and (3.3).
This finalizes the proof of Theorem 3.3.

4. Applications

In this section, we indicate applications of Theorems 3.1, 3.2, and 3.3 to obtain the stability, the almost coercive stability, and the coercive stability estimates for the solutions of these difference schemes for the approximate solution of nonlocal mixed problems. First, let be the unit open cube in the -dimensional Euclidean space with boundary . In the boundary value problem for the multidimensional elliptic-parabolic equation is considered. Problem (4.1) has a unique smooth solution for the smooth functions, and .

The discretization of problem (4.1) is carried out in two steps. In the first step, the grid sets are defined. To the differential operator generated by problem (4.1), we assign the difference operator by the formula acting in the space of grid functions satisfying the conditions for all With the help of we arrive at the nonlocal boundary value problem for an infinite system of ordinary differential equations.

Replacing problem (4.4) by the difference scheme (1.2), one can obtain the second order of accuracy difference scheme

Let us give a corollary of Theorems 3.1 and 3.2.

Theorem 4.1. Let and be sufficiently small positive numbers. Then, solutions of difference scheme (4.5) satisfy the following stability and almost coercivity estimates: Here, is independent of not only but also , and

The proof of Theorem 4.1 is based on Theorems 3.1 and 3.2, the estimate the symmetry properties of the difference operator defined by formula (4.3) in , and the following theorem.

Theorem 4.2. For the solution of the elliptic difference problem the following coercivity inequality holds [41]:

Let us give a corollary of Theorem 3.3.

Theorem 4.3. Let and be sufficiently small positive numbers. Then, solutions of difference scheme (4.5) satisfy the following coercivity stability estimates: where is independent of not only but also and

The proof of Theorem 4.3 is based on the abstract Theorems 3.3 and 4.2, and the symmetry properties of the difference operator defined by the formula (4.3).

Second, the mixed boundary value problem for the elliptic-parabolic equation is considered. Problem (4.11) has a unique smooth solution for , the smooth functions, and

Note that in a similar manner one can construct the difference schemes of the second order of accuracy with respect to one variable for approximate solutions of the boundary value problem (4.11). Abstract theorems given above permit us to obtain the stability, the almost stability and the coercive stability estimates for the solutions of these difference schemes.

5. Numerical Results

We consider the nonlocal boundary value problem for the elliptic-parabolic equation.

The exact solution of this problem is

Now, we give the results of the numerical analysis. The errors computed by of the numerical solutions are given inTable 1.

tab1
Table 1: Comparison of the errors.

Thus, the second order of accuracy difference scheme is more accurate than the first order of accuracy difference scheme.

Acknowledgments

The authors would like to thank the referees and Professor P. E. Sobolevskii (Jerusalem, Israel) for helpful suggestions to the improvement of this paper.

References

  1. O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, NY, USA, 1968. View at MathSciNet
  2. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, vol. 23 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 1968.
  3. 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. View at Zentralblatt MATH · View at MathSciNet
  4. M. I. Vislik, A. D. Myshkis, and O. A. Oleinik, “Partial differential equations,” in Mathematics in USSR in the Last 40 Years, vol. 1, pp. 563–599, Fizmatgiz, Moscow, Russia, 1959.
  5. D. Bazarov and H. Soltanov, Some Local and Nonlocal Boundary Value Problems for Equations of Mixed and Mixed-Composite Types, Ylim, Ashgabat, Turkmenistan, 1995.
  6. A. Ashyralye, “A note on the nonlocal boundary value problem for elliptic-parabolic equations,” Nonlinear Studies, vol. 13, no. 4, pp. 327–333, 2006. View at Zentralblatt MATH · View at MathSciNet
  7. M. Sapagovas, “On the stability of a finite-difference scheme for nonlocal parabolic boundary-value problems,” Lithuanian Mathematical Journal, vol. 48, no. 3, pp. 339–356, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  8. M. S. Salakhitdinov, Uravneniya Smeshanno-Sostavnogo Tipa, Fan, Tashkent, Uzbekistan, 1974. View at MathSciNet
  9. R. Ewing, R. Lazarov, and Y. Lin, “Finite volume element approximations of nonlocal reactive flows in porous media,” Numerical Methods for Partial Differential Equations, vol. 16, no. 3, pp. 285–311, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. J. R. Cannon, S. Pérez Esteva, and J. van der Hoek, “A Galerkin procedure for the diffusion equation subject to the specification of mass,” SIAM Journal on Numerical Analysis, vol. 24, no. 3, pp. 499–515, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. J. R. Cannon, “The solution of the heat equation subject to the specification of energy,” Quarterly of Applied Mathematics, vol. 21, pp. 155–160, 1963. View at MathSciNet
  12. N. Gordeziani, P. Natalini, and P. E. Ricci, “Finite-difference methods for solution of nonlocal boundary value problems,” Computers & Mathematics with Applications, vol. 50, no. 8-9, pp. 1333–1344, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. M. Dehghan, “On the solution of the diffusion equation with a nonlocal boundary condition,” Numerical Methods for Partial Differential Equations, vol. 21, no. 1, pp. 24–40, 2005.
  14. R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, vol. 1–11, Masson, Paris, France, 1988.
  15. A. Ashyralyev, “On well-posedness of the nonlocal boundary value problems for elliptic equations,” Numerical Functional Analysis and Optimization, vol. 24, no. 1-2, pp. 1–15, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. A. Ashyralyev, “High-accuracy stable difference schemes for well-posed NBVP,” in Modern Analysis and Applications, vol. 191 of Oper. Theory Adv. Appl., pp. 229–252, Birkhäuser, Basel, Switzerland, 2009. View at Zentralblatt MATH · View at MathSciNet
  17. D. Guidetti, B. Karasözen, and S. Piskarev, “Approximation of abstract differential equations,” Journal of Mathematical Sciences, vol. 122, no. 2, pp. 3013–3054, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. A. Ashyralyev, S. Piskarev, and L. Weis, “On well-posedness of difference schemes for abstract parabolic equations in Lp([0,T];E) spaces,” Numerical Functional Analysis and Optimization, vol. 23, no. 7-8, pp. 669–693, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. V. L. Makarov and D. T. Kulyev, “Solution of a boundary value problem for a quasilinear equation of parabolic type with nonclassical boundary condition,” Differential Equations, vol. 21, no. 2, pp. 296–305, 1985. View at MathSciNet
  20. I. P. Gavrilyuk and V. L. Makarov, “Exponentially convergent parallel discretization methods for the first order evolution equations,” Computational Methods in Applied Mathematics, vol. 1, no. 4, pp. 333–355, 2001. View at Zentralblatt MATH · View at MathSciNet
  21. I. P. Gavrilyuk and V. L. Makarov, “Algorithms without accuracy saturation for evolution equations in Hilbert and Banach spaces,” Mathematics of Computation, vol. 74, no. 250, pp. 555–583, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. I. P. Gavrilyuk and V. L. Makarov, “Exponentially convergent algorithms for the operator exponential with applications to inhomogeneous problems in Banach spaces,” SIAM Journal on Numerical Analysis, vol. 43, no. 5, pp. 2144–2171, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. D. Gordeziani, H. Meladze, and G. Avalishvili, “On one class of nonlocal in time problems for first-order evolution equations,” Zhurnal Obchyslyuval'nö ta Prykladnö Matematyky, vol. 88, no. 1, pp. 66–78, 2003.
  24. D. G. Gordeziani and G. A. Avalishvili, “Time-nonlocal problems for Schrödinger-type equations. I. Problems in abstract spaces,” Differential Equations, vol. 41, no. 5, pp. 703–711, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. D. Gordeziani, G. Avalishvili, and M. Avalishvili, “Hierarchical models of elastic shells in curvilinear coordinates,” Computers & Mathematics with Applications, vol. 51, no. 12, pp. 1789–1808, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. R. P. Agarwal, M. Bohner, and V. B. Shakhmurov, “Maximal regular boundary value problems in Banach-valued weighted space,” Boundary Value Problems, vol. 2005, no. 1, pp. 9–42, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. V. B. Shakhmurov, “Coercive boundary value problems for regular degenerate differential-operator equations,” Journal of Mathematical Analysis and Applications, vol. 292, no. 2, pp. 605–620, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. A. Favini, V. Shakhmurov, and Y. Yakubov, “Regular boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces,” Semigroup Forum, vol. 79, no. 1, pp. 22–54, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. A. V. Gulin and V. A. Morozova, “On the stability of a nonlocal difference boundary value problem,” Differential Equations, vol. 39, no. 7, pp. 962–967, 2003 (Russian). View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. A. V. Gulin, N. I. Ionkin, and V. A. Morozova, “On the stability of a nonlocal two-dimensional difference problem,” Differential Equations, vol. 37, no. 7, pp. 970–978, 2001 (Russian). View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, Mass, USA, 2000. View at MathSciNet
  32. A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and Their Applications, 16, Birkhäuser, Basel, Switzerland, 1995. View at MathSciNet
  33. A. K. Ratyni, “On the solvability of the first nonlocal boundary value problem for an elliptic equation,” Differential Equations, vol. 45, no. 6, pp. 862–872, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  34. T. A. Jangveladze and G. B. Lobjanidze, “On a variational statement of a nonlocal boundary value problem for a fourth-order ordinary differential equation,” Differential Equations, vol. 45, no. 3, pp. 335–343, 2009.
  35. A. Ashyralyev and O. Gercek, “Nonlocal boundary value problems for elliptic-parabolic differential and difference equations,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 904824, 16 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  36. P. E. Sobolevskii, “On the stability and convergence of the Crank-Nicolson scheme,” in Variational-Difference Methods in Mathematical Physics, pp. 146–151, Vychisl.Tsentr Sibirsk. Otdel. Akademii Nauk SSSR, Novosibirsk, Russia, 1974.
  37. P. E. Sobolevskiĭ, “The theory of semigroups and the stability of difference schemes,” in Operator Theory in Function Spaces, pp. 304–337, Nauka, Novosibirsk, Russia, 1977. View at MathSciNet
  38. P. E. Sobolevskiĭ, “The coercive solvability of difference equations,” Doklady Akademii Nauk SSSR, vol. 201, pp. 1063–1066, 1971. View at MathSciNet
  39. A. O. Ashyralyev and P. E. Sobolevskiĭ, “Coercive stability of a Crank-Nicolson difference scheme in spaces C˜0α,” in Approximate Methods for Investigating Differential Equations and Their Applications, pp. 16–24, Kuĭbyshev. Gos. Univ., Kuybyshev, Russia, 1982. View at MathSciNet
  40. A. O. Ashyralyev and P. E. Sobolevskiĭ, “Correct solvability of the Crank-Nicholson scheme for parabolic equations,” Izvestiya Akademii Nauk Turkmenskoĭ SSR. Seriya Fiziko-Tekhnicheskikh, Khimicheskikh i Geologicheskikh Nauk, no. 6, pp. 10–16, 1981. View at MathSciNet
  41. P. E. Sobolevskii, Difference Methods for the Approximate Solution of Differential Equations, Voronezh State University Press, Voronezh, Russia, 1975.