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Simplest Test for the Three-Dimensional Dynamical Inverse Problem (The BC-Method)
- Source :
- Journal of Mathematical Sciences. 252:576-591
- Publication Year :
- 2021
- Publisher :
- Springer Science and Business Media LLC, 2021.
-
Abstract
- A dynamical system $$ {\displaystyle \begin{array}{ll}{u}_{tt}-\Delta u-\nabla 1\mathrm{n}\;\rho \cdot \nabla u=0& in\kern0.6em {\mathrm{\mathbb{R}}}_{+}^3\times \left(0,T\right),\\ {}{\left.u\right|}_{t=0}={\left.{u}_t\right|}_{t=0}=0& in\kern0.6em \overline{{\mathrm{\mathbb{R}}}_{+}^3},\\ {}{\left.{u}_z\right|}_{z=0}=f& for\kern0.36em 0\le t\le T,\end{array}} $$ is under consideration, where ρ = ρ(x, y, z) is a smooth positive function; f = f(x, y, t) is a boundary control; u = uf (x, y, z, t) is a solution. With the system one associates a response operator R : f ↦ uf|z = 0. The inverse problem is to recover the function ρ via the response operator. A short representation 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 to use 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.
- Subjects :
- Statistics and Probability
Applied Mathematics
General Mathematics
Operator (physics)
010102 general mathematics
Boundary (topology)
Function (mathematics)
Inverse problem
Positive function
01 natural sciences
010305 fluids & plasmas
Combinatorics
0103 physical sciences
Nabla symbol
0101 mathematics
Dynamical system (definition)
Realization (systems)
Mathematics
Subjects
Details
- ISSN :
- 15738795 and 10723374
- Volume :
- 252
- Database :
- OpenAIRE
- Journal :
- Journal of Mathematical Sciences
- Accession number :
- edsair.doi...........034629463f9d5b6e3442911062b75e8b