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 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. 1. Introduction Inverse source transient heat problem has been studied by a huge number of authors. In relation to books with specific chapters in the subject, we can give special attention to Anger [1] and Isakov [2]. Those gives specific results for the problem of source reconstruction in models with different operators and overspecification of boundary conditions, and specifically demonstrates an uniqueness theorem related with the moving characteristic source studied in this work. Early works by Cannon and Pérez Esteva [3] studied stationary support reconstruction under hypotheses of a known intensity function, . Some years later, Cannon and Pérez Esteva studied the same problem in a three-dimensional case [4]. More recently, Lefevre and Le Niliot [5] used the boundary elements method for identification of static and moving point sources. Also, it is important to mention the authors Hettlich and Rundell [6] who model the unmoving characteristic domain and El Badia and Ha Duong [7], whose stationary source reconstruction and the transient point sources reconstructions have a fundamental influence in the present work. The present work has come from the investigation of the stationary source reconstruction by the fundamental solution method in Alves et al. [8]. The adoption of the reciprocity gap functional method to solve the stationary source in the Laplace Poisson equation, Roberty and Alves [9], and the solution of the full identification of sources with the Helmholtz Poisson model, Alves et al. [10], have developed to the modeling adopted to the transient heat transient characteristic source reconstruction in this work. The model is based on the modified Helmholtz