The discrete and the continuous: which comes first?

The notion of a continuum and its description with a differential equation are ruling paradigms of mathematical physics. Over the past half a century, the digital computer has enabled numerical solution of complex physical problems defined over discrete domains. Commonly, such discretely defined problems are treated as approximations to continuum problems described by differential equations. The observational world is discrete and finite. If so, how may one assume that the discrete representation is an approximation to an idealized continuum and the associated differential equation? Which comes first, the discrete or the continuous? To find answers, it is necessary to examine the history of difference equations and differential equations. Accordingly, we examine eighteenth century investigations of probability, followed by the birth of the heat equation during early nineteenth century.

Following the introduction of mathematical probability as a tool for describing outcomes of games of chance, the study of probability engaged the attention of distinguished mathematicians during the eighteenth century. Initially, these problems were investigated using combinatoric methods. For example, James Bernoulli considered a problem involving two events, one of 1 them having probability of occurrence p, and the other having probability of q = 1-p. He showed that the probability that the first event will occur m times and fail n times is equal to a certain term in the expansion of (p+q) , namely, ì

where ì = (m+n). Unfortunately, in the absence of computing devices, expressions such as (1) could not be numerically evaluated when ì was large.

To overcome this difficulty, they devised a strategy of evaluating these ratios using definite integrals. For this purpose, the technique of génératrice function was introduced during the middle of the 18 century. The central idea was to set up a power series expansion in which the th required probability will form a coefficient. Having set up such an equation, the next step was to evaluate a desired coefficient by setting up an appropriate difference equation expressing the incremental change in the coefficient as ì is increased to (ì+1).

Pursuing this approach, Pierre Simon Laplace considered a power series involving products of 2 two variables t and t’ and established the following finite difference expressions for the x,x’ coefficient y of the product t t’ :

x x’ Note that x and x’ are integers, and Äx = Äx’ = 1. Therefore, the right hand sides of (2a) and (2b) denote typical finite difference expressions with (Äx) and Äx’ implied as denominators. where y is relative frequency (probability density) representing y . Joseph Fourier built on 4 Laplace’s solution to (4) and showed that that (4) is satisfied by normal distribution.

Clearly, the principal idea underlying differential equation ( 4) is a difference equation.

At the beginning of the nineteenth century, Fourier embarked on setting up a partial differential This finite difference equation describes one-dimensional transient heat flow in a prism in sufficient detail to obtain numerical solutions when boundary conditions and initial conditions are prescribed. Obviously, no computing devices capable of handling the numerical calculations involved in implementing (7) were available at the time of Fourier.

To convert (7) proposed the concept of a resistance and expressed electrical flux in terms of a difference equation, rather than using a gradient as Fourier did. Fick who conducted salt-diffusion assumed to be exactly balanced by resistive forces, and flux was effectively given by (10).

Together, the contributions of Ohm, Fick, and Maxwell enable a fully self-consistent finitedifference statement of the heat flow problem, independent of the differential equation. Recall that in setting up difference equation (7) for the prism, average temperature was associated with the isothermal surface exactly midway between interfaces bounding an element. However, when the area of cross section is variable, linear temperature variation along flow path cannnot be assumed, and average temperature cannot be associated with an isothermal surface midway between bounding interfaces. With this recognition, consider now three adjoining elements L, M, and N along a tube with variable cross section, as shown in Figure 3. For element M bounded by interfaces at x and x , average temperature at an instant t is given, analogous to (6) by, M M Given this, x will be that location at which the isotherm with magnitude T intersects the abcissa. Clearly, this position will be determined by the functional dependence of S on x. For M L,M any prescribed variation S(x), x can be determined using (10), assuming known values of T L,M M,N M,N at x and T at x (ref. 11) .

Subject to these considerations, finite difference equation ( 6) for element M is to be replaced by,

where average temperature ÄT is associated with x as described above.

Just as (7),

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