Originally Posted by

**dwsmith** A rod of length L with insulated lateral surface has thermal constants c, p, k. One end of the rod is kept at temperature zero. The other end is connected to a black box which manages to maintain that end at the instantaneous average temperature of the entire rod. Initially, the rod is at temperature U. Formulate the initial-boundary value problem for the temperature in the rod. Is it a linear problem?

The solution is

$\displaystyle \displaystyle\text{B.C.} \ \ u(0,t)=0, \ \ \ u(L,t)=\frac{1}{L}\int_0^L u(x,t)dx \ \ 0<t$

I don't understand how this part (see below) is obtained.

$\displaystyle \displaystyle u(L,t)=\frac{1}{L}\int_0^L u(x,t)dx $