Application of integral equations with superstrong singularity to steady state heat conduction

Abstract In this paper, we present the theory of dual integral equations for steady state heat conduction. There are four kernel functions with different orders of singularity in the two equations. Using the first equation with weaker singularity, the conventional direct boundary integral equation method (BIEM) was proposed long ago. An important characteristics of the first equation is that its kernels are of the Riemann and Cauchy types. The purpose of this paper is to present a method based on the second equation with stronger singularity kernels to solve the steady state heat conduction problems. Whereas the kernels of the second equation are of the Cauchy and Hadamard types. It is further shown that combination of the two equations can be used to solve problems with degenerate boundary which have long suffered from lack of a general formulation of the BIEM. For concreteness, an illustrative example is performed numerically to see the validity of the theory.