Abstract. This paper presents a Galerkin approach for solving a regularized version of the Cauchy singular, linear integro-differential equation• Jy=o x-y subject to 0(0) = 0(1) = 0. This equation has appeared in both combined infrared gaseous radiation and molecular conduction, and elastic contact studies. A regularized formulation is produced which suggests the use of an expansion technique where the orthogonal basis functions are chosen as the Chebychev polynomials of the first kind. Accurate results, requiring a minimal computational cost, are formally documented and compared to a purely numerical solution.