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Advances in Mathematical Modeling: Dynamical Equations on Time Scales. Ian A. Gravagne School of Engineering and Computer Science Baylor University, Waco, TX. Outline. Background and Motivation Intro to Time Scales Mathematical Basics Software and Simulation Wrap Up. Background.

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Advances in Mathematical Modeling:

Dynamical Equations on Time Scales

Ian A. Gravagne

School of Engineering and Computer Science

Baylor University, Waco, TX

outline
Outline
  • Background and Motivation
  • Intro to Time Scales
    • Mathematical Basics
    • Software and Simulation
  • Wrap Up
background
Background

“ A major task of mathematics today is to harmonize the continuous and the discrete, to include them in one comprehensive mathematics, and to eliminate obscurity from both.” E.T. Bell, 1937

time scales

Where 99.9 % of engineering has

taken place up to now…

… Time Scales!
  • Body of theory springs from Ph.D. dissertation of S. Hilger in 1988.
  • Captured interest of math community in 1993. First comprehensive monograph on subject published in 2002.
  • Definition: a time scale is a closed subset of the real numbers: special case of a measure chain.

R

h

hZ

a

b

Pab

H0

Cantor sets, limit points, etc!

terminology
Terminology

Forward Jump Operator:

Backward Jump Operator:

Graininess:

operators

(The delta-derivative only exists for and . This offer expires 11/21/03.)

(Hilger integrals only exist if and over .)

Operators
  • Derivatives:
  • Integrals:
diff int rules
Diff/Int Rules
  • Product Rule for differentiation
  • Chain Rule for differentiation

Derivatives and Integrals are linear and homogeneous.

  • No more “rules of thumb” for differentiation!!
  • Integration by Parts
  • Very few closed-form indefinite integrals known.
differential equations

The TS exponential exists iff

If then

“Differential” Equations
  • The first (and arguably most important) dynamical equation to examine is

The solution is

properties of ts exp
Properties of TS exp

Why do we need ?

Operators form a Lie Group on the Regressive Set with

identity

higher order systems
Higher Order Systems
  • As expected, solutions to higher order linear systems are sums of

Leads to logical definitions

  • Alternatively, systems of linear equations are also well-defined:
  • Need
properties of ts sin cos
Properties of TS sin, cos…

Notes:

Thought of the day: the “natural” trig functions (i.e. above) are defined as the solutions to a 2nd (or 4th) order undamped diff. eqs. They cannot alias no matter how high the “frequency”!

other ts work
Other TS work
  • We have only scratched the surface of existing work in Time Scales:
  • Nabla derivatives:
  • PDE’s:
  • Generalized Laplace Tranform:
  • Ricatti equations, Green’s functions, BVPs, Symplectic systems, nonlinear theory, generalized Fourier transforms.

OK, OK… But what do these things look like??

ts toolbox
TS Toolbox
  • Worked with John Davis, Jeff Dacunha, Ding Ma over summer ’03 to develop first numerical routines to:
  • Construct and manipulate time scales
  • Perform basic arithmetic operations
  • Calculate
  • Solve arbitrary initial-value ODEs
  • Visualize functions on timescales
  • Routines were written in MATLAB.
time scale objects
Time Scale Objects
  • It quickly became apparent that we would need to use MATLAB’s object-oriented capabilities:
  • A time scale cannot be effectively stored as a simple vector or array.
  • Need to overload arithmetic functions, syntax
  • Is T=[0,1,2,3,4,5,6,7,8,9,10]
  • an isolated time scale?
  • a discretization of a continuous interval?
  • a mixture?
  • Need more information: where are the breaks between intervals, and what kind of intervals are they: discrete or continuous.
  • Package this info up into an object…
time scale objects 2
Time Scale Objects 2

Solution:

T.data=[0,0.1,0.2,0.3,0.4,0.5,1,1.5,2,2.1,2.3,2.4,2.5]

T.type=[6 ,0

8 ,1

13,0]

Shows whether interval is discrete (1) or continuous (0)

Shows final ordinal for last point in interval

overloads
Overloads

Now we can overload common functions, e.g. + - * / ^ as well as syntax, e.g. [ ], ( ), : etc…

graphics
Graphics

The “tsplot” function plots time scale images, and colors the intervals differently.

the ts exponential
The TS exponential

TS exponential on the time scale

If then

at

more ts exp

Im

AC

MC

AD

-10

Definition:

The Hilger Circle is

More TS Exp

TS exponential on the first 20 harmonics.

sin cos
Sin, Cos

Sin, Cos on a logarithmic time scale.

slide23
Fin!
  • Dynamical Equations on Time Scales == powerful tool to model systems with mixtures of continuous/discrete dynamics or discrete dynamics of non-uniform step size.
  • Mathematics very advanced in some ways, but in other ways still in relative infancy.
  • Need to overcome “rut thinking”