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

Survey of Methods for solving a 2x2 dynamical system1. 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 MatrixDrill ProblemsIn 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 modelsThe job-site cable hoist example [delayed] Sliding plates example [delayed] Home heating example [more coming]

Sample exam 3 solutions to problems 1,2.

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. 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

Exam 3 from 1-5pm Wed, then 6:50am Thu.

Friday exams are special cases, scheduled individually.

Non-Homogeneous SystemsDirect 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: Systems theory and examples (785.8 K, pdf, 16 Nov 2008)Manuscript: 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)Slides

References for Eigenanalysis and Systems of Differential Equations.: 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)Manuscript: 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)Slides: Systems of DE examples and theory (785.8 K, pdf, 16 Nov 2008)Manuscript: Home heating, attic, main floor, basement (73.8 K, pdf, 30 Nov 2009)Slides: Lawrence Page's pagerank algorithm (0.7 K, txt, 06 Oct 2008)Text: History of telecom companies (1.1 K, txt, 05 Oct 2008)Text

Systems of Differential Equations references: Systems of DE examples and theory (785.8 K, pdf, 16 Nov 2008)Manuscript: 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)Slides