Math 269B, 2012 Winter, Homework 3
Professor Joseph Teran
Jerey Lee Hellrung, Jr.
February 17, 2012
1
Theory
1. (Strikwerda 5.1.2.) Show that the modied leapfrog scheme (5.1.6) is stable for
0< 1
0 < a2 2
if
satisfying
1
2
and
1
a2 2 < 1.
2
Note that th
Math 269B, 2012 Winter, Homework 3 (Solutions)
Professor Joseph Teran
Jerey Lee Hellrung, Jr.
February 17, 2012
1
Theory
1. (Strikwerda 5.1.2.) Show that the modied leapfrog scheme (5.1.6) is stable for
0< 1
0 < a2 2
if
satisfying
1
2
and
1
a2 2 < 1.
2
Math 269B, 2012 Winter, Homework 4 (Solutions)
Professor Joseph Teran
Jerey Lee Hellrung, Jr.
March 14, 2012
1
Theory
1. Solve the heat equation ut = buxx on a interval I R with periodic boundary conditions. How does
u(t, x)dx vary with time t?
I
Solution
Math 269B, 2012 Winter, Final
Professor Joseph Teran
Jerey Lee Hellrung, Jr.
March 14, 2012
1
Theory
1. Suppose u : [0, ) R R satises the inviscid Burgers equation,
0 = ut +
1 2
u
2
x
= ut + uux ,
u(0, x) = u0 (x).
(1)
Use the method of characteristics to
Math 269B, 2012 Winter, Homework 1 (Solutions)
Professor Joseph Teran
Jerey Lee Hellrung, Jr.
January 30, 2012
1
Theory
1. (Strikwerda 1.1.3.) Solve the initial value problem for
ut +
1
ux = 0
1 + cos x
1
2
Show that the solution is given by u(t, x) = u0
Math 269B, 2012 Winter, Homework 2 (Solutions)
Professor Joseph Teran
Jerey Lee Hellrung, Jr.
February 13, 2012
1
Theory
1. (Strikwerda 2.1.9.) Finite Fourier Transforms. For a function vm dened on the integers, m =
0, 1, . . . , M 1, we can dene the Four
Math 269B, 2012 Winter, Final (Solutions)
Professor Joseph Teran
Jerey Lee Hellrung, Jr.
March 26, 2012
1
Theory
1. Suppose u : [0, ) R R satises the inviscid Burgers equation,
0 = ut +
1 2
u
2
x
= ut + uux ,
u(0, x) = u0 (x).
(1)
Use the method of charac
Math 269B, 2012 Winter, Homework 2
Professor Joseph Teran
Jerey Lee Hellrung, Jr.
February 1, 2012
1
Theory
1. (Strikwerda 2.1.9.) Finite Fourier Transforms. For a function vm dened on the integers, m =
0, 1, . . . , M 1, we can dene the Fourier transform
Math 269B, 2012 Winter, Homework 4
Professor Joseph Teran
Jerey Lee Hellrung, Jr.
February 29, 2012
1
Theory
1. Solve the heat equation ut = buxx on a interval I R with periodic boundary conditions. How does
u(t, x)dx vary with time t?
I
2. (Strikwerda 6.
Math 269B, 2012 Winter, Homework 1
Professor Joseph Teran
Jerey Lee Hellrung, Jr.
January 18, 2012
1
Theory
1. (Strikwerda 1.1.3.) Solve the initial value problem for
ut +
1
ux = 0
1 + cos x
1
2
Show that the solution is given by u(t, x) = u0 (), where is