Dr. Ezzat G Bakhoum

In 1891, J.J. Thomson--the discoverer of the electron--stated a formula that relates the first derivative of the electric field intensity to the mean curvature of an equipotential surface. That formula was later proved by others, but remained unexploited in any practical purpose to this date. This dissertation presents a numerical method based on Thomson's formula for the rapid solution of Laplace's equation, the governing equation of field theory. The presented method is based on geometric construction principles. Specifically, the method uses the concept of representing equipotential surfaces by polynomials for the rapid tracing of these surfaces; and is therefore fundamentally different from previously-known techniques which are based on discretizing the domain or the boundary of the problem. The new method is especially suited for problems which have complicated or irregular boundaries as well as problems in exterior domains. Previously, such types of problems have required a number of computations of O(N.M), where N is the number of points taken on the boundary of the problem and M is the number of points inside the domain at which the solution is to be computed. The new method requires an O(M) computations only; and is therefore significantly faster than the previous techniques. Applications include problems of electrostatics, cosmology, biomedical engineering, nuclear and particle physics, etc.