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## Description

• Part I (50%)

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

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

1. Exercise 5.10.4 a,b,c,d

1. Exercise 5.4.30.

1. 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.

1. 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)?

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

1. 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.

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