Edwards-Penney, sections 4.1, 4.2, 5.1 to 5.6 The textbook topics, definitions and theorems

Edwards-Penney 4.1 to 4.3 (10.7 K, txt, 05 Jan 2015)

Edwards-Penney 5.1 to 5.6 (24.0 K, txt, 06 Jan 2015)

Numerical Methods for Systems: Section 4.3 Review of Scalar methods Euler, Heun, RK4 Change the methods to the vector case, based upon the statement of the IVP in Picard's Theorem. It amounts to putting arrows over y and f. Vector Methods Euler, Heun, RK4 Manuscript: Vector Methods EXAMPLE. Solve the vector problem u'=Au, u(0)=u_0 with A=Matrix([[3,-2],[5,-4]]) and u_0 = <3,6> using Euler's method and Heuns's method with step size h=0.1. This is the scalar problem x'=3x-2y, y'=5x-4y, x(0)=3, y(0)=6. Details: PDF: maple Example 1, section 4.3

Review of Eigenanalysis Eigenvalue Eigenvector Eigenpair Fourier's Model Diagonalizable matrix Main theorem for solving u'=Au by eigenanalysis Example 1. Consider the 2x2 system x'=x+3y, y'=2y, x(0)=1, y(0)=-1. Method 1: Linear integrating factor method for triangular systems Method 2: Cayley-Hamilton-Ziebur, textbook shortcuts applied to 2x2 Method 3: Eigenanalysis Method 4: Laplace resolvent Method 5: Exponential matrix, maple or Putzer's method: 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: Matrix Exponential, Putzer Formula, Variation Parameters (122.0 K, pdf, 14 Mar 2016)Slides

Applications Brine tank models. Recirculating brine tanks. Pond pollution. Home heating. Earthquakes. Railway cars. All are 2x2 or 3x3 or nxn system applications that can be solved by Laplace methods. We investigate 3 fundamental methods: Eigenanalysis, Laplace, Cayley-Hamilton-ZieburMethods to solve dynamical systemsExample 2. Consider the 2x2 system x'=x-5y, y'=x-y, x(0)=1, y(0)=2. Cayley-Hamilton-Ziebur method. Textbook shortcut preferred, section 4.2 example 1 Laplace resolvent. Eigenanalysis method. Exponential matrix using maple Putzer's method to compute the exponential matrix [slides, not in the textbook]

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 THEOREM. 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.http://www.math.utah.edu/~gustafso/s2017/2280/maple/numerical-4.3-systems-example1.pdf 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. Next week: Brine tank models. Recirculating brine tanks. Pond pollution. Home heating. Earthquakes. Railway cars. All are 2x2 or 3x3 or nxn system applications that can be solved by Laplace methods. We investigate 3 fundamental methods: Eigenanalysis, Laplace, Cayley-Hamilton-Ziebur

References for Eigenanalysis and Systems of Differential Equations.: Algebraic eigenanalysis (173.4 K, pdf, 14 Mar 2016)Sildes: What's eigenanalysis? (124.0 K, pdf, 14 Nov 2007)Slides: Advanced topics in Linear Algebra, Cayley-Hamilton, Jordan form (334.7 K, pdf, 19 Mar 2016)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: Cayley-Hamilton-Ziebur for second order systems (130.4 K, pdf, 22 Mar 2017)Slides: Systems of DE examples and theory (730.9 K, pdf, 09 Apr 2014)Manuscript: Home heating, attic, main floor, basement (99.3 K, pdf, 14 Mar 2016)Slides: Introduction to dynamical systems (144.9 K, pdf, 14 Mar 2016)Slides: Matrix Exponential, Putzer Formula, Variation Parameters (122.0 K, pdf, 14 Mar 2016)Slides

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.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 Equations: Laplace second order systems (273.7 K, pdf, 14 Mar 2016) andSlides: Cayley-Hamilton-Ziebur for second order systems (130.4 K, pdf, 22 Mar 2017) The model u'' = Ax + F(t) Coupled Spring-Mass System. Problem 5.3-6 A:=matrix([[-6,4],[2,-4]]); eigenvals(A); lambda1= -2, lambda2= -8SlidesZiebur'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 Ch5u(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 5.3-24Cayley-Hamilton-Ziebur method Laplace Resolvent method for second order Eigenanalysis method section 5.3

Matrix Exponential Topics: Matrix Exponential, Putzer Formula, Variation Parameters (122.0 K, pdf, 14 Mar 2016) Fundamental matrix Phi(t) Matrix Variation of Parameters Definition of the exponential matrix exp(At) 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.Slides