Problem # $1$

Suppose heat is lost from the lateral surface of a thin rod of length $L$ into a

surrounding medium at temperature zero. If the linear law of heat transfer

applies, then the heat equation takes on the form:

$k{u}_{}-hu=u_t,00$

Heat transfer from

lateral surface of

the rod

Find the temperature $u(x,t)$ if the initial temperature is $f(x)$ throughout and

the ends $x=0$ and $x=L$ are insulated.