On the application of numerical methods to Hallen's equation

The so-called Hallen integral equation for the current on a finite linear antenna center-driven by a delta-function generator takes two forms depending on the choice of kernel. The two kernels are usually referred to as the exact and the approximate or reduced kernel. With the approximate kernel, the integral equation has no solution. Nevertheless, the same numerical method is often applied to both forms of the integral equation. In this paper, the behavior of the numerical solutions thus obtained is investigated, and the similarities and differences between the two numerical solutions are discussed. The numerical method is Galerkin's method with pulse functions. We first apply this method to the two corresponding forms of the integral equation for the current on a linear antenna of infinite length. In this case, the method yields an infinite Toeplitz system of algebraic equations in which the width of the pulse basis functions enters as a parameter. The infinite system is solved exactly for nonzero pulse width; the exact solution is then developed asymptotically for the case where the pulse width is small. When the asymptotic expressions for the case of the infinite antenna are used as a guide for the behavior of the solutions of the finite antenna, the latter problem is greatly facilitated. For the approximate kernel, the main results of this paper carry over to a certain numerical method applied to the corresponding equation of the Pocklington type.