电邮这篇文章 AN INVERSE PROBLEM FOR ELECTRODYNAMIC EQUATIONS WITH A NONLINEAR CURRENT DEPENDENCE OF A TENSION
- 作者: Romanov V.G1
-
隶属关系:
- Sobolev Institute of Mathematics of Siberian Branch of RAS
- 期: 卷 61, 编号 5 (2025)
- 页面: 628–639
- 栏目: PARTIAL DERIVATIVE EQUATIONS
- URL: https://journals.rcsi.science/0374-0641/article/view/299162
- DOI: https://doi.org/10.31857/S0374064125050056
- EDN: https://elibrary.ru/HAUVPB
- ID: 299162
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详细
The system of Maxwell equations in which a current depends nonlinearly of the electrical tension is considered. In the studying case, it is determined of 4 coefficients depended of space variables. These coefficients are supposed to be finite functions with a support located within ball 𝐵(𝑅) of radius 𝑅. For electrodynamic equations a problem of falling down of a plane running wave with a strong front on the inhomogeneity localized in ball 𝐵(𝑅) is posed. A formula for calculation of an amplitude of this wave is derived. In the sequel, an inverse problem of finding 4 coefficients whose determine the current is considered. For this goal the amplitudes formula for different directions of falling waves is used for points at a part of the boundary of 𝐵(𝑅). It is demonstrated that this inverse problem is decomposed at 4 separated problems. One of them is the usual X-ray tomography problem, when the remain 3 others problems are identical problems of the integral geometry for a family of strait lines. In the latter problems, integrals of an unknown function is given along strait lines with a weight function which depends on the finding coefficients after solving the tomography problem. Arising problems of the integral geometry is studied and stability estimate of its solutions is found.
作者简介
V. Romanov
Sobolev Institute of Mathematics of Siberian Branch of RAS
Email: romanov@math.nsc.ru
Novosibirsk, Russia
参考
- Kurylev, Y. Inverse problems for Lorentzian manifolds and non-linear hyperbolic equations / Y. Kurylev, M. Lassas, G. Uhlmann // Invent. Math. — 2018. — V. 212. — P. 781—857.
- Lassas, M. Inverse problems for semilinear wave equations on Lorentzian manifolds / M. Lassas, G. Uhlmann, Y. Wang // Commun. Math. Phys. — 2018. — V. 360. — P. 555—609.
- Lassas, M. Inverse problems for linear and non-linear hyperbolic equations / M. Lassas // Proc. Intern. Congress Math. — 2018. — V. 3. — P. 3739—3760.
- Hintz, P. Reconstruction of Lorentzian manifolds from boundary light observation sets / P. Hintz, G. Uhlmann // Intern. Math. Res. Notices. — 2019. — V. 22. — P. 6949—6987.
- Hintz, P. An inverse boundary value problem for a semilinear wave equation on Lorentzian manifolds / P. Hintz, G. Uhlmann, J. Zhai // Intern. Math. Res. Notices. — 2022. — V. 17. — P. 3181–3211.
- Uniqueness, reconstruction and stability for an inverse problem of a semi-linear wave equation / M. Lassas, T. Liimatainen, L. Potenciano-Machado, T. Tyni // J. Differ. Equat. — 2022. — V. 337. — P. 395–435.
- Detection of Hermitian connections in wave equations with cubic non-linearity / X. Chen, M. Lassas, L. Oksanen, G. P. Paternain // J. Eur. Math. Soc. — 2022. — V. 24, № 7. — P. 2191–2232.
- Wang, Y. Inverse problems for quadratic derivative nonlinear wave equations / Y. Wang, T. Zhou // Commun. Partial Differ. Equat. — 2019. — V. 44, № 11. — P. 1140–1158.
- Barreto, A.S. Interactions of semilinear progressing waves in two or more space dimensions / A.S. Barreto // Inverse Probl. Imaging. — 2020. — V. 14, № 6. — P. 1057—1105.
- Uhlmann, G. On an inverse boundary value problem for a nonlinear elastic wave equation / G. Uhlmann, J. Zhai // J. Math. Pures Appl. — 2021. — V. 153. — P. 114–136.
- Barreto, A.S. Recovery of a cubic non-linearity in the wave equation in the weakly Nonlinear regime / A.S. Barreto, P. Stefanov // Commun. Math. Phys. — 2022. — V. 392. — P. 25–53.
- Романов, В.Г. Обратная задача для полулинейного волнового уравнения / В.Г. Романов // Докл. РАН. Математика, информатика, процессы управления. — 2022. — Т. 504, № 1. — С. 36–41.
- Романов, В.Г. Обратная задача для волнового уравнения с нелинейным поглощением / В.Г. Романов // Сиб. мат. журн. — 2023. — Т. 64, № 3. — С. 635–652.
- Романов, В.Г. Оценка устойчивости в обратной задаче для нелинейного гиперболического уравнения / В.Г. Романов // Сиб. мат. журн. — 2024. — Т. 65, № 3. — С. 560–576.
- Романов, В.Г. Обратная задача для волнового уравнения с двумя нелинейными членами / В.Г. Романов // Дифференц. уравнения. — 2024. — Т. 60, № 4. — С. 508–520.
- Romanov, V.G. An inverse problem for a nonlinear hyperbolic equation / V.G. Romanov, T.V. Bugueva // Eurasian J. Math. Comp. Appl. — 2024. — V. 12, № 2. — P. 134–154.
- Romanov, V.G. An one-dimensional inverse problem for the wave equation / V.G. Romanov, T.V. Bugueva // Eurasian J. Math. Comp. Appl. — 2024. — V. 12, № 3. — P. 135–162.
- Мухометов, Р.Г. Задача восстановления двумерной римановой метрики и интегральная геометрия / Р.Г. Мухометов // Докл. АН СССР. — 1977. — Т. 232, № 1. — С. 32–35.
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