CheckHeatEqn[u_] :=D[u, t] ==  D[u, {x, 2}]
u[k_, x_, t_] := E^(-k^2 t) Sin[ k x ];
v[x_, t_] := Pi-x + u[1, x, t] +  u[2, x,t] + 3/2 u[3, x, t];
CheckHeatEqn[u[1, x, t]]
CheckHeatEqn[v[x, t]]

Manipulate[Plot[ u[1, x, t], {x, 0, 2 Pi}, PlotRange-> {-1.1, 1.1}] , {t, 0, 5}]

Plot3D[ u[1, x, t], {x, 0, 2 Pi}, {t, 0, 2}, PlotRange -> All]

Manipulate[Plot[ v[x,t], {x, 0, 2 Pi}, PlotRange -> {-6, 6}] , {t, 0, 3}]

Plot3D[ v[ x, t], {x, 0, 2 Pi}, {t, 0, 0.5}, PlotRange -> All]