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 theorems

Edwards-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)

Exam 3 ReviewSample exam 3Diagonalization TheoryIn 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 systemsConsider 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 system1. 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.

Exam 3 ReviewSample 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 SystemsHow 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 EquationsThe model u'' = Ax + F(t) Coupled Spring-Mass System. Problem 7.4-6 A:=matrix([[-6,4],[2,-4]]); eigenvals(A); lambda1= -2, lambda2= -8Ziebur's Methodroots 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 atomsMaple routines for second orderde1:=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 7.4u(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 7.4-24Cayley-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.

Extra Credit Maple Project: Tacoma narrows. Explore an alternative explanation for what caused the bridge to fail, based on the hanging cables.: Maple Lab 9. Tacoma Narrows (0.0 K, pdf, 31 Dec 1969)MAPLEExtra 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 TopicsEquilibria. Stability. Instability.

Wednesday-Friday: Fourier's History. Rod Problem. Fourier Series. Section 9.1, 9.2Chapter 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:= 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. Letn=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. 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: Systems theory and examples (730.9 K, pdf, 10 Apr 2014)Manuscript: 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 classificationSlidesSlides on Dynamical Systems: Systems theory and examples (730.9 K, pdf, 10 Apr 2014)Manuscript: 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)Slides

References for Eigenanalysis and Systems of Differential Equations.: 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)Slides: Systems of DE examples and theory (730.9 K, pdf, 10 Apr 2014)Manuscript: Home heating, attic, main floor, basement (99.3 K, pdf, 14 Mar 2016)Slides: Lawrence Page's pagerank algorithm (0.0 K, txt, 31 Dec 1969)Text: History of telecom companies (0.0 K, txt, 31 Dec 1969)Text

Systems of Differential Equations applications: Cable hoist example (73.2 K, pdf, 21 Aug 2008)Slides: Sliding plates example (105.8 K, pdf, 21 Aug 2008)SlidesExtra Credit Maple Project: Tacoma narrows. Explore an alternative explanation for what caused the bridge to fail, based on the hanging cables.: Maple Lab 9. Tacoma Narrows (0.0 K, pdf, 31 Dec 1969)MAPLELaplace theory references: 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)Slides: Maple Lab 7. Laplace applications (0.0 K, pdf, 31 Dec 1969)MAPLE: DE systems, examples, theory (730.9 K, pdf, 10 Apr 2014)Manuscript: 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)Slides: Heaviside's method 2008 (352.3 K, pdf, 07 Jan 2014)Manuscript: Laplace theory 2008 (497.3 K, pdf, 19 Mar 2014)Manuscript: Ch10 Laplace solutions 10.1 to 10.4 (1068.7 K, pdf, 22 Feb 2015)Transparencies: Laplace theory problem notes (0.0 K, txt, 31 Dec 1969)Text: Final exam study guide (0.0 K, txt, 31 Dec 1969)Text