Solution of time domain problems without using the time variable

Abstract

In this presentation, we propose a numerical method to obtain an unconditionally stable solution for any time domain problems without using the time variable. The classes of problems not only include integral equations, but also deal with finite element time domain and finite difference time domain techniques. This novel method does not utilize the customary marching-on in time solution method often used to solve a hyperbolic partial differential equation. Instead we solve the wave equation by expressing the transient behaviors in terms of associate Laguerre polynomials. By using these orthonomal basis functions for the temporal variation, the time derivatives can not only be handled analytically, but also they can be eliminated completely from the final computations. Since these weighted associate Laguerre polynomials converge to zero as time progresses, the induced electric currents when expanded in a series of weighted Laguerre polynomials also converge to zero. In order to solve the wave equation, we introduce two separate testing procedures, a spatial and temporal testing. By introducing first use the temporal testing procedure, the marching-on in time procedure is replaced by a recursive relation between the different orders of the weighted Laguerre polynomials. The other novelty of this approach is that through the use of the entire domain Laguerre polynomials for the expansion of the temporal variation of the current, the spatial and the temporal variables can be separated. Examples will be presented using all the three classes of techniques.