The discrete and the continuous: which comes first?

Abstract

In solving diffusion problems, it is common to consider the finite difference equation to be an approximation to the differential equation. Nevertheless, history shows that the finite difference equation is primitive and that the differential equation is its idealized representation designed to obtain solutions in algebraic form. The difference equation is logically consistent within itself, independent of the differential equation. The difference equation and the differential equation together constitute two powerful complementary tools, one providing numerical solutions to problems of arbitrary complexity on a case by case basis, and the other providing insights into classes of problems under idealized conditions.