Math Homework #3 Solution

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Description

Part I (50%)

  1. Exercise 5.10.4.d

  1. Exercise 5.10.7

  1. Exercise 5.11.10

  1. Exercise 5.11.11

Part II (50%)

Consider the following IVP

(

y0(t) =

20y + 20t2

+ 2t;

0

t

1;

y(0) =

1=3

with the exact solution y(t) = t2 + 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

  1. Euler’s method

  1. Runge-Kutta method of order four

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

  1. Milne-Simpson predictor-corrector method which combines the explicit Milne’s method

4h

wi+1 = wi 3 + 3 [2f(ti; wi) f(ti 1; wi 1) + 2f(ti 2; wi 2)];

and the implicit Simpson’s method

h

wi+1 = wi 1 + 3 [f(ti+1; wi+1) + 4f(ti; wi) + f(ti 1; wi 1)]:

Compare the results to the actual solution in plots, compute jwi yij, 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 le lastname_firstname_hw3.zip containing the following les:

A MATLAB function abm4.m that implements Adams fourth-order predictor-corrector method, a MAT-LAB 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