calka1[x_,t_]:=NIntegrate[(E^(-y^2)),{y,(x-1)/(2(Sqrt[t])),x/(2(Sqrt[t]))}]
calka2[x_,t_]:=NIntegrate[(E^(-y^2)),{y,x/(2(Sqrt[t])),(x+1)/(2(Sqrt[t]))}]

u[x_,t_]:=(Sqrt[t])/(Sqrt[Pi])( (E^(-((x+1)^2)/(4t)))-2(E^(-(x^2)/(4t)))+(E^(-((x-1)^2)/(4t))))+((1-x)/(Sqrt[Pi]))calka1[x,t]+((1+x)/(Sqrt[Pi]))calka2[x,t]

Animate[Plot[u[x,t],{x,-1.5,1.5},PlotRange->{0,1}],{t,0,0.01}]