Math 4501: Numerical Applied Mathematics
		  Reading and Homework Assignment 12
			  Prof. Wickerhauser


You are encouraged to collaborate on homework, and to work additional
exercises from the indicated problem sections, although the homework
grade will be based only on the exercises listed below.  Please upload
your solutions to Canvas/Assignments using GradeScope.  LATE HOMEWORK
WILL NOT BE ACCEPTED.

Check your solutions to problems marked with an asterisk(*) against the
values in the textbook's Answers section, pp.646--672.

Read Chapter 9, Sections 9.4 to 9.9 of the text.

Do Algorithm 13 of Section 9.7, p.526. NOTE: use h=0.1 and M=50


Problem H12.1.  Write a computer program to solve the following initial value
problem on the interval [0,1]:

        y'(t) = 2*t*y(t), 0<t<1;     y(0)=1,

using the fourth-order Runge-Kutta method.  Have the program prompt the user
for the number of steps N, then use h=1/N as the step size.  Make a table of
the values y(1) for N=10, N=100, and N=1000.  Compare the results with the
exact value e=2.71828 18284 59045 23536 02874....



Problem H12.2.  Write a computer program to solve the initial value problem
of H12.1, using the Adams-Bashforth-Moulton predictor-corrector method.  Use
the fourth-order Runge-Kutta method to produce the initial values y_1, y_2,
y_3, f_1, f_2, and f_3.  Make a table of the values y(1) for N=10, N=100, and
N=1000 and compare the results with the exact value e.



Problem H12.3.  Write a computer program to solve the boundary value problem

    x''(t) = 2*t*x'(t)/(1+t*t) - 2*x(t)/(1+t*t) + 1;     x(0)=1, x(4)=-1,

over the interval [0,4].  Use shooting with any method (Euler, Runge-Kutta,
Heun, etc.) and a step size of 0.1 to compute a solution.



Problem H12.4.  Write a computer program to solve the boundary value problem
of Problem H12.3 using the finite difference method.  Choose h=0.1 and
compare the value at t=2 with the one given by the shooting method.