Chapter 4: Systems Based on Numbers

Section 9: Partial Differential Equations

Klein-Gordon equation

The behavior of the Klein-Gordon equation D[u[t, x], t, t] == D[u[t, x], x, x] - u[t, x] is visually very similar to that shown for the sine-Gordon equation. For the Klein-Gordon equation, however, there is an exact solution:

u[t, x] = If[x^2 > t^2, 0, BesselJ[0, Sqrt[t^2-x^2]]]

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