I am trying to solve the one-dimensional transient heat equation with a specified flux in one end ($x=0$) and perfect insulation on the other ($x=L$):
$$\frac{\partial T(x,t)}{\partial x} = \alpha \frac{\partial^2 T(x,t)}{\partial x^2}$$ $$\frac{dT(0,t)}{dx}=W$$ $$\frac{dR(L,t)}{dx}=0$$
The initial temperature is constant throughout the system:
$$T(x,0)=1$$
By separation of variables I can get the general heat equation solution:
$$T(x,t) = Ae^{-\lambda \alpha t}\left[ B\sin(\sqrt{\lambda}x) + C\cos(\sqrt{\lambda}x)\right]$$
but I am stuck trying to apply the boundary conditions, specifically the constant flux. Every time I try to solve this I get "constants" that vary with time. It has been many moons since I have solved an equation like this, and I expected to get some complicated formula for $\lambda$, but unfortunately I can't even get that far.
I suppose the underlying question is, "Can the general heat equation solution above be solved to fit the initial and boundary conditions also specified above?".
Separation of variables requires that the boundary conditions are homogeneous. So the first thing is to find a single inhomogeneous solution, not necessarily with $T(x,0)=0$, and subtract off. Maybe guess a solution of the form $T(x,t) = a x^2 + b t$.