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)

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, see below 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 A second bit of information is there, about metformin, which is the diabetic drug (pill) most often prescribed for Type II diabetes. I didn't know it is also given to hypoglycemics, to control symptoms of low blood sugar. That is non-intuitive, something to catalog, along with how hypoglycemia can mutate into Type II diabetes, later in life. [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.: Extra Credit Maple Lab 6. Tacoma Narrows (31.9 K, pdf, 17 Oct 2016)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.: Extra Credit Maple Lab 7. Earthquake (32.3 K, pdf, 17 Oct 2016)MAPLE

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.

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Study for the Final Exam: Final exam study guide (0.0 K, txt, 31 Dec 1969)Text