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GCOE Seminars
Seminar information archive ～04/24｜Next seminar｜Future seminars 04/25～
2014/02/28
13:00-14:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Patricia Gaitan (Aix-Marseille University)
Probing for inclusions for heat conductive bodies. Time independent and time dependent cases (ENGLISH)
Patricia Gaitan (Aix-Marseille University)
Probing for inclusions for heat conductive bodies. Time independent and time dependent cases (ENGLISH)
[ Abstract ]
This work deals with an inverse boundary value problem arising from the equation of heat conduction. Mathematical theory and algorithm is described in dimensions 1{3 for probing the discontinuous part of the conductivity from local temperature and heat ow measurements at the boundary. The ap- proach is based on the use of complex spherical waves, and no knowledge is needed about the initial temperature distribution. In dimension two we show how conformal transformations can be used for probing deeper than is pos- sible with discs. Results from numerical experiments in the one-dimensional case are reported, suggesting that the method is capable of recovering loca- tions of discontinuities approximately from noisy data.
For moving inclusions, we consider an inverse boundary value problem for the heat equation on the interval (0; 1), where the heat conductivity (t; x) is piecewise constant and the point of discontinuity depends on time : (t; x) = k2 (0 < x < s(t)), (t; x) = 1 (s(t) < x < 1). Firstly we show that k and s(t) on the time interval [0; T] are determined from the partial Dirichlet- to-Neumann map : u(t; 1) ! @xu(t; 1); 0 < t < T, u(t; x) being the solu- tion to the heat equation such that u(t; 0) = 0, independently of the initial data u(0; x). Secondly we show that the partial Dirichlet-to-Neumann map u(t; 0) ! @xu(t; 1); 0 < t < T, u(t; x) being the solution to the heat equation such that u(t; 1) = 0, determines at most two couples (k; s(t)) on the time interval [0; T], independently of the initial data u(0; x).
This work deals with an inverse boundary value problem arising from the equation of heat conduction. Mathematical theory and algorithm is described in dimensions 1{3 for probing the discontinuous part of the conductivity from local temperature and heat ow measurements at the boundary. The ap- proach is based on the use of complex spherical waves, and no knowledge is needed about the initial temperature distribution. In dimension two we show how conformal transformations can be used for probing deeper than is pos- sible with discs. Results from numerical experiments in the one-dimensional case are reported, suggesting that the method is capable of recovering loca- tions of discontinuities approximately from noisy data.
For moving inclusions, we consider an inverse boundary value problem for the heat equation on the interval (0; 1), where the heat conductivity (t; x) is piecewise constant and the point of discontinuity depends on time : (t; x) = k2 (0 < x < s(t)), (t; x) = 1 (s(t) < x < 1). Firstly we show that k and s(t) on the time interval [0; T] are determined from the partial Dirichlet- to-Neumann map : u(t; 1) ! @xu(t; 1); 0 < t < T, u(t; x) being the solu- tion to the heat equation such that u(t; 0) = 0, independently of the initial data u(0; x). Secondly we show that the partial Dirichlet-to-Neumann map u(t; 0) ! @xu(t; 1); 0 < t < T, u(t; x) being the solution to the heat equation such that u(t; 1) = 0, determines at most two couples (k; s(t)) on the time interval [0; T], independently of the initial data u(0; x).
