The Scientific World Journal

Volume 2014, Article ID 841602, 22 pages

http://dx.doi.org/10.1155/2014/841602

## On the Solution of NBVP for Multidimensional Hyperbolic Equations

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

Received 16 August 2013; Accepted 10 February 2014; Published 25 May 2014

Academic Editors: A. Ibeas, L. Kong, and F. Mukhamedov

Copyright © 2014 Allaberen Ashyralyev and Necmettin Aggez. 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 multidimensional hyperbolic equations with nonlocal integral and Neumann or nonclassical conditions. For the approximate solution of this problem first and second order of accuracy difference schemes are presented. Stability estimates for the solution of these difference schemes are established. Some numerical examples illustrating applicability of these methods to hyperbolic problems are given.

#### 1. Introduction

In the last decades, for the development of numerical methods and theory of solutions of the hyperbolic problems with nonlocal integral, Neumann and nonclassical conditions have been an important research topic in many natural phenomena. Solutions of this type of hyperbolic problems were investigated in [1–13]. These problems were studied in various directions: qualitative properties of solutions, spectral problems, various statements of boundary value problems, and numerical investigations.

For example, in [5] the nonlocal boundary value problem was investigated. The stability estimates for the solution of the problem were established. The first order of accuracy difference schemes for the approximate solution of this problem was presented. The stability estimates for the solution of these difference schemes were established. Theoretical statements were supported by numerical examples.

The well-posedness of the Cauchy problem, Goursat problem, and boundary value problem for multidimensional hyperbolic equations have been studied extensively in a large cycle of papers (see, e.g., [14–21] and the references therein).

Actually, in paper [14], the Goursat problem for a linear multidimensional hyperbolic equation was investigated. Uniqueness of the solution and weak solvability of the Goursat problem were established.

In paper [15], the existence or nonexistence of global solutions of a multidimensional version of the first Darboux problem for wave equations with power nonlinearity in the conic domain was investigated.

In [16, 17], the solvability of an initial-boundary value problem for second order linear hyperbolic equations with a condition on the lateral boundary connecting the values of the solution or the conormal derivative of the solution with the values of some integral operator of the solution was studied. The existence and uniqueness theorems for regular solutions were proved.

In [18–20], the difference schemes for multidimensional hyperbolic equations were investigated. These methods were stable under the inequalities and contain the connection between the grid step sizes of time and space variables.

In [21], the authors develop a finite difference method (FDM) for a multidimensional coupled system of nonlinear parabolic and hyperbolic equations and prove the existence, stability, and uniqueness of its solution by a set of theorems. Finally, the proposed method was illustrated by a number of numerical experiments.

The study of difference schemes for hyperbolic equations with nonlocal conditions without using any necessary condition concerning the grid step sizes is of great interest. Such a difference scheme for solving the initial-value problem for abstract hyperbolic equations was studied for the first time in [22]. In the present paper, the following multidimensional hyperbolic equation with nonlocal integral and Neumann or nonclassical conditions under the assumption is considered. Here, is the unit open cube in the -dimensional Euclidean space with boundary , , , , , and are given smooth functions, and . is the normal vector to .

The first and second order of accuracy difference schemes for multidimensional hyperbolic problem (2) are presented. The schemes are shown to be absolutely stable. It is naturally seen that the second order difference schemes are much more advantageous than the first order ones.

#### 2. Stability of First Order of Accuracy Difference Scheme

For approximately solving problem (2), first order of accuracy difference scheme is considered. A study of discretization of the nonlocal boundary value problem also permits one to include general difference schemes in applications, if differential operator in space variables is replaced by difference operator that acts in a Hilbert space and is uniformly self-adjoint positive definite in for .

The stability estimates of solution of difference scheme (7) are established under the assumption

Lemma 1. *The following estimates hold [23]:
**
where
*

*Lemma 2. The operator
has an inverse
and the following estimate is satisfied:
*

*Proof. *Using formula (11) and the triangle inequality, we can write
Applying the triangle inequality and estimates (9), we get
Thus, estimate (13) follows from this estimate. Lemma 2 is proved. The following theorem on the stability estimates for the solution of difference scheme (7) is established.

*Theorem 3. Let , , and (8) hold. Then, for the solution of difference scheme (7) the following stability estimates hold:
where is independent of , , and , .*

*Proof. *First, we obtain formula for the solution of difference scheme (7). For the solution of difference scheme
the following formulas
were obtained in [22]. Applying formula (20) and nonlocal boundary conditions in (7), we can write formula for and
Hence, for the solution of nonlocal boundary value problem (7) we have formulas (20), (21), and (22).

Second, let us investigate stability of difference scheme (7). In [22], for the solution of (19) stability estimates
were established.

First of all, let us find estimate for . By using formula (21) and estimates (9), we obtain
And, applying to formula (22), we get
Using the triangle inequality, formula (27), and estimates (9), it follows that
So, estimate (16) follows from estimates (23), (26), and (28). Second, applying to formula (21) and using estimates (9), we get estimate
By using formula (22) and estimates (9), we obtain
Using estimates (24), (29), and (30), we obtain estimate (17) for the solution of (7). Third, applying to formula (21) and using Abel’s formula, we can write formula for
It follows from formula (31) and estimates (9) that
Applying to formula (22) and using Abel’s formula, we get
and using the triangle inequality and estimates (9), we obtain the estimate
Thus, estimate (18) follows from estimates (23), (32), and (34). This is the end of the proof of Theorem 3.

*3. Stability of Second Order of Accuracy Difference Scheme*

*3. Stability of Second Order of Accuracy Difference Scheme*

*Now, we consider the second order accuracy difference scheme for approximate solution of boundary value problem (2)
The stability of solutions of this difference scheme is investigated under the assumption
*

*Lemma 4. The following estimates hold [23]:
where
*

*Lemma 5. Suppose that assumption (36) holds. Then, the operator
has an inverse and the following estimate is satisfied:
*

*Proof. *Using formula (39), the triangle inequality, and estimates (37), we obtain
Estimate (40) follows from this estimate. Lemma 5 is proved.

*Theorem 6. Let , , and assumption (36) hold. For the solution of difference scheme (35) the following stability estimates
are valid, where is independent of , , and , .*

*Proof. *We obtain formula for the solution of difference scheme (35). For the solution of difference scheme
the following formulas
were obtained in [22]. Applying formula (46) and nonlocal boundary conditions in (35), we obtain formulas
Hence, for the solution of nonlocal boundary value problem (35) we have formulas (46), (47), and (48).

Now, let us investigate the stability of difference scheme (35). In [22], for the solution of (45) the following stability estimates
were established. Now, from formula (47) and estimates (37) it follows that
Applying to formula (48), we get
and using estimates (37), we obtain
So, using estimates (49), (52), and (54), we obtain (42) for the solution of (35). Applying to formula (47), we get
From the last formula and estimates (37) it follows that
Using formula (48), the triangle inequality, and estimates (37), we obtain