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\noindent
Math 436

\noindent
December 11, 2000

$$
\bf{FINAL}
$$

\bigskip

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. (25 points) a) Solve:  
\begin{eqnarray*} 
&\ &u_{tt}=u_{xx},\qquad t>0,\ 0<x<1,\\
&\ & u(x,0)=0, \qquad 0<x<1,\\
&\ & u_t(x,0)=x(1-x), \qquad 0<x<1,\\
&\ &u(0,t)=0=u(1,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 $u(\frac 34,\frac 54)\/$.  (Your answer should be a
number.)

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2. (10 points)  Find the solution  $u(\rho, \phi)\/$ 
of Laplace's equation in the disk  $\rho <1\/$
and satisfying the boundary condition  
$$
u(1,\phi)=54917 \sin 6327145 \phi
-72901 \cos 143587 \phi.
$$

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3. (20 points) a) Solve: 
\begin{eqnarray*}
&\ &u_t=\nabla ^2 u, \qquad   
0<x<\pi, \ 0<y<\pi,\ 0<t,\\
&\ &u(0,y,t)=0=u(\pi,y,t)=u(x,0,t)=u(x,\pi,t),\\
&\ &u(x,y,0)=xy(\pi - x)(\pi -y).
\end{eqnarray*}


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

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4. (20 points) a) Solve:
\begin{eqnarray*}
 u_t&=&u_{xx}, \qquad t>0,\  -\infty < x<\infty,\\
u(x,0)&=& \left\{\begin{array}{ll} 0,& \qquad x<-5\\
     x, & \qquad -5<x<8\\
       0, &\qquad 8<x.
  \end{array}
   \right.
\end{eqnarray*}



 
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\quad b) Find  $u(0,t)\/$.  (Your answer should be a function of  $t\/$
which does not involve an integral.)
 
\medskip
 
\quad c) What is
$ \displaystyle{\lim_{t\to 0^+} u(-5,t)}\/$?


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