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).
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.