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\noindent
Math 436

\noindent
December 16, 1999

$$
\bf{FINAL}
$$

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Show all
your work.  
If you find you are doing a horrendous calculation, you are
making a mistake or at least making the problem unnecessarily
difficult.  (Some calculations are necessary.)

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1. (30 points) a) Solve:  
\begin{eqnarray*} 
&\ &u_{tt}=u_{xx},\qquad t>0,\ 0<x<2,\\
&\ & u(x,0)=0, \qquad 0<x<2,\\
&\ & u_t(x,0)=x(2-x), \qquad 0<x<2,\\
&\ &u(0,t)=0=u(2,t), \qquad t>0.
\end{eqnarray*}


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\quad b) Show that your answer solves the problem.
 
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\quad c) Find $u(\frac 74,\frac 94)\/$.  (Your answer should be a
number.)

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2. (15 points)  Find the solution  $u(\rho, \phi)\/$ 
of Laplace's equation in the disk  $\rho <1\/$
and satisfying the boundary condition  
$$
u(1,\phi)=68924 \sin 9815423 \phi
-79021 \cos 4230874 \phi.
$$

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3. (30 points) a) Solve: 
\begin{eqnarray*}
&\ &u_t=\nabla ^2 u, \qquad   
0<x<2\pi, \ 0<y<2\pi,\ 0<t,\\
&\ &u(0,y,t)=0=u(2\pi,y,t)=u(x,0,t)=u(x,2\pi,t),\\
&\ &u(x,y,0)=xy(2\pi - x)(2\pi -y).
\end{eqnarray*}


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\quad b) Show that your answer  solves the problem.

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