Boundary function method for inverse geometry problem in two-dimensional anisotropic heat conduction equation

This paper develops a novel approach for detecting unknown boundaries in the two-dimensional anisotropic heat conduction equations based on the boundary function method, in which a partial homogenization function satisfied the over-specified Cauchy data on an arc is derived to effectively solve the inverse geometry problem. After the homogenized technique, the governing equation is transformed into the one in a reduced domain, whose numerical solution is expanded by a sequence of boundary functions, automatically satisfying the homogeneous boundary conditions on the arc. The nonlinear equation will be formed and then solved by the Newton iterative method. Two numerical examples are provided to demonstrate the ability and accuracy of the proposed scheme.