The Scientific World Journal

Volume 2014 (2014), Article ID 841602, 22 pages

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

Research Article

On the Solution of NBVP for Multidimensional Hyperbolic Equations

Allaberen Ashyralyev and Necmettin Aggez

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

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

Thus, estimate (43) follows from estimates (50), (56), and (57). Applying to (47) and using Abel’s formula, we obtain

From the last formula and estimates (37) it follows that

Applying to (48) and using Abel’s formula, we get

Using the triangle inequality and estimates (37), we obtain

As a result, estimate (44) follows from estimates (50), (59), and (61). Theorem 6 is proved.

4. Application

The discretization of hyperbolic equation (2) with Neumann and integral or nonclassical and integral boundary conditions is carried out in two steps. First, let us define the grid sets

We introduce the Banach space of the grid functions

defined on , equipped with the norm

To the differential operator generated by (2), we assign the difference operator by the formula

acting in the space of grid functions , satisfying the condition for all or , and , , . is the approximation of . It is known that is a self-adjoint positive definite operator in . With the help of we arrive at the nonlocal boundary value problem

for an infinite system of ordinary differential equations. Second, we replace problem (66) by the difference scheme

of the first order accuracy in . For the stability of first order of accuracy difference scheme, the following theorem is presented.

Theorem 7. Let and be sufficiently small numbers. Then, the solutions of difference scheme (67) satisfy the following stability estimates:

Here, is independent of , ,, , and , .

The proof of Theorem 7 is based on the symmetry property of difference operator defined by formula (65) and on the following theorem on coercivity inequality of the elliptic difference problem.

Theorem 8. For the solutions of the elliptic difference problem

the following coercivity inequality holds [23]:

In addition, the second order of accuracy difference scheme

for approximately solving hyperbolic equation (2) with nonlocal integral and Neumann or nonclassical conditions is presented. The following theorem on the stability of (71) is obtained.

Theorem 9. Let and be sufficiently small numbers. Then, the solution of difference scheme (71) satisfies the following stability estimates:

Here, does not depend on ,  ,  , and , .

The proof of Theorem 9 is based on the symmetry property of difference operator defined by formula (65) and on Theorem 8 on coercivity inequality of elliptic difference problem (69).

5. Numerical Examples

In this section, we apply finite difference schemes (67) and (71) to three examples which are one-dimensional hyperbolic equation with nonlocal and nonclassical conditions.

Example 10. The nonlocal boundary value problem

for one-dimensional hyperbolic equation with variable coefficients is considered. The exact solution of this problem is

For the approximate solution of the problem (73), we apply finite difference schemes (67) and (71).

First, we obtain the first order of accuracy difference scheme

The system can be written in the matrix form

Here,

and is the identity matrix.

Consider

This type system was used by [24] for difference equations. For the solution of matrix equation (76), we will use modified Gauss elimination method. We seek a solution of the matrix equation by the following form:

where , are square matrices and are column matrices. is identity and is zero matrices, and

Second, applying formulas

and using the second order of accuracy implicit difference scheme (71), we get second order of accuracy difference scheme

for the approximate solutions of nonlocal boundary value problem (73). We have again system of linear equations. We can write the system as a matrix equation (76). Here,

For the solution of the matrix equation (76), we used the same algorithm as in the first order of accuracy difference scheme. Here, ,

Example 11. The nonlocal boundary value problem

is considered. Here, we use the same procedure as in the first example. The exact solution of this problem is

Using the same manner, we can construct first order of accuracy difference scheme and it can be written in the matrix form

Here, matrices , , , and are given in the previous example, and

For the solution of matrix equation (87), we will use modified Gauss elimination method. We seek a solution of the matrix equation by the following form:

where ,    are square matrices and are column matrices. and are zero matrices, and

By using the second order of accuracy implicit difference scheme (71), we can write the matrix form

Here, matrices , , , and are given in the previous example, and also is given in the first order accuracy difference scheme. For the solution of the matrix equation (91), we used the same algorithm as in the first order of accuracy difference scheme, where , and are zero matrices.

Example 12. In this example, the nonlocal boundary value problem

for one-dimensional hyperbolic equation is considered. The exact solution of this problem is

First, we use the first order of accuracy implicit difference scheme (67) for the approximate solutions of nonlocal boundary value problem (92) and we obtain the matrix equation

Here, matrices , , , and are the same as in the first example, and

For the solution of matrix equation (94), we will use modified Gauss elimination method. We seek a solution of the matrix equation by the following form:

where , is identity and is zero matrices, and

Second, for the approximate solutions of nonlocal boundary value problem (92), we use second order of accuracy difference scheme (71) and the formulas

The system can be written in the following matrix form:

Here, and zero matrices and , , , and are given in the first example, and also is given in the first order accuracy difference scheme. For the solution of matrix equation (99), we will use modified Gauss elimination method. We seek a solution of the matrix equation in the following form:

where , are square matrices and    are column matrices defined by

Using formulas (100) and (101), we can compute , .

Now, let us give the results of numerical analysis. The numerical solutions are recorded for different values of , and represents the exact solution, and represents the numerical solution at . For their comparison, the errors are computed by

Thus, the results given in Tables 1, 2, and 3 show that the second order of accuracy difference scheme (71) is more accurate comparing with the first order of accuracy difference scheme (67).

Table 1: Errors for the approximate solution of problem (73).

Difference schemes

Difference scheme (67)
0.1983
0.0449
0.0201

Difference scheme (71)
0.056
0.0014
0.0003994

Table 2: Errors for the approximate solution of problem (87).

Difference schemes

Difference scheme (67)
0.1813
0.0516
0.0202

Difference scheme (71)
0.0049
0.001
0.0002381

Table 3: Errors for the approximate solution of problem (94).

Difference schemes

Difference scheme (67)
0.1498
0.0713
0.0347

Difference scheme (71)
0.0482
0.019
0.0047

6. Conclusion

In this paper, we presented first and second order stable difference schemes for solving the second order multidimensional hyperbolic equation with nonlocal integral and Neumann or nonclassical boundary conditions. Stability of the difference schemes do not depend on any additional condition between and . The numerical results given in the previous sections demonstrate the efficiency and good accuracy of these schemes. Finally we would like to mention that this technique can be applied to get the highest order stable difference schemes.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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