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 $h$.  
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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 $h$.  
  
 
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

Revision as of 08:03, 10 June 2015

Dynamic equations are 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 $h$.

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

$$y^{\Delta}(t)=p(t)y(t); y(s)=1.$$

  • Let $\mathbb{T} \subset (0,\infty)$. The Euler-Cauchy Equations are defined by a second-order linear dynamic equation: let $a,b \in \mathbb{R}$,

$$t \sigma(t)y^{\Delta \Delta}+ aty^{\Delta} + by=0.$$

See Also

Second order dynamic equations