We consider an inverse transmission scattering problem. This problem consists in determining an interface between two-layered media by farfield measurements. We prove that the interface is uniquely determined by the measurements of the far field pattern associated to incoming plane waves at a fixed frequency. For the reconstruction of the interface we solve a non linear integral equation using a truncated Newton-CG algorithm.

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We consider an axisymmetric inverse problem for the heat equation inside the cylinder a ≤ r ≤ b. We wish to determine the surface temperature on the interior surface {r = a} from the Cauchy data on the exterior surface {r = b}. This problem is ill-posed. Using the Laplace transform, we solve the direct problem. Then the inverse problem is reduced to a Volterra integral equation of the first kind. A standard Tikhonov regularization method is applied to the approximation of this integral equation when the data is not exact. Some numerical examples are given to illustrate the stability of the proposed method. MSC: 35K05; 65N06; 35R30; 44A10

Abstract. We present here a theoretical study of eigenmodes in axisymmetric elastic layers. The mathematical modelling allows us to bring this problem to a spectral study of a sequence of unbounded self-adjoint operators An, n ∈ N, in a suitable Hilbert space. We show that the essential spectrum ofAn is an interval of type [γ, +∞[ and that, under certain conditions on the coefficients of the medium, the discrete spectrum is non empty.Résumé. Nous présentons ici uneétude théorique des modes propres dans une coucheélastique axisymétrique. La modélisation mathématique permet de ramener ce problèmeà l'étude spectrale

We consider an inverse problem for the Poisson equation [Formula: see text] in the square [Formula: see text] which consists of determining the source [Formula: see text] from boundary measurements. Such problem is ill-posed. We restrict ourselves to a class of functions [Formula: see text]. To illustrate our method, we first assume that [Formula: see text] and [Formula: see text] are known functions with partial data at the boundary. For the reconstruction, we consider approximations by the Fourier series, therefore we obtain an ill-posed linear system which requires a regularization strategy. In the general case, we propose an iterative algorithm based on the full data at the boundary. Finally, some numerical results are presented to show the effectiveness of the proposed reconstruction algorithms.