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 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

Discrete + Continuous = …

… Time Scales! Cantor sets, limit points, etc! R hZhZ P ab H0H0 h b a Where 99.9 % of engineering has taken place up to now… Body of theory springs from Ph.D. dissertation of S. Hilger in Captured interest of math community in First comprehensive monograph on subject published in Definition: a time scale is a closed subset of the real numbers: special case of a measure chain.

Terminology Forward Jump Operator: Backward Jump Operator: Graininess: t 1 is isolated t 2 is left-scattered (right-dense) t 3 is dense t 4 is right-scattered (left-dense)

Operators Derivatives: (The delta-derivative only exists for and. This offer expires 11/21/03.) Integrals: (Hilger integrals only exist if and over.)

Diff/Int Rules Product Rule for differentiation Chain Rule for differentiation No more rules of thumb for differentiation!! Very few closed-form indefinite integrals known. Integration by Parts Derivatives and Integrals are linear and homogeneous.

Differential Equations The first (and arguably most important) dynamical equation to examine is The solution is The TS exponential exists iff If then

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

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

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??

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 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 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 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 final ordinal for last point in interval Shows whether interval is discrete (1) or continuous (0)

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

Overloads 2

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

The TS exponential TS exponential on the time scale If then at

More TS Exp TS exponential on the first 20 harmonics. AD ACMC Definition: The Hilger Circle is -10 Im

Sin, Cos Sin, Cos on a logarithmic time scale.

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