Abstract
This paper presents a general theory of the new integral equation method for solutions of boundary value problems with an operator form, which are given as some mathematical model in engineering sciences. The method presented herein is based on the inverse formulation of our original boundary value problem and the fundamental solution for the corresponding singular differential equation of a main operator in the governing equation. With this method the boundary value problems are transformed into a set of integral equations which differs from ordinary integral equations or boundary ones. Two integral equation formulations are derived corresponding to the original boundary value problem and its equivalent canonical one. Using the canonical decomposition of a main self-adjoint and positive operator, it is shown that the canonical problem is transformed into a set of coupled integral equations. This approach is useful for problems with the higher order operator. Finally, the discretized expressions of approximate solutions for a set of our integral equations are given by means of a numerical procedure relying on some discretization of not only boundary but also interior regions and making use of the standard manner on the unknowns in F.E.M and B.E.M.
