Darboux transformations and linear parabolic partial differential equations

A method is provided to solve boundary value problems to parabolic partial differential equations of the form: u(t) = u(x)x + f (x)u, provided f (x) is obtained as twice the second derivative of the logorithm of the wronskian of seperable solutions to the heat equation and the boundary conditions result in a regular Sturm Liouville problem upon doing seperation of variables. Darboux transformations are used to obtain a complete set of eigenfunctions for the boundary value problem allowing for a solution in terms of an eigenfunction expansion.