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) = x.
2
Will have du/dt = u if dx/dt = 2xt; that is, if
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 breaking time t and the position of the shock
for t > t
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 values of a does this problem have a unique solution?
By
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, y, 0) = (x, y ), ut (x, y, 0) = (x, y ),
0 x a, 0 y b
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) sin(nx) dx +
0
1
sin(x) sin(nx) dx
0
sin(x) sin(nx) d
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 (x) = exp(sin x).
Expanding in a Taylor series, we have
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 .
Since du/dt = ut + ux (dx/dt)+ uy (dy/dt), will have
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) = sin(x).
Since du/dt = ut + ux (dx/dt), will have du/dt =
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 MAY NOT
use: ANYTHING ELSE; e.g., no other books or in