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.