Integrate[Exp[-r^2],{r,0,x/Sqrt[4 k t]}]

Plot[Erf[z],{z,-10,10}]

w[x_,t_,k_,u0_]=u0/2(Erf[x/Sqrt[4k t]]+1)

w[2,5,1,1]

N[%]

Animate[Plot[w[x,t,1,1],{x,-10,10},PlotRange->{0,1}],{t,0,10},AnimationRunning->False]