On the image method of separable variable problems

Abstract

The image method is a classical analysis method for mathematical physics problems but restricted by domain geometry and boundary conditions. In this paper, the scope of the image method is investigated and extended to separable variable problems. The general description of the image method is established and the image source distribution is determined for separable variable problems. The image source distribution function is proved to be the out of domain part of the eigenfunction series of the source distribution function. This continuation from the source distribution function to the image source distribution function is also given as the inner product with the eigenfunction series of the Dirac delta function. Three numerical examples with complex boundary conditions or geometry domains are prepared to verify the generality and effectiveness of the conclusion.