A Solution of some Problems in Potential Theory

Summary. In the present paper, we give an approximate analytic solution of the Dirichlet and Neumann problems in the two-dimensional case for a contour of any shape. The formalism is based on a representation of potential functions as a sum of elementary interpolating functions and uses the theory of generalized inverse matrices. Formulae are given in the cases of Cartesian, polar and elliptic co-ordinates. The fact that an analytic expression for continuation is obtained enables one to compute and draw equipotential lines as well as field lines; besides, any further computations that one might want to perform on either the field or the potential can be handled in an analytic way. The formalism can be extended from the Laplace to the Helmholtz equation.
We give examples in the case of magnetostatics, treating in more detail the problem of the distortion of magnetic field lines by an inclusion. We also show how the method allows the computation of conformal mappings in otherwise intricate situations.