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| − | Dynamic equations are analogues of differential equations on a [[time scale]]. 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 $h$.
| + | 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
| + | =See Also= |
| − | $$y^{\Delta} = f(t,y,y^{\sigma})$$
| + | [[Delta exponential dynamic equation]]<br /> |
| − | 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.
| + | [[First order dynamic equations]]<br /> |
| − | | + | [[Second order dynamic equations]]<br /> |
| − | Suppose there are [[continuity | rd-continuous]] functions $p_i \colon \mathbb{T} \rightarrow \mathbb{R}$. 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 dynamic equation.
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| − | ==Examples of Dynamic Equations==
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| − | *The [[Exponential_functions | exponential functions]] are defined by first-order linear initial value problem, for [[Regressive_function | regressive]] $p$: for $s,t \in \mathbb{T}$,
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| − | $$y^{\Delta}(t)=p(t)y(t); y(s)=1.$$
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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
