Potential function construction by use of an adaptive mesh algorithm for a class of singular integral equations Original Research Article

An effective adaptive mesh algorithm for the numerical solution of Fredholm singular integral equations of the first kind is presented. These integrals occur in engineering and mathematical physics. Important examples include their use as boundary conditions for certain elliptic partial differential equations. In this latter case, an ability to solve such boundary integral equations can lead to a closed form answer for the differential equation itself. The method is to fashion a potential function, as an integral, which is the solution to the partial differential equation over its region. The answer from the Fredholm integral equation appearing on the boundary is needed since it forms part of the integrand of the potential function. This paper gives both the adaptive mesh algorithm for solving the integral equation and examples of the use of first kind Fredholm singular integral equations on the boundary to compute integral potential functions. The method herein has special value when jumps and discontinuities in derivatives are present in the boundary conditions.