An energetic boundary functional method for solving the inverse heat conductivity problems in arbitrary plane domains

Abstract

In the paper, we solve Caldero´n’s inverse conductivity problems in a rectangular plane domain and in an arbitrary plane domain, to recover an unknown heat conductivity function σ(x, y) inside a material object. Firstly, the homogenization functions are derived, by which the Dirichlet as well as the Neumann boundary conditions over-specified on the boundary become zeros for the new variable. Secondly, an integral identity expressing the conservation of energy is derived; hence, we can set up an energy functional to effectively solve the inverse heat conductivity problem. Thirdly, a sequence of boundary functions in terms of the modified Pascal triangle are given, which satisfy the homogeneous boundary conditions automatically. A new methodology of energetic boundary functional is constructed in a linear space, wherein each energetic boundary function preserves the energy identity. The linear system used to recover the unknown heat conductivity function σ(x, y) with energetic boundary functions as the bases is derived, and then the iterative algorithm is developed which converges very fast. The accuracy and robustness of the energetic boundary functional method (EBFM) are confirmed by comparing the recovered results under a large noisy disturbance to the exact heat conductivity functions σ(x, y).