Chapter 12.3, Problem 7E

A thin wire coinciding with the x-axis on the interval [–L, L] is bent into the shape of a circle so that the ends x = –L and x = L are joined. Under certain conditions, the temperature u(x, t) in the wire satisfies the boundary-value problem

$\begin{array}{l}k\frac{{\partial }^{2}u}{\partial x^2}=\frac{\partial u}{\partial t},-L<x<L,t>0,\\ u(-L,t)=u(L,t),t>0\\ {\frac{\partial u}{\partial x}|}_{x=-L}={\frac{\partial u}{\partial x}|}_{x=L},t>0\\ u(x,0)=f(x),-L<x<L.\end{array}$

Find the temperature u(x, t).