## Description

## Part I (50%)

(1) Exercise 5.10.4.d

(2) Exercise 5.10.7

(3) Exercise 5.11.10

(4) Exercise 5.11.11

## Part II (50%)

Consider the following IVP

(

y

0

(t) = −20y + 20t

2 + 2t, 0 ≤ t ≤ 1;

y(0) = 1/3

with the exact solution y(t) = t

2 + 1/3e

−20t

. Use the time step sizes h = 0.2, 0.125, 0.1, 0.02 for all methods.

Solve the IVP using the following methods

(a) Euler’s method

(b) Runge-Kutta method of order four

(c) Adams fourth-order predictor-corrector method (see ALGORITHM 5.4 p.311)

(d) Milne-Simpson predictor-corrector method which combines the explicit Milne’s method

wi+1 = wi−3 +

4h

3

[2f(ti

, wi) − f(ti−1, wi−1) + 2f(ti−2, wi−2)],

and the implicit Simpson’s method

wi+1 = wi−1 +

h

3

[f(ti+1, wi+1) + 4f(ti

, wi) + f(ti−1, wi−1)].

Compare the results to the actual solution in plots, compute |wi − yi

|, and specify which methods become

unstable. Based on the values of h that were chosen, can you make a statement about the region of absolute

stability for Euler’s method and Runge-Kutta method of order four?

Requirements Submit to CCLE a file lastname_firstname_hw3.zip containing the following files:

• A MATLAB function abm4.m that implements Adams fourth-order predictor-corrector method, a MATLAB function ms.m that implements Milne-Simpson predictor-corrector method, and a MATLAB script

main.m that solves the given IVP and plots the approximated solutions versus the exact one. (Please

include euler.m and rk4.m for completeness.)

• A PDF report that shows the plots and answers the above questions.

1