Nonlocal boundary value hyperbolic problems involving integral conditions
© Ashyralyev and Aggez; licensee Springer 2014
Received: 15 June 2014
Accepted: 19 August 2014
Published: 25 September 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 –.
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.
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
Estimate (4) follows from this estimate. Lemma 2.2 is proved. □
where the function is not only continuous but also continuously differentiable on , and .
are valid, where M does not depend on, , φ, and ψ.
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).
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).
under assumption (2). Here, (), , () and , , are given smooth functions and .
where M does not depend on, , and (, ).
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.
This work is supported by the Scientific Research Fund of Fatih University (Project No: P50041203-B).
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