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.

CORRECTNESS OF THE MIXED PROBLEM FOR ONE CLASS OF DEGENERATE MULTIDIMENSIONAL HYPERBOLO-PARABOLIC EQUATIONS

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 medium. If the medium is non-conductive, we get degenerate multi-dimensional hyperbolic equations. If the medium has a high conductivity, then we go to degenerate multidimensional parabolic equations. Consequently, the analysis of electromagnetic fields in complex media (for example, if the conductivity of the medium changes) reduces to degenerate multidimensional hyperbolic-parabolic equations. Also, it is known that the oscillations of elastic membranes in space according to the Hamilton principle can be modeled by degenerating multidimensional hyperbolic equations. Studying the process of heat propagation in a medium filled with mass leads to degenerate multidimensional parabolic equations. Consequently, by studying the mathematical modeling of the process of heat propagation in oscillating elastic membranes, we also come to degenerate multidimensional hyperbolic-parabolic equations. When studying these applications, it is necessary to obtain an explicit representation of the solutions of the studied problems. The mixed problem for degenerate multidimensional hyperbolic equations was previously considered. As far as is known, these questions for degenerate multidimensional hyperbolic-parabolic equations have not been studied. In this paper, unique solvability is shown and an explicit form of the classical solution of the mixed problem for one class of degenerate multidimensional hyperbolic-parabolic equations is obtained.

IXED PROBLEM FOR THREE-DIMENSIONAL HYPERBOLIC EQUATIONS WITH DEGENERATION OF TYPE AND ORDER

It is known that in space during mathematical modeling of electromagnetic fields in space, the nature of the electromagnetic process is determined by the properties of the medium. If the medium is non-conductive, then we get degenerating multidimensional hyperbolic equations. Therefore, the analysis of electromagnetic fields in complex environments (for example, if the conductivity of the medium changes) is reduced to degenerating multidimensional hyperbolic equations. It is also known that oscillations of elastic membranes in space according to the Hamilton principle can be modeled by degenerating multidimensional hyperbolic equations. Therefore, by studying mathematical modeling of the process of heat propagation in oscillating elastic membranes, we also come to degenerating multidimensional hyperbolic equations. When studying these applications, it becomes necessary to obtain a clear representation of the solutions to the investigated problems. The mixed problem for degenerating multidimensional hyperbolic equations in generalized spaces is well researched. This task is also studied in the works of S. A. Aldashev, where it is shown that its correctness significantly depends on the height of the cylindrical region under consideration. A.V. Bitsadze drew attention to the importance of studies of multidimensional hyperbolic equations with degeneration of type and order. Mixed problems for these equations have not previously been studied. In this work, the solvability of a mixed problem is shown and a clear form of a classical solution for three-dimensional hyperbolic equations with degeneration of type and order is obtained.