Local boundary value problem for a class of third-order elliptic-hyperbolic type equation

In recent years, non-classical equations of mathematical physics have been attracting more and more attention of specialists; this is due to both theoretical and practical interest. Third-order equations are found in various problems of physics, mechanics, and biology. For example, in the theory of transonic flows, the propagation of plane waves in a viscoelastic solid, and the prediction and regulation of groundwater. One of the important classes of non-classical equations of mathematical physics is the equations of composite and mixed-composite type, which the main parts contain operators of elliptic, elliptic-hyperbolic and parabolic-hyperbolic types. Correct boundary value problems for equations of elliptic-hyperbolic and parabolic-hyperbolic types of the third order, in case that the main part of the operator contains the derivative with respect to x or y, is first studied by A.B. Bitsadze, M.S. Salakhitdinova and T.D. Djuraev, in addition to the fact that these equations are found in various problems of mechanics. For example, the propagation of a plane wave in a viscoelastic solid. In these works on the investigation of boundary value problems, a representation of general solution of a mixed-composite type equation in the form of a sum of functions was used. Such representation takes place only for equations, which are composed of the product of permutable differential operators. In this paper, we study boundary value problem for a third-order equation with an elliptic-hyperbolic operator in the main part. The equation under consideration is composed of the product of non-permutable differential operators, therefore the well-known representations of the general solution introduced by A.V. Bitsadze and M.S. Salakhitdinova are not applied. To study the considered third-order equation of the mixed type, we applied a method, which does not require a special representation of the general solution of the equation. This method determines the study of an equation of elliptic-hyperbolic type of the second order with unknown right-hand sides, which is of interest for solving important inverse problems of mechanics and physics. The existence and uniqueness theorems of the classical solution of the problem are proved. The proof is based on the extremum principle for a third-order equation and on the theory of singular and Fredholm integral equations.