Math Homework #3 Solution

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

(

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

• Euler’s method

• Runge-Kutta method of order four

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

• Milne-Simpson predictor-corrector 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

^wi+1 ^{= w}i    1 ⁺ ₃ ^[f(ti+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_ij, 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.

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