A thin wire coinciding with the x-axis on the interval [L, L] is bent into the shape of

Chapter 13, Problem 7

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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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