Ill-posed boundary-value problems
type: Item
instance of: scholarly article
title: Ill-posed boundary-value problems
zbMATH Open document ID: 1104.35003
author: S. E. Temirbolat
published in: Inverse and Ill-Posed Problems Series
publication date: 2006-08-04T00:00:00Z
review text: This monograph is an attempt at extending well-known facts to new classes of problems and at working out approaches to the solution of these problems. To be more exact, the monograph is devoted to the questions of consistency of ill-posed boundary-value problems for systems of various types of first-order differential equations with constant coefficients and to the methods of their solution. The introductory chapter gives some facts of the theory of matrices and ordinary differential equations necessary for understanding of the book. In particular, canonical direct and inverse problems and the relation between an arbitrary ordinary equation and a first-order system are given. The first chapter studies well-posedness for an arbitrary system of ordinary equations and the occurrence of ill-posed problems among them. The author considers the first boundary-value problem on the ray \(t>0\) and on the interval. For the general boundary-value problem the author analyse boundary conditions and linearization algorithm for its solving. In the second chapter a similar approach is applied to various kinds of parabolic systems and solution methods for ill-posed problems are given. In the section ``Parabolic systems'' the author considers parabolic systems in Petrovskii's sense, and in Shirota's sense, the power of parabolic systems, and gives several examples. In the section ``Boundary-value problems'' the author considers the mixed problem, the unique solvability of the Cauchy problem, and the problem without initial data. Section ``The first boundary-value problem'' is devoted to the parabolic system in Petrovskii's sense and Eidelmann's hypothesis, to other types of parabolic systems, to the solution of conditionally well-posed problems for these systems. In the section ``Mixed problem of heat and mass exchange'' the author considers the violation of the Lopatinskii condition, and presents cases of ill-posed and well-posed problems. In the third chapter the above-described me…
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 35R25
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 35K50
Mathematics Subject Classification ID: 35L50
Mathematics Subject Classification ID: 35F15
zbMATH DE Number: 5043473
zbMATH Keywords: systems of the first-order differential equations
zbMATH Keywords: parabolic systems in Petrovskii's sense
zbMATH Keywords: parabolic systems in Shirota's sense
zbMATH Keywords: mixed problem
zbMATH Keywords: Cauchy problem
zbMATH Keywords: Eidelmann's hypothesis
zbMATH Keywords: Lopatinskii condition
zbMATH Keywords: acoustics
zbMATH Keywords: gas dynamics
reviewed by: A. L. Karchevsky
MaRDI profile type: Publication
description: scientific article