Abstract and Applied Analysis

VolumeΒ 2012Β (2012), Article IDΒ 237657, 12 pages

http://dx.doi.org/10.1155/2012/237657

## Well-Posedness of the First Order of Accuracy Difference Scheme for Elliptic-Parabolic Equations in HΓΆlder Spaces

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

Received 30 March 2012; Accepted 17 April 2012

Academic Editor: AllaberenΒ Ashyralyev

Copyright Β© 2012 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 first order of accuracy difference scheme for the
approximate solution of abstract nonlocal boundary value problem , , , , for differential equations in a Hilbert space with a self-adjoint positive definite operator *A* is considered. The well-posedness of this difference scheme in HΓΆlder spaces without a weight is established. Moreover, as applications, coercivity estimates in HΓΆlder norms
for the solutions of nonlocal boundary value problems for elliptic-parabolic equations are obtained.

#### 1. **Introduction**

Nonlocal boundary value problems for partial differential equations have been applied by various researchers in order to model numerous processes in different fields of applied sciences when they are unable to determine the boundary values of the unknown function (see, e.g., [1β15] and the references therein).

Well-posedness of difference schemes of elliptic-parabolic equations with nonlocal boundary conditions in HΓΆlder spaces with a weight was studied in [16β19].

In paper [20], the well-posedness of abstract nonlocal boundary value problem in HΓΆlder spaces without a weight was established. The coercivity inequalities for solutions of the boundary value problem for elliptic-parabolic equations were obtained.

In the present paper, the first order of accuracy difference scheme for the approximate solution of problem (1.1) is considered. The well-posedness of difference scheme (1.2) in HΓΆlder spaces without a weight is established. As an application, coercivity inequalities for solutions of difference scheme for elliptic-parabolic equations are obtained.

Throughout the paper, denotes a Hilbert space and is a self-adjoint positive definite operator with for some . Then, it is wellknown that is a self-adjoint positive definite operator and . Furthermore, and which are defined on the whole space , are bounded operators, where is the identity operator.

#### 2. **Well-Posedness of (1.2)**

First of all, let us start with some auxiliary lemmas that are used throughout the paper.

Lemma 2.1. *The following estimates are satisfied [19, 21, 22]:
**
for some , which is independent of is a positive small number.*

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

With the help of the self-adjoint positive definite operator in a Hilbert space , the Banach space consists of those for which the norm (see [22, 23]): is finite. By the definition of , for all .

Lemma 2.2. *For , the norms of the spaces and are equivalent (see [24]).*

Theorem 2.3. *Suppose , , and . Boundary value problem (1.2) is wellposed in HΓΆlder space and the following coercivity inequality holds:
**
where is independent of not only , , andββ but also of and .*

*Proof. *First of all, let us get the formulae for solution of problem (1.2). By [21, 25],
is the solution of boundary value difference problem:
is the solution of inverse Cauchy problem:

Combining the conditions and formulas (2.6), (2.8), we get formulas

Operator equation
follows from formulas (2.10), (2.11), and the condition . As the operator
has an inverse
it follows that
for the solution of operator equation (2.12). Hence, we have formulas (2.10), (2.11), and (2.15) for the solution of difference problem (1.2).

Using formulae (2.10) and (2.15), we can get

Finally, we will get coercivity estimate (2.5). It is based on estimates
for the solution of boundary value difference problem (2.7),
for the solution of inverse Cauchy difference problem (2.9), and
for the solution of problem (1.2). Estimates (2.18) and (2.19) were established in [21, 25], respectively.

Estimates (2.20) are derived from the formulas (2.16) and (2.17) for the solution of problem (1.2), estimates (2.1) and following estimates
which were established in [26]. This finalizes the proof of Theorem 2.3.

#### 3. **An Application**

In this section, an application of the abstract Theorem 2.3 is considered. First, let be the unit open cube in the -dimensional Euclidean space with boundary . In , the mixed boundary value problem for multidimensional mixed equation: is considered. Here, , and are given smooth functions and .

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

are defined. To the differential operator generated by problem (3.1), the difference operator is assigned by 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.

In the second step, problem (3.4) is replaced by difference scheme (1.2) (see [21]): To formulate the result, we introduce the Hilbert spaces , and of the grid functions defined on , equipped with the norms:

Theorem 3.1. *Let and be sufficiently small numbers. Then, the solutions of difference scheme (3.5) satisfy the following coercivity stability estimate:
**
where is not dependent on , and .*

The proof of Theorem 3.1 is based on Theorem 2.3, the symmetry properties of the difference operator defined by formula (3.3), and along with the following theorem on the coercivity inequality for the solution of elliptic difference equation in .

Theorem 3.2. *For the solution of elliptic difference problem:
**
the following coercivity inequality holds [27]:
**
Here, depends neither on nor .*

#### Acknowledgments

The author would like to thank Professor Allaberen Ashyralyev (Fatih University, Turkey) for his inspirational contributions and to the anonymous referees whose careful reading of the paper and valuable comments helped to improve it.

#### References

- M. S. Salakhitdinov,
*Equations of Mixed-Composite Type*, Fan, Tashkent, Uzbekistan, 1974. - T. D. Dzhuraev,
*Boundary Value Problems for Equations of Mixed and Mixed Composite Types*, Fan, Tashkent, Uzbekistan, 1979. - D. G. Gordeziani,
*On Methods of Resolution of a Class of Nonlocal Boundary Value Problems*, Tbilisi University Press, Tbilisi, Georgia, 1981. - V. N. Vragov,
*Boundary Value Problems for Nonclassical Equations of Mathematical Physics*, Textbook for Universities, NGU, Novosibirsk, Russia, 1983. - D. Bazarov and H. Soltanov,
*Some Local and Nonlocal Boundary Value Problems for Equations of Mixed and Mixed-Composite Types*, Ylim, Ashgabat, Turkmenistan, 1995. - A. M. Nakhushev,
*Equations of Mathematical Biology*, Vysshaya Shkola, Moskow, Russia, 1995. - S. N. Glazatov, βNonlocal boundary value problems for linear and nonlinear equations of variable type,β
*Sobolev Institute of Mathematics SB RAS*, no. 46, p. 26, 1998. View at Google Scholar - A. Ashyralyev and H. A. Yurtsever, βOn a nonlocal boundary value problem for semilinear hyperbolic-parabolic equations,β
*Nonlinear Analysis-Theory, Methods and Applications*, vol. 47, no. 5, pp. 3585β3592, 2001. View at Google Scholar - D. Guidetti, B. Karasözen, and S. Piskarev, βApproximation of abstract differential equations,β
*Journal of Mathematical Sciences*, vol. 122, no. 2, pp. 3013β3054, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - J. I. Díaz, M. B. Lerena, J. F. Padial, and J. M. Rakotoson, βAn elliptic-parabolic equation with a nonlocal term for the transient regime of a plasma in a stellarator,β
*Journal of Differential Equations*, vol. 198, no. 2, pp. 321β355, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. Ashyralyev, βA note on the nonlocal boundary value problem for elliptic-parabolic equations,β
*Nonlinear Studies*, vol. 13, no. 4, pp. 327β333, 2006. View at Google Scholar Β· View at Zentralblatt MATH - A. S. Berdyshev and E. T. Karimov, βSome non-local problems for the parabolic-hyperbolic type equation with non-characteristic line of changing type,β
*Central European Journal of Mathematics*, vol. 4, no. 2, pp. 183β193, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. Ashyralyev and Y. Ozdemir, βOn stable implicit difference scheme for hyperbolic-parabolic equations in a Hilbert space,β
*Numerical Methods for Partial Differential Equations*, vol. 25, no. 5, pp. 1100β1118, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - V. B. Shakhmurov, βRegular degenerate separable differential operators and applications,β
*Potential Analysis*, vol. 35, no. 3, pp. 201β222, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - J. Martín-Vaquero, A. Queiruga-Dios, and A. H. Encinas, βNumerical algorithms for diffusion-reaction problems with non-classical conditions,β
*Applied Mathematics and Computation*, vol. 218, no. 9, pp. 5487β5495, 2012. View at Google Scholar - A. Ashyralyev and H. Soltanov, βOn elliptic-parabolic equations in a Hilbert space,β in
*Proceedings of the IMM of CS of Turkmenistan*, pp. 101β104, Ashgabat, Turkmenistan, 1995. - A. Ashyralyev and O. Gercek, βOn second order of accuracy difference scheme of the approximate solution of nonlocal elliptic-parabolic problems,β
*Abstract and Applied Analysis*, vol. 2010, Article ID 705172, 17 pages, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. Ashyralyev and O. Gercek, βFinite difference method for multipoint nonlocal elliptic-parabolic problems,β
*Computers & Mathematics with Applications*, vol. 60, no. 7, pp. 2043β2052, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. Ashyralyev and O. Gercek, βNonlocal boundary value problems for elliptic-parabolic differential and difference equations,β
*Discrete Dynamics in Nature and Society*, vol. 2008, Article ID 904824, 16 pages, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - A. Ashyralyev, βOn the well-posedness of the nonlocal boundary value problem for elliptic-parabolic equations,β
*Electronic Journal of Qualitative Theory of Differential Equations*, vol. 49, pp. 1β16, 2011. View at Google Scholar - P. E. Sobolevskii, βThe theory of semigroups and the stability of difference schemes,β in
*Operator Theory in Function Spaces*, pp. 304β337, Nauka, Novosibirsk, Russia, 1977. View at Google Scholar - A. Ashyralyev and P. E. Sobolevskii,
*Well-Posedness of Parabolic Difference Equations*, vol. 69 of*Operator Theory: Advances and Applications*, Birkhäuser, Basel, Switzerland, 1994. - H. Triebel,
*Interpolation Theory, Function Spaces, Differential Operators*, vol. 18, North-Holland, Amsterdam, The Netherlands, 1978. - A. Ashyralyev,
*Method of positive operators of investigations of the high order of accuracy difference schemes for parabolic and elliptic equations [Doctor of Sciences Thesis]*, Kiev, Ukraine, 1992. - P. E. Sobolevskii, βThe coercive solvability of difference equations,β
*Doklady Akademii Nauk SSSR*, vol. 201, pp. 1063β1066, 1971. View at Google Scholar - 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. - P. E. Sobolevskii,
*On Difference Methods for the Approximate Solution of Differential Equations*, Voronezh State University Press, Voronezh, Russia, 1975.