% maple Numerical Methods S2015
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\begin{document}
\hfill {\bf
Name~}\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}
%\hfill  {Class Time~}\underline{~~~~~~~~}\newline

\begin{center}
\bf Math 2250
\bf Numerical Methods Project\\
\bf S2015
\end{center}


{\bf References}: Edwards-Penney, Sections 2.4,2.5, 2.6.
This document: \href{http://www.math.utah.edu/~gustafso/s2015/2250/numericalDEproject-S2015.pdf}
{numericalDEproject-S2015.pdf}

Other related and required documents
are available at the \href{http://www.math.utah.edu/~gustafso/s2015/2250/}{course web site},
in particular
\href{http://www.math.utah.edu/~gustafso/s2015/2250/numericalDE2008.pdf}{Numerical Methods Slides},
\href{http://www.math.utah.edu/~gustafso/s2015/2250/numericalManuscript.pdf}{Numerical Methods Manuscript},
\href{http://www.math.utah.edu/~gustafso/s2015/2250/2250numerical-hints.txt}{Sample Maple Code for Euler, Heun, RK4}.



\smallskip

\Prob{Symbolic Solution}{(E \& P Exercises 2.4-6, 2.5-6, 2.6-6 Symbolic Solution)}
The symbolic solution of $y'=-2xy$, $y(0)=2$ is $y=2e^{-x^2}$.
Display details for the derivation of this symbolic solution, using methods from Edwards-Penney section 1.4 or 1.5. Present a full 2-panel answer check.


The answer $y=2e^{-x^2}$ can be used to make the following table, which is used in the problems below.

\begin{tt}
\begin{tabular}{|l||l|l|l|l|l|l|l|l|l|}
\hline
$x$ & 0.00 & 0.10 & 0.20 & 0.25 & 0.30 & 0.40 & 0.50 \\ \hline
$2e^{-x^2}$ & 2.000000000 & 1.980099667 & 1.921578878 & 1.878826126 &
1.827862371 & 1.704287578 & 1.557601566 \\ \hline
\end{tabular}
\end{tt}

%  y:=x->2*exp(-x^2); [seq([0.05*i,y(0.05*i)],i=0..10)];

\def\AA{\fbox{$\phantom{L}$~~~~}}

\Prob{Euler's Method}{(E \& P Exercise 2.4-6)} Consider the initial value
problem $y'=-2xy$, $y(0)=2$ with symbolic solution $y=2e^{-x^2}$.
Apply Euler's method to produce two dot tables, as shown below. The
first has three pairs, $h=0.25$. The second has six pairs, $h=0.1$.
Reproduce the summary of results below, writing $4$ small digits (Geek pen required) into the blank rectangle \AA. The 4th digit can be rounded or not.

The work for $h=0.25$ is to be entirely hand-written, with calculator assist. Answer checks and the work for $h=0.1$ may use technology.

\begin{tt}
Table $h=0.25$: [0,2], [0.25,2], [0.5,1.75\AA]

Table $h=0.10$: [0,2], [0.1,2.0], [0.2,1.96], [0.3,1.88\AA], [0.4,1.76\AA], [0.5,1.627\AA]

\def\AA{\fbox{$\phantom{L}$~~~~~~}}
\begin{tabular}{|l|l|l|}
\hline
 Actual $y(.5)$  & Approx $y(.5)$, $h=0.25$  & Approx $y(.5)$, $h=0.10$\\ \hline
 1.557601566      &   1.75\AA     & 1.627\AA        \\ \hline
\end{tabular}
\end{tt}

\Prob{Heun's Method (Modified Euler)}{(E \& P Exercise 2.5-6)} Consider the initial value
problem $y'=-2xy$, $y(0)=2$ with symbolic solution $y=2e^{-x^2}$.
Apply Heun's method (Improved Euler) to produce one dot table of six
rows, $h=0.1$. Reproduce the table below and fill in missing digits.
Hand-written work is expected with computer assist. Hand-written work with calculator assist should end after the estimate for $y(0.1)$. Technology should be used for the remaining answers. Four digits are expected in \AA.

\def\AA{\fbox{$\phantom{L}$~~~~~~~}}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
 $x$ &
 0.0 &
 0.1 &
 0.2 &
 0.3 &
 0.4 &
 0.5
\\ \hline
 Actual $y(x)$  &
 2.000000000      &
 1.980099667      &
 1.921578878      &
 1.827862371      &
 1.704287578      &
 1.557601566
\\ \hline
 Approx $y(x)$  &
 2.0000000        &
 1.980\AA        &
 1.921\AA        &
 1.827\AA        &
 1.704\AA        &
 1.557\AA
\\ \hline
\end{tabular}


\Prob{RK4 Method}{(E \& P Exercise 2.6-6)} Consider the initial value
problem $y'=-2xy$, $y(0)=2$ with symbolic solution $y=2e^{-x^2}$.
Apply the RK4 method to produce one dot table of three rows,
$h=0.25$. Reproduce the table below, filling in the missing digits.
Hand-written work should complete the estimate of $y(0.25)$ to 8 digits, using calculator assist. The remaining steps should use technology, with a plan to reproduce the hand-written result. Four digits are expected in \AA.

\def\AA{\fbox{$\phantom{L}$~~~~~~~}}
\begin{tabular}{|l|l|l|l|}
\hline
 $x$ &
 0.00 &
 0.25 &
 0.50
\\ \hline
 Actual $y(x)$  &
 2.000000000      &
 1.878826126      &
 1.557601566
\\ \hline
 Approx $y(x)$  &
 2.000000000        &
 1.878\AA &
 1.557\AA
\\ \hline
\end{tabular}

\bigskip

{\bf Submit this printed page}, with blanks filled in, as the first page of your report. Attach hand-written solutions next. Append computer results last, then staple. Due in Week 4.

\end{document}
