Difference between revisions of "Dynamic equation"
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Suppose that $f \colon \mathbb{T} \times \mathbb{R}^2 \rightarrow \mathbb{R}$. Then the equation | Suppose that $f \colon \mathbb{T} \times \mathbb{R}^2 \rightarrow \mathbb{R}$. Then the equation | ||
$$y^{\Delta} = f(t,y,y^{\sigma})$$ | $$y^{\Delta} = f(t,y,y^{\sigma})$$ | ||
| − | is called a first order dynamic equation. | + | 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. |
Suppose there are [[continuity | rd-continuous]] functions $p_i \colon \mathbb{T} \rightarrow \mathbb{R}$. Define the operator | Suppose there are [[continuity | rd-continuous]] functions $p_i \colon \mathbb{T} \rightarrow \mathbb{R}$. Define the operator | ||
Revision as of 03:46, 25 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. Our goal is generally to find all functions $y \colon \mathbb{T} \rightarrow \mathbb{R}$ that satisfies the 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.
