Abstract and Applied Analysis

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

http://dx.doi.org/10.1155/2012/230190

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

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:

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:

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,

Furthermore, positive constants will be indicated by

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 *

Now, we study well-posedness of problem (1.3). Let

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

*Proof. *By [22], we have

Theorem 2.4. *Assume that *

*Proof. *We have

Theorem 2.5. *Let assumptions of Theorem 2.5 be satisfied. Then, boundary value problem (1.3) is well-posed in Hölder spaces *

*Proof. *By [22, 24], we have

#### 3. An Application

In this section, an application of these abstract Theorems 2.3, 2.4, and 2.5 is considered. In

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

To the differential operator

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 *

The proof of Theorem 3.1 is based on Theorem 2.3, Theorem 2.4, the symmetry property of the difference operator

Theorem 3.2. *For the solution of the following elliptic difference problem:
**
the following coercivity inequality holds [25]:
*

Theorem 3.3. *Let *

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

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