On the Use of Entire-Domain Basis Functions in Galerkin Methods Applied to Certain Integral Equations for Wire Antennas With the Approximate Kernel

When the approximate kernel is used, Hallen's integral equation for the current distribution on a center-driven, straight-wire antenna does not have a solution. This is true for at least two types of feed: the delta-function generator and the frill generator. For the case of subsectional basis functions in Galerkin's method, recent papers have shown that the main associated difficulty is the unavoidable appearance of oscillations near the center and/or the ends of the antenna. In this paper, we investigate the nature of the difficulties for the case of entire-domain, cosine basis functions. We find that the difficulties are similar to those of the subsectional case, something that we had not expected beforehand. In particular, undesirable oscillations appear when the number of basis functions is greater than a number dependent on the length-to-radius ratio, giving one a simple rule for choosing the number of basis functions so as to yield smooth solutions. We also compare results to "true" solutions, study the separate but important effects of roundoff, and give extensions to equations of the Pocklington type.