Chapter 4: Systems Based on Numbers

Section 9: Partial Differential Equations

Klein–Gordon equation

The behavior of the Klein–Gordon equation δ[u[t, x], t, t] == δ[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[x2 > t2, 0, BesselJ[0, Sqrt[t2-x2]]]

Image Source Notebooks:

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