A comparative investigation of certain difference equations and related differential equations: Implications for model-building

Abstract

Many mathematical models for physical and biological problems have been and will be built in the form of differential equations or systems of such equations. With the advent of digital computers one has been able to find (approximate) solutions for equations that used to be intractable. Many of the mathematical techniques used in this area amount to replacing the given differential equations by appropriate difference equations, so that extensive research has been done into how to choose appropriate difference equations whose solutions are “good” approximations to the solutions of the given differential equations.

The present paper investigates a different, although related problem. For many physical and biological phenomena the “continuum” type of thinking, that is at the basis of any differential equation, is not natural to the phenomenon, but rather constitutes an approximation to a basically discrete situation: in much work of this type the “infinitesimal step lengths” handled in the reasoning which leads up to the differential equation, are not really thought of as infinitesimally small, but as finite; yet, in the last stage of such reasoning, where the differential equation rises from the differentials, these “infinitesimal” step lengths are allowed to go to zero: that is where the above-mentioned approximation comes in. Under this kind of circumstances, it seems more natural tobuild themodel as adiscrete difference equation (recurrence relation) from the start, without going through the painful, doubly approximative process of first, during the modeling stage, finding a differential equation to approximate a basically discrete situation, and then, for numerical computing purposes, approximating that differential equation by a difference scheme.

The paper pursues this idea for some simple examples, where the old differential equation, though approximative in principle, had been at least qualitatively successful in describing certain phenomena, and shows that this idea, though plausible and sound in itself, does encounter some difficulties. The reason is that each differential equation, as it is set up in the way familiar to theoretical physicists and biologists, does correspond to a plethora of discrete difference equations, all of which in the limit (as step length→0) yield the same differential equation, but whose solutions, for not too small step length, are often widely different, some of them being quite irregular. The disturbing thing is that all these difference equations seem to adequately represent the same (physical or biological) reasoning as the differential equation in question. So, in order to choose the “right” difference equation, one may need to draw upon more detailed (physical or) biological considerations. All this does not say that one should not prefer discrete models for phenomena that seem to call for them; but only that their pursuit may require additional (physical or) biological refinement and insight.

The paper also investigates some mathematical problems related to the fact of many difference equations being associated with one differential equation.