The following simulation in Fig.10.1 of one-dimensional equilibration or diffusion process shows for example the time and space dependence of the temperature after heating a homogeneous thermally insulated thin wire at a point.
According to the above mentioned special solution an approximated delta function at the origin is used as initial function, that spreads in Gaussian shape under conservation of the area under the curve (the amount of heat). The arrows indicate the -width, the number field the respective point in time. The diffusion constant can be adjusted with the slider over a wide range of values.
The description pages contain further hints.