  We study the diffusion equation with an appropriate change of variables. This
equation is in general a partial differential equation (PDE). With the
self-similar and related Ansat\"atze we transform the PDE of diffusion to an
ordinary differential equation. The solutions of the PDE belong to a family of
functions which are presented for the case of infinite horizon. In the
presentation we accentuate the physically reasonable solutions. We also study
time dependent diffusion phenomena, where the spreading may vary in time. To
describe the process we consider time dependent diffusion coefficients. The
obtained analytic solutions all can be expressed with Kummer's or
Whittaker-type of functions.
