Advanced Methods for Partial Differential Equations
AMATH 569

Spring 2012
AMath 569, Spring 2013
Sample Solutions for Practice Problems for Midterm, Mon., May 6.
Covers: Chapters 13 in the Notes or text.
1. Use the method of characteristics to solve
ut + 2xtux = u,
u(x, 0)
Advanced Methods for Partial Differential Equations
AMATH 569

Spring 2012
AMath 569, Spring 2013
Sample Solutions for Midterm, Mon., May 6.
1. Use the method of characteristics to solve
ut + uux
0,
x,
u(x, 0) =
0
= 0,
x<0
0x1
x>1
Sketch the characteristics, determine the b
Advanced Methods for Partial Differential Equations
AMATH 569

Spring 2012
AMath 569, Spring 2013
Partial Solutions for Assignment 7.
Reading: Chapter 7 in the Notes or text.
1. Exercise 1 on p. 141 in the Notes.
u + a2 u = f (x), 0 < x < 1,
u(0) = u(1) = 0.
(a) For which va
Advanced Methods for Partial Differential Equations
AMATH 569

Spring 2012
AMath 569, Spring 2013
Sample Solutions for Assignment 5.
Reading: Chapter 4 in the Notes or Secs. 4.14.2 in the text.
1. Exercise 1 on p. 77 in the Notes.
utt = uxx + uyy ,
0 < x < a, 0 < y < b,
u(x
Advanced Methods for Partial Differential Equations
AMATH 569

Spring 2012
AMath 569, Spring 2013
Sample Solutions for Assignment 4.
Reading: Chapter 3 in the Notes or text.
1. Do exercise 1 on p. 61 of the text.
The Fourier coecients for  sin x are
1
1
=
1
=
0
bn =
sin(x
Advanced Methods for Partial Differential Equations
AMATH 569

Spring 2012
AMath 569, Spring 2013
Sample Solutions for Assignment 3.
Reading: Section 3.1 in the Notes or Sections 3.13.3 in the text.
1. Exercise 1 on p. 61 of the Notes.
The function that I came up with is
f
Advanced Methods for Partial Differential Equations
AMATH 569

Spring 2012
AMath 569, Spring 2013
Sample Solutions for Assignment 2.
Due Friday, Apr. 19.
Reading: Chapter 2 in the Notes.
1. Exercise 8, p. 31.
ut + c1 (u)ux + c2 (u)uy = 0
u(x, y, 0) = u0 (x, y ),
(x, y ) R2 .
Advanced Methods for Partial Differential Equations
AMATH 569

Spring 2012
AMath 569, Spring 2013
Sample Solutions for Assignment 1.
Due Friday, Apr. 12.
Reading: Ch. 1 of text and Ch. 1 of Notes. Sec. 2.1 in the Notes.
1. Exercise 1 on p. 29.
ut + tuux = 0, x R
u(x, 0) = si
Advanced Methods for Partial Differential Equations
AMATH 569

Spring 2012
AMath 569, Spring 2013
TakeHome Final.
Due Monday, June 10, by 5pm. You may use: The textbook, course notes, your own notes
from class and solutions to homework problems from the course web page. You