# 1D Heat Equation

Having been introduced solely to the basics of solving 1D heat equations in my analysis course, I have trouble solving this particular one:

$\frac{\partial T}{\partial t} (x,t) = \frac{K}{c_\rho \rho}\frac{\partial^2 T}{\partial x^2} (x,t); x \in [0,L], t > 0$

given BCs:

$T(0,t) = T_0 + q \frac{2}{\sqrt\pi} \sqrt t; T(x,0) = T_0 ; \frac{\partial T}{\partial x}|_{(0,t)} = -\frac{q}{K}$

where $T_0, q, K, c_\rho,\rho$ are constants.

In this particular (real-life) case, x represents the width of the bar whose upper surface is heated due to friction. Firstly I am unsure if this equation is even solvable, as we have not imposed anything on the other end of the bar (as we simply dont know anything about it) (and instead imposed an additional Neumann condition on the known end). If it is, how can I go about solving it analytically (if this is feasible)?

EDIT

Apparently the equation above does not seem well-posed. If I remove the Neumann boundary condition for $(0,t)$ and instead specify that $T(L,t) = T_L = const$, will it be possible to introduce a function to render the BCs homogenous and hence use separation of variables?

• @Harry49 I assume that this is due to not specifying a BC for the other end? – Ruslan Mushkaev Jan 7 '18 at 20:44
• @Harry49 Assuming that I remove the Neumann condition and instead specify that $T(L,t) = T_{L}=const$, is it possible to introduce a function to render the boundary conditions homogenous? – Ruslan Mushkaev Jan 7 '18 at 21:06
• Since the boundary conditions are time dependent, the method of separation of variables fails. One can have a look at these related posts (1), (2). – Harry49 Jan 8 '18 at 18:04