One dimensional diffusion equation

∂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)