ESAM 311-3 Problem Set 6
Due in class on Friday, May 22
Evaluate the following integrals using residues. Please make sure to give a reason for eliminating
the contribution from any large contour.
1.
R x2
0
1+x6
dx
2.
R
x
1+x3
dx
3.
R
4.
R
5.
R x sin x
6.
ESAM 311-3 Problem Set 8
This homework will not be collected. However, a problem like this may appear on the final exam
for the course. These problems are taken directly from Chapter 10 problems on Temperatures
in a quadrant and there are some hints that
ESAM 311-3 Problem Set 7
Due in class on Monday, June 1
Evaluate the following integrals using residues. Please make sure to give a reason for eliminating
the contribution from any large contour.
1.
R x1/3
0
(1+x)2
dx
(Hint: integrate around the appropria
ESAM 311-3 Problem Set 1
Read Chapter 1 of the text and do the following problems.
Note: Solutions must be neatly written and your arguments and the steps of your calculations must be easy for
the grader to follow for full credit. Each student must write
ESAM 311-3 Problem Set 2
Read Chapters 2 and 3 of the text and do the following problems.
Note: Solutions must be neatly written and your arguments and the steps of your calculations must be easy for
the grader to follow for full credit. Each student must
ESAM 311-3 Problem Set 5
Due in class Friday, May 15.
Read Chapters 6 of the text and do the following problems.
Note: Solutions must be neatly written and your arguments and the steps of your calculations must be easy for the grader to follow for full cr
ESAM 311-3 Problem Set 4
Due in class Friday, May 8.
Read Chapters 5 of the text and do the following problems.
Note: Solutions must be neatly written and your arguments and the steps of your calculations must be easy for the grader to follow for full cre
ESAM 311-3 Problem Set 3
Due in class Friday, April 24.
Read Chapter 4 of the text and do the following problems.
1. a) Each of the following equations describes an arc in the complex plane for 0 t 1.
Sketch the arc and indicate its direction of traversal
ES_APPM 322 Homework 4 Solutions
1. Classification of Fixed Points
Identify the type of fixed point and sketch the phase portrait for linear system x = Lx for the 3 cases
1 2
0
2
1
2
L1 =
L2 =
L3 =
1
2
1
5
1
3
Tr
Det
Tr2
4Det
L1
L2
L3
3
5
2
4
2
7
1
17
8
S
322 Applied Dynamical Systems
Syllabus
Class Information
Lectures are TuTh 3:30-5 in M416 (applied math conference room)
Professor: Hermann Riecke, Office Tech M458, [email protected]
TA: Cangjie Xu, Office Tech M463, [email protected]
Applied Nonlinear Dynamics 322
Spring 2016
Hermann Riecke
Problem Set 4
Thursday, April 21, 2016
due Thursday, April 28, 2016
1. Classification of Fixed Points
Identify the type of fixed point and sketch the phase portrait for the linear system
x = Lx
for
Applied Nonlinear Dynamics 322
Spring 2016
Hermann Riecke
Problem Set 1
Thursday, March 31, 2016
due Thursday, April 7, 2016
The numbered problems are taken from Nonlinear Dynamics and Chaos by S.H. Strogatz
(2nd edition).
1. Fixed Points and Stability: 2
Applied Dynamical Systems
Midterm, May 6, 2014
Name:
Spring 2014
Riecke
Score
Instructions:
1. Print your name on this page in the space provided.
2. If you need extra space, use the back of a page
and make a note that you have continued elsewhere.
3. Thi
Applied Nonlinear Dynamics 322
Spring 2016
Hermann Riecke
Problem Set 5
Friday, April 29, 2016
due Tuesday, May 10, 2016
1. Poincare-Bendixson Theorem
Consider the dynamical system
x = y + x3 x x2 + y 2
2
y = x + y 3 y x2 + y
,
2 2
(1)
.
(2)
Perform a lin
Applied Nonlinear Dynamics 322
Spring 2016
Hermann Riecke
Problem Set 2
Thursday, April 7, 2016
due Thursday, April 14, 2016
1. Vector Fields and Bifurcations
For each of the following dynamical systems sketch the bifurcation diagram of the
fixed points a
Applied Nonlinear Dynamics 322
Spring 2016
Hermann Riecke
Problem Set 3
Thursday, April 14, 2016
due Thursday, April 21, 2016
1. Scaling at SNIC
Consider the following dynamical system on a circle
d
= r + sin2n
dt
with n integer. For which value of r doe
Some additional plots for Problem 3, illustrating the dynamics:
Problem 2
(a)
The plots of V and w are for initial conditions at (0, 0).
Different initial conditions converge to the invariant circle.
(b)
Each row shows the resulting phase portrait for a d
Chapter 1
Introduction to spectral theory
1.1
An ODE problem
Before dealing with partial differential equations, lets revisit ordinary differential equations, e.g.,
d xE
D AxE
dt
with x.0/
E
D xE0 :
(1.1.1)
Lets consider the simple case where A is a symme
Chapter 3
Greens Functions and Distribution theory
3.1
Introduction to the Greens function method for solving ODEs1
Consider the 1-D wave equation for a stretched string with harmonic forcing
't t
c 2 'xx D F .x/e
i !t
Lets look for harmonic solutions by
Chapter 6
Regular and singular boundary value problems
Now that weve learned how to use the Greens function to derive eigenfunction expansions and
transforms associated with boundary value problems, its worth revisiting some of the basic theory.
One of th
Chapter 2
The basic differential equations
Before delving into solution methods, its worth motivating things a bit. We do this by considering
some partial differential equations that come up in various applications.
2.1
Electrostatics
Maxwells electromagn
Chapter 4
Inner Products and Adjoints
4.1
Eigenfunction Expansions
The next thing we need to do is better understand the expansions of the type,
f .x/ D
1
X
fn sin.nx/ ;
nD1
fn D
2
f ./ sin.n/ d ;
0
and how they are connected with boundary value problems.
Chapter 5
Spectral Theory of 2nd-order Differential Operators1
5.1
The Greens function method
Recall that one way of looking at an eigenvalue problem for an operator L is to consider the forced
problem
.L C /u D f
with u confined to some domain D by the b
411-1 Final Exam Solutions
1. Consider the half-plane problem
r 2 g D .x x0 / .y y0 / for
y; y0 > 0 and
1 < x; x0 < 1 ;
with gy D 0 on y D 0. In how many ways can this problem be solved using transforms or eigenfunction expansions? (Explain.) Pick one and
411-1 Homework 3
1. What is .x 2
2. What is
a2 / equal to in the sense of distributions?
d
H.x 2
dx
3. Solve x 2
a2 / equal to in the sense of distributions?
du
D 0 in the sense of distributions.
dx
[Hint: first show that a test function f is of the form
411-1 Homework 2
1. The goal of this problem is to develop a general way to determine the specific form of the
Laplacian in different coordinate systems. [Note: you only have to do the parts (a)-(g); the
notes in between these is information given to you.
411-1 Homework 4
1. By modifying the method used in class (or using some other way, such as the one outlined
on page 119 of Stakgold), show that
Z 1
sin kx
lim
f .x/ dx D f .0/:
k!1
x
1
Solution: Heres one way to do this, I think, when f .x/ is a testing
411-1 Homework 1
1. If z D x C iy, D C i and 2 D z, determine and in terms of x and y. (Hint: first eliminate
to find a quadratic equation for 2 in terms of x and y.) This gives a way to calculate complex
square roots without using the polar form. Use it
Fall 2015
ESAM 441-1 PS2
1. The goal of this problem is to develop a general way to determine the specific form of the
Laplacian in different coordinate systems. [Note: you only have to do the parts (a)-(g);
the notes in between these is information given
411-1 Homework 5
1. Use two different spectral representations to find the solution of Laplaces equation for
'.x; y/ in the rectangular region 0 < x < 1, 0 < y < b, with the boundary conditions '.x; 0/ D 0, '.x; b/ D 0 and '.0; y/ D 1. Show from either of