Volume 11, Issue 1

An approach of the research of multidimensional inverse problems based on special representations of solutions of evolution equations is suggested in this paper.

We consider the inverse problem of crack determination related to the non-destructive thermal testing of materials process. Using arbitrary transient heat flux applied to the external boundary and measuring the induced planar cracks.

We consider the inverse problem of identifying a Robin coefficient on some part of the boundary of a smooth 2D domain from overdetermined data available on the other part of the boundary, for Laplace equation in the domain. Using tools from complex analysis and analytic functions theory, we provide a constructive and convergent identification scheme for this inverse problem, together with numerical experiments.

We study the problem of finding the memory term of a hyperbolic equation from the Dirichlet-to-Neumann map given on a finite time interval. We prove that this map determines uniquely some characteristics of the memory function and thereby memory functions of a special form.

We are concerned with the problem of recovering the kernel k , depending on time and having a special spatial symmetry, in the parabolic integro-differential equation (1.1) and related to a domain Ω which is union of level sets of each function k ( t , ·). We single out a special class of differential operators A and two pieces of suitable additional information for which the problem of identifying k can be uniquely solved locally in time.

In this paper we study the problem of estimating a two-dimensional parameter in the wave equation from overdetermined observational boundary data. The inverse problem is reformulated as an integral equation and two numerical algorithms, the projection method and the Landweber iteration method are investigated. By the projection method the inverse problem is reduced to a finite dimensional system of integral equations. We prove convergence of the projection method. Moreover, we show that the Landweber iteration method is a stable and convergent numerical method for solving this parameter estimation problem.