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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