Chapter 4: Systems Based on Numbers

Section 9: Partial Differential Equations

Other PDEs

The pictures above show three PDEs that have been studied in recent years. All are of the so-called parabolic type, so that, unlike my equation, they have no limit on the rate of information propagation, and thus a solution in any region immediately depends on values on the boundary—which in the pictures below is taken to be periodic. (The deterministic Kardar-Parisi-Zhang equation D[u[t,x],t] == a D[u[t,x],x,x]+ b/2 D[u[t,x],x]^2 yields behavior like Burger's equation, but symmetrical. Note that Abs[u] is plotted in the second picture, while for the last equation a common less symmetrical form replaces the last term by u[t, x] D[u[t, x],x].)

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