## 2280 Lectures Week 12 S2018

Last Modified: March 26, 2018, 11:35 MDT.    Today: December 14, 2018, 00:27 MST.

### Week 12: Systems, Sections 5.1 to 5.6

``` Edwards-Penney, sections 4.2, 5.1 to 5.6, 6.1, 6.2
The textbook topics, definitions and theoremsEdwards-Penney 5.1 to 5.6 (24.0 K, txt, 06 Jan 2015)Edwards-Penney 6.1 to 6.4 (11.8 K, txt, 05 Apr 2015)```
```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 [upcoming week 13]
Slides: Cayley-Hamilton-Ziebur method for solving vector-matrix system u'=Au. (143.5 K, pdf, 16 Mar 2018) 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)```
```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-Ziebur

Methods to solve dynamical systems
Example 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 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
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.
Sildes: Algebraic eigenanalysis (173.4 K, pdf, 14 Mar 2016) Slides: 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. (143.5 K, pdf, 16 Mar 2018) 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) 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)   Slides: Introduction to dynamical systems (0.0 K, pdf, 31 Dec 1969)   Slides: Matrix Exponential, Putzer Formula, Variation Parameters (0.0 K, pdf, 31 Dec 1969)
```