The discrete and the continuous: which comes first?

TL;DR

This paper explores the fundamental relationship between difference equations and differential equations, emphasizing their complementary roles in solving diffusion problems and highlighting the historical and logical precedence of difference equations.

Contribution

It clarifies the conceptual hierarchy and historical development of difference and differential equations, advocating for their combined use in mathematical modeling.

Findings

Difference equations are primitive and historically prior to differential equations.

Differential equations serve as idealized representations for algebraic solutions.

Both tools are essential and complementary in solving complex diffusion problems.

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.