Finite difference schemes for a non-linear heat equation with functional dependence

We consider a heat equation with a non-linear right-hand side which depends on certain Volterra-type functionals. We analyse the question of convergence for some finite difference schemes by means of discrete inverse formulae instead of a maximum principle. It is due to this new technique that one can efficiently approximate these heat transport and diffusion-reaction equations which require some functional (delayed-integral) values of the gradient, too. Numerical experiments confirm our theoretical analysis and encourage engineers to apply difference schemes to parabolic equations and systems.