The elliptic type equation-Δu+k²u=f (1) is derived from heat conduction equation by finit difference discreting. The ordinary boundary integral equation of (1) is of the second kind for the unknown potential, but is the first kind for the unknown exterior normal derivative of potential. In this paper, a new type of boundary integral equation has been derived by conservation integral method, which is of the second kind for the unknown exterior normal derivative of potential and is the frist kind for the unknown poential, and the two equations method for the mixed boundary value prblem of (1) is presented, which use the ordinary boundary integral equation at the collocation points in the boundary segment in which the potential is unknown, and use the new type of boundary integral equation at the collocation points in the boundary segment in which the exterior normal derivative is unknown. Some numerical examples indicate this method has higer accuracy than ordinary boundary element method.

Boundary element method (BEM) is usually done by only one boundary integral equation (BIE).The ordinary BIE for potential in the plane is of the second kind for potential U, but is of the first kind for potential derivative ∂_Ω.In order to result in diagonal dominance of the control system of equations of BEM for mixed boundary value problem, we present an improvements that using the ordinary control system on the NEUMANN boundary segment,but the control system of the conjugate harmonic function on the DIRICHLET boundary segment.Some examples for computing the capacitance density of two-wire transmission lines are given.which indicate the method of conjugate function has higher accuracy than ordinary BEM.