Our unique inverse methodology for finding unknown boundary conditions for Laplace equation utilizing the Boundary Element Method (BEM) has been extended to the solution of two-dimensional inverse (ill-posed) Poisson problem of steady heat conduction with heat sources and sinks. The procedure is simple, reliable, non-iterative and cost effective. Accurate results in two-dimensional heat conduction with arbitrary distributions of heat sources have been obtained for several test cases where boundary conditions were unknown on certain boundaries. Because of its non-iterative, direct nature, our algorithm does not amplify errors in the over-specified input data supplied to parts of the boundary. Furthermore, it does not require regularization schemes, extrapolation to the boundary or mollification to suppress the amplification of input errors. Instead, a straight-forward modification to the BEM produces a single, highly singular solution matrix which we solved using a singular value decomposition matrix solver. Our method for the solution of ill-posed boundary condition problems governed by the Poisson equation also accepts input data at isolated interior points.
