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%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Sun Jun 13 00:58:59 1999}
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Use the method of separation of variables to show that the 1-D heat
conduction problem: find $u=u(x,y)$ so that
\begin{eqnarray*}
\frac{\partial u}{\partial t}(x,t)-\alpha \frac{\partial ^{2}u}{\partial
x^{2}}(x,t) &=&0,\;0<x<l,\;t>0 \\
u(0,t) &=&0,\;t\geq 0 \\
u(l,t) &=&0,\;t\geq 0 \\
u(x,0) &=&f(x),\;0\leq x\leq l
\end{eqnarray*}

has the solution
\[
u(x,t)=\sum_{n=1}^{\infty }c_{n}\sin \frac{n\pi x}{l}\exp \left[ -\alpha
\left( \frac{n\pi }{l}\right) ^{2}t\right] 
\]

where
\[
c_{n}=\frac{2}{l}\int_{0}^{l}f(x)\sin \frac{n\pi x}{l}dx.
\]

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