Analytic continuation considerations when using generalized formulations for scattering problems

A generalized operator equation has been developed for a simple scattering problem for which a closed-form solution exists. The operator equation has been solved numerically via the method of moments using spatially impulsive fictitious sources as expansion functions together with a simple point-matching testing procedure. The study focused on the convergence and accuracy of the solution and examined how they are dependent on the location and distance of the fictitious sources relative to the area containing the singularities of the actual field simulated by these sources. As expected, it was found that when the actual field simulated by the fictitious impulsive sources has singularities lying between the physical boundary and the closed surface over which the sources are placed, the impulsive expansion does not yield a uniformly convergent solution. In this case, instabilities are encountered as the number of sources increases, in the sense that a small improvement of the boundary error requires a considerable change in the currents. The moment matrix is then difficult to invert and easily susceptible to large round-off errors. Conversely, if the actual field has no singularities lying between the physical boundary and the closed surface over which the sources are placed, the impulsive expansion does yield a uniformly convergent solution to any degree of precision.< >