|
This article is cited in 1 scientific paper (total in 1 paper)
Scientific Part
Mathematics
The uniqueness of the solution of an initial boundary value problem for a hyperbolic equation with a mixed derivative and a formula for the solution
V. S. Rykhlov Saratov State University, 83 Astrakhanskaya St., Saratov 410012, Russia
Abstract:
An initial boundary value problem for an inhomogeneous second-order hyperbolic equation on a finite segment with constant coefficients and a mixed derivative is investigated. The case of fixed ends is considered. It is assumed that the roots of the characteristic equation are simple and lie on the real axis on different sides of the origin. The classical solution of the initial boundary value problem is determined. The uniqueness theorem of the classical solution is formulated and proved. A formula is given for the solution in the form of a series whose members are contour integrals containing the initial data of the problem. The corresponding spectral problem for a quadratic beam is constructed and a theorem is formulated on the expansion of the first component of a vector-function with respect to the derivative chains corresponding to the eigenfunctions of the beam. This theorem is essentially used in proving the uniqueness theorem for the classical solution of the initial boundary value problem.
Key words:
hyperbolic equation, second order, constant coefficients, mixed derivative in the equation, finite segment, initial boundary value problem, fixed ends, classic solution, uniqueness of the solution, formula for the solution, expansion of the first component of the vector-function in a series.
Received: 15.04.2022 Accepted: 01.09.2022
Citation:
V. S. Rykhlov, “The uniqueness of the solution of an initial boundary value problem for a hyperbolic equation with a mixed derivative and a formula for the solution”, Izv. Saratov Univ. Math. Mech. Inform., 23:2 (2023), 183–194
Linking options:
https://www.mathnet.ru/eng/isu977 https://www.mathnet.ru/eng/isu/v23/i2/p183
|
Statistics & downloads: |
Abstract page: | 95 | Full-text PDF : | 40 | References: | 18 |
|