u[x_,t_]:=1/2+Erf[x/Sqrt[4*t]]/2;

Plot[{HeavisideTheta[x]},{x,-8,8},PlotRange->{0,1.2},PlotStyle->Thick]

Plot[{u[x,0.01]},{x,-8,8},PlotRange->{0,1.2},PlotStyle->Thick]

Plot[{u[x,0.1]},{x,-8,8},PlotRange->{0,1.2},PlotStyle->Thick]

Plot[{u[x,0.3]},{x,-8,8},PlotRange->{0,1.2},PlotStyle->Thick]

Plot[{u[x,0.7]},{x,-8,8},PlotRange->{0,1.2},PlotStyle->Thick]

Plot[{u[x,1]},{x,-8,8},PlotRange->{0,1.2},PlotStyle->Thick]

Plot[{u[x,2]},{x,-8,8},PlotRange->{0,1.2},PlotStyle->Thick]

Plot[{u[x,4]},{x,-8,8},PlotRange->{0,1.2},PlotStyle->Thick]

Plot[{u[x,6]},{x,-8,8},PlotRange->{0,1.2},PlotStyle->Thick]

Plot[{u[x,8]},{x,-8,8},PlotRange->{0,1.2},PlotStyle->Thick]

Plot[{u[x,0.01],u[x,0.1],u[x,0.3],u[x,1],u[x,2],u[x,4],u[x,8]},{x,-8,8},PlotRange->{0,1.2},PlotStyle->Thick]

Plot3D[u[x,y],{x,-8,8},{y,0,5},PlotPoints->60,Mesh->None,Boxed->True,Axes->True]

u[x_,t_]:=Exp[-x^2/(4*t)]/Sqrt[4*Pi*t];

Plot[{u[x,0.02]},{x,-8,8},PlotRange->{0,2},PlotStyle->Thick]

Plot[{u[x,0.1]},{x,-8,8},PlotRange->{0,2},PlotStyle->Thick]

Plot[{u[x,0.3]},{x,-8,8},PlotRange->{0,1.2},PlotStyle->Thick]

Plot[{u[x,0.7]},{x,-8,8},PlotRange->{0,1.2},PlotStyle->Thick]

Plot[{u[x,1]},{x,-8,8},PlotRange->{0,1.2},PlotStyle->Thick]

Plot[{u[x,2]},{x,-8,8},PlotRange->{0,1.2},PlotStyle->Thick]

Plot[{u[x,4]},{x,-8,8},PlotRange->{0,1.2},PlotStyle->Thick]

Plot[{u[x,8]},{x,-8,8},PlotRange->{0,1.2},PlotStyle->Thick]

Plot[{u[x,0.02],u[x,0.1],u[x,0.3],u[x,1],u[x,2],u[x,4],u[x,8]},{x,-8,8},PlotRange->{0,2},PlotStyle->Thick]

Plot3D[u[x,y],{x,-8,8},{y,0.02,5},PlotRange->{0,2},PlotPoints->60,Mesh->None,Boxed->True,Axes->True]