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  \begin{huge}
    MATH 609-602: Numerical Methods
  \end{huge}
\end{center}

\begin{tabular}{ll}
Lecturer: & Prof. Wolfgang Bangerth \\
& Blocker Bldg., Room 507D \\
& (979) 845 6393 \\
& \texttt{bangerth@math.tamu.edu}\\[5pt]
Teaching Assistant: & Seungil Kim \\
& Blocker Bldg., Room 507A \\
& (979) 862 3259 \\
& \texttt{sgkim@math.tamu.edu}
\end{tabular}

\section*{Homework assignment 10 -- due Tuesday 11/22/2005}

\paragraph{Problem 1 (Numerical solution of a scalar ODE).}
Consider the following scalar ordinary differential equation (ODE):
\begin{align*}
  x'(t) &= \frac{1}{3x(t)^2},
  \\
  x(0) &= \frac 1{10^{1/3}}.
\end{align*}
The solution of this equation is $x(t)=(t+\tfrac 1{10})^{1/3}$.

Compute approximations to $x(4)$ using the 
\begin{itemize}
\item first order Taylor expansion method,
\item second order Taylor expansion method,
\item implicit Euler method,
\item trapezoidal method,
\end{itemize}
each with step sizes $h=2,1,\frac 12, \frac 14, \ldots, \frac 1{32}$. Compute
their respective errors $e=|x_N-x(4)|$ where $x_N$ is the approximation to
$x(4)$ at the end of the last time step, and compute the convergence
rates. Compare the accuracy of all these methods for the same step size $h$.
\points{7}


\paragraph{Problem 2 (Numerical solution of a vector-valued ODE).}
A rocket that is shot up vertically experience upward acceleration from its
engines, and downward acceleration due to gravity. Its height therefore
satisfies Newton's law
\begin{align}
  \label{eq:newton}
  d''(t) = \frac{F(t)}{m(t)},
\end{align}
where $d(t)$ denotes the distance from the earth's center. Assume that the
rocket is initially at rest at $d(0)=6371000$. After ignition, the engines
produce a constant thrust for 10 minutes before shutting down:
\begin{align*}
  T(t) = \left\{
    \begin{array}{ll}
      12 & \text{for $t<600$,} \\
      0 & \text{for $t\ge 600$}.
    \end{array}\right.
\end{align*}
On the other hand, gravity generates the force
\begin{align*}
  G(t) = -(6371000)^2 \frac{10 m(t)}{d(t)^2}.
\end{align*}
The total force is $F(t)=T(t)+G(t)$. The mass of the rocket decreases while
fuel is burnt in the engines according to
\begin{align*}
  m(t) = \left\{
    \begin{array}{ll}
      1-\frac {0.9 t}{600} & \text{for $t<600$,} \\
      0.1 & \text{for $t\ge 600$}.
    \end{array}\right.
\end{align*}

Compute the altitude of the rocket for times between $t=0$ and $t=36000$ using
the explicit Euler method. Try to determine the altitude up to an accuracy of
100 meters.
\points{5}


\paragraph{Problem 3 (Some parameter determination with ODEs).}
In skydiving, freefall is a balance between the force gravity exerts on the
skydiver, and the counteracting air friction. At usual altitudes, gravity does
not depend on the height of a person (and is approximately $10ms^{-2}$, and
air friction increases like the square of the velocity. We can therefore
describe the falling velocity by the ODE
\begin{align*}
  v'(t) &= 10 - av(t)^2, 
  \\
  v(0) &= 0.
\end{align*}
Using your own ODE solver, compute an approximate value for the coefficient $a$
such that the speed of the skydiver after 10 seconds is $v(10)=50$ (that's a
realistic free fall velocity in meters per second: approximately 115 mph).

(For this question, creativity in finding a way to approximate $a$ is
encouraged---the way counts, not the result up to 6 digits; in return, no
ideas on how to achieve the goal will be provided.)
\points{3}

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