Applications of nonstandard analysis to partial differential equations—I. the diffusion equation

AbstractThis paper begins by developing a basis for using ∗ finite difference equations to model physical phenomena. Under appropriate conditions the solution F of a ∗ finite difference equation has S-continuous “finite difference derivatives” up to order r. In these circumstances we can show that the standard function °F is a Cr-function and satisfies the differential equation corresponding to the original finite difference equation. The second part of the paper illustrates these techniques by applying them to the heat equation. In particular, we obtain a very nice model of the heat equation with initial conditions corresponding to all the heat concentrated at a single point