Some exact results for the Smoluchowski equation for a parabolic potential with time dependent delta function sink

The Smoluchowski equation with a time dependent delta function sink is solved exactly for many special cases. In all other cases the problem can be reduced to an integral equation. It is shown that by knowing the probability distribution at the position of sink, one can derive analytical expression for probability distribution everywhere. Thus the problem is reduced from a PDE in two variables to an integral equation of one. As far as the authors knowledge, we are the first one to provide an exact analytical solution of Smoluchowski equation for a parabolic potential with time dependent sink.