Difference between revisions of "Dynamic equation"
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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 $h$. | ||
| + | Suppose that $f \colon \mathbb{T} \times \mathbb{R}^2 \rightarrow \mathbb{R}$. Then the equation | ||
| + | $$y^{\Delta} = f(t,y,y^{\sigma})$$ | ||
| + | is called a first order dynamic equation. | ||
| + | |||
| + | Suppose there are rd-continuous functions $p_i \colon \mathbb{T} \rightarrow \mathbb{R}$. Define the operator | ||
| + | $$Ly=y^{\Delta^n}+\displaystyle\sum_{k=1}^n p_i y^{\Delta^{n-i}}.$$ | ||
| + | We say that the equation $Ly=f$ is an $n$th order dynamic equation. | ||
Revision as of 21:31, 19 May 2014
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 $h$.
Suppose that $f \colon \mathbb{T} \times \mathbb{R}^2 \rightarrow \mathbb{R}$. Then the equation $$y^{\Delta} = f(t,y,y^{\sigma})$$ is called a first order dynamic equation.
Suppose there are rd-continuous functions $p_i \colon \mathbb{T} \rightarrow \mathbb{R}$. Define the operator $$Ly=y^{\Delta^n}+\displaystyle\sum_{k=1}^n p_i y^{\Delta^{n-i}}.$$ We say that the equation $Ly=f$ is an $n$th order dynamic equation.
