Multidimensional inverse and ill-posed problems for differential equations
type: Item
instance of: scholarly article
title: Multidimensional inverse and ill-posed problems for differential equations
zbMATH Open document ID: 0846.35144
author: Yu. E. Anikonov
published in: Inverse and Ill-Posed Problems Series
publication date: 1996-02-13T00:00:00Z
review text: The book is divided into four chapters dealing with inverse problems: 1) for second-order differential equations; 2) for kinetic equations; 3) related to scattering theory and the determination of unknown bodies; 4) related to integral geometry. 1) This chapter is mostly devoted to the question of uniqueness. The term to be recovered is usually a coefficient or the right-hand side of a linear differential equation. Hyperbolic problems are dealt with by the technique of the so-called quasimonotonic operators, while in the parabolic case the coefficients are assumed to be independent of one variable. When this is time, the inverse problem consists in recovering the coefficient of \(D_t u\), assuming that the equation is satisfied in the whole space-time cylinder \(D\times R\), \(D\) being a domain in \(\mathbb{R}^n\). Such a setting, which gives rise to a severely ill-posed direct problem, turns out to be appropriate in the identification case, when, in addition to the boundary data, we are given also an ``initial'' condition at some time \(t= t_0\). Using a Fourier \(t\)-transform and assuming that the boundary data admit a Fourier \(t\)-transform vanishing outside a given ball, the inverse problem can be reduced to a nonlinear integro-differential elliptic direct problem. The same can be done when the unknown coefficient is independent of one spatial variable, say \(z\), and the elliptic differential operator can be split as the sum of two differential operators acting on different variables. Another important technique developed by the author consists in deriving explicit representations for the unknowns in terms of the solution to some parabolic direct problem, when the data admit a factorization. In some specific cases it is possible to derive representations in terms of data, only. 2) After having introduced the general integrodifferential kinetic equation, the author begins to consider the case when such equation reduces to a differential one. For some simple o…
reviewed by: Alfredo Lorenzi
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 65-02
Mathematics Subject Classification ID: 34-02
Mathematics Subject Classification ID: 34A55
Mathematics Subject Classification ID: 53C65
Mathematics Subject Classification ID: 65R30
Mathematics Subject Classification ID: 65M30
Mathematics Subject Classification ID: 86A22
Mathematics Subject Classification ID: 44A12
zbMATH DE Number: 844030
zbMATH Keywords: integrodifferential equations
zbMATH Keywords: domain identification
zbMATH Keywords: solution in closed form
zbMATH Keywords: second-order differential equations
zbMATH Keywords: kinetic equations
zbMATH Keywords: scattering theory
zbMATH Keywords: integral geometry
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 844030