I thought about the following PDE (the $u(x=0,t)$ is not a boundary condition! It really is a part of the PDE):
$$\dfrac{\partial u(x,t)}{\partial t}=\alpha \dfrac{\partial^2 u(x,t)}{\partial x^2}+u(x=0,t).$$
How can i determine the symmetries of this PDE with Maple? And how are such PDEs called?