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Damping of a discontinuity; problem and model

Problem. Two pieces of a material, at different temperatures, are brought in contact at

$t=0$ . Assume the end points of the pieces are kept at the initial temperature. How does the heat flow from the hot to the cold piece?

Solution.

Assume a 1D model is sufficient (insulated rod):

$u(x,0)=\left\lbrace \begin{array}{ll} U_L, & x < L/2\\ U_R,& x\geq L/2 \end{array}\right.$

$\frac{\partial u}{\partial t} = \dfc \frac{\partial^2 u}{\partial x^2},\quad u(0,t)=U_L,\ u(L,t)=U_R$