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2250-1 7:30am Lecture Record Week 13 S2012

Last Modified: April 10, 2012, 13:04 MDT.    Today: October 23, 2017, 18:16 MDT.

Week 13, April 9 to 13: Sections 6.1,6.2,7.2, 7.3

Thu, April 12: Exam 3, part I: problems 1,2

Topics from chapter 10, EPbvp7.6. Problems 1,2 are on Laplace theory,
but there is some contact with chapter 5, sections 5.1 to 5.4. See the online
sample exam for details. Problems 3,4,5 will be on April 19.

Sample Exam 3 for S2012
   
PDF: sample exam 3, all five problems. (182.4 K, pdf, 14 Nov 2010)

Mon and Tue, Apr 9-10: Sections 6.1, 6.2, 7.1

REVIEW from Last Week
Conversion Methods to Create a First Order System
    The position-velocity substitution.
    How to convert second order systems.
    How to convert nth order scalar differential equations.
    Non-homogeneous terms and the vector matrix system
         u' = Au + F(t)
    Non-linear systems and the vector-matrix system
         u' = F(t,u)
    Answer checks for u'=Au
      Example: The system u'=Au, A=matrix([[2,1],[0,3]]);   

Systems of two differential equations
   The Laplace resolvent method for systems.
        Solving the resolvent equation for L(x), L(y).
          Cramer's Rule
          Matrix inversion
          Elimination
   Example: Solving a 2x2 dynamical system
     Study of u'=Au, u(0)=vector([1,2]), A=matrix([[2,1],[0,3]]).
     Dynamical system scalar form is
         x' = 2x + y,
         y' = 3y,
         x(0)=1, y(0)=2.
        The equations for L(x), L(y)
             (s-2)L(x)  +  (-1)L(y)=1,
                (0)L(x)  + (s-3)L(y)=2
        REMARK: Laplace resolvent method shortcut.
        How to solve the [resolvent] equations for L(x), L(y).
          Cramer's Rule
          Matrix inversion
          Elimination
          Answers: L(x) = delta1/delta, L(y)=delta2/delta
                       delta=(s-2)(s-3), delta1=s-1, delta2=2(s-2)
                       L(x) = -1/(s-2)+2/(s-3), L(y)=2/(s-3)
        Backward table and Lerch's theorem
        Answers: x(t) = - e^{2t} + 2 e^{3t},
                     y(t) = 2 e^{3t}.
      Edwards-Penney Shortcut Method in Example 5, 7.1. Uses Chapter 1+5 methods.
      This is the Cayley-Hamilton-Ziebur method. See below.
        Solve w'+p(t)w=0 as w = constant / integrating factor.
        Then  y' -2y=0 ==> y(t) = 2 exp(3t)
        Stuff y(t) into the first DE to get the linear DE
           x' - 2x = 2 exp(3t)
        Superposition: x(t)=x_h(t)+x_p(t),
           x_h(t)=c exp(2t),
           x_p(t) = d1 exp(t) = 2 exp(3t) by undetermined coeff.
        Then x(t)= - exp(2t) + 2 exp(3t).

Cayley-Hamilton Theorem
   A matrix satisfies its own characteristic equation.
   ILLUSTRATION: det(A-r I)=0 for the previous example
     is (2-r)(3-r)=0 or r^2 -5r + 6=0. Then C-H says
        A^2 - 5A + 6I = 0.
Cayley-Hamilton-Ziebur Method
  ZIEBUR'S LEMMA.
        The components of u in u'=Au are linear combinations of
        the atoms created by Euler's theorem applied to the
        roots of the characteristic equation det(A-rI)=0.
  THEOREM. Solve u'=Au without complex numbers or eigenanalysis.
        The solution of u'=Au is a linear combination of atoms
        times certain constant vectors [not arbitrary vectors].
             u(t)=(atom_1)vec(c_1)+ ... + (atom_n)vec(c_n)

  PROBLEM: Solve by Cayley-Hamilton-Ziebur the 2x2 dynamical system
             x' = 2x + y,
             y' = 3y,
             x(0)=1, y(0)=2.
        The characteristic equation is (2-lambda)(3-lambda)=0
          with roots lambda = 2,3
        Euler's theorem implies the atoms are exp(2t), exp(3t).
        Ziebur's Theorem says that
           u(t) = exp(2t) vec(u_1) + exp(3t) vec(u_2)
        where vectors u_1, u_2 are to be determined from the matrix
        A = matrix([[2,1],[0,3]]) and initial conditions x(0)=1, y(0)=2.

  ZIEBUR ALGORITHM.
        To solve for u_1, u_2 in the example, differentiate the
        equation u(t) = exp(2t) u_1 + exp(3t) u_2 and set t=0
        in both relations. Then u'=Au implies
             u_0 =    u_1  +   u_2,
            Au_0 = 2 u_1 + 3 u_2.
        These equations can be solved by elimination.
        The answer:
            u_1 = (3 u_0 -Au_0), u_2 = (Au_0 - 2 u_0)
                = vector([-1,0])     = vector([2,2])
        Vectors u_1, u_2 are recognized as eigenvectors of A for
        lambda=2 and lambda=3, respectively, after studying chapter 6.

  ZIEBUR SHORTCUT [Edwards-Penney textbook method, Example 5 in 7.1]
        Start with Ziebur's theorem, which implies that
           x(t) = k1 exp(2t) + k2 exp(3t).
        Use the first DE to solve for y(t):
           y(t) = x'(t) - 2x(t)
                =  2 k1 exp(2t) + 3 k2 exp(3t)
                         - 2 k1 exp(2t) - 2 k2 exp(3t))
                =   k2 exp(3t)
        For example, x(0)=1, y(0)=2 implies k1 and k2 are
        defined by
           k1 + k2 = 1,
                k2 = 2,
        which implies k1 = -1, k2 = 2, agreeing with a previous
        solution formula.
  
EIGENANALYSIS WARNING
  Reading Edwards-Penney Chapter 6 may deliver the wrong ideas
  about how to solve for eigenpairs. The examples emphasize a
  clever shortcut, which does not apply in general to solve for
  eigenpairs.

     HISTORY. Chapter 6 originally appeared in the 2280 book
     as a summary, which assumed a linear algebra course. The
     chapter was copied without changes into the Edwards-Penney
     Differential Equations and Linear Algebra textbook, which you
     currently own. The text contains only shortcuts. There is
     no discussion of a general method for finding eigenpairs.
     You will have to fill in the details by yourself. The online
     lecture notes and slides were created to fill in the gap.

Lecture: Fourier's Model. Intro to Eigenanalysis, Ch6.
  Examples and motivation.
     Ellipse, rotations, eigenpairs.
     General solution of a differential equation u'=Au and eigenpairs.
  Fourier's model.
  History.
    J.B.Fourier's 1822 treatise on the theory of heat.
    The rod example.
      Physical Rod: a welding rod of unit length, insulated on the
                    lateral surface and ice packed on the ends.
    Define f(x)=thermometer reading at loc=x along the rod at t=0.
    Define u(x,t)=thermometer reading at loc=x and time=t>0.
    Problem: Find u(x,t).
      Fourier's solution assume that
      f(x) = 17 sin (pi x) + 29 sin(5 pi x)
           = 17 v1 + 29 v2
      Packages v1, v2 are vectors in a vector space V of functions on [0,1].
      Fourier computes u(x,t) by re-scaling v1, v2 with numbers Lambda_1,
      Lambda_2 that depend on t. This idea is called Fourier's Model.

      u(x,t) = 17 ( exp(-pi^2 t) sin(pi x)) + 29 ( exp(-25 pi^2 t) sin (5 pi x))
             = 17 (Lambda_1 v1) + 29 (Lambda_2 v2)

  Eigenanalysis of u'=Au is the identical idea.
     u(0) = c1 v1 + c2 v2  implies
     u(t) = c1 exp(lambda_1 t) v1 + c2 exp(lambda_2 t) v2
     Fourier's re-scaling idea from 1822, applied to u'=Au,
       replaces v1 and v2 in the expression
                c1 v1 + c2 v2
       by their re-scaled versions to obtain the answer
                c1 (Lambda1 v1) + c2 (Lambda2 v2)
       where
         Lambda1 = exp(lambda_1 t), Lambda2 = exp(lambda_2 t).
Main Theorem on Fourier's Model

  THEOREM. Fourier's model
     A(c1 v1 + c2 v2) = c1 (lambda1 v1) + c2 (lambda2 v2)
  with v1, v2 a basis of R^2 holds [for all constants c1, c2]
    if and only if
  the vector-matrix system
    A(v1) = lambda1 v1,
    A(v2) = lambda2 v2,
  has a solution with vectors v1, v2 independent
    if and only if
  the diagonal matrix D=diag(lambda1,lambda2) and
  the augmented matrix P=aug(v1,v2) satisfy
     1. det(P) not zero [then v1, v2 are independent]
     2. AP=PD

  THEOREM. The eigenvalues of A are found from the determinant
    equation
                        det(A -lambda I)=0,
    which is called the characteristic equation.
  THEOREM. The eigenvectors of A are found from the frame
    sequence which starts with B=A-lambda I [lambda a root of
    the characteristic equation], ending with last frame rref(B).

    The eigenvectors for lambda are the partial derivatives of
    the general solution obtained by the Last Frame Algorithm,
    with respect to the invented symbols t1, t2, t3, ...
 
Algebraic Eigenanalysis Section 6.2.
  Calculation of eigenpairs to produce Fourier's model.
    Connection between Fourier's model and a diagonalizable matrix.
    How to find the variables lambda and v in Fourier's model using
      determinants and frame sequences.
  Solved in class: examples similar to the problems in 6.1 and 6.2.
    Web slides and problem notes exist for the 6.1 and 6.2 problems.
  Examples where A has an eigenvalue of multiplicity greater than one.


Wed, Apr 11: Eigenanalysis. First Order Systems. Sections 6.1, 6.2, 7.1, 7.3

 Projection: glass-breaking video. Wine glass experiment. Tacoma narrows.
       
Video: Wine glass breakage (QuickTime MOV) (96.8 K, mov, 31 Mar 2008)
Video: Wine glass experiment (12mb mpg, 2min) (12493.8 K, mpg, 01 Apr 2008)
Video: Tacoma Narrows Bridge Nov 7, 1940 (18mb mpg, 4min) (18185.8 K, mpg, 01 Apr 2008)
Video: Resonance #17, Wine Glass and Tacoma Narrows (29min Annenburg CPB)
Eigenanalysis Examples
   Problems 6.1, 6.2. See also FAQ online.
Solving DE System u' = Au by Eigenanalysis
  Example: Solving a 2x2 dynamical system
     Study of u'=Au, u(0)=vector([1,2]), A=matrix([[2,3],[0,4]]).
     Dynamical system scalar form is
         x' = 2x + 1y,
         y' = 3y,
         x(0)=1, y(0)=2.
     Find the eigenpairs (2, v1), (3,v2) where v1=vector([1,0])
     and v2=vector([1,1]).
     THEOREM. The solution of u' = Au in the 2x2 case is
         u(t) = c1 exp(lambda1 t) v1 + c2 exp(lambda2 t) v2
     APPLICATION:
       u(t) = c1 exp(2t) v1 + c2 exp(4t) v2
                        [ 1 ]            [ 1 ]
       u(t) = c1 e^{2t} [   ] + c2 e^4t} [   ]
                        [ 0 ]            [ 1 ]
       which means
          x(t) = c1 exp(2t) + 3 c2 exp(4t),
          y(t) = 2 c2 exp(4t).
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
Examples
    Given the eigenpairs of A, find A via AP=PD.
    Given P, D, then find A.
    Given A, then find P, D.
Cayley-Hamilton topics, Section 6.3.
    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.
     Powers of A and the meaning of A^n x_0 for the telecom example.
 Google Algorithm
   Lawrence Page's pagerank algorithm, google web page rankings. 
   Eigenanalysis and powers of A. 
Text: Lawrence Page's pagerank algorithm (0.7 K, txt, 06 Oct 2008)
    References for Eigenanalysis and Systems of Differential Equations.
    Manuscript: Algebraic eigenanalysis (187.6 K, pdf, 04 Mar 2012)
    Manuscript: What's eigenanalysis 2008 (174.2 K, pdf, 04 Mar 2012)
    Manuscript: What's eigenanalysis, draft 1 (152.2 K, pdf, 01 Apr 2008)
    Manuscript: 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 (785.8 K, pdf, 16 Nov 2008)
    Slides: Home heating, attic, main floor, basement (109.8 K, pdf, 04 Mar 2012)
    Text: Lawrence Page's pagerank algorithm (0.7 K, txt, 06 Oct 2008)
    Text: History of telecom companies (1.4 K, txt, 30 Dec 2009)
    Systems of Differential Equations references
    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.
    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 (108.1 K, pdf, 04 Mar 2012)
    MAPLE: Maple Lab 7. Laplace applications (60.6 K, pdf, 09 Dec 2011)
    Manuscript: DE systems, examples, theory (785.8 K, pdf, 16 Nov 2008)
    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 (109.8 K, pdf, 04 Mar 2012)
    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 (186.8 K, pdf, 20 Oct 2009)
    Manuscript: Laplace theory (350.5 K, pdf, 06 Mar 2009)
    Transparencies: Ch10 Laplace solutions 10.1 to 10.4 (1068.7 K, pdf, 28 Nov 2010)
    Text: Laplace theory problem notes (8.9 K, txt, 18 Nov 2010)
    Text: Final exam study guide (8.2 K, txt, 11 Dec 2011)