On a boundary value problem for a parabolic-hyperbolic equation of the fourth order

Abstract

In this paper a boundary value problem for a fourth-order equation of parabolic-hyperbolic type within a pentagonal domain was investigated. In the equation under consideration, one characteristic aligns with the Ox axis while the other aligns with the Oy axis. Initially, the problem was examined within the lower triangle of the specified domain. Utilizing a differential equation solution construction method, a solution to the formulated problem was derived. Subsequently, within the rectangles of the domain, employing the continuation method, two relationships between the solution’s traces were established. Moreover, from the parabolic segment of the domain, two additional relationships between unknown traces will be derived. Solving this system of four equations enables determination of these traces, thereby resolving the problem.