For a nonlinear differential equation whose main part is the wave operator, we considerthe inverse problem of determining the coefficient of the nonlinear term in the equation. It isassumed that the desired coefficient is a continuous compactly supported function in R3. For the original equation, we consider plane waves incident on theinhomogeneity at different angles. In the inverse problem, it is assumed that the solutionscorresponding to these waves can be measured at points on the boundary of a certain ballcontaining the inhomogeneity at times close to the wave front arrival at these points and for acertain range of angles of incidence of the plane wave. It is shown that the solutions of thecorresponding direct problems for the differential equation are bounded in some neighborhood ofthe wave front, and an asymptotic expansion of the solution is found in this neighborhood. On thebasis of this expansion, it is established that the information specified in the inverse problemallows reducing the problem of finding the desired function to the problem of X-ray tomographywith incomplete data. A theorem on the uniqueness of the solution of the inverse problem isstated and proved. It is shown that in an algorithmic sense, this problem is reduced to thewell-known moment problem.
Предметные области OECD FOS+WOS
- 1.01 МАТЕМАТИКА
- 2.03 МЕХАНИКА И МАШИНОСТРОЕНИЕ