<< RealTime3D`

e[x_, a_]  :=  (E^(-(x/a)^2))

x0  =   .15; f[x_]  :=  e[x - x0,  .2] ; 
    u[t_, x_] :=  f[x (E^(-t))]

Plot3D[u[t, x], {x, 0, 1}, {t, 0, 3}, BoxRatios  -> {1, 1,  .2}, 
      Boxed -> False, Axes -> False, Mesh -> False, 
      PlotPoints -> 150]

\(\(\(\ \)\(\(δt\  = \  .02\)\(;\)\(\ \)\(\(Plot[
          u[#, x], {x, 0, 1}, PlotRange\  -> {\(- .2\), 1.2}, 
          AspectRatio\  -> \  .3, 
          PlotStyle\  -> \ {Thickness[ .004], Hue[ .45]}] &\)\  /@ 

      Range[0, 3, δt]\)\(;\)\(
\)\(
\)\(
\)\(
\)\)\)\)