  In the present report, we have introduced the Fredholm integral method to
solve the Smoluchowski equation in the Laplace domain. We get an exact
semi-analytical solution for the linear potential energy curve in the dynamic
diffusion process, and the survival probability is calculated by the numerical
inverse Laplace transform method. We apply our method in two different physical
contexts for finding different observable like average rate constant in
electronic relaxation in solution and quantum yields in a photosynthetic system
or doped molecular crystal.
