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Use the method of separation of variables to show that the
1-D heat conduction problem: find $u = u(x,t)$ 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 \geq 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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