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Week 14: 30 Nov,  01 Dec,  02 Dec,  03 Dec,  04 Dec,

2250 Lecture Record Week 14 F2009

Last Modified: December 06, 2009, 17:10 MST.    Today: October 17, 2017, 14:36 MDT.

Week 14, Nov 30, Dec 1-4: Sections 7.3,7.4,8.1,8.2,9.1

30 Nov: Second Order Systems Section 7.4

 Survey of Methods for solving a 2x2 dynamical system
  1. Cayley-Hamilton method for u'=Au
    Solution: u(t)=(atom_1)vec(u_1)+ (atom_2)vec(u_2)
    Atoms: They are constructed by Euler's theorem from roots of det(A-rI)=0
    Vectors: Symbols vec(u_1), vec(u_2) are not arbitrary. They are
         determined from A and u(0). Algorithm outlined earlier for 2x2.
  2. Laplace resolvent L(u)=(s I - A)^(-1) u(0)
  3. Eigenanalysis  u(t) = exp(lambda_1 t) v1 + exp(lambda_2 t) v2
  4. 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
  5. ALL METHODS apply to nxn matrices A.



REVIEW: First Order linear systems:
    Brine tank models.
    Recirculating brine tanks.
    Pond pollution.
    Home heating.
    All are 2x2 or 3x3 or nxn system applications
    All answers are vector linear combinations of atoms
    Solve them by
      Eigenanalysis, Cayley-Hamilton-Ziebur,
      Putzer's method, Laplace resolvent method,
      Exponential Matrix



Drill Problems

   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

   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.

 Methods to solve dynamical systems like
    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
   Spectral methods [ch8; not studied in 2250]


Engineering models
   The job-site cable hoist example [delayed]
   Sliding plates example  [delayed]
   Home heating example  [more coming]


01 Dec: Second Order Systems. Section 7.4


Sample exam 3 solutions to problems 1,2.


02 Dec: Second Order Systems. Section 7.4

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.

  The model u'' = Ax + F(t)
  Coupled Spring-Mass System. Problem 7.4-6
    A:=matrix([[-6,4],[2,-4]]);
  Railway cars. Problem 7.4-24
  Earthquake model
     Cayley-Hamilton-Ziebur method
     Laplace Resolvent method for second order
     Maple routines for second order


Sample exam 3 solutions to problems 3,4

02-03-04 Dec: Exam days.



Exam 3 from 1-5pm Wed, then 6:50am Thu.
Friday exams are special cases, scheduled individually.

04 Dec: Non-Homogeneous Systems. Section 8.2

Non-Homogeneous Systems
    Direct solution methods with the Laplace Resolvent
    Computer Algebra System methods
    Variation of Parameters Formula for systems



Exercise solutions: ch7 and ch8.
    Slides on Dynamical Systems
    Manuscript: Systems theory and examples (785.8 K, pdf, 16 Nov 2008)
    Slides: Laplace second order systems, spring-mass,boxcars, earthquakes (248.9 K, pdf, 01 Nov 2009)
    Slides: Introduction to dynamical systems (126.2 K, pdf, 30 Nov 2009)
    Slides: Phase Portraits for dynamical systems (205.5 K, pdf, 11 Dec 2009)
    Slides: Stability for dynamical systems (125.7 K, pdf, 30 Nov 2009)
    References for Eigenanalysis and Systems of Differential Equations.
    Manuscript: Algebraic eigenanalysis (135.8 K, pdf, 07 Apr 2008)
    Manuscript: What's eigenanalysis 2009 (129.5 K, pdf, 07 Apr 2008)
    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. (111.4 K, pdf, 30 Nov 2009)
    Slides: Laplace resolvent method (56.4 K, pdf, 01 Nov 2009)
    Slides: Laplace second order systems (248.9 K, pdf, 01 Nov 2009)
    Manuscript: Systems of DE examples and theory (785.8 K, pdf, 16 Nov 2008)
    Slides: Home heating, attic, main floor, basement (73.8 K, pdf, 30 Nov 2009)
    Text: Lawrence Page's pagerank algorithm (0.7 K, txt, 06 Oct 2008)
    Text: History of telecom companies (1.1 K, txt, 05 Oct 2008)
    Systems of Differential Equations references
    Manuscript: Systems of DE examples and theory (785.8 K, pdf, 16 Nov 2008)
    Slides: Laplace resolvent method (56.4 K, pdf, 01 Nov 2009)
    Slides: Laplace second order systems (248.9 K, pdf, 01 Nov 2009)
    Slides: Home heating, attic, main floor, basement (73.8 K, pdf, 30 Nov 2009)
    Slides: Cable hoist example (73.2 K, pdf, 21 Aug 2008)
    Slides: Sliding plates example (105.8 K, pdf, 21 Aug 2008)