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We study the diffusion equation with an appropriate change of variables. Thisequation is in general a partial differential equation (PDE). With theself-similar and related Ansat\"atze we transform the PDE of diffusion to anordinary differential equation. The solutions of the PDE belong to a family offunctions which are presented for the case of infinite horizon. In thepresentation we accentuate the physically reasonable solutions. We also studytime dependent diffusion phenomena, where the spreading may vary in time. Todescribe the process we consider time dependent diffusion coefficients. Theobtained analytic solutions all can be expressed with Kummer's orWhittaker-type of functions.

Advanced self-similar solutions of regular and irregular diffusion equations