The Problem of Determining the Coefficient of the Nonlinear Term in a Quasilinear Wave Equation Full article
| Journal |
Journal of Applied and Industrial Mathematics
ISSN: 1990-4789 , E-ISSN: 1990-4797 |
||||
|---|---|---|---|---|---|
| Output data | Year: 2022, Volume: 16, Number: 3, Pages: 550–562 Pages count : 13 DOI: 10.1134/S1990478922030188 | ||||
| Tags | nonlinear wave equation, inverse problem, tomography | ||||
| Authors |
|
||||
| Affiliations |
|
Funding (1)
| 1 | Sobolev Institute of Mathematics | FWNF-2022-0009 |
Abstract:
For a nonlinear differential equation whose main part is the wave operator, we consider the inverse problem of determining the coefficient of the nonlinear term in the equation. It is assumed that the desired coefficient is a continuous compactly supported function in R3. For the original equation, we consider plane waves incident on the inhomogeneity at different angles. In the inverse problem, it is assumed that the solutions corresponding to these waves can be measured at points on the boundary of a certain ball containing the inhomogeneity at times close to the wave front arrival at these points and for a certain range of angles of incidence of the plane wave. It is shown that the solutions of the corresponding direct problems for the differential equation are bounded in some neighborhood of the wave front, and an asymptotic expansion of the solution is found in this neighborhood. On the basis of this expansion, it is established that the information specified in the inverse problem allows reducing the problem of finding the desired function to the problem of X-ray tomography with incomplete data. A theorem on the uniqueness of the solution of the inverse problem is stated and proved. It is shown that in an algorithmic sense, this problem is reduced to the well-known moment problem.
Cite:
Romanov V.G.
, Bugueva T.V.
The Problem of Determining the Coefficient of the Nonlinear Term in a Quasilinear Wave Equation
Journal of Applied and Industrial Mathematics. 2022. V.16. N3. P.550–562. DOI: 10.1134/S1990478922030188 Scopus РИНЦ OpenAlex
The Problem of Determining the Coefficient of the Nonlinear Term in a Quasilinear Wave Equation
Journal of Applied and Industrial Mathematics. 2022. V.16. N3. P.550–562. DOI: 10.1134/S1990478922030188 Scopus РИНЦ OpenAlex
Original:
Романов В.Г.
, Бугуева Т.В.
Задача об определении коэффициента при нелинейном члене квазилинейного волнового уравнения
Сибирский журнал индустриальной математики. 2022. Т.25. №3. С.154--169. DOI: 10.33048/SIBJIM.2022.25.313 РИНЦ
Задача об определении коэффициента при нелинейном члене квазилинейного волнового уравнения
Сибирский журнал индустриальной математики. 2022. Т.25. №3. С.154--169. DOI: 10.33048/SIBJIM.2022.25.313 РИНЦ
Dates:
| Submitted: | Feb 9, 2022 |
| Accepted: | Jun 22, 2022 |
| Published print: | Dec 18, 2022 |
| Published online: | Dec 18, 2022 |
Identifiers:
| Scopus: | 2-s2.0-85144179394 |
| Elibrary: | 59144937 |
| OpenAlex: | W4313039738 |