A Review of the Theory

Abstract

The modern theory of boundary integral equations began with Fredholm [1], who established the existence of solutions on the basis of his limiting discretisation procedure. It was not envisaged by Fredholm or his immediate successors that solutions could actually be constructed in this way. However the advent of fast digital computers, some 50 years later, opened up the possibility of implementing the discretisation process arithmetically and so enabled numerical solutions of tolerable accuracy to be attempted. This possibility in turn gave a considerable impetus to the development of new and improved boundary integral formulations. In 1962, Hess and Smith [2, 3] formulated a Fredholm integral equation of the second kind for the distribution of simple sources over a surface of revolution. By solving this equation numerically, they were able to compute the perturbation of a uniform potential flow by the surface. In 1963, Jaswon and Ponter [4] threw the torsion problem on to the boundary by formulating an integral equation of the second kind for the warping function, which was solved numerically as a means of computing the torsional rigidity and boundary shear stress for cross-sections inaccessible to other methods of attack. This was one of the first published papers which effectively exploited Green’s formula on the boundary, by emphasising its role as a functional relation between the boundary values and normal derivatives of an arbitrary harmonic function. Also in 1963, Jaswon [5] formulated the electrostatic capacitance problem in terms of a Fredholm integral equation of the first kind for the charge distribution, a formulation which had been noted and discarded by Volterra [6] because of apparent difficulties with the two-dimensional theory.