Simplest Test for the Two-Dimensional Dynamical Inverse Problem (BC-Method)

Abstract

The dynamical system

\( {\displaystyle \begin{array}{ll}{u}_{tt}-\Delta u-\nabla \ln \rho \cdot \nabla u=0& in\kern1em {\mathbb{R}}_{+}^2\times \left(0,T\right),\\ {}u\left|{}_{t=0}\right.={u}_t\left|{}_{t=0}\right.=0& in\kern1em {\mathbb{R}}_{+}^2,\\ {}{u}_y\left|{}_{y=0}\right.=f& for\kern1em 0\le t\le T,\end{array}} \)

is under consideration, where \( {\mathbb{R}}_{+}^2:= \left\{\left(x,y\right)\in {\mathbb{R}}^2\left|y\right.>0\right\} \); ρ = ρ(x, y) is a smooth positive function; f = f(x, t) is a boundary control; u = u^f(x, y, t) is a solution. With the system one associates a response operator \( R:f\mapsto {u}^f\left|{}_{y=0}\right. \). The inverse problem is to recover the function ρ via the response operator. A short presentation of the local version of the BC-method, which recovers ρ via the data given on a part of the boundary, is provided.

If ρ is constant, the forward problem is solved in explicit form. In the paper, the corresponding representations for the solutions and response operator are derived. A way of making use of them for testing the BC-algorithm, which solves the inverse problem, is outlined. The goal of the paper is to extend the circle of the BC-method users, who are interested in numerical realization of methods for solving inverse problems.