Dynamic equations on time scales

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The study of dynamic equations on time scales is an area of mathematics that tries to unify the study of differential and difference equations. It goes back to its founder Stefan Hilger in 1988, but it has recently received a lot of attention. Since much of the work is fairly recent, it is not yet clear how this compares to other approaches.

[edit] Dynamic equations

Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their continuous counterparts. The study of dynamic equations on time scales reveals such discrepancies, and helps avoid proving results twice — once for differential equations and once again for difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so-called time scale, which may be an arbitrary closed subset of the reals. In this way, results not only related to the set of real numbers or set of integers but those pertaining to more general time scales are obtained.

The three most popular examples of calculus on time scales are differential calculus, difference calculus, and quantum calculus. Dynamic equations on a time scale have a potential for applications, such as in population dynamics. For example, it can model insect populations that are continuous while in season, die out in say winter, while their eggs are incubating or dormant, and then hatch in a new season, giving rise to a nonoverlapping population. Since then several authors have expounded on various aspects of this new theory.

[edit] Calculus on time scale

A time scale or measure chain T is a closed subset of the real line R.

Define

σ(t) = inf{s an element of T, s > t} (forward shift operator)

ρ(t) = sup{s an element of T, s < t} (backward shift operator)

Let t be an element of T: t is

left dense if ρ(t) = t,

right dense if σ(t) = t,

left scattered if ρ(t) < t,

right scattered if σ(t) > t,

dense if left dense or right dense.

Define the graininess μ of a measure chain T by

μ(t) = σ(t) − t.

Take a function

f : T → R,

(where R could be any Banach space, but set it to be the real line for simplicity).

Definition: generalised derivative or fdelta(t)

For every ε > 0 there exists a neighbourhood U of t such that

|f(σ(t)) − f(s) − fdelta(t)(σ(t) − s)| ≤ ε|σ(t) − s|

for all s in U.

Take T = R. Then σ(t) = t,μ(t) = 0, fdelta = f′ is the derivative used in standard calculus. If T = Z (the integers), σ(t) = t + 1, μ(t)=1, fdelta = Δf is the forward difference operator used in difference equations.