Abstract and Applied Analysis

Volume 2012 (2012), Article ID 230190, 13 pages

doi:10.1155/2012/230190

Research Article

On the Second Order of Accuracy Stable Implicit Difference Scheme for Elliptic-Parabolic Equations

Allaberen Ashyralyev and Okan Gercek

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

Received 7 April 2012; Accepted 24 April 2012

Academic Editor: Ravshan Ashurov

Copyright © 2012 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.

We are interested in studying a second order of accuracy implicit difference scheme for the solution of the elliptic-parabolic equation with the nonlocal boundary condition. Well-posedness of this difference scheme is established. In an application, coercivity estimates in Hölder norms for approximate solutions of multipoint nonlocal boundary value problems for elliptic-parabolic differential equations are obtained.

1. Introduction

Methods of solutions of nonlocal boundary value problems for mixed-type differential equations have been studied extensively by various researchers (see, e.g., [1–19] and the references therein).

In [20], we considered the well-posedness of the following multipoint nonlocal boundary value problem:

in a Hilbert space with the self-adjoint positive definite operator under assumption

The well-posedness of multipoint nonlocal boundary value problem (1.1) in Hölder spaces with a weight was established. Moreover, coercivity estimates in Hölder norms for the solutions of nonlocal boundary value problems for elliptic-parabolic equations were obtained.

In [21], we studied the well-posedness of the first order of accuracy difference scheme for the approximate solution of boundary value problem (1.1) under assumption (1.2).

Throughout this work, we consider the following second order of accuracy difference scheme:

for the approximate solution of boundary value problem (1.1) under assumption (1.2).

The well-posedness of difference scheme (1.3) in Hölder spaces with a weight is established. As an application, the stability, almost coercivity stability, and coercivity stability estimates for solutions of second order of accuracy difference scheme for the approximate solution of the nonlocal boundary elliptic-parabolic problem are obtained.

2. Main Theorems

Throughout the paper, is a Hilbert space and we denote , where is a self-adjoint positive definite operator. Then, it is clear that is the self-adjoint positive definite operator and where , and , which is defined on the whole space , is a bounded operator. Here, is the identity operator. The following operators

exist and are bounded for a self-adjoint positive operator . Here,

Furthermore, positive constants will be indicated by which can differ in time. On the other hand is used to focus on the fact that the constant depends only on and the subindex is used to indicate a different constant.

First of all, let us start with some auxiliary lemmas from [16, 22–24] that are essential below.

Lemma 2.1. For a self-adjoint positive operator A, the following estimates are satisfied:

From these estimates, it follows that

Lemma 2.2. For any and , the solution of problem (1.3) exists, and the following formulas hold:

Now, we study well-posedness of problem (1.3). 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 following norms:

Theorem 2.3. Nonlocal boundary value problem (1.3) is stable in space.

Proof. By [22], we have

for the solution of the following boundary value problem:

By [24], we have

for the solution of an inverse Cauchy difference problem:

Then, the proof of Theorem 2.3 is based on stability inequalities (2.7) and (2.9) and on the following estimates:

for the solution of boundary value problem (1.3). Estimates (2.11) follow from estimates (2.3) and (2.4) and formula (2.5). This finishes the proof of Theorem 2.3.

Theorem 2.4. Assume that and . Then, for the solution of difference problem (1.3), the following almost coercivity inequality holds:

Proof. We have

for the solution of boundary value problem (2.8) (see [22]), and we get

for the solution of inverse Cauchy difference problem (2.10) (see [24]). Then, the proof of Theorem 2.4 is based on almost coercivity inequalities (2.13) and (2.14) and on the following estimates:

for the solution of boundary value problem (1.3). Proofs of these estimates follow the scheme of the papers [23, 24] and rely on both formula (2.5) and estimates (2.3) and (2.4). Theorem 2.4 is proved.

Theorem 2.5. Let assumptions of Theorem 2.5 be satisfied. Then, boundary value problem (1.3) is well-posed in Hölder spaces ,  and  , and the following coercivity inequalities hold:

Proof. By [22, 24], we have

for the solution of boundary value problem (2.8), and

for the solution of inverse Cauchy difference problem (2.10), respectively. Then, the proof of Theorem 2.5 is based on coercivity inequalities (2.17)–(2.19) and the following estimates:

for the solution of difference scheme (1.3). Proofs of these estimates follow the scheme of the papers [22, 24] and rely on both estimates (2.3) and (2.4) and formula (2.5). This concludes the proof of Theorem 2.5.

3. An Application

In this section, an application of these abstract Theorems 2.3, 2.4, and 2.5 is considered. In , let us consider the following boundary value problem for multidimensional elliptic-parabolic equation:

where , and are given smooth functions. Here, is the unit open cube in the -dimensional Euclidean space with boundary , and .

The discretization of problem (3.1) is carried out in two steps. In the first step, let us define the following grid sets:

We introduce the Hilbert spaces , and of the grid functions defined on , equipped with the following norms:

To the differential operator generated by problem (3.1), we assign the difference operator by formula

acting in the space of grid functions , satisfying the conditions for all . With the help of , we arrive at the following nonlocal boundary value problem:

for an infinite system of ordinary differential equations.

In the second step, we replace problem (3.5) by difference scheme (1.3) accurate to the following second order (see [22, 24]):

Theorem 3.1. Let and be sufficiently small positive numbers. Then, solutions of difference scheme (3.6) satisfy the following stability and almost coercivity estimates:

The proof of Theorem 3.1 is based on Theorem 2.3, Theorem 2.4, the symmetry property of the difference operator defined by formula (3.4), the estimate

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

Theorem 3.2. For the solution of the following elliptic difference problem:

the following coercivity inequality holds [25]:

Theorem 3.3. Let and be sufficiently small positive numbers. Then, solutions of difference scheme (3.6) satisfy the following coercivity stability estimates:

The proof of Theorem 3.3 is based on the abstract Theorem 2.5, Theorem 3.2, and the symmetry property of the difference operator defined by formula (3.4).

References

D. G. Gordeziani, On Methods of Resolution of a Class of Nonlocal Boundary Value Problems, Tbilisi University Press, Tbilisi, Georgia, 1981.

A. M. Nakhushev, Equations of Mathematical Biology, Textbook for Universities, Vysshaya Shkola, Moscow, Russia, 1995.

M. S. Salakhitdinov and A. K. Urinov, Boundary Value Problems for Equations of Mixed Type with a Spectral Parameter, Fan, Tashkent, Uzbekistan, 1997.

A. Ashyralyev and H. A. Yurtsever, “On a nonlocal boundary value problem for semi-linear hyperbolic-parabolic equations,” Nonlinear Analysis. Theory, Methods and Applications, vol. 47, no. 5, pp. 3585–3592, 2001.

J. I. Diaz, M. B. Lerena, J. F. Padial, and J. M. Rakotoson, “An elliptic-parabolic equation with a nonlocal term for the transient regime of a plasma in a stellarator,” Journal of Differential Equations, vol. 198, no. 2, pp. 321–355, 2004.

D. Guidetti, B. Karasozen, and S. Piskarev, “Approximation of abstract differential equations,” Journal of Mathematical Sciences, vol. 122, no. 2, pp. 3013–3054, 2004.

R. P. Agarwal, M. Bohner, and V. B. Shakhmurov, “Maximal regular boundary value problems in Banach-valued weighted space,” Boundary Value Problems, no. 1, pp. 9–42, 2005.

A. Ashyralyev, G. Judakova, and P. E. Sobolevskii, “A note on the difference schemes for hyperbolic-elliptic equations,” Abstract and Applied Analysis, vol. 2006, Article ID 14816, 13 pages, 2006.

A. S. Berdyshev and E. T. Karimov, “Some non-local problems for the parabolic-hyperbolic type equation with non-characteristic line of changing type,” Central European Journal of Mathematics, vol. 4, no. 2, pp. 183–193, 2006.

A. Ashyralyev, C. Cuevas, and S. Piskarev, “On well-posedness of difference schemes for abstract elliptic problems in Lp([0, T]; E) spaces,” Numerical Functional Analysis and Optimization, vol. 29, no. 1-2, pp. 43–65, 2008.

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.

M. De la Sen, “About the stability and positivity of a class of discrete nonlinear systems of difference equations,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 595367, 18 pages, 2008.

P. V. Vinogradova, “Error estimates for a projection-difference method for a linear differential-operator equation,” Differential Equations, vol. 44, no. 7, pp. 970–979, 2008.

A. Ashyralyev and Y. Ozdemir, “On stable implicit difference scheme for hyperbolic-parabolic equations in a Hilbert space,” Numerical Methods for Partial Differential Equations, vol. 25, no. 5, pp. 1100–1118, 2009.

A. Castro, C. Cuevas, and C. Lizama, “Well-posedness of second order evolution equation on discrete time,” Journal of Difference Equations and Applications, vol. 16, no. 10, pp. 1165–1178, 2010.

A. Ashyralyev and O. Gercek, “On second order of accuracy difference scheme of the approximate solution of nonlocal elliptic-parabolic problems,” Abstract and Applied Analysis, vol. 2010, Article ID 705172, 17 pages, 2010.

A. Ashyralyev, “On the well-posedness of the nonlocal boundary value problem for elliptic-parabolic equations,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 49, pp. 1–16, 2011.

V. B. Shakhmurov, “Regular degenerate separable differential operators and applications,” Potential Analysis, vol. 35, no. 3, pp. 201–222, 2011.

J. Martin-Vaquero, A. Queiruga-Dios, and A. H. Encinas, “Numerical algorithms for diffusion-reaction problems with non-classical conditions,” Applied Mathematics and Computation, vol. 218, no. 9, pp. 5487–5495, 2012.

O. Gercek, Difference Schemes of Nonlocal Boundary Value Problems for Elliptic-Parabolic Differential Equations [Ph.D. thesis], Yildiz Teknik University, Istanbul, Turkey, 2010.

A. Ashyralyev and O. Gercek, “On multipoint nonlocal elliptic-parabolic difference problems,” Vestnik of Odessa National University, vol. 15, no. 10, pp. 135–156, 2010.

P. E. Sobolevskii, “The coercive solvability of difference equations,” Doklady Akademii Nauk SSSR, vol. 201, pp. 1063–1066, 1971.

A. Ashyralyev and P. E. Sobolevskii, New Difference Schemes for Partial Differential Equations, Birkhäuser, Basle, Switzerland, 2004.

A. Ashyralyev and P. E. Sobolevskii, Well-Posedness of Parabolic Difference Equations, Birkhäuser, Basle, Switzerland, 1994.

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