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.
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