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Three partial differential equations that have historically been studied extensively. Just like in other pictures in this book, position goes across the page, and time down the page. In each equation u is the gray level at a particular point, D[u, t] is the rate of change (derivative) of the gray level with time, and D[u,t,t] is the rate of change of that rate of change (second derivative). Similarly, D[u,x] is the rate of change with position in space, and D[u,x,x] is the rate of change of that rate of change. On this page and the ones that follow [165, 166] the initial conditions used are u=Exp[-x^2], D[u,t]=0.

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