Difference between revisions of "Dynamic equation"

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Let $\mathbb{T}$ be a [[time scale]]. Dynamic equations are [[time scale analogue|time scale analogues]] of differential equations and difference equations. If we have $\mathbb{T}=\mathbb{R}$ then the resulting theory of dynamic equations is the thoery of differential equations. If $\mathbb{T}=h\mathbb{Z}$ then the resulting theory of dynamic equations is the theory of difference equations of stepsize $h$.
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Let $\mathbb{T}$ be a [[time scale]]. Dynamic equations are a generalization and extension of [[differential equation|differential equations]] and [[difference equation|difference equations]].
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Suppose that $f \colon \mathbb{T} \times \mathbb{R}^2 \rightarrow \mathbb{R}$. Then the equation
 
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$$y^{\Delta} = f(t,y,y^{\sigma})$$
 
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is called a first order dynamic equation. Our goal is generally to find all functions $y \colon \mathbb{T} \rightarrow \mathbb{R}$ that satisfies the equation.
 
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Let the functions $p_i \colon \mathbb{T} \rightarrow \mathbb{R}$ be [[rd continuous]]. Define the operator
 
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$$Ly=y^{\Delta^n}+\displaystyle\sum_{k=1}^n p_i y^{\Delta^{n-i}}.$$
 
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We say that the equation $Ly=f$ is an $n$th order linear dynamic equation.
 
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==Examples of Dynamic Equations==
 
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*The [[Exponential_functions | exponential functions]] are defined by a first-order linear initial value problem, for [[Regressive_function | regressive]] $p$: let $s,t \in \mathbb{T}$,
 
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$$y^{\Delta}(t)=p(t)y(t); y(s)=1.$$
 
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*Let $\mathbb{T} \subset (0,\infty)$. The [[Euler_Cauchy_Equations | Euler-Cauchy Equations]] are defined by a second-order linear dynamic equation: let $a,b \in \mathbb{R}$,
 
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$$t \sigma(t)y^{\Delta \Delta}+ aty^{\Delta} + by=0.$$
 
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*[[Abel dynamic equation]]
 
  
 
=See Also=
 
=See Also=
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[[Delta exponential dynamic equation]]<br />
 
[[First order dynamic equations]]<br />
 
[[First order dynamic equations]]<br />
 
[[Second order dynamic equations]]<br />
 
[[Second order dynamic equations]]<br />

Latest revision as of 16:32, 9 February 2017

Let $\mathbb{T}$ be a time scale. Dynamic equations are a generalization and extension of differential equations and difference equations.

See Also

Delta exponential dynamic equation
First order dynamic equations
Second order dynamic equations

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