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Nonlocal boundary value hyperbolic problems involving integral conditions
Boundary Value Problems volume 2014, Article number: 205 (2014)
Stability estimates for the solution of the nonlocal boundary value problem with two integral conditions for hyperbolic equations in a Hilbert space H are established. In applications, stability estimates for the solution of the nonlocal boundary value problems for hyperbolic equations are obtained.
It is well known that nonlocal boundary value problems with integral conditions are widely used for thermo-elasticity, chemical engineering, heat conduction, and plasma physics –. Some problems arising in dynamics of ground waters are defined as hyperbolic equations with nonlocal conditions  and . The authors of  investigate nonclassical problems for multidimensional hyperbolic equation with integral boundary conditions and the uniqueness of classical solution. In  a linear second-order hyperbolic equation with forcing and integral constraints on the solution is converted to a nonlocal hyperbolic problem. Using the Riesz representation theorem and the Schauder fixed point theorem, existence and uniqueness of a generalized solution are proved. The solutions of hyperbolic equations with nonlocal integral conditions were investigated in –. The method of operators as a tool for investigation of the solution to hyperbolic equations in Hilbert and Banach spaces has been used extensively in –.
In  the nonlocal boundary value problem
was investigated. Stability estimates for the solution of the problem were established. First order of accuracy difference schemes for the approximate solutions of the problem were presented. Stability estimates for the solution of these difference schemes were established. Theoretical statements were supported by numerical examples.
In the present paper, we consider the nonlocal boundary value problem with integral conditions
in a Hilbert space H with a self-adjoint positive definite operator A. We are interested in studying the stability of solutions of problem (1) under the assumption
is twice continuously differentiable on the interval and continuously differentiable on the segment .
The element belongs to for all , and the function is continuous on the segment .
satisfies the equation and nonlocal boundary conditions (1).
2 The main theorem
Let H be a Hilbert space, A be a positive definite self-adjoint operator with , where . Throughout this paper, is a strongly continuous cosine operator-function defined by
Then, from the definition of sine operator-function ,
it follows that
The following estimates hold:
Suppose that assumption (2) holds. Then the operator T,
has the inverse
and the following estimate is satisfied:
Applying the triangle inequality and estimates (3), we obtain
Estimate (4) follows from this estimate. Lemma 2.2 is proved. □
has a unique solution,
where the function is not only continuous but also continuously differentiable on , and .
Using (6) and the nonlocal boundary condition
Differentiating both sides of (6), we obtain
Using this formula and the integral condition
are valid, where M does not depend on, , φ, and ψ.
We take the estimates
Applying to (10), we get
Second, applying operator to (9), we get
Third, applying A to (9) and using Abel’s formula, we have
Now, we consider the applications of Theorem 2.1. First, a nonlocal boundary value problem for a hyperbolic equation
under assumption (2) is considered. Problem (23) has a unique smooth solution for (2), smooth functions (), , , () and (), σ a positive constant and under some conditions. This allows us to reduce problem (23) to nonlocal boundary value problem (1) in the Hilbert space with a self-adjoint positive definite operator defined by (23).
For the solution of problem (23), we have the following stability inequalities:
where M does not depend on, , and.
The proof of Theorem 3.1 is based on Theorem 2.1 and the symmetry properties of the space operator generated by problem (23).
Problem (23) can be written in the abstract form
in the Hilbert space of all square integrable functions defined on with a self-adjoint positive definite operator defined by the formula
with the domain
Second, let Ω be the unit open cube in the m-dimensional Euclidean space with boundary S, . In , let us consider a boundary value problem for the multidimensional hyperbolic equation
under assumption (2). Here, (), , () and , , are given smooth functions and .
Let us introduce the Hilbert space of all square integrable functions defined on , equipped with the norm
For the solution of problem (27), the following stability inequalities hold:
where M does not depend on, , and (, ).
For the solution of the elliptic differential problem
the following coercivity inequality holds:
where M is independent of ω.
This work is devoted to the study of the stability of the nonlocal boundary value problem with integral conditions for hyperbolic equations. For the solution of nonlocal boundary problem (1) in a Hilbert space H with a self-adjoint positive definite operator A, Theorem 2.1 is established. Two applications of Theorem 2.1 are given. Of course, stable two-step difference schemes for approximate solution of problem (1) can be presented. The methods given above permit us to establish the stability of these difference schemes. Applying , we can give a numerical support of the theoretical results.
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This work is supported by the Scientific Research Fund of Fatih University (Project No: P50041203-B).
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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Ashyralyev, A., Aggez, N. Nonlocal boundary value hyperbolic problems involving integral conditions. Bound Value Probl 2014, 205 (2014). https://doi.org/10.1186/s13661-014-0205-4
- hyperbolic equation
- nonlocal boundary value problems