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\centerline{\large \bf MA 573 --- PROJECT 4}

\bigskip
\noindent
{\large \bf Due: Monday, October 16}

\bigskip
\noindent
{\large \bf Problem 1.}

\medskip
In class, we showed that the solution to the diffusion model
\begin{equation}
 \begin{array}{l}
    {\ds \frac{ \d \rho}{\d t} = D \frac{1}{r^2} \frac{\d }{\d r}\left(r^2 \frac{\d \rho}{\d r} \right) } \\
    \noalign{\bigskip}
    {\ds \rho(0,r) = M \delta_0 (r)}
 \end{array}
 \label{eq1}
 \end{equation}
 is
 \begin{equation}
   \rho(t,r) = \frac{M}{\sqrt{(4 \pi D t)^3}} e^{-r^2/4Dt} .
 \label{eq2}
 \end{equation}
We also collected the data in Table~1 where times are reported in seconds and distances in both inches
and meters.  You should use the metric measurements for your computations.
 
For this project, you can assume that $\rho(t,r) = 1$ when the durian smell was first noted.   For each $(t,r)$ pair, (\ref{eq2}) then has the form
$$
  1 - M \cdot f(D) = 0,
$$
where $f(D)$ is the Gaussian function evaluated for specific values of $t$ and $r$. 

\begin{description}
\item{(i)}  We will initially solve for $D$ and take $M=1000$.  The easiest way to do this is to plot the equation for a range of $D$-values and  zoom in on the region where it crosses the axis.  This zooming process can be done with arbitrary accuracy
so you can get a good estimate for $D$.  Note that there will likely be two roots so you can choose the first.
Report the five values that you recover for $D$ and discuss assumptions and sources of error.

\item{(ii)}  Try varying the value of $M$ and discuss its effect on your solution.  You do not need to do this
here, but you would typically estimate it in addition to $D$. 

\item{(iii)} The diffusion constant for certain gasses is reported to be $2 \times 10^{-5}$ m$^2$/s.  How do
your results compare to this value?
\end{description}


\begin{table}[!hh]
\begin{center}
\begin{tabular}{ccc}
$t$~(s) & $r$~(inches) &  $r$~(m) \\ \hline
 2.1 & 62 &   \\
 27.1 & 88 &  \\
 41.4 &  115 &  \\
 96.6 &  153 &   \\
 112 &  170 &   \\  \hline
\end{tabular}
\end{center}

\vspace{-.2truein}
\caption{Times and distances noted for the durian scent.}
\label{table1}
\end{table}
 

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