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.

Examples of Dynamic Equations