A thin wire coinciding with the x-axis on the interval [L, L] is bent into the shape of
Chapter 13, Problem 7(choose chapter or problem)
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
\(\begin{aligned}
&k \frac{\partial^{2} u}{\partial x^{2}}=\frac{\partial u}{\partial t},-L<x<L, t>0 \\
&u(-L, t)=u(L, t), t>0 \\
&\left.\frac{\partial u}{\partial x}\right|_{x=-L}=\left.\frac{\partial u}{\partial x}\right|_{x-L}, t>0 \\
&u(x, 0)=f(x),-L<x<L
\end{aligned}\)
Find the temperature u(x, t).
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