Ill-posed boundary-value problems
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 methods are used to study problems for hyperbolic-type equations. In the section ``Hyperbolicity and mixed problem'' the author presents the problem statement. Finally, in the sections ``Two-dimensional acoustic problem'' and ``Linearized plane problem of gas dynamics'' the author describes applications.
systems of the first-order differential equations
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