I have recently read a paper on heating a metal bar. The heat equation is used to analyze the system but as I can't find the right boundary conditions used to solve the heat equation. I'm hoping that by explaining the experiment here someone could point out what I'm missing.
A metal rod (of length $L$) is held at a constant temperature $T=T_0$ at the $x=L$. At the end $x=0$ heat is generated that results in a constant heat flux of $\vec{q}=\frac{\dot{Q}}{A}$.
Solving the equation for the initial conditions, they get that $T(x,0)=T_0+(L-x)\frac{\dot{Q}}{AK}$.
I've been able to follow the paper this far where I get:
\begin{array}{lkl} \frac{\partial^2 T}{\partial x^2} = \frac{1}{D}\frac{\partial T}{\partial t} & \textrm{Heat equation} \\ T(L,t)=T_0 & \textrm{Boundary condition} \\ T(x,0)=T_0+(L-x)\frac{\dot{Q}}{KA} & \textrm{Initial condition} \end{array}
I only know how to solve PDEs with the variable separation method and I don't see how I could apply it here. I feel like I'd need another boundary condition.
I'm not looking for a solution as I'd like to get it myself but I don't even know where to start or where to get that extra boundary condition.
Thanks for the help and sorry for any spelling or grammar mistakes I might have made as english is not my first language
In case anyone is wondering the name of the paper is "Measurement of the thermal properties of a metal using a relaxation method" by John N. Fox and Richard H. Mc.Master