Tricomi problem analogue for a second order mixed type equation

Zh. A. Balkizov

Vladikavkazskij matematičeskij žurnal, Tome 26 (2024) no. 4, pp. 44-54

Résumé

In this paper, we investigated an analogue of the Tricomi problem for an equation of parabolic-hyperbolic type of the second order with a heat conduction operator in the parabolicity region and with a degenerate hyperbolic operator of the first kind in the hyperbolicity region. The type change line is characteristic for a parabolic equation and non-characteristic for a hyperbolic one. The problem is studied when the values of the sought function are given on non-characteristic segments of the boundary of a parabolic parabolic equation, and a value is also given on the characteristic of the hyperbolic equation. Previously, similar problems were the subject of research by many authors, such as, for example, the works of A. M. Nakhushev and Kh. G. Bzhikhatlova, M. S. Salakhitdinov and A. S. Berdysheva. In this paper, we prove uniqueness and existence theorems for a regular solution to the problem under study. When proving the uniqueness theorem for the solution of the problem, modern methods of the theory of fractional calculus are used, and the existence is proven using the method of integral equations. In the case of when the coefficients of the equation under consideration are constant, the solution is found and written out explicitly. The theorems proved generalize the previously obtained results both in terms of sufficiency conditions for the uniqueness of a solution to an analogue of the Tricomi problem, and in terms of the existence theorem.