On an analogue of the Tricomi problem for a «pointwise» loaded hyperbolic-parabolic equation

Для нагруженного уравнения гиперболо-параболического типа исследуется однозначная разрешимость аналога задача Трикоми. Нагрузка определена в фиксированных точках области искомых решений, в том числе и во внутренних точках. Найдены условия существования и единственности регулярного решения задачи. The unique solvability of an analogue of the Tricomi problem is investigated for a loaded hyperbolic-parabolic equation. The load is determined at boundary and interior fixed points of the domain in which the solutions are sought. Sufficient conditions are found for the existence and uniqueness of solutions.

TRICOMI PROBLEM FOR MULTIDIMENSIONAL MIXED HYPERBOLIC-PARABOLIC EQUATION

It is known that in mathematical modeling of electromagnetic fields in space, the nature of the electromagnetic process is determined by the properties of the media. If the medium is non-conducting, then we obtain multidimensional hyperbolic equations. If the mediums conductivity is higher, then we arrive at multidimensional parabolic equations. Consequently, the analysis of electromagnetic fields in complex media (for example, if the conductivity of the medium changes) reduces to multidimensional hyperbolic-parabolic equations. When studying these applications, one needs to obtain an explicit representation of solutions to the problems under study. Boundary-value problems for hyperbolic-parabolic equations on a plane are well studied; however, their multidimensional analogs have been analyzed very little. The Tricomi problem for the above equations has been previously investigated, but this problem in space has not been studied earlier. This article shows that the Tricomi problem is not uniquely solvable for a multidimensional mixed hyperbolic-parabolic equation. An explicit form of these solutions is given.