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The reservoir on the left is being kept at temp ##T_s##, and it is a fluid that has very high convective coefficient ##h##. As a result, the boundary condition at the LHS wall is given by:

$$ T(0,t) = T_s $$

at time ##t=0## the wall and the cold reservoir are brought into thermal contact with the warm reservoir. The wall and the blue reservoir are both initially at the same uniform temperature ##T(x,0) = T_o##.

The blue reservoir and the wall start to heat up. The boundary condition on the RHS wall is given by:

$$ -k \left. \frac{\partial T}{ \partial x} \right|_{L} = h \left( T(L,t) - T_{air}(t) \right) $$

The heat equation for the plane wall is given by:

$$ \frac{\partial^2 T}{\partial x ^2} = \rho c_p \frac{\partial T}{\partial t}$$

So we have three non-homogenous boundary conditions, what is the idea to solve the PDE? Apparently, separation of variables does not work because of the non-zero boundary conditions.

The only thing I know (almost nothing) about solving PDE's I've learned from YouTube videos... Why does having non-zero boundary conditions throw a fork in it, and how big of a fork is it?

Thanks for any guidance!