Google Search in:

2250-1 7:30am Lectures Week 14 S2014

Last Modified: April 24, 2014, 11:16 MDT.    Today: December 11, 2017, 00:44 MST.

Week 14: Sections 7.3, 7.4, 9.1, 9.2

 Edwards-Penney, sections 6.2, 6.3, 7.3, 7.4, 9.1, 9.2
  The textbook topics, definitions and theorems
Edwards-Penney 6.1, 6.2 (7.6 K, txt, 19 Dec 2013)
Edwards-Penney 7.1, 7.2, 7.3, 7.4 (25.6 K, txt, 19 Dec 2013)
Edwards-Penney 9.1,9.2,9.3,9.4 (12.1 K, txt, 19 Dec 2013)

Monday and Tuesday: Sections 6.2, 6.3, 7.3

Exam 3 Review
      Sample exam 3, Problem 2
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
Drill Problems
   1. Problem: Given P and D, find A in the relation AP=PD.
   2. Problem: Given Fourier's model, find A.
   3. Problem: Given A, find Fourier's model.
   4. Problem: Given A, find all eigenpairs.
   5. Problem: Given A, find packages P and D such that AP=PD.
   6. Problem: Give an example of a matrix A which has no Fourier's model.
   7. Problem: Give an example of a matrix A which is not diagonalizable.
   8. Problem: Given 2 eigenpairs, find the 2x2 matrix A.
Cayley-Hamilton topics, Section 6.3.
   Power Method
    Computing powers of matrices.
    Stochastic matrices.
    Example of 1984 telecom companies ATT, MCI, SPRINT with discrete
    dynamical system u(n+1)=A u(n). Matrix A is stochastic.
     EXAMPLE:
                    [ 6  1  5 ]               [ a(t) ]
             10 A = [ 2  7  1 ]        u(t) = [ m(t) ]
                    [ 2  2  4 ]               [ s(t) ]

     Meaning: 60% stay with ATT and 20% switch to MCI, 20% switch to SPRINT.
              70% stay with MCI and 20% switch to SPRINT, 10% switch to ATT.
              40% stay with SPRINT and 50% switch to ATT, 10% switch to MCI.
   Google Algorithm
        Lawrence Page's pagerank algorithm, google web page rankings.
  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, not studied in 2250]
   Spectral methods [ch8; not studied in 2250]
 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 7 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 7.4

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 Deprecated 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 (17.2 K, pdf, 15 Dec 2013) 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.
  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 (288.1 K, pdf, 04 Mar 2012)
Slides: Introduction to dynamical systems (158.0 K, pdf, 04 Mar 2012)
Slides: Phase Portraits for dynamical systems (239.3 K, pdf, 04 Mar 2012)
Slides: Stability for dynamical systems (170.8 K, pdf, 04 Mar 2012)
Slides: Nonlinear classification spiral, node, center, saddle (75.3 K, pdf, 12 Dec 2009)
Slides: Matrix Exponential, Putzer Formula, Variation Parameters (130.1 K, pdf, 04 Mar 2012)
References for Eigenanalysis and Systems of Differential Equations.
Slides: Algebraic eigenanalysis (187.6 K, pdf, 04 Mar 2012)
Slides: What's eigenanalysis 2008 (174.2 K, pdf, 04 Mar 2012)
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. (152.9 K, pdf, 04 Mar 2012)
Slides: Laplace resolvent method (88.1 K, pdf, 04 Mar 2012)
Slides: Laplace second order systems (288.1 K, pdf, 04 Mar 2012)
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, 10 Apr 2014)
Text: Lawrence Page's pagerank algorithm (0.7 K, txt, 06 Oct 2008)
Text: History of telecom companies (1.9 K, txt, 04 Apr 2013)
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. (200.2 K, pdf, 04 Mar 2012)
Slides: Intro to Laplace theory. Calculus assumed. (163.0 K, pdf, 19 Mar 2012)
Slides: Laplace rules (160.3 K, pdf, 04 Mar 2012)
Slides: Laplace table proofs (169.6 K, pdf, 04 Mar 2012)
Slides: Laplace examples (149.1 K, pdf, 04 Mar 2012)
Slides: Piecewise functions and Laplace theory (108.5 K, pdf, 03 Mar 2013)
MAPLE: Maple Lab 7. Laplace applications (151.6 K, pdf, 18 Mar 2014)
Manuscript: DE systems, examples, theory (730.9 K, pdf, 10 Apr 2014)
Slides: Laplace resolvent method (88.1 K, pdf, 04 Mar 2012)
Slides: Laplace second order systems (288.1 K, pdf, 04 Mar 2012)
Slides: Home heating, attic, main floor, basement (99.3 K, pdf, 10 Apr 2014)
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, 28 Nov 2010)
Text: Laplace theory problem notes (17.7 K, txt, 18 Mar 2014)
Text: Final exam study guide (8.0 K, txt, 20 Apr 2014)