Conditional Correctness and Approximate Solution of Boundary Value Problem for the System of Second Order Mixed-type Equations

Abstract

The general theory of boundary value problems for the mixed type equations with variable coefficients and with a manifold of type change have been the subject of research M. A. Lavryent’yev, A.V. Bitsadze, M. M. Smirnov, M. S. Salakhitdinov, T. D.Djuraev, V.N. Vragov, K. B. Sabitov, A. I. Kozhanov and many others [1, 2]. These type of the equations have many different applications, for example, the problems encountered in applications, in particular the problem of transonic flow of a compressible medium, and without torque shell theory. Many important practical applications, such as jet aircraft and astronautics, rocketry, gas-dynamic lasers, caused an avalanche growth of research in the field of boundary value problems for equations of mixed type (see. [3, 4]). Here we consider the system of mixed type equations. Systematic study of such equations began from the work of F. Trikomi and S. Gellerstedt [1, 2]. The theory of the solvability of boundary value problems for linear models described by a such equations has been constructed in the papers S. A. Tersenov, I. E. Egorov, A. A. Kerefov, N. V. Kislov, S.G. Pyatkov and others [5–7]. The problem considered in this paper belongs to the class of ill-posed problems of mathematical physics. Namely, in this problem the solution does not continuously depend on the initial data. Ill-posed problems for such equations were considered in [8–11]. In this paper, we establishe the conditional correctness of this problem and construct the approximate solution of the problem by regularization and quasi-inverse methods.