Difference between revisions of "Shifting problem"
From timescalewiki
(→Examples) |
|||
| Line 1: | Line 1: | ||
| − | Let $\mathbb{T}$ be a time scale, $t_0 \in \mathbb{T}$, and $f \colon [t_0,\infty) \cap \mathbb{T} \rightarrow \mathbb{C}$. The shifting problem is the following [[partial dynamic equation]] for $t,s \in \mathbb{T}$: | + | Let $\mathbb{T}$ be a time scale with $\sup \mathbb{T}=\infty$, $t_0 \in \mathbb{T}$, and $f \colon [t_0,\infty) \cap \mathbb{T} \rightarrow \mathbb{C}$. The shifting problem is the following [[partial dynamic equation]] for $t,s \in \mathbb{T}$: |
$$\left\{ \begin{array}{ll} | $$\left\{ \begin{array}{ll} | ||
\dfrac{\partial \hat{f}}{\Delta t}(t,\sigma(s))=-\dfrac{\partial \hat{f}}{\Delta s}(t,s)& ; t \geq s \geq t_0, \\ | \dfrac{\partial \hat{f}}{\Delta t}(t,\sigma(s))=-\dfrac{\partial \hat{f}}{\Delta s}(t,s)& ; t \geq s \geq t_0, \\ | ||
| Line 40: | Line 40: | ||
|} | |} | ||
</center> | </center> | ||
| + | |||
| + | =References= | ||
| + | *{{PaperReference|The convolution on time scales|2007|Martin Bohner|author2=Gusein Sh. Guseinov|prev=|next=}}: Definition 2.1 | ||
=See also= | =See also= | ||
[[Convolution]]<br /> | [[Convolution]]<br /> | ||
Revision as of 14:49, 21 January 2023
Let $\mathbb{T}$ be a time scale with $\sup \mathbb{T}=\infty$, $t_0 \in \mathbb{T}$, and $f \colon [t_0,\infty) \cap \mathbb{T} \rightarrow \mathbb{C}$. The shifting problem is the following partial dynamic equation for $t,s \in \mathbb{T}$: $$\left\{ \begin{array}{ll} \dfrac{\partial \hat{f}}{\Delta t}(t,\sigma(s))=-\dfrac{\partial \hat{f}}{\Delta s}(t,s)& ; t \geq s \geq t_0, \\ \hat{f}(t,t_0)=f(t)&; t \geq t_0. \end{array} \right.$$ The solution $\hat{f}$ of the shifting problem is called the shift of $f$ (also called the delay of $f$).
Contents
Properties
Delta integral of certain shift of f is delta integral of f
Delta partial derivative of shift along diagonal
Examples
| $\mathbb{T}$ | $\hat{f}(t,s)=$ |
| $\mathbb{R}$ | $f(t-s)$ |
| $\mathbb{Z}$ | $f(t-s+t_0)$ |
| $h\mathbb{Z}$ | |
| $\mathbb{Z}^2$ | |
| $\overline{q^{\mathbb{Z}}}, q > 1$ | |
| $\overline{q^{\mathbb{Z}}}, q < 1$ | |
| $\mathbb{H}$ |
References
- Martin Bohner and Gusein Sh. Guseinov: The convolution on time scales (2007): Definition 2.1
