A note on initial and boundary conditions.

  What truly makes the PDE different from the initial and boundary conditions is the region of applicability. The PDE applies at all points. (At least, if we ignore the possibility that some points could be singular). The initial and boundary conditions only apply at a small subset of points in the x,t-plane of figure 2. It is the PDE that must be integrated into the field. You cannot, say, take the Neumann boundary condition into the interior. Clearly, it is in general not true that $\partial T / \partial x = 0$ in the interior.