The non-linear form of the heat equation can be written as:
$\rho(T) c_p(T) \frac{\partial T}{\partial t}= \frac{\partial}{\partial z} \left ( k(T) \frac{\partial T}{\partial z} \right).$
Assuming the following boundary and initial conditions:
$ T(z,0) = T_0 ,$
$-k \left. \frac{\partial T}{\partial z} \right\vert_{z=0} =q(t), $
$\left. \frac{\partial T}{\partial z} \right\vert_{z=L} = 0,$
is a solution possible? I have a solution for the associated linear problem, which relied on the use of Duhamel's theorem but I am struggling to find mention of a non-linear solution. I am currently using a finite difference scheme to model this problem but was interested in the existence/viability of an exact answer.
