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\begin{document}
{\LARGE
\begin{center}
MATH 220 E1 ~~~~~~ QUIZ \#13  ~~~~~  (20 points)
\end{center}
(12 minutes, 16 April 1999 \{13th Week\}, PDEs: 10.2,10.3,10.5$^{-}$)
\\*[0.25em]
NAME:
\\*[0.25em]
SOCIAL SECURITY NUMBER (SSN):
\\*[0.25em]
DISCUSSION TIME AND DAY (DT\&D):}
\\*[0.5em]
{\B 1. Formally Solve IVP/BVP for Heat Equation:}
\EQ{
\mathrm{PDE:}~~
\frac{\partial u}{\partial t}
= 3^2*\frac{\partial^2 u}{\partial x^2}~,
~~ 0<x<\pi/2~, ~ t>0~;\\*[-1.5em]
}
\EQ{
\mathrm{BCs:}~~
u(0,t) = 0 = u(\pi/2,t), ~~ t>0~;\\*[-2.5em]
}
\EQ{
\mathrm{IC:}~~
u(x,0)=10*\sin(2x)-0.1*\sin(4x)\\*[0.5em]
+0.001*\sin(8x)
~, ~~ 0 <x< \pi / 2~;
}
{\B by Separation of Variables and Fourier Series Methods.
\underline{\B You Must Show Your Work.}}
\\*[0.5em]
{\I Fourier Sine Coefficient for $f(x)$ on $0<x<L$ (if needed):}
\EQ{
\mathrm{FSC:}~~
b_n=\frac{2}{L}\int_0^L
f(x)\sin\left(\frac{n\pi x}{L}\right) dx ;
}
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