Numerical inversion of a Laplace transform solution of a diffusion equation with a mixed derivative term

We consider a diffusion equation with a mixed derivative term modelling the transient flow of a second grade fluid. The same diffusion equation with a mixed derivative term also comes up in the modelling of heat conduction in which the history of the temperature gradient is included. The equations differ in the sign of the coefficient of the mixed derivative term. We show that the method of lines solution is stable for the transient flow of a second grade fluid and unstable for heat conduction with the temperature gradient history included. A Laplace transform solution for the heat conduction case is then investigated. The Laplace transform solution is inverted numerically using a Fourier based method. A stable solution is obtained. The solutions are compared and discussed.