Open AccessStability estimation of the generalized solution to the direct problem for the acoustic equation
Author(s) -
С. И. Кабанихин,
A. T. Nurseitova,
Syrym Kasenov
Publication year - 2021
Publication title -
journal of physics. conference series
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.21
H-Index - 85
eISSN - 1742-6596
pISSN - 1742-6588
DOI - 10.1088/1742-6596/2092/1/012005
Subject(s) - mathematics , boundary value problem , inverse problem , boundary (topology) , domain (mathematical analysis) , mathematical analysis , well posed problem , stability (learning theory) , surface (topology) , initial value problem , computer science , geometry , machine learning
The initial-boundary value problem for the acoustic equation with data on a timelike surface is considered in this paper. Such a problem arises, for example, if it is required to determine the acoustic pressure inside the region from a fixed response to part of the boundary from the source involved at the same boundary. It is assumed that the medium is at rest up to a certain instant of time and the parameters of the medium, for example, acoustic density, are known. The problem is considered in a triangular domain. The advisability of this was shown in the second half of the last century in the works of Romanov V.G. (for example, [1]), where it was proved that the solution to the direct problem of acoustic is representable as the sum of a singular and a continuous terms. The author has written out the form of the singular part, investigated the problem in an integral statement, and also proved conditional well-posedness theorems for three cases: for a small parameter of the domain, for small data, and for the source representability of the sought solution. It is known that the initial-boundary value problem for the acoustic equation with data on a timelike surface is ill-posed. In this paper, the original ill-posed problem is reduced to an inverse problem with respect to some direct (well-posed) problem. The theorem is proved and a stability estimate of the generalized solution to the direct problem is obtained.