Abstract: The inverse problem of determining the variable conductivity coefficient in the system of electrodynamic equations with nonlinear conductivity is considered. The required coefficient is assumed to be a smooth compactly supported function of space variables in R3. A plane wave with a sharp front traveling from the homogeneous space in some direction ν is incident on an inhomogeneity. The direction ν is a parameter of the problem. The magnitude of the electric strength vector for some range of incident wave directions and for times close to those at which the wave arrives at points of the ball surface containing the inhomogeneity is given as information for solving the inverse problem. It is shown that this information reduces the inverse problem to the X-ray tomography problem, for which numerical solution algorithms are well developed. The contents of this part of my talk corresponds to the paper [1]. Some inverse problems of recovering coefficients in a semilinear wave equation is studied in papers [2] and [3]. These problems are related to finding a coefficient under nonlinearity in the lower term. In paper [4] a semilinear wave equation contains a damping term and a term with a quadratic nonlinearity. The inverse problem consists in recovering coefficients under these terms as functions of the space variable x ∈ R3. A forward problem with a point source for the original equation is stated. The given information for the inverse problem is the trace of a solution of the forward problem on a surface of a ball for various positions of the point source. A structure of the forward problem is studied. As a result, the inverse problem reduces to two problems, one of them is the problem of X-ray tomography, the other one is the problem of integral geometry with a special given weight function. The latter problem is studied and a stability estimate for the solution of this problem is stated. This work was performed within the state assignment of the Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Science, project no. FWNF-2022-0009. References: 1. Romanov V. G. An Inverse Problem for Electrodynamic Equations with Nonlinear Conductivity // Doklady Mathematics, 2023, vol. 107, no. 1, pp. 53-56. DOI: 10.1134/S1064562423700503 2. Romanov V. G., Bugueva T. V. The Problem of Determining the Coefficient of the Nonlinear Term in a Quasilinear Wave Equation // J. of Appl. and Indust. Math., 2022, vol. 16, no. 3, pp. 550-562. DOI: 10.1134/S1990478922030188 3. Romanov V. G. An Inverse Problem for a Semilinear Wave Equation // Doklady Math., 2022, vol. 105, no. 3, pp. 166-170. DOI: 10.1134/S1064562422030097 4. Romanov V. G. An Inverse Problem for a Nonlinear Wave Equation with Damping // Eurasian J. of Mathematical and Computer Applications, 2023, vol. 11, no. 2, pp. 99-115. DOI: 10.32523/2306-6172-2023-11-2-99-115.

Cite: Romanov V.G.
Inverse problems for some nonlinear equations
In compilation Сборник тезисов Международной конференции "Quasilinear Equations, Inverse Problems and Their Applications". 2023.

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