Carleman estimate for a strongly damped wave equation and applications to an inverse problem

In this paper, we establish a Carleman estimate for a strongly damped wave equation in order to solve a coefficient inverse problems of retrieving a stationary potential from a single time‐dependent Neumann boundary measurement on a suitable part of the boundary. This coefficient inverse problem is for a strongly damped wave equation. We prove the uniqueness and the local stability results for this inverse problem. The proof of the results relies on Carleman estimate and a certain energy estimates for hyperbolic equation with strongly damped term. Moreover, this method could be used for a similar inverse problem for an integro‐differential equation with hyperbolic memory kernel. Copyright © 2012 John Wiley & Sons, Ltd.     

On the inverse spectral problem for the string equation

The theory of Hilbert spaces of entire functions is used to deduce the changes in the mass distribution of a string effected by a small change in its spectral function. The formula of Gelfand-Levitan emerges as a special case.     

Periodic problem for the nonlinear damped wave equation with convective nonlinearity

We study the nonlinear damped wave equation with a linear pumping and a convective nonlinearity. We consider the solutions, which satisfy the periodic boundary conditions. Our aim is to prove global existence of solutions to the periodic problem for the nonlinear damped wave equation by applying the energy-type estimates and estimates for the Green operator. Moreover, we study the asymptotic profile of global solutions.     

Numerical solution of the inverse problem for the wave equation

Translated from Metody Matematicheskogo Modelirovaniya i Vychislitel'noi Diagnostiki, pp. 6–9, Izd. Moskovskogo Universiteta, Moscow, 1990.     

On an inverse spectral problem for a quadratic Jacobi matrix pencil

Given two monic polynomials P2n and P2n−2 of degree 2n and 2n−2 (n?2) with complex coefficients and with disjoint zero sets. We give necessary and sufficient conditions on these polynomials such that there exist two n×n Jacobi matrices B and C for which

     

The inverse problem for differential pencils on a star-shaped graph with mixed spectral data

The partial inverse problem for differential pencils on a star-shaped graph is studied from mixed spectral data.More precisely,we show that if the potentials on all edges on the star-shaped graph but one are known a priori then the unknown potential on the remaining edge can be uniquely determined by partial information on the potential and a part of eigenvalues.     

Solution method for the inverse boundary-value problem for the wave equation

Existence and uniqueness issues are considered for the inverse boundary-value problem of determining the variable coefficient of the one-dimensional wave equation on a half-line. A Tikhonov-regularizing algorithm is constructed to find an approximate solution using input data that are specified with an error.Translated from Metody Matematicheskogo Modelirovaniya, Avtomatizatsiya Obrabotki Nablyudenii i Ikh Primeneniya, pp. 80–88, 1986.     

半线性波方程的一个反问题

利用压缩映像原理讨论了一类半线性波方程确定未知系数的反问题,文中给出了该问题解的存在性、唯一性和稳定性。     

Eigenvalue asymptotics, inverse problems and a trace formula for the linear damped wave equation

We determine the general form of the asymptotics for Dirichlet eigenvalues of the one-dimensional linear damped wave operator. As a consequence, we obtain that given a spectrum corresponding to a constant damping term this determines the damping term in a unique fashion. We also derive a trace formula for this problem.     

Asymptotics of solutions to the periodic problem for the nonlinear damped wave equation

We study large time asymptotic behavior of solutions to the periodic problem for the nonlinear damped wave equation

Finite dimensional spectral inverse problem for a bundle of hermitian quadratic forms

The paper is devoted to recovering the coefficients of a pair of Hermitian quadratic forms c(x, x) and (x, x) in a special basis, in which the matrix of the form c(x, x) is tridiagonal and the matrix of the form m(x, x) is diagonal. The form c(x, x) is positive definite, and the form m(x, x) is nondegenerate, but is not positive difinite in contrast with a well-known case. The data of the inverse problem include the spectrum λ₁...,λ_n of the bundle II_λ and the set of numbers ρ₁...,ρ_n connected with the main normalized elements of the bundle. A procedure for solving the inverse problem is described. The characteristic conditions for λ₁...,λ_n; are found that provide its solution. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 33–36, 1990. Translated by T. N. Surkova.     

Spectral problem for polynomial matrix pencils. 2

For an arbitrary polynomial pencil of matrices A_i of dimensions m×n one presents an algorithm for the computation of the eigenvalues of the regular kernel of the pencil. The algorithm allows to construct a regular pencil having the same eigenvalues as the regular kernel of the initial pencil or (in the case of a dead end termination) allows to pass from the initial pencil to a pencil of smaller dimensions whose regular kernel has the same eigenvalues as the initial pencil. The problem is solved by reducing the obtained pencil to a linear one. For solving the problem in the case of a linear pencil one considers algorithms for pencils of full column rank as well as for completely singular pencils.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 111, pp. 109–116, 1981.