Moving Heat Source Reconstruction from the Cauchy Boundary Data

We consider the problem of reconstruction of an unknown characteristic transient thermal source inside a domain. By introducing the definition of an extended dirichlet-to-Neumann map in the time-space cylinder and the adoption of the anisotropic Sobolev-Hilbert spaces, we can treat the problem with methods similar to those used in the analysis of the stationary source reconstruction problem. Further, the finite difference θ scheme applied to the transient heat conduction equation leads to a model based on a sequence of modified Helmholtz equation solutions. For each modified Helmholtz equation the characteristic star-shape source function may be reconstructed uniquely from the Cauchy boundary data. Using representation formula, we establish reciprocity functional mapping functions that are solutions of the modified Helmholtz equation to their integral in the unknown characteristic support