Computer Assignment 1, Math 222, Fall 2002

Euler and Improved Euler Methods

You may use whatever language or software you wish to complete this assignment, but it is strongly recommended that you use MATLAB. If you need assistance with MATLAB, there are two TAs, George Gaines and Yuriy Mileyko who can help you. They are available in Room 48 of the Student Mall (under the parking deck) four afternoons per week as follows: Monday 3-5 pm (Gaines); Tuesday 2:30-4:30 pm (Mileyko); Thursday 2-4 pm (Mileyko); Friday 2-4 pm (Gaines). These hours are also posted on the door of Room 48 and will be posted on the Department website shortly.

(a) Find the exact solution y = y(t) of the IVP

(1-t)dy/dt = y ², y(0) = y₀, y₀ ³0.

(b) Write programs to implement the Euler and Improved Euler methods to obtain approximate numerical solutions_.y_n and Y_n , respectively, of the IVP in (a) for t ³ 0. Your program should save t_n, y(t_n), y_n, Y_n and the approximation errors e_n = ½y_n - y(t_n) ½ and E_n = ½ Y_n - y(t_n) ½.

(c) Produce graphs like figures 6.1.6 and 6.2.9 in the textbook for each of the two methods for the t variable ("time") steps h = 1/N with N = 10 and N = 100 for t ³ 0 for each of the initial conditions (i) y₀ = 0; (ii) y₀ = 0.1; and (iii) y₀ = 1.0. There should be altogether twelve graphs.

(d) Graph the errors e_n and E_n versus t_n for the approximation errors for each of the cases considered in (c). You should produce a graph with two labeled error curves for each method and each case. Do your graphs behave badly in any regions? Explain why the bad behavior, if any, is to be expected or not.

Computer Assignment #2 in PDF file below.