% Euler's Method % MATH 266B Exercise 4 \documentclass[12pt]{article} \usepackage{times} \pagestyle{empty} \addtolength{\textwidth}{1.2in} \addtolength{\textheight}{1.2in} \addtolength{\oddsidemargin}{-.58in} \addtolength{\evensidemargin}{-.58in} \renewcommand{\baselinestretch}{1.0} \parindent = 0cm \parskip = .1cm \begin{document} \begin{center} {\Large Euler's Method \footnote{Copyright \copyright 2003 Winfried Just, Department Mathematics, Ohio University. All rights reserved.}} \end{center} In this exercise, we will do some experiments to visualize the workings of Euler's method. You may submit it on Tuesday for three bonus points. You should prepare a worksheet with the answers to all questions posed below for submission.\\ For this exercise, you need to download the file Euler.m from my web page and save it in the ``work'' directory of your \textsc{MatLab} folder without changing its name (and without changing the extension .m)!\\ First let us explore Euler's method for the autonomous differential equation $\frac{dy}{dt} = y$. Suppose you want to approximate $y(2)$ for the solution $y(t)$ of this equation that takes the initial value $y(0) = 1$.\\ Enter:\\ \verb$>> Euler$ \\At the next prompt enter:\\ \verb$>> y$\\ As we discussed in class, the solution $y(t)$ that we want to approximate is given by the formula\\ $y(t) = e^t$. Therefore, at the next prompt enter:\\ \verb$>> exp(t)$\\ At the next prompts enter \verb& 0 & for \verb&a&, \verb& 1 & for \verb& y(a)&, and \verb& 2 & for \verb& b &. Finally, the program will prompt you to enter $\Delta t$. First try $\Delta t = 0.5$. The figure shows you the graph of the function and the graph of the approximations. Look closely and convince yourself that the approximations are a piecewise linear function. \textsc{MatLab's} Command Window will show you the error $y(2) - \hat{y}(2)$ and prompt you for another $\Delta t$. \begin{enumerate} \item What errors does \textsc{MatLab} give you for $\Delta t = 0.5$, $\Delta t = 0.1$, and $\Delta t = 0.01$?\\ Now let us consider the initial value problem $\frac{dy}{dt} = 2ty$; $y(0) = 1$. It can be easily verified that the solution of this initial value problem is $y(t) = e^{t^2}$. Let us use the program ``Euler'' to find approximations of $y(2)$ for this solution. Follow similar steps as before. Recall that you should enter the right-hand side of your differential equation as:\\ \verb$>> 2*t*y$ \item What errors does \textsc{MatLab} give you for $\Delta t = 0.5$, $\Delta t = 0.1$, and $\Delta t = 0.01$? \item Why are the errors so much bigger than for the previous example? \end{enumerate} \end{document}