# 2250 8:05am Lectures Week 14 S2016

Last Modified: January 08, 2016, 12:27 MST.    Today: September 23, 2018, 14:02 MDT.

### Week 14: Sections Ch6 and Ch9

``` Edwards-Penney, sections 6.1, 6.2, 6.3, 6.4,7.3, 7.4, 9.1, 9.2, 9.3, 9.4
The textbook topics, definitions and theoremsEdwards-Penney 6.1, 6.2, 6.3, 6.4 (11.8 K, txt, 05 Apr 2015)Edwards-Penney 9.1,9.2,9.3,9.4 (0.0 K, txt, 31 Dec 1969)```

## Monday and Tuesday: Sections 6.2, 6.3, 7.3

```Exam 3 Review
Sample exam 3
Diagonalization Theory
In the case of a 2x2 matrix A,
FOURIER'S MODEL is
A(c1 v1 + c2 v2) = c1(lambda1 v1) + c2(lambda2 v2)
where v1,v2 are a basis for the plane
equivalent to DIAGONALIZATION
AP=PD, where D=diag(lamba1,lambda2), P=augment(v1,v2),
where det(P) is not zero
equivalent to EIGENPAIR EQUATIONS
A(v1)=lambda1 v1,
A(v2)=lambda2 v2,
where vectors v1,v2 are independent
```
```  Methods to solve dynamical systems
Consider the 2x2 system
x'=x-5y, y'=x-y, x(0)=1, y(0)=2.
Cayley-Hamilton-Ziebur method.
Laplace resolvent.
Eigenanalysis method.
Exponential matrix using maple
Putzer's method to compute the exponential matrix, slides
Spectral methods [not studied in 2280]
```
``` Survey of Methods for solving a 2x2 dynamical system
1. Cayley-Hamilton-Ziebur method for u'=Au
Solution: u(t)=(atom_1)vec(d_1)+ (atom_2)vec(d_2)
Atoms: They are constructed by Euler's theorem from roots of det(A-rI)=0
Vectors vec(d_1),vec(d_2) are found from the equation
[d1 | d2]=[u(0) | Au(0)](W(0)^T)^(-1)
where W(t) is the Wronskian matrix of the two atoms.
2. Laplace resolvent L(u)=(s I - A)^(-1) u(0)
See slides for details about the resolvent equation.
3. Eigenanalysis  u(t) = exp(lambda_1 t) v1 + exp(lambda_2 t) v2
See chapter 5 in Edwards-Penney for examples and details.
This method fails when matrix A is not diagonalizable.
EXAMPLE. Solve a homogeneous system u'=Au, u(0)=vector([1,2]),
A=matrix([[2,3],[0,4]]) using
Zeibur's method, Laplace resolvent and eigenanalysis.
```

## Tuesday and Wednesday: Second Order Systems. Section 5.4

```Exam 3 Review
Sample exam 3, Problem 3
Eigenvalues
A 4x4 matrix.
Block determinant theorem.
Eigenvectors for a 4x4.
B:=matrix([[5,0,0,0],[0,5,0,0],[0,0,0,3],[0,0,-3,0]]);
lambda=5,5,3i,-3i
v1=[1,0,0,0], v2=[0,1,0,0], v3=[0,0,i,-1], v4=[0,0,i,1]
One panel for lambda=5
First frame is A-5I with 0 appended
Find rref
Apply last frame algorithm
Scalar general solution
Take partials on t1, t2 to find v1,v2
Eigenpairs are (5,v1), (5,v2)
One panel for lambda=3i
Same outline as lambda=5
Get one eigenpair (3i,v3)
Other eigenpair=(-3i,v4) where v4 is the conjugate of v3.
Final exam: Second shifting theorem in Laplace theory.
Second Order Systems
How to convert mx''+cx'+kx=F0 cos (omega t) into a
dynamical system  u'=Au+F(t).
Electrical systems u'=Au+E(t) from LRC circuit equations.
Electrical systems of order two: networks
Mechanical systems of order two: coupled systems
Second order systems u''=Au+F
Examples are railway cars, earthquakes,
vibrations of multi- component systems,
electrical networks.
Second Order Vector-Matrix Differential Equations
The model u'' = Ax + F(t)
Coupled Spring-Mass System. Problem 7.4-6
A:=matrix([[-6,4],[2,-4]]); eigenvals(A);
lambda1= -2, lambda2= -8
Ziebur's Method
roots for Ziebur's theorem are plus or minus sqrt(lambda)
Roots = sqrt(2)i,  sqrt(8)i, -sqrt(2)i, -sqrt(8)i
Atoms = cos (sqrt(2)t), sin(sqrt(2)t), cos(sqrt(8)t), sin(sqrt(8)t)
Vector x(t) = vector linear combination of the above 4 atoms
Maple routines for second order
de1:=diff(x(t),t,t)=-6*x(t)+4*y(t); de2:=diff(y(t),t,t)=2*x(t)-4*y(t);
dsolve({de1,de2},{x(t),y(t)});
x(t) = _C1*sin(sqrt(2)*t)+_C2*cos(sqrt(2)*t)+_C3*sin(2*sqrt(2)*t)+_C4*cos(2*sqrt(2)*t),
y(t) = _C1*sin(sqrt(2)*t)+_C2*cos(sqrt(2)*t)-(1/2)*_C3*sin(2*sqrt(2)*t)-(1/2)*_C4*cos(2*sqrt(2)*t)}
Eigenanalysis method section 7.4
u(t) = (a1 cos(sqrt(2)t) + b1 sin(sqrt(2)t)) v1 + (a2 cos(sqrt(8)t) + b2 sin(sqrt(8)t)) v2
where (-2,v1), (-8,v2) are the eigenpairs of A.  The two vector terms in u(t) are called
the natural modes of oscillation. The natural frequencies are sqrt(2), sqrt(8).
Eigenanalysis of A gives v1=[1,1], v2=[2,-1].
Railway cars. Problem 7.4-24
Cayley-Hamilton-Ziebur method
Laplace Resolvent method for second order
Eigenanalysis method section 7.4
```
```Some Extra Topics
Putzer's method for the 2x2 matrix exponential.
Solution of u'=Au is: u(t) = exp(A t)u(0)
THEOREM: exp(A t) = r1(t) I + r2(t) (A-lambda_1 I),
Lambda Symbols: lambda_1 and lambda_2 are the roots of det(A-lambda I)=0.
The DE System:
r1'(t) = lambda_1 r1(t),         r1(0)=0,
r2'(t) = lambda_2 r2(t) + r1(t), r2(0)=0
See the slides and manuscript on systems for proofs and details.
THEOREM. The formula can be used as
e^{r1 t} - e^{r2 t}
e^{At} = e^{r1 t} I  +  ------------------- (A-r1 I)
r1 - r2
where r1=lambda_1, r2=lambda_2 are the eigenvalues of A.

EXAMPLE. Solve a homogeneous system u'=Au, u(0)=vector([1,2]),
A=matrix([[2,3],[0,4]]) using the matrix exponential,
Zeibur's method, Laplace resolvent and eigenanalysis.
EXAMPLE. Solve a non-homogeneous system u'=Au+F(t), u(0)=vector([0,0]),
A=matrix([[2,3],[0,4]]), F(t)=vector([3,1]) using variation
of parameters.
```

## Friday and Monday: Dynamical Systems. Sections 9.1, 9.2

```Extra Credit Maple Project: Tacoma narrows. Explore an alternative
explanation for what caused the bridge to fail, based on the hanging cables.
MAPLE: Maple Lab 9. Tacoma Narrows (0.0 K, pdf, 31 Dec 1969)
Extra Credit Maple Project: Earthquakes. Explore a 5-story or 7-story building
and the resonant frequencies of oscillation of the building which might make it destruct
during an earthquake. See Edwards-Penney, application section in 7.4.
```
```Dynamical Systems Topics
Equilibria.
Stability.
Instability.Wednesday-Friday: Fourier's History. Rod Problem. Fourier Series. Section 9.1, 9.2
Chapter 9 Edwards-Penney BVP textbook
Fourier Series Methods

9.1 Periodic Functions and Trigonometric Series

DEF. Periodic Function. f(t+p)=f(t) for all t. p=period.

Orthogonality Relations

DEF.  Two functions u, v are said to be othogonal on [a,b] provided integral(u*v,a..b)=0.
A list of functions is said to be othogonal on [a,b] provided any two of them are orthgonal on [a,b].

THEOREM. The trigonometric list of sin(nt), cos(mt), n=1..infinity, m=0..infinity, is orthogonal on [-Pi,Pi].
The trigonometric list is independent on [-Pi,Pi], because it is a list of Euler atoms.

DEF. A Fourier series is a formal sum of trigonometric terms from the trig list.
A Fourier sine series is a Fourier series with no cosine terms.
A Fourier cosine series is a Fourier series with no sine terms.

Fourier Coefficient Formulas

Let f(x) be defined on [-Pi,Pi]. Define

a[m] = (1/Pi)*integral(f(t)*cos(mt),-Pi..Pi), m=0..infinity

b[n] = (1/Pi)*integral(f(t)*sin(nt),-Pi..Pi), n=1..infinity

Classical Fourier Series

(1/2)*a[0] + SUM( a[m]*cos(m*x), m=1..infinity)  +  SUM( b[n]*sin(n*x), n=1..infinity)

THEOREM. The formulas for a[m], b[n] need not be memorized. They arise from one idea:

(2)  Multiply the equation in (1) by one trigonometric atom. Integrate over [-Pi,Pi].
(3)  Orthogonality implies that the integrated series has exactly one nonzero term!
Divide to find the corresponding coefficient a[m] or b[n].

DEF:   = integral(u*v,a..b). It has these INNER PRODUCT properties. The vector space V
together with these properties is called an INNER PRODUCT SPACE.

(1)  =  +   linear in the first argument
(2)  = c*
(3) =  symmetry
(4)  = 0 if and only if u=0

DEF:  On inner product space V, the NORM is defined by |u| =sqrt(),
or equivalently, |u|^2 = .

DEF: A VECTOR is a package in a set V. Set V, called a VECTOR SPACE, is equipped with
addition and scalar multiplication, such that the two closure laws hold and the 8 properties
are valid (group under addition, scalar disribution laws).

THEOREM. Let  n=1..infinity be a list of orthogonal functions on [a,b].
Let f = SUM(c[n]*f[n],n=1..infinity). Then

c[n] = / = integral(f*f[n],a..b) / integral(f[n]*f[n],a..b)

EXAMPLE. Find a[m], b[n] for the square wave
f(x) = -1 on (-Pi,0), f(x) = 1 on (0,Pi), f(x)=0 for x=-Pi,0,Pi.
Plot the Fourier series F(x) of f(x) on -2Pi to 2Pi.

ANSWER. a[m]=0 for all m, because f(x) is odd.
b[n] = 4/(n*Pi) for n odd
b[n] = 0 for n even
GIBB's OVERSHOOT.

At discontinuities of f(x), F(x) has a strange behavior, called Gibb's Overshoot.
This can be seen by plotting a truncated Fourier series near discontinuities of f(x).

EXAMPLE.  Find the Fourier series of f(x) on [-Pi,Pi], where f(x) = x*pulse(x,0,Pi)
except that f(x)=Pi/2 at x=Pi and x=-Pi.
a[m] = [(-1)^m-1]/(m^2*Pi^2)  for m=1..infinity
b[n] = (-1)^n(-1)/n  for n=1..infinity

9.2  Fourier Convergence Theorem

THEOREM. Let f(x) be smooth on [-Pi,Pi] and F(x) its formal Fourier series,
built with the Fourier coefficient formulas.
Then f(x) = F(x) for all x in [-Pi,Pi].

THEOREM. The convergence theorem above continues to hold if f(x) is only piecewise smooth,
but the equation f(x) = F(x) only holds at points of continuity of f(x).
At other points, there is the equation (f(x+)+f(x-))/2  =  F(x).

THEOREM.  The series convergence is uniform if f(x) is smooth. It is not uniform for the Gibb's example.

Slides on Dynamical Systems
Manuscript: Systems theory and examples (730.9 K, pdf, 10 Apr 2014)   Slides: Laplace second order systems, spring-mass,boxcars, earthquakes (273.7 K, pdf, 14 Mar 2016)   Slides: Introduction to dynamical systems (144.9 K, pdf, 14 Mar 2016)   Slides: Phase Portraits for dynamical systems (221.2 K, pdf, 14 Mar 2016)   Slides: Stability for dynamical systems (158.2 K, pdf, 14 Mar 2016)   Slides: Nonlinear classification spiral, node, center, saddle (97.9 K, pdf, 14 Mar 2016)   Slides: Matrix Exponential, Putzer Formula, Variation Parameters (122.0 K, pdf, 14 Mar 2016)
Asymptotic stability.
Classification of equilibria for u'=Au when
det(A) is not zero, for the 2x2 case.
Impact of Cayley-Hamilton-Ziebur on classification

Slides on Dynamical Systems
Manuscript: Systems theory and examples (730.9 K, pdf, 10 Apr 2014)   Slides: Laplace second order systems, spring-mass,boxcars, earthquakes (273.7 K, pdf, 14 Mar 2016)   Slides: Introduction to dynamical systems (144.9 K, pdf, 14 Mar 2016)   Slides: Phase Portraits for dynamical systems (221.2 K, pdf, 14 Mar 2016)   Slides: Stability for dynamical systems (158.2 K, pdf, 14 Mar 2016)   Slides: Nonlinear classification spiral, node, center, saddle (97.9 K, pdf, 14 Mar 2016)   Slides: Matrix Exponential, Putzer Formula, Variation Parameters (122.0 K, pdf, 14 Mar 2016)
References for Eigenanalysis and Systems of Differential Equations.
Slides: Algebraic eigenanalysis (173.4 K, pdf, 14 Mar 2016) Slides: What's eigenanalysis 2008 (161.5 K, pdf, 14 Mar 2016) Slides: What's eigenanalysis, draft 1 (152.2 K, pdf, 01 Apr 2008) Slides: What's eigenanalysis, draft 2 (124.0 K, pdf, 14 Nov 2007) Slides: Cayley-Hamilton-Ziebur method for solving vector-matrix system u'=Au. (137.4 K, pdf, 14 Mar 2016) Slides: Laplace resolvent method (81.0 K, pdf, 14 Mar 2016) Slides: Laplace second order systems (273.7 K, pdf, 14 Mar 2016) Manuscript: Systems of DE examples and theory (730.9 K, pdf, 10 Apr 2014) Slides: Home heating, attic, main floor, basement (99.3 K, pdf, 14 Mar 2016) Text: Lawrence Page's pagerank algorithm (0.0 K, txt, 31 Dec 1969) Text: History of telecom companies (0.0 K, txt, 31 Dec 1969)
Systems of Differential Equations applications
Slides: Cable hoist example (73.2 K, pdf, 21 Aug 2008) Slides: Sliding plates example (105.8 K, pdf, 21 Aug 2008)
Extra Credit Maple Project: Tacoma narrows. Explore an alternative
explanation for what caused the bridge to fail, based on the hanging cables.
MAPLE: Maple Lab 9. Tacoma Narrows (0.0 K, pdf, 31 Dec 1969)
Laplace theory references
Slides: Laplace and Newton calculus. Photos. (188.3 K, pdf, 14 Mar 2016) Slides: Intro to Laplace theory. Calculus assumed. (144.9 K, pdf, 13 Mar 2016) Slides: Laplace rules (144.9 K, pdf, 14 Mar 2016) Slides: Laplace table proofs (151.9 K, pdf, 14 Mar 2016) Slides: Laplace examples (133.7 K, pdf, 14 Mar 2016) Slides: Piecewise functions and Laplace theory (98.5 K, pdf, 14 Mar 2016) MAPLE: Maple Lab 7. Laplace applications (0.0 K, pdf, 31 Dec 1969) Manuscript: DE systems, examples, theory (730.9 K, pdf, 10 Apr 2014) Slides: Laplace resolvent method (81.0 K, pdf, 14 Mar 2016) Slides: Laplace second order systems (273.7 K, pdf, 14 Mar 2016) Slides: Home heating, attic, main floor, basement (99.3 K, pdf, 14 Mar 2016) Slides: Cable hoist example (73.2 K, pdf, 21 Aug 2008) Slides: Sliding plates example (105.8 K, pdf, 21 Aug 2008) Manuscript: Heaviside's method 2008 (352.3 K, pdf, 07 Jan 2014) Manuscript: Laplace theory 2008 (497.3 K, pdf, 19 Mar 2014) Transparencies: Ch10 Laplace solutions 10.1 to 10.4 (1068.7 K, pdf, 22 Feb 2015) Text: Laplace theory problem notes (0.0 K, txt, 31 Dec 1969) Text: Final exam study guide (0.0 K, txt, 31 Dec 1969)

```