$30.00
Description
Part I (50%)

Exercise 5.10.4.d

Exercise 5.10.7

Exercise 5.11.10

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

Euler’s method

RungeKutta method of order four

Adams fourthorder predictorcorrector method (see ALGORITHM 5.4 p.311)

MilneSimpson predictorcorrector method which combines the explicit Milne’s method
4h
w_{i+1} = w_{i} _{3} + _{3} [2f(t_{i}; w_{i}) f(t_{i} _{1}; w_{i} _{1}) + 2f(t_{i} _{2}; w_{i} _{2})];
and the implicit Simpson’s method
h
^{w}i+1 ^{=} ^{w}i 1 ^{+} _{3} ^{[f(t}i+1^{; w}i+1^{) + 4f(t}i^{; w}i^{) +} ^{f(t}i 1^{; w}i 1^{)]:}
Compare the results to the actual solution in plots, compute jw_{i} y_{i}j, 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 RungeKutta 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 fourthorder predictorcorrector method, a MATLAB function ms.m that implements MilneSimpson predictorcorrector 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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