  Frequency domain Mie solutions to scattering from spheres have been used for
a long time. However, deriving their transient analogue is a challenge as it
involves an inverse Fourier transform of the spherical Hankel functions (and
their derivatives) that are convolved with inverse Fourier transforms of
spherical Bessel functions (and their derivatives). Series expansion of these
convolutions are highly oscillatory (therefore, poorly convergent) and
unstable. Indeed, the literature on numerical computation of this convolution
is very sparse. In this paper, we present a novel quasi-analytical approach to
computing transient Mie scattering that is both stable and rapidly convergent.
The approach espoused here is to use vector tesseral harmonics as basis
function for the currents in time domain integral equations together with a
novel addition theorem for the Green's functions that renders these expansions
stable. This procedure results in an orthogonal, spatially-meshfree and
singularity-free system, giving us a set of one dimensional Volterra Integral
equations. Time-dependent multipole coefficients for each mode are obtained via
a time marching procedure. Finally, several numerical examples are presented to
show the accuracy and stability the proposed method.
