Volume 16, Issue 3

The solvability of the coefficient inverse problem for parabolic high-order equation is considered. As the overdetermination conditions the final overdetermination is taken. The existence and uniqueness theorems are proved.

We study the singular perturbation approach proposed by Lavrentiev for the regularization of systems of Volterra integral equations of the first kind, in the case that the kernel K( t ) is not invertible for t = 0 and without assuming K( t ) ~ t v I. We single out a class of kernels, which we call “diagonally dominant”. We show that when the kernel belongs to this class then it is possible to regularize the problem using a multiscale singular perturbation method.

In this paper we recover two unknown kernels related to a thermal body Ω with memory consisting of two different sub-bodies Ω 1 and Ω 2 , when the boundaries of Ω 1 and Ω 2 have a common (closed) surface Γ intersecting the boundary ∂Ω of Ω. The additional measurements are performed on two (accessible) subsets of ∂Ω 1 and ∂Ω 2 . For this problem we prove existence, uniqueness and continuous dependence on the data in the framework of Sobolev spaces of L 2 -type in space.

The paper is devoted to the analysis of linear ill-posed operator equations A x = y with an injective, compact linear forward operator A : X → Y mapping between Hilbert spaces X , Y . We are going to combine the ideas of [S. I. Kabanikhin, Convergence rate estimation of gradients methods via conditional stability of inverse and ill-posed problems. J. Inv. Ill-Posed Problems 13 (2005), 259–264.] and [S. I. Kabanikhin, Conditional stability stopping rule for gradient methods applied to inverse and ill-posed problems. J. Inv. Ill-Posed Problems 14 (2006), 805–812.] concerning the conditional stability on some subset M ⊂ X and the ideas of [B. Hofmann and P. Mathé, Analysis of profile functions for general linear regularization methods. SIAM J. Numer. Anal. 45 (2007), 1122–1141.] concerning profile functions in order to find convergence rates for general linear regularization methods. Moreover, an extension to Lavrentiev regularization is given.

The present paper proves uniqueness theorem in an inverse problem for the kinetic equation of the motion of particles in plasma.

In the present paper, the inverse problem with unknown composite right-hand side for second-order linear hyperbolic equations is examined. Solution existence, uniqueness, and stability for the boundary-value problem are established.