\documentclass[11pt]{article} \usepackage{fullpage,amsmath} \newcommand{\T}{^{\mathsf{T}}} % matrix transposition \newcommand{\bfy}{\mathbf{y}} \newcommand{\rmd}{\mathrm{d}} \begin{document} \begin{center} {\Large COMPUTER SCIENCE 614}\\ {\Large Numerical Solution of Ordinary Differential Equations} \\ {\large SPRING 2012} \\ ASSIGNMENT \# 1 (26 points) \\ {\small January 13} \end{center} %\paragraph{Announcement} %\begin{itemize} %\item[] %\end{itemize} \paragraph{Due Friday, January 27 in class} This assignment covers Sections 1.0-1.2, 2.0, 2.1, 2.3, 2.5, 12.0, 12.1 of the textbook. {\em Rules.} \begin{itemize} \item Your solution to a problem, or to a problem part in the case of a problem with labeled parts, must not cross page boundaries. \item Your notation should be chosen with sufficient care that you do not make statements that seriously'' incorrect. In particular, you must never equate two quantities that are not equal. \end{itemize} \begin{enumerate} \item (3 points) The following reactions are those of an autocatalytic reaction between 3 chemical species $A$, $B$, and $C$: \begin{eqnarray*} A & \stackrel{k_1}{\rightarrow} & B, \\ B + B & \stackrel{k_2}{\rightarrow} & C + B, \\ B + C & \stackrel{k_3}{\rightarrow} & A + C. \end{eqnarray*} Write this as a system of first order ODEs in standard form. Be careful. \item (2 points) Do Exercise~1.10 of the textbook. \item (6 points) The solution of $y' = y(1 - y)$, $y(0) = c$ is $y(t) = \frac{c}{\exp(-t) + c(1 -\exp(-t))}.$ Determine the linearized ODE for a solution with a slightly different initial value. To be specific, let $y(t;\varepsilon)$ satisfy the given ODE with $y(0;\varepsilon) = c +\varepsilon$, and determine the ODE that is satisfied by the linear term $z(t)$ in the expansion $y(t;\varepsilon) = y(t) +\varepsilon z(t) +\mathcal{O}(\varepsilon^2)$. The right-hand side function of your ODE for $z(t)$ must be a closed-form expression in terms of $t$ and $z$. \item (3 points) Do Exercise~12.8 of the textbook. \item (12 points) The Euler method with step size $h$ applied to $y' = y$, $y(0) = 1$, produces a numerical solution $\{y_n\}$, which is identical (at grid points) to the analytical solution for $y' = c(h)y$ where $c(h)$ is to be determined. \begin{enumerate} \item Determine $c(h)$. \item Let $h = t/n$ for some fixed $t > 0$. Obtain an expression for the global error $y_n - y(t_n)$ in terms of $h$ and $t$ only. \item Use your answer to part~(b) to obtain the first 3 terms in an expansion for the global error in powers of $h$. \end{enumerate} \end{enumerate} \end{document}