Efficient numerical techniques for solving Pocklington's equation and their relationships to other methods

It is shown that testing Pocklington's equation with piecewise sinusoidal functions yields an integro-difference equation whose numerical solution is identical to that of the point-matched Hallen's equation when a common set of basis functions is used with each. For any choice of basis functions, the integro-difference equation has the simple kernel, the fast convergence, the simplicity of point-matching, and the adequate treatment of rapidly varying incident fields, but none of the additional unknowns normally associated with Hallen's equation. Furthermore, for the special choice of piecewise sinusoids as the basis functions, the method reduces to Richmond's piecewise sinusoidal reaction matching technique, or Galerkin's method. It is also shown that testing with piecewise linear (triangle) functions yields an integro-difference equation whose solution converges asymptotically at the same rate as that of Hallen's equation. The resulting equation is essentially that obtained by approximating the second derivative in Pocklington's equation by its finite difference equivalent. The authors suggest a simple and highly efficient method for solving Pocklington's equation. This approach is contrasted to the point-matched solution of Pocklington's equation and the reasons for the poor convergence of the latter are examined.