Volume 9, Issue 4

- The Cauchy problems for operator analogues of the Laplace and Poisson equations with data known on a part of the domain boundary are considered. For solutions of these problems formulas of Carleman type are derived. The formulas obtained can be used in investigation of two-dimensional tomography problems with incomplete data.

- Much work has been done on inverse scattering problems for one-dimensional acoustic media. Unfortunately, the linear method of the Gel’fand - Levitan integral equations is developed only for inversion from reflection data in case of negligible dissipative effects. On the contrary, measurement data, in geophysical prospecting particularly, are often influenced by these effects. The main result of this paper is a necessary and sufficient condition of solvability of the inverse dissipative scattering problem with data given on a finite time interval. We also consider a bilinear procedure for recovering the dissipation coefficient from the reflection data. The approach is based on a step-by-step split method for numerical solution of a system of the Gel’fand - Levitan integral equations under nonlocal boundary conditions. Besides, we develop a method of inversion of difference schemes for solving the inverse dissipative scattering problems.

- We present a class of iterative methods for solving nonlinear ill-posed operator equations with differentiable operators in a Banach space. The methods are based on regularization of the linearized equation by using an appropriate regularization scheme at each iteration. We establish that the iterations converge locally with power rate provided that the solution admits of a sourcewise representation. We also prove that the condition of sourcewise representability is very close to a necessary condition for this kind of estimates.

- We consider the problem of reconstruction of the temperature from measurements of the temperature and heat flux on a part of the boundary and present iterative methods for solving this problem. The characteristic feature of the methods is the fact that in each iteration step, well-posed problems for the same equation are solved. The regularizing character of the methods comes from suitable choice of boundary conditions. This fact allows one to use standard numerical packages. Some numerical experiments are given.

- We consider the inverse problem of identification of memory kernels in one-dimensional heat flow are dealt with where the kernel is represented by a finite sum of products of known spatially-dependent functions and unknown time-dependent functions. As additional conditions for the inverse problems observations of both heat flux and temperature are prescribed. Using the Laplace transform method we prove an existence and uniqueness theorem for the memory kernel.

- In this paper we prove the existence of regular solutions for some nonlinear coefficient inverse problems for elliptic equations.

- In this paper inverse problems for the Sturm-Liouville operator are studied. A set of values of the logarithmic derivative of eigenfunctions at some internal point and parts of two spectra are taken as data. Uniqueness theorems are obtained. The approach that was used in investigation of problems with partially known potential is employed.