Abstract and Applied Analysis

Volume 2013 (2013), Article ID 959216, 15 pages

http://dx.doi.org/10.1155/2013/959216

Research Article

On Stability of a Third Order of Accuracy Difference Scheme for Hyperbolic Nonlocal BVP with Self-Adjoint Operator

Allaberen Ashyralyev¹ and Ozgur Yildirim²

¹Department of Mathematics, Fatih University, Buyukcekmece, 34500 Istanbul, Turkey

²Department of Mathematics, Yildiz Technical University, Esenler, 34210 Istanbul, Turkey

Received 13 May 2013; Accepted 20 July 2013

Academic Editor: Abdullah Said Erdogan

Copyright © 2013 Allaberen Ashyralyev and Ozgur Yildirim. 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 third order of accuracy absolutely stable difference schemes is presented for nonlocal boundary value hyperbolic problem of the differential equations in a Hilbert space with self-adjoint positive definite operator . Stability estimates for solution of the difference scheme are established. In practice, one-dimensional hyperbolic equation with nonlocal boundary conditions is considered.

1. Introduction

In modeling several phenomena of physics, biology, and ecology mathematically, there often arise problems with nonlocal boundary conditions (see [1–5] and the references given therein). Nonlocal boundary value problems have been a major research area in the case when it is impossible to determine the boundary conditions of the unknown function. Over the last few decades, the study of nonlocal boundary value problems is of substantial contemporary interest (see, e.g., [6–14] and the references given therein).

We consider the nonlocal boundary value problem

for hyperbolic equations in a Hilbert space with self-adjoint positive definite linear operator with domain .

A function is called a solution of problem (1) if the following conditions are satisfied.

(i) is twice continuously differentiable on the segment . The derivatives at the endpoints of the segment are understood as the appropriate unilateral derivatives.(ii)The element belongs to for all and the function is continuous on the segment .(iii) satisfies the equations and the nonlocal boundary conditions (1).

Here, , and are smooth functions.

In the study of numerical methods for solving PDEs, stability is an important research area (see [6–27]). Many scientists work on difference schemes for hyperbolic partial differential equations, in which stability was established under the assumption that the magnitudes of the grid steps and with respect to the time and space variables are connected. This particularly means that when .

We are interested in studying high order of accuracy unconditionally stable difference schemes for hyperbolic PDEs.

In the present paper, third order of accuracy difference scheme generated by integer power of for approximately solving nonlocal boundary value problem (1) is presented. The stability estimates for solution of the difference scheme are established.

In [8], some results of this paper, without proof, were presented.

The well posedness of nonlocal boundary value problems for parabolic equations, elliptic equations, and equations of mixed types have been studied extensively by many scientists (see, e.g., [11–14, 19–32] and the references therein).

2. Third Order of Accuracy Difference Scheme Subject to Nonlocal Conditions

In this section, we obtain stability estimates for the solution of third order of accuracy difference scheme

for numerical solution of nonlocal boundary value problem (1). Here,

We study the stability of solutions of difference scheme (2) under the following assumption:

We give a lemma that will be needed in the sequel which was presented in [18]. First, let us present the following operators:

and its conjugate ,

and its conjugate ,

and its conjugate , and

and its conjugate .

We consider the following operators:

and its conjugate ,

and its conjugate .

Lemma 1. The following estimates hold:

Now let us give, without proof, the second lemma.

Lemma 2. The following estimates hold:

Throughout the section, for simplicity, we denote

Lemma 3. Suppose that assumption (4) holds. Then, the operator has an inverse . From symmetry and positivity properties of operator , the following estimate is satisfied:

Proof. Using the definitions of , estimates (11), and the following simple estimates,

and the triangle inequality, we get

where

Since , the operator has a bounded inverse and

Lemma 3 is proved.

Now, let us obtain formula for the solution of problem (2). Using the results of [18], one can obtain the following formula:

for the solution of difference scheme

Applying formula (19) and nonlocal boundary conditions

one can write

Using formulas in (22), we obtain

So, formulas (19) and (23) give a solution of problem (2).

Unfortunately, the estimates for , , and cannot be obtained under the conditions

Nevertheless, we have the following theorem.

Theorem 4. Suppose that assumption (4) holds and , . Then, for solution of difference scheme (2), the following stability estimates hold:

where does not depend on , , , , and , .

Proof. Using formulas in (23) and estimates (11), (12), and (14), we obtain

Applying to formulas in (23), we get

Now, applying Abel’s formula to (23), we obtain the following formulas:

Next, let us obtain the estimates for and . First, applying to formula (28) and using estimates (11), (12), and (14) and the triangle inequality, one can obtain

Second, applying to formula (29) and using estimates (11), (12), and (14) and the triangle inequality, we get

Now, we will prove estimates (25). Using formula (19), estimates (11), (12), (26), and (27), and the triangle inequality, we obtain

for any . Applying to (19), we get

for . Now, applying Abel’s formula to (19), we have

Applying to formula (34) and using estimates (11) and (12) and the triangle inequality, we obtain

for . Theorem 4 is proved.

Note that the stability estimates obtained previously permit us to get the convergence estimate of difference scheme (2) under the smoothness property of solution (1). Actually, under the condition , we can obtain the third order of accuracy for the error of difference scheme (2). Since , this condition is satisfied under the given data , , , and .

Now, let us give application of this abstract result for nonlocal boundary value problem

for hyperbolic equation. Problem (36) has a unique smooth solution , and the smooth functions , , and . This allows us to reduce mixed problem (36) to nonlocal boundary value problem (1) in a Hilbert space with a self-adjoint positive definite operator defined by (36).

The discretization of problem (36) is carried out in two steps. In the first step, let us define the grid space

We introduce Hilbert space , , and of the grid functions defined on , and we assign the difference operator by the formula

acting in the space of grid functions satisfying the conditions , .

With the help of , we arrive at the nonlocal boundary value problem

for a system of ordinary differential equations.

In the second step, we replace problem (2) with difference scheme (40)

Theorem 5. Let and be sufficiently small numbers. Then, the solution of difference scheme (40) satisfies the following stability estimates:

Here, does not depend on , and .

The proof of Theorem 5 is based on the proof of abstract Theorem 4 and the symmetry property of operator defined by (38).

Acknowledgments

The authors would like to thank Professor Pavel E. Sobolevskii and reviewers for their helpful comments.

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