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\begin{center}
{\Large COMPUTER SCIENCE 614}\\
{\Large Numerical Solution of Ordinary Differential Equations} \\
{\large SPRING 2012} \\
ASSIGNMENT \#5 (31 points) \\
{\small March 10}
\end{center}

\paragraph{Due Monday, April 2 in class}
This assignment covers Sections 7.2, 9.1--9.2 of the class notes,
which corresponds to Chapters 13 and 14 of the textbook.

\begin{enumerate}

\item (5 points)  % 7.2.2
 From first principles, determine the Stratonowich integral
 \[\int_0^T W(t)\!\circ\!\rmd W(t).\]
 The Stratonovich integral uses the midpoints rather than the left endpoints
 of the subintervals of the partitions whose limit defines the integral.
 You may refer to any results already in the class notes.

\item (7 points)  % 7.2.5
 \begin{enumerate}
 \item Using the Ito formula, rewrite $\int_0^T W(t)\rmd W(t)$
  as an expression in terms of $T$ and $\int_0^T f(t)\rmd W(t)$
  where $f(t)$ is some function independent of $W(t)$ for which
  $\int_0^T f(t)^2\rmd t$ is known.
 \item Using two applications of the Ito formula,
  rewrite $\int_0^T W(t)^2\rmd W(t)$
  as an expression in terms of $T$
  and two integrals $\int_0^T f_k(t)\rmd W(t)$, $k = 1, 2$,
  where the $f_k(t)$ are functions independent of $W(t)$ for which
  the integrals $\int_0^T f_i(t)f_j(t)\rmd t$ are known.
 \end{enumerate}

\item (4 points)  % 7.2.6
 For the algorithm for one step of the second order Taylor method
 given in Section~7.2.6 of the class notes,
 show how to generate $R_1$ and $R_2$ from two independent standard
 Gaussian random variables $Z_1$ and $Z_2$.
 Be aware that $R_1$ and $R_2$ are not necessarily independent.

\item (6 points)  % 9.1
 Exercise 13.11 from Griffiths \& Higham has five parts.
 Do parts 3 and 4.
 Note that ``derive'' means discovering the answer,
 not merely verifying that it is correct.

\item (9 points)  % 9.2
 Two particles in motion interacting through a central force satisfy
 \[{\vecr_i\,}' = \frac1{m_i}\vecp_i,
 \quad{\vecp_i\,}' = -\frac{\partial}{\partial\vecr_i}U(\|\vecr_2 -\vecr_1\|),
 \quad i = 1, 2.\]
 \begin{enumerate}
 \item Show that the total angular momentum $\sum_i\vecr_i\times\vecp_i$
  is conserved.
 \item One step of the symplectic Euler method is given by
  \[\vecp_i^{\,1} =\vecp_i + h\vecF_i,
   \quad\vecr_i^{\,1} =\vecr_i +\frac{h}{m_i}\vecp_i^{\,1},\]
  where $\vecF_i = - (\partial/\partial\vecr_i)U(\|\vecr_2-\vecr_1\|)$
  and the superscript 1 denotes the value at the new time point.
  Show that this conserves total angular momentum.
 \end{enumerate}
 You might find it useful to structure your answer using lemmas.

\end{enumerate}

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