Determining electrical and heat transfer parameters using coupled boundary measurements

Details Determining electrical and heat transfer parameters using coupled boundary measurements Determining electrical and heat transfer parameters using coupled boundary measurements

2010-12-14

arxiv.org/abs/1012.3099v1

Mathematics

Analysis of PDEs

Scientific paper

Let $\Omega\subset\R^n$, $n\ge 3$, be a smooth bounded domain and consider a coupled system in $\Omega$ consisting of a conductivity equation $\nabla \cdot \gamma(x) \nabla u(t,x)=0$ and an anisotropic heat equation $\kappa^{-1}(x)\partial_t\psi(t,x)=\nabla\cdot (A(x)\nabla \psi(t,x))+(\gamma\nabla u(t,x))\cdot \nabla u(t,x), \quad t\ge 0$. It is shown that the coefficients $\gamma$, $\kappa$ and $A=(a_{jk})$ are uniquely determined from the knowledge of the boundary map $u|_{\partial\Omega}\mapsto \nu\cdot A\nabla \psi|_{\partial\Omega}$, where $\nu$ is the unit outer normal to $\partial\Omega$. The coupled system models the following physical phenomenon. Given a fixed voltage distribution, maintained on the boundary $\partial\Omega$, an electric current distribution appears inside $\Omega$. The current in turn acts as a source of heat inside $\Omega$, and the heat flows out of the body through the boundary. The boundary measurements above then correspond to the map taking a voltage distribution on the boundary to the resulting heat flow through the boundary. The presented mathematical results suggest a new hybrid diffuse imaging modality combining electrical prospecting and heat transfer-based probing.

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