Volume 12, Issue 4

In this paper the inverse problem of finding the minor coefficient and the inverse problem of finding the right-hand side member of special type in a second order parabolic equation with changing time direction are considered. The existence and uniqueness of solutions to these problems are investigated.

We consider the problem of detecting corrosion damage on an inaccessible part of a specimen. We adopt a model in which the corrosion damage is represented by an unknown boundary coefficient appearing in a third kind boundary value problem for Laplace's equation. The associated inverse problem consists in the determination of the unknown boundary coefficient from an overdetermined data on an accessible part of the boundary. In the present paper we are interested in the derivation of stability estimates for this inverse problem. We first prove a local Lipschitz stability estimate for an arbitrary smooth domain. Next, we establish a log-log stability estimate in the case of a rectangular domain.

In this paper we present a new formulation for the nonlinear Cauchy problem for elliptic equations. This formulation allows us to give a method for solving this class of problems. The nonlinear problem is reduced to a linear Cauchy problem for the Laplace equation coupled with a sequence of nonlinear scalar equations. We solve the linear problem using the iterative method introduced in [7]. The linear dense systems obtained from a boundary element approximation are solved using an efficient implementation which accelerate the iterative process. Various types of convergence, comparison results and effects of small perturbations on boundary data are investigated. The numerical results show that the method produces a stable good approximate solution.

The inverse problem for linear operator-differential equation of the first order with degenerate operator near derivative, with unknown function in the right-hand side of the equation and generally speaking with integral condition of overdetermination are investigated. It is shown that under the assumptions on the operator in equations that guarantee the existence of analytic resolving semigroup of homogeneous equation the satisfying of some concordance condition for initial and final datas is necessary for the solvability of the problem. But if one put initial data only on the image of the resolving semigroup then the conditions of well-posedness of the equations can be found.

We study the following inverse problem of finding a solution u and a coefficient q : where G ⊂ is a bounded domain with boundary Γ and L is a second order elliptic operator. It is demonstrated that the problem is solvable locally in time or in the case when the norms of the data are sufficiently small.

We prove that in dimension two potential scattering the leading order singularities of the unknown potential are obtained exactly by the Born approximation which corresponds to the scattering data with fixed energy. The proof is based on the new estimates for the Green-Faddeev's function in L p (), 1 < p ≤ 2. These estimates allow us to consider the potentials with stronger singularities than in previous publications and with noncompact support.

The uniqueness of reconstruction of a connected compact analytic embedded in manifold without boundary from some projection and section functionals of its polar of the convex hull is discussed in the paper.

We consider the inverse problem for a first-order homogeneous system of linear ordinary differential equations (LODE), where Y ( t ) is a vector-function with n components and A is an unknown matrix of dimensionality n × n with constant complex coefficients and certain restrictions imposed on its eigenvalues. The boundary conditions are C k := Y ( t k ), t k = t 0 + kd , d > 0, k = 0, 1, … , N , N ≥ n . Here is a given system of vectors in . This problem is equivalent to the problem of extrapolating a vector-function composed of quasi-polynomials representing solutions of some LODEs with constant coefficients of order n . The zone of solution stability of the system against small-amplitude input data oscillations is described. The results obtained are used to construct an approximation algorithm for a real vector-function of one variable set at a finite number of nodes of a uniform grid (modified Prony algorithm).