The notion of equations [versus constraints]

In the mathematical framework traditionally used in the exact sciences, laws of nature are usually represented not by explicit rules for evolution, but rather by abstract equations. And in general what such equations do is to specify constraints that systems must satisfy. Sometimes these constraints just relate the state of a system at one time to its state at a previous time. And in such cases, the constraints can usually be converted into explicit evolution rules. But if the constraints relate different features of a system at one particular time, then they cannot be converted into evolution rules. In computer programs and other kinds of discrete systems, explicit evolution rules and implicit constraints usually work very differently. But in traditional continuous mathematics, it turns out that these differences are somewhat obscured. First of all, at a formal level, equations corresponding to these two cases can look very similar. And secondly, the equations are almost always so difficult to deal with at all that distinctions between the two cases are not readily noticed.

In the language of differential equations—the most widely used models in traditional science—the two cases we are discussing are essentially so-called initial value and boundary value problems, discussed on page 923. And at a formal level, the two cases are so similar that in studying partial differential equations one often starts with an equation, and only later tries to work out whether initial or boundary values are needed in order to get either any solution or a unique solution. For the specific case of second-order equations, it is known in general what is needed. Elliptic equations such as the Laplace equation need boundary values, while hyperbolic and parabolic equations such as the wave equation and diffusion equation need initial values. But for higher-order equations it can be extremely difficult to work out what initial or boundary values are needed, and indeed this has been the subject of much research for many decades.

Given a partial differential equation with initial or boundary values, there is then the question of solving it. To do this on a computer requires constructing a discrete approximation. But it turns out that the standard methods used (such as finite difference and finite element) involve extremely similar computations for initial and for boundary value problems, leaving no trace of the significant differences between these cases that are so obvious in the discrete systems that we discuss in most of this book.