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2250 8:05am Lectures Week 14 S2017

Last Modified: April 10, 2017, 12:41 MDT.    Today: November 25, 2017, 04:10 MST.

Week 14: Partial Differential Equations, Ch9

 Edwards-Penney, sections 9.1, 9.2, 9.3, 9.4
  The textbook topics, definitions and theorems
Edwards-Penney 9.1,9.2,9.3,9.4 (0.0 K, txt, 31 Dec 1969)

Monday: Exam 3 Review

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, see below
   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
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             [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.
  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
      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.

Extra Credit Maple Project: Tacoma narrows. Explore an alternative
explanation for what caused the bridge to fail, based on the hanging cables.

MAPLE: Extra Credit Maple Lab 6. Tacoma Narrows (31.9 K, pdf, 17 Oct 2016) 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.
MAPLE: Extra Credit Maple Lab 7. Earthquake (32.3 K, pdf, 17 Oct 2016)

Monday-Tuesday: 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:

        (1)  Start with f(x) = trigonometric series 
        (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:  < u,v > = 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) < u,v+w > = < u,w > + < v,w >  linear in the first argument
         (2) < c*u,v > = c*< u,v >
         (3) < u,v >=< v,u >  symmetry
         (4) < u,u > = 0 if and only if u=0

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

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 < f[n] > 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] = < f,f[n] >/< f[n],f[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.
          ANSWER.  a[0] = Pi/2
                   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, 09 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, 09 Apr 2014)
Slides: Home heating, attic, main floor, basement (99.3 K, pdf, 14 Mar 2016)
Systems of Differential Equations applications
Slides: Cable hoist example (73.2 K, pdf, 21 Aug 2008)
Slides: Sliding plates example (105.8 K, pdf, 20 Aug 2008)
Study for the Final Exam
Text: Final exam study guide (0.0 K, txt, 31 Dec 1969)