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_xx + f(x)u, provided f(x) is obtained as twice the second derivative of the logarithm of the wronskian of separable solutions to the heat equation and the boundary conditions result in a regular Sturm Liouville problem upon doing separation 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.