Heat equation IBVP with non-homogeneous Neumann BC's

Question

I have an IBVP heat equation with non-homogeneous Neumann boundary conditions $$\frac{\delta u}{\delta x}(0,t)=A, \frac{\delta u}{\delta x}(L,t)=B, t>0$$

Part $(a)$ was to prove, using the steady state/transient solution, that $A=B$ in order for the IBVP to have a solution, which I have already done. Part $(b)$ asks to find the function $v(x)$ (which comes from $u(x,t) = v(x) +w(x,t)$). I am a little bit lost at this point. I have the BVP $v''(x) = 0, v'(0)=A=v'(L)=B$

I would really appreciate if someone could point me in the right direction for finding v(x). Thank you.

Answer 1

Hint: note that the function $v(x)$ is related only to space. What happens to the original EDP when the term transient disappears? Solving this problem with the given boundary conditions you can find the function you want.

Answer 2

Try solving for $v(x)$ and $v'(x)$ from $v''(x)$ (i.e. using integration) and plug in what you found in (a)

Extra Hint:

You're going to get something along the lines of $$v''(x)=0 \\v'(x)= C \\v(x)= Cx+D$$

Final Hint:

Your end solution for v(x) will still involve a D but not a C