Advances in Mathematical Modeling:

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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
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Outline

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

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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

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Discrete Continuous
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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!
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Terminology
Forward Jump Operator Backward Jump
Operator Graininess
t1 is isolated
t2 is left-scattered (right-dense)
t3 is dense
t4 is right-scattered (left-dense)
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Operators

• Derivatives

• Integrals

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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.

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Differential Equations

• The first (and arguably most important)
  dynamical equation to examine is

The solution is
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Properties of TS exp






Why do we need ?
Operators form a Lie Group on the
Regressive Set with identity
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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

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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!
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Other TS work

• We have only scratched the surface of existing
  work in Time Scales
• Nabla derivatives
• PDEs
• Generalized Laplace Tranform
• Ricatti equations, Greens functions, BVPs,
  Symplectic systems, nonlinear theory, generalized
  Fourier transforms.

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

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Time Scale Objects

• It quickly became apparent that we would need to
  use MATLABs object-oriented capabilities
• A time scale cannot be effectively stored as a
  simple vector or array.
• Need to overload arithmetic functions, syntax
• Is T0,1,2,3,4,5,6,7,8,9,10
• 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

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Time Scale Objects 2
Solution T.data0,0.1,0.2,0.3,0.4,0.5,1,1.5,2,2
.1,2.3,2.4,2.5 T.type6 ,0 8 ,1
13,0
Shows whether interval is discrete (1) or
continuous (0)
Shows final ordinal for last point in interval
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Overloads
Now we can overload common functions, e.g. -
/ as well as syntax, e.g. , ( ), etc
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Overloads 2
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Graphics
The tsplot function plots time scale images,
and colors the intervals differently.
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The TS exponential
TS exponential on the time scale
If then at
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More TS Exp
TS exponential on the first 20 harmonics.
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Sin, Cos
Sin, Cos on a logarithmic time scale.
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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