A Technique for Solving the Singular Integral Equations of Potential Theory

Abstract

The singular integral equations of Potential Theory are investigated using ideas from both classical and contemporary mathematics. The goal of this semi-analytic approach is to produce numerical schemes that are both general and computationally simple. Previous works based on classical methods have yielded solutions only for very special cases while contemporary methods such as finite differences, finite elements and boundary element techniques are computationally extensive. Since the two-dimensional integral equations of interest exhibit structural invariance under a wide class of conformal mappings initial emphasis is placed on circular domains. By Fourier expansion with respect to the angular variable, such two-dimensional integral equations yield simultaneous systems of one-dimensional integral equations that, in many cases, uncouple. Integral transform techniques and classical function theory are used to identify the eigenfunctions associated with the dominant parts of the onedimensional singular equations. Hilbert spaces spanned by these eigenfunctions are then constructed and an operator theory developed for the general class of integral equations. Numerical algorithms are derived for both Galerkin and collocation solution techniques with convergence proved in the Galerkin case and collocation method verified experimentally. A generalization of the Hilbert space theory is then applied to the two-dimensional case with eigenfunctions created by combining the angular Fourier terms with the radial eigenfunctions of the dominant one-dimensional parts. Numerical algorithms based Galerkin and collocation methods are again derived and used to solve the two-dimensional equations. The techniques developed are used to solve a number of both previously known and new problems in Electrostatics and Fracture Mechanics. Simple layer potential representations yield weakly singular integral equations for the induced charge on disc shaped conductors that are placed in an electrostatic field. Similarly, double layer potentials yield hyper-singular integral equations for the crack opening displacement of penny shaped cracks in an elastic solid under various loading conditions. Conformal mapping techniques for solving problems on non-circular domains are also briefly discussed as are extensions to fields that are governed by the Helmholtz Equation.