Math Homework #2 Solution

Description

 

• Part I (50%)

 

• Show that the Modi ed Euler method is of order two.

 

• Use Theorem 5.20 to show that the Runge-Kutta method of order four is consistent.

 

• Exercise 5.10.4 a,b,c,d

 

• Exercise 5.4.30.

 

• Exercise 5.4.32.

 

• Part II (50%)

 

Consider the following well-posed IVP:

 

8	y0(t) = 1 +	y	;  1	t  2;	(1)
		t
<

 

• y(1) = 2;

 

with the exact solution y(t) = t ln t + 2t. Choose the step sizes h = 0:2; 0:1; 0:05, respectively.

 

• Use Taylor’s method of order two to approximate the solution. Discuss the behavior of the approximated solution as a function of h, and compare it with the exact solution in plots of t versus y. Estimate the

 

order of the method from the error. Which value of h do you need to choose (approximately) to achieve an accuracy of 10 ⁴ for y(2)?

 

• Use Midpoint method (p.286) to redo Part (a).

 

• Compare the results and running times¹ of Part (a) and (b). What does the comparison of error and running time tell us about the e ciency of the two methods?

 

Requirements

 

Submit the code le to CCLE : A MATLAB (or other software) function taylor2.m that implements Taylor’s method of order two, a MATLAB function (or other software) midpt.m that implements Mid-point method, and a MATLAB (or other software) script main.m that solves the IVP (1) and plots the approximated solutions versus the exact one.

 

Print a PDF report to your TA.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

• tic and toc can be used to record the running time. See http://www.mathworks.com/help/matlab/ref/tic.html and http: //www.mathworks.com/help/matlab/ref/tic.html for more details.

 

 

 

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