Math 472: Computer Assignment 2 due Monday, Oct.24, 2005
1. In the 1968 Olympic games in Mexico City, Bob Beamon established a world record with a
long jump of 8.90 meters. This was 0.80 meters longer than the previous world record. Since
1968, Beamons ju
Math 472: Computer Assignment 1 due Monday, Oct. 3, 2005
1. Consider the following system of second-order initial-value problems:
x(t)
,
+ y (t)2 )3/2
y (t)
,
y (t) =
2 + y (t)2 )3/2
(x(t)
x (t) =
(x(t)2
x(0) = 1, x (0) = 0,
(1)
y (0) = 0, y (0) = 1.
(2
Math 472: Computer Assignment 6 due Wednesday, Dec.7, 2005
1. (a) Use the linear transformation from Problem 2 in Assignment 8 to modify the Matlab program PSBVP.m so that you can solve linear 2-pt BVPs on arbitrary intervals [a, b] with
boundary conditio
Math 472: Computer Assignment 5 due Wednesday, Nov.30, 2005
1. Execute the command plot(sin(1:3000),.) in Matlab. What do you see? What does this
have to do with aliasing? Give a quantitative answer, explaining exactly what frequency is being
aliased by y
Math 472: Computer Assignment 4 due Monday, Nov.21, 2005
1. Write a Matlab function [t,y] = bvpsolve(u,v,w,a,b,alpha,beta,m) to solve a linear twopoint boundary value problem of the form
y (t) = u(t) + v (t)y (t) + w(t)y (t)
y (a) = ,
y (b) =
with the ni
Math 472: Computer Assignment 3 due Monday, Nov.7, 2005
1. Consider the IVP
y (t) = y (t) + 2et cos(2t),
y (0) = 0
et sin(2t).
with exact solution y (t) =
(a) Write a Matlab program based on Eulers method that uses Richardson extrapolation to
obtain a met
Math 472: Assignment 1 due Wednesday, Sept. 14, 2005
2
1. Show that the function f (t, x) = x2 et sin t is Lipschitz continuous for x [0, 2].
2. (a) Approximate the function f (x) = ex/2 over the interval [1, 9] by a fourth-degree polynomial
in two ways:
Math 472: Assignment 2 due Monday, Sept. 26, 2005
1. Present a detailed discussion of the end of the proof of convergence of Eulers method (equations
(7) and (8).
2. Do Exercise 1.1 in the textbook.
3. Do Exercise 1.4 in the textbook.
Math 472: Assignment 7 due Wednesday, Nov. 30, 2005
1. Provide the details of the case j = 0 in the formula
Dj 0 =
d
t
sinc
dt
h
=
t=tj =jh
0,
j=0
(1)j
jh ,
otherwise,
for the entries in the k = 0 column of the dierentiation matrix D on unbounded grids.
2
Math 472: Assignment 8 due Wednesday, Dec. 7, 2005
(2)
2
1. In the periodic case, determine the Fourier dierentiation matrices DN , DN , and DN for N = 2
(2)
2
and N = 4, and conrm that in both cases DN = DN .
2. Find the linear transformation that is nee
Math 472: Assignment 6 due Monday, Nov. 14, 2005
1. Verify that the function y (t) = c sin t is a solution of the boundary value problem
y (t) + y (t) = 0
y (0) = 0,
y ( ) = 0
for any constant c. Comment.
2. Find the solution at t =
1
2
of the linear two-
Math 472: Assignment 5 due Monday, Oct. 31, 2005
1. Do Exercise 4.4 in the textbook.
2. Do parts (b) and (c) of Exercise 5.3 in the textbook.
Misprint: The formulas for part (a) should read
y (tn+1 ) y n+1 =
y (tn+1 ) xn+1 =
3. Do Exercise 5.4 in the tex
Math 472: Assignment 3 due Monday, Oct. 10, 2005
1. Do Exercise 2.1 (for the two three-step methods only) in the textbook.
2. Which of the following multistep methods is convergent?
(a) y n+2 y n = h [f (tn+2 , y n+2 ) 3f (tn+1 , y n+1 ) + 4f (tn , y n )]
Math 472: Assignment 4 due Monday, Oct. 17, 2005
1. Do Exercise 2.6 in the textbook.
2. Do Exercise 3.4 in the textbook.
3. By considering the scalar equation y (t) = f (t), i.e., f is independent of y , show that in this case
the classical fourth-order R