The Tricomi--Neymann problem for a three-dimensional mixed-type equation with singular coefficients

A. K. Urinov ; K. T. Karimov

Vladikavkazskij matematičeskij žurnal, Tome 25 (2023) no. 4, pp. 120-134

Résumé

In this paper, we study the Tricomi-Neumann problem for a three-dimensional mixed-type equation with three singular coefficients in a mixed domain, for which the elliptic part consists of a quarter of a cylinder, and the hyperbolic part of a rectangular prism. The unique solvability of the formulated problem in the class of regular solutions is proved. In this case, the Fourier method based on the separation of variables was used. After the separation of variables in the hyperbolic part of the mixed domain, eigenvalue problems for one-dimensional and two-dimensional equations appear. By solving these problems, the eigenfunctions of the corresponding problems are found. To solve the two-dimensional problem, the formula that gives the solution to the Cauchy–Goursat problem is used. As a result, solutions of eigenvalue problems for the three-dimensional equation in the hyperbolic part are found. With the help of these eigenfunctions and the gluing condition, a non-local problem appears in the elliptic part of the mixed domain. To solve the problem in the elliptic part, the problem was reflected in the cylindrical coordinate system, and then, by separating the variables, the eigenvalue problems for two ordinary differential equations were obtained. Based on the property of completeness of systems of eigenfunctions of these problems, a uniqueness theorem is proved. The solution of the problem under study is constructed as the sum of a double series. When substantiating the uniform convergence of the constructed series, asymptotic estimates for the Bessel functions of the real and imaginary arguments were used. Based on them, estimates were obtained for each member of the series, which made it possible to prove the convergence of the resulting series and its derivatives up to the second order inclusive, as well as the existence theorem in the class of regular solutions.