Chapter 4: Systems Based on Numbers

Section 9: Partial Differential Equations

Existence and uniqueness [in PDEs]

Unlike systems such as cellular automata, PDEs do not have a built-in notion of "evolution" or "time". Instead, as discussed on page 940, a PDE is essentially just a constraint on the values of a function at different times or different positions. In solving a PDE, one is usually interested in determining values that satisfy this constraint inside a particular region, based on information about values on the edges. It is then a fundamental question how much can be specified on the edges in order to obtain a unique solution. If too little is specified, there may be many possible solutions, while if too much is specified there may be no consistent solution at all. For some very simple PDEs, the conditions for unique solutions are known. So-called hyperbolic equations (such as the wave equation, the sine-Gordon equation and my equation) work a little like cellular automata in that in at least one dimension information can propagate only at a limited speed, say c. The result is that in such equations, giving values for u[t, x] at t=0 for -s<x<s will uniquely determine u[t,x] at larger t for -s+c t < x < s-c t. In other PDEs, such as so-called elliptic ones, there is no such limit on the rate of information propagation, and as a result, it is immediately necessary to know values of u[t, x] at all x, and on the boundaries of the region, in order to determine u[t, x] for any t>0.

From Stephen Wolfram: A New Kind of Science [citation]