scholarly journals Property C and an inverse problem for a hyperbolic equation

1991 ◽  
Vol 156 (1) ◽  
pp. 209-219 ◽  
Author(s):  
A.G Ramm ◽  
Rakesh
Keyword(s):  
Inverse Problem ◽  
Hyperbolic Equation

On the stability of recovering two sources and initial status in a stochastic hyperbolic-parabolic system

2021 ◽  
Author(s):  
Bin Wu ◽  
Jijun Liu
Keyword(s):  
Inverse Problem ◽  
Parabolic Equation ◽  
Hyperbolic Equation ◽  
Wave Field ◽  
Time Derivative ◽  
Stability Index ◽  
Carleman Estimates ◽  
Conditional Stability ◽  
Coupling System ◽  
The Stability

Abstract Consider an inverse problem of determining two stochastic source functions and the initial status simultaneously in a stochastic thermoelastic system, which is constituted of two stochastic equations of different types, namely a parabolic equation and a hyperbolic equation. To establish the conditional stability for such a coupling system in terms of some suitable norms revealing the stochastic property of the governed system, we first establish two Carleman estimates with regular weight function and two large parameters for stochastic parabolic equation and stochastic hyperbolic equation, respectively. By means of these two Carleman estimates, we finally prove the conditional stability for our inverse problem, provided the source in the elastic equation be known near the boundary and the solution be in a prior bound set. Due to the lack of information about the time derivative of wave field at final moment, the stability index with respect to the wave field at final time is found to be halved, which reveals the special characteristic of our inverse problem for the coupling system.

scholarly journals INVERSE PROBLEM WITH INTEGRAL IN TIME OVERDETERMINATION AND NONLOCAL BOUNDARY CONDITIONS FOR HYPERBOLIC EQUATION

2017 ◽  
Vol 22 (1-2) ◽  
pp. 27-32
Author(s):  
A. V. Duzheva
Keyword(s):  
Inverse Problem ◽  
Boundary Conditions ◽  
Hyperbolic Equation ◽  
Boundary Problem ◽  
Existence Result ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Nonlocal Boundary Conditions ◽  
Time Variable ◽  
Initial Boundary Problem

In this article, we consider a question of sovability of an inverse problem for a linear hyprbolic equation. Properties of the solution of an associated nonlocal initial-boundary problem with displacement in boundary conditions are used to develop an existence result for the identification of the unknown source. Overdetermination is represented as integral with respect to time-variable.

scholarly journals Recovery of unknown coefficients in a two-dimensional hyperbolic equation with additional conditions

2021 ◽  
Author(s):  
Elvin Azizbayov ◽  
He Yang ◽  
Yashar Mehraliyev
Keyword(s):  
Inverse Problem ◽  
Boundary Value Problem ◽  
Hyperbolic Equation ◽  
Original Problem ◽  
Boundary Value ◽  
Auxiliary Problem ◽  
Fourier Method ◽  
Two Dimensional ◽  
Inverse Boundary ◽  
Inverse Boundary Value Problem

In this paper, a nonlocal inverse boundary value problem for a two-dimensional hyperbolic equation with overdetermination conditions is studied. To investigate the solvability of the original problem, we first consider an auxiliary inverse boundary value problem and prove its equivalence (in a certain sense) to the original problem. Then using the Fourier method, solving an equivalent problem is reduced to solving a system of integral equations and by the contraction mappings principle the existence and uniqueness theorem for auxiliary problem is proved. Further, on the basis of the equivalency of these problems the uniquely existence theorem for the classical solution of the considered inverse problem is proved and some considerations on the numerical solution for this inverse problem are presented with the examples.

scholarly journals On the definition of a time-dependent lower coefficient in a third-order hyperbolic equation

2021 ◽  
pp. 9-18
Author(s):  
Б.С. Аблабеков ◽  
А.К. Жороев
Keyword(s):  
Inverse Problem ◽  
Hyperbolic Equation ◽  
Unknown Coefficient ◽  
Existence And Uniqueness Theorem ◽  
Third Order ◽  
Volterra Type ◽  
System Of Integral Equations ◽  
Additional Information ◽  
Order Hyperbolic Equation ◽  
Definition Of

В работе рассматривается обратная задача для гиперболического уравнения третьего порядка. Ставится обратная задача, состоящая в определении неизвестного коэффициента, зависящего от времени. В качестве дополнительной информации для решения обратной задачи задаются значения решения задачи во внутренней точке. Доказывается теорема существования и единственности решения обратной задачи. Доказательство основано на выводе нелинейной системы интегральных уравнений типа Вольтерра второго рода и доказательстве его разрешимости. The paper deals with an inverse problem for a hyperbolic equation of the third order. An inverse problem is posed, which consists in determining an unknown coefficient that depends on time. As additional information for solving the inverse problem, we set the values of the solution to the problem at an interior point, and prove the existence and uniqueness theorem for the solution of the inverse problem. The proof is based on the derivation of a nonlinear system of integral equations of the Volterra type of the second kind and the proof of its solvability.

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