∂Φ/∂t = a ∂2Φ/∂t2
With normalized delta function as initial function
Φ(0,0) = δ(0) = 0 for x≠0 and ∫δdx = 1
the analytic solution is a normalized Gaussian function.
Φ(x,t) = exp(-x2/at) / sqrt (4πat)
1/e- width: at
maximum amplitude: 1 / sqrt (4πat)