AME 40510
Homework 12
Due: Wednesday, 4 May 2011, in class
Reading: J. E. Pearson, 1993, Complex patterns in a simple system, Science, 261(5118): 189-192.
1. Consider the Gray-Scott model for reaction-diusion on the domain x [0, 2.5], y [0, 2.5],
t [0, 15

AME 40510
Examination 1
Prof. J. M. Powers
9 March 2011
Happy 111th birthday, Howard Hathaway Aiken (center),
mathematician in uenced by Babbage, who suggested the
use of Hollerith punch cards in computing machines he was
developing at Harvard and IBM, in

AME 40510
Homework 7
Due: Wednesday, 23 March 2011, in class
Reading: Chapter 11
1. Consider the Lorenz equations:
dx
dt
dy
dt
dz
dt
=
(y x),
=
rx y xz,
=
bz + xy,
x(0) = 1,
y (0) = 1,
z (0) = 1.
Using any convenient numerical method for solution of syst

AME 40510
Homework 6
Due: Wednesday, 2 March 2011, in class
Reading: Chapter 5
1. BF, 3a, p. 314.
2. BF, 9, p. 338.
3. BF, 5ab, p. 347.
4. BF, 9, p. 354.
5. Consider the dierence equation
yn+1 = yn hyn ,
y0 = 1 ,
n [0, . . . , 6].
(a) Find the rst order o

AME 40510
Homework 5
Due: Wednesday, 23 February 2011, in class
Reading: Chapter 5
1. Write your own Runge-Kutta-Fehlberg code to integrate a single non-linear ordinary dierential
eqution in fortran or matlab. Use the algorithm in the text to guide you. I

AME 40510
Homework 2
Due: Wednesday, 2 February 2011, in class
Reading: Chapter 4
1. Using the notation from the course notes, show that
= (1 )1/2 ,
hD = 2sinh1
1
.
2
2. BF, 11b, p. 183. Also, plot on log-log paper the error in approximating df /dx at x

AME 40510
Homework 3
Due: Wednesday, 9 February 2011, in class
Reading: Chapter 4
1. BF, 24, p. 212.
2. BF, 5, p. 219, compute only for 1d.
3. BF, 5d, p. 227. For the Simpsons rule portion, plot on log-log paper the error in the approximation as a functio

AME 40510
Homework 4
Due: Wednesday, 16 February 2011, in class
Reading: Chapter 5
1. BF, 6a, p. 274; additionally, plot on log-log coordinates the error in y at t = 1 as a function
of h for several orders of magnitude of h. Drive your solution to machine

AME 40510
Homework 8
Due: Wednesday, 30 March 2011, in class
Reading: Chapter 12
1. BF, 5a, p. 736.
2. Repeat the previous problem letting h vary over as many orders of magnitude for which
computational results are practical. For each value of h, employ a

AME 40510
Homework 9
Due: Wednesday, 6 April 2011, in class
Reading: Chapter 12
1. Numerically solve the linear advection equation
u
u
+a
= 0,
t
x
u(x, 0) = 1 + H (x + 1) H (x 1).
for a = 1, x [5, 5], t [0, 1]. Here H ( ) is the Heaviside step function, H

1) Discretizing the advection equation using three different schemes we have:
Upwind:
uin1 uin
u n uin1
a i
0
t
x
Lax-Friedrichs:
uin 1
1n
ui1 uin1 uin1 uin1
2
a
0
t
2x
Lax-Wendroff:
1n
ui uin1 uin1 uin
2
1)
a
0
1
x
t
2
uin122 uin122
uin 1 uin
1
1
2)
a

AME 40510
Homework 11
Due: Wednesday, 20 April 2011, in class
Reading: Chapter 6, 7, 12
1. BF, 6 (consider only 2d), p. 460;
2. BF, 9, p. 460.
3. BF, 1, p. 723; Let h = k , and let both vary over a few orders of magnitude. Plot the maximum
error in the so

AME 40510
Homework 1
Due: Wednesday, 26 January 2011, in class
Reading: Chapter 3
1. Take the dimension of the matrix A to be N
to initialize all of the elements of the matrix.
product ij . So if N = 3,
1
A = 2
3
N and use a variety of software programs