An initial-boundary problem for a hyperbolic equation with~three lines of degenerating of the second kind

A. K. Urinov ; D. A. Usmonov

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 26 (2022) no. 4, pp. 672-693

Résumé

In present paper, a hyperbolic partial differential equation of the second kind degenerates on the sides and on the base of the rectangle has been considered. For the considered equation, an initial-boundary problem with non-local boundary conditions has been formulated. The uniqueness, existence, and stability of the solution to the stated problem were investigated. The uniqueness of the solution to the problem was proved by the method of energy integrals. The existence of obtained solution was investigated by the Fourier method based on the separation of variables. To do this first, a spectral problem for an ordinary differential equation was investigated, which arises from the formulated problem in virtue of the separation of variables. It was proved that this spectral problem can have only positive eigenvalues. Then, Green's function of the spectral problem has been constructed, and equivalently reduced to an integral Fredholm equation of the second kind with a symmetric kernel. Hence, on the basis of the theory of integral equations, it has been concluded that there is a countable number of eigenvalues and eigenfunctions of the spectral problem. Using the properties of constructed Green's function of the spectral problem, some lemmas, using to prove the existence of a solution to the problem on the uniform convergence of some bilinear series, were proved. Lemmas on the order of the Fourier coefficients of a given function have also been proved. The solution of the considered problem is derived as the sum of a Fourier series with respect to the system of eigenfunctions of the spectral problem. The uniform convergence of this series and the series obtained from it by term-by-term differentiation is proved using the lemmas mentioned above. At the end of the paper, two estimates are obtained for the solution to the problem, one of which is in the space of square summable functions with weight, and the other is in the space of continuous functions. These inequalities imply the stability of the solution in the corresponding spaces.

Détail

Keywords: equation of hyperbolic type, degenerate equation of the second kind, initial-boundary value problem, spectral problem, Green's function, integral equation, Fourier series, method of separation of variables, method of energy integrals, uniqueness, existence and stability of the solution.