The first can be used to solve TDIEs (as examined next) and the second can be used to post-process data.
In this section, TDIEs governing the time-dependent scattering from dielectric spheres are to be solved.
As in usual discretization of integral equations of time-harmonic case, after substituting the representations of currents defined in (9) into (5) and (7) and using expansion for the tangential trace of dyadic Green's function, one can test the TDIEs with [[PSI].sup.*.sub.nm] and [[PHI].sup.*.sub.nm], respectively, using the Galerkin testing scheme.
All the reductions of the semidiscrete version of original TDIEs are based on the orthogonality of VSH, and the spatial convolution is taken care of analytically.
However, ongoing work is focused on studying late time stability for this system and extending it to analyzing stabilities of TDIEs as applied to electromagnetics--a long standing problem where significant literature exists [56-62].
Recently, for acoustic scattering, Laplace transform based approach [41, 42] and TDIE based approach  have been proposed to solve the time-dependent scattering problem.