Math Help - Set up boundary value problem

1. Set up boundary value problem

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

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

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

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

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

$\displaystyle u(L,t)=\frac{1}{L}\int_0^L u(x,t)dx$
The right hand side is the instantaneous average temperature of the rod, which is the temperature at the end of the rod at t.

CB

3. The right hand side is the instantaneous average temptation of the rod,...
Those game little rod-tempters!

4. Originally Posted by CaptainBlack
The right hand side is the instantaneous average temperature of the rod, which is the temperature at the end of the rod at t.

CB
I don't know how I would have come up with that integral though just from reading that passage though.

5. The average value of an integrable function, f(x), on the interval [a, b], is defined as $\frac{\int_a^b f(x)dx}{b- a}$. That is the extension, through the "Riemann sums" limiting process, of the average value of n numbers, $f(x_0)= f(a), f(x_1), ..., f(x_n)= f(b)$.