Abstract
In the present paper, the finitedifference method for the initialboundary value problem for a hyperbolic system of equations with nonlocal boundary conditions is studied. The positivity of the difference analogy of the space operator generated by this problem in the space C with maximum norm is established. The structure of the interpolation spaces generated by this difference operator is investigated. The positivity of this difference operator in Hölder spaces is established. In applications, stability estimates for the solution of the difference scheme for a hyperbolic system of equations with nonlocal boundary conditions are obtained. A numerical example is applied.
MSC: 35L40, 35L45.
Keywords:
hyperbolic system of equations; nonlocal boundary value problems; difference schemes; interpolation spaces; positivity of the difference operator; stability estimates1 Introduction
Nonlocal problems are widely used for mathematical modeling of various processes of physics, ecology, chemistry, and industry, when it is impossible to determine the boundary or initial values of the unknown function. The method of operators as a tool for the investigation of the solution of local and nonlocal problems for partial differential equations in Hilbert and Banach spaces has been systematically developed by several authors (see, e.g., [127]). It is well known that (see, e.g., [2832] and the references given therein) many application problems in fluid mechanics, physics, mathematical biology, and chemistry were formulated as nonlocal mathematical models. Note that such problems were not well studied in general.
In the paper [33], the initialboundary value problem
for the hyperbolic system of equations with nonlocal boundary conditions was considered. Here
, (), , () are given smooth functions and they satisfy all compatibility conditions which guarantee the problem (1) has a smooth solution and . As noted in the paper [33], the problem of sound waves [34] and the problem of the expansion of electricity oscillations [35] can be replaced by the problem (1). Note that, we have the nonclassical initialboundary value problem (1) with boundary conditions , , , , . These conditions are given on two boundary points. It is clear that it is impossible to determine the boundary values of the unknown function. So, these conditions are not local.
Let E be a Banach space and be a linear unbounded operator densely defined in E. We call A a positive operator in the Banach space if the operator has a bounded inverse in E for any , and the following estimate holds:
Throughout the present paper, M is defined as a positive constant. However, we will use to stress the fact that the constant depends only on .
For a positive operator A in the Banach space E, let us introduce the fractional spaces () consisting of those for which the norm
is finite.
Let us introduce the Banach space () of all continuous vector functions defined on and satisfying a Hölder condition for which the following norm is finite:
Here is the Banach space of all continuous vector functions defined on with norm
We consider the space operator A generated by the problem (1) defined by the formula
with domain
The Green’s matrix function of A was constructed. The positivity of the operator A in the Banach space was established. It was proved that for any the norms in spaces and are equivalent. The positivity of A in the Hölder spaces of , was proved. In applications, stability estimates for the solution of the problem (1) for the hyperbolic system of equations with nonlocal boundary conditions were obtained.
In the present paper, the finitedifference method for the initial value problem for the hyperbolic system of equations with nonlocal boundary conditions is applied. The positivity of the difference analogy of the space operator A defined by equation (1) in the difference analogy of spaces is established. The structure interpolation spaces generated by this difference operator is studied. The positivity of this difference operator in Hölder spaces is established. In practice, stability estimates for the solution of the difference scheme for the hyperbolic system of equations with nonlocal boundary conditions are obtained. The method is illustrated by numerical example.
The organization of the present paper as follows. Section 1 is an introduction where we provide the history and formulation of the problem. In Section 2, the Green’s matrix function of the difference space operator is presented and positivity of this operator in the difference analogy of spaces is proved. In Section 3, the structure of fractional spaces generated by this difference operator is investigated and positivity of this difference operator in Hölder spaces is established. In Section 4, stable difference schemes for the approximate solution of the problem (1) are constructed. A theorem on the stability for the first order of accuracy in the t difference scheme is proved. In Section 5, a numerical application is given. Finally, Section 6 is for our conclusion.
2 The Green’s matrix function of difference space operator and positivity
Let us introduce the Banach spaces () and of all mesh vector functions defined on
with the following norms:
We consider the difference space operator generated by the problem (1) defined by the formula
acting on the space of mesh vector functions defined on , satisfying the conditions
Here . We will study the resolvent of the difference space operator , i.e.
or
Lemma 2.1For any, equation (7) is uniquely solvable and the following formula holds:
where
Here
Proof Using the resolvent equation (7), we get
From that follows the following recursive formula:
Hence
From this formula and the nonlocal boundary condition it follows that
Then,
Using the resolvent equation (7), we get
From that follows the system of recursion formulas
Hence
From this formula and the nonlocal boundary condition it follows that
Therefore,
Applying equation (12), we get
From the last two formulas it follows that
Lemma 2.1 is proved. □
Lemma 2.2The following pointwise estimates hold; see equation (7):
Proof It is easy to see that the estimates of equations (14), (15), and (16) follow from the triangle inequality. Applying the triangle inequality, we get
If . Then, using the estimates of equations (14), (15), (16), and inequality (18), we get
If . Then, using the estimates of equations (14), (15), (16), and inequality (18), we get
Here . Then, using the estimates of equations (14), (15), (16), and inequality (18), we get
Lemma 2.2 is proved. □
Theorem 2.1The operatorhas a bounded inverse infor anyand the following estimate holds:
Proof Using the formula equation (13) and the triangle inequality, we get
for any . Using the estimate of equation (15), we get
Using the estimate of equation (16), we get
Using the estimate of equation (17), we get
Therefore,
From this it follows that
Theorem 2.1 is proved. □
3 The structure of fractional spaces and positivity of in Hölder spaces
Clearly, the operator and its resolvent commute. By the definition of the norm in the fractional space , we get
Thus, from Theorem 2.1 it follows that is a positive operator in the fractional spaces . Moreover, we have the following result.
Theorem 3.1For, the norms of the spacesand the Hölder spaceare equivalent uniformly with respect toh. Here
Proof For any we have the obvious equality
By equation (8), we can write
Applying equation (21) and the following obvious equalities:
and using the nonlocal boundary conditions
we get
Using this formula, the triangle inequality and the definition of spaces and , we get
Here
Using the estimates
and the estimates of equations (14), (15), (16), and (17), we get
Then
Let us prove the opposite inequality. For any positive operator we can write
From the relation and formula (21) it follows that
Consequently,
whence
Let
Note that it suffices to consider the case when . Applying the scheme of the paper [33] and using equations (8), (9), (10), (11), and the estimates of equations (14), (15), (16), and (17), we can establish the following estimate:
for . Applying the triangle inequality and the estimate of equation (22), we get
This means that the following inequality holds:
Theorem 3.1 is proved. □
Since the is a positive operator in the fractional spaces , from the result of Theorem 3.1 it follows that it is also a positive operator in the Hölder space . Namely, we have the following.
Theorem 3.2The operatorhas a bounded inverse inuniformly with respect tohfor anyand the following estimate holds:
4 Applications
In this section we consider the application of results of Sections 2 and 3. For a positive operator A in E the following result was established in papers [36,37].
Theorem 4.1LetAbe a positive operator inE. Then it obeys the following estimate:
wheredoes not depend onτandk. Hereis the Padé approximation ofnear.
For a numerical solution of the initialboundary value problem (1) the following difference scheme is presented:
We introduce the Banach space of all continuous abstract mesh vector functions
defined on with values in E, equipped with the norm
Note that the problem (24) can be written in the form of the abstract Cauchy problem
in a Banach space with a positive operator defined by (5). Here is the given abstract vector function defined on with values in E, is the element of . It is well known that (see, for example [3]) the formula
gives a solution of the problem (25) in .
Theorem 4.2For the solution of the problem (25) the stability inequality holds:
The proof of Theorem 4.2 is based on the positivity of the operator , equation (26) and the estimate of equation (23).
Applying the results of Theorems 2.1 and 4.2, we get the following theorem.
Theorem 4.3The solution of the problem (24) satisfies the following estimate:
Applying results of Theorems 3.2, 4.1, and 4.2, we get the following theorem.
Theorem 4.4Assume that
Then the solution of the problem (24) satisfies the following estimate:
Finally, one has not been able to obtain a sharp estimate for the constants figuring in the stability estimates. Therefore, our interest in the present paper is studying the difference scheme equation (24) by numerical experiments. Applying this difference scheme, the numerical method is proposed in the following section for the numerical solution of the hyperbolic system of equations with nonlocal boundary conditions. The method is illustrated by a numerical example.
5 Numerical results
For the numerical result, the initial value problem
for the hyperbolic system of equations with nonlocal boundary conditions is considered. Applying the difference scheme equation (24), we obtain
where
We get the system of equations in the matrix form
where
Thus, we have the firstorder difference equation with respect to k matrix coefficients. To solve this difference equation we have the following procedure:
For their comparison, the errors are computed by
of the numerical solutions. The numerical solutions are recorded for different values of ; , represent the numerical solutions of these difference schemes at . The errors are given in Table 1 for , and , respectively.
Table 1. Difference scheme
6 Conclusion
In the present study, the finitedifference method for the initialboundary value problem for the hyperbolic system of equations with nonlocal boundary conditions is studied. The positivity of the difference analogy of the space operator generated by this problem in the space with maximum norm is established. The structure interpolation spaces generated by this difference operator is investigated. The positivity of this difference operator in Hölder spaces is established. In practice, stability estimates for the solution of the difference scheme for the hyperbolic system of equations with nonlocal boundary conditions are obtained. A numerical example is applied. Moreover, applying this approach we can construct the stable difference schemes for numerical solutions of the initialboundary value problem
for the hyperbolic system of semilinear equations with nonlocal boundary conditions. Of course, convergence estimates for the solution of these difference schemes can be obtained.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Each of the authors read and approved the final version of the manuscript.
Acknowledgements
We would like to thank the reviewers whose careful reading, helpful suggestions, and valuable comments helped us to improve the manuscript.
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