Difference between revisions of "Shifting problem"

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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 $t,s \in \mathbb{T}$:
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__NOTOC__
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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, \\
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=Properties=
 
=Properties=
[[Delta integral of certain shift of f is delta integral of f]]
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[[Delta integral of certain shift of f is delta integral of f]]<br />
 
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[[Delta partial derivative of shift along diagonal]]<br />
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
<strong>Theorem:</strong> Suppose that $\hat{f}$ has partial $\Delta$-derivatives of all orders. Then
 
$$\dfrac{\partial^k \hat{f}}{\Delta^k t} (t,t)=f^{\Delta^k}(t_0).$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong>  █
 
</div>
 
</div>
 
 
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
 
<strong>Theorem:</strong> Define $u_a(t)= \left\{\begin{array}{ll} 0 &; t < a \\
 
1 &; t \geq a \end{array} \right..$ Then
 
$$\mathscr{L}_{\mathbb{T}}\{u_s \hat{f}(\cdot,s) \}(z) = e_{\ominus z}(s,t_0)\mathscr{L}_{\mathbb{T}}\{f\}(z).$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong>  █
 
</div>
 
</div>
 
  
 
=Examples=
 
=Examples=
 +
<center>
 
{| class="wikitable"
 
{| class="wikitable"
|+Time Scale Shift
+
|+Time Scales Shift
 
|-
 
|-
|$\mathbb{T}$ | Shift $\hat{f}(t,s)$
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| $\mathbb{T}$
|
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| $\hat{f}(t,s)=$
 
|-
 
|-
 
|[[Real_numbers | $\mathbb{R}$]]
 
|[[Real_numbers | $\mathbb{R}$]]
|$\hat{f}(t,s)=f(t-s)$
+
|$f(t-s)$
 
|-
 
|-
 
|[[Integers | $\mathbb{Z}$]]
 
|[[Integers | $\mathbb{Z}$]]
|$\hat{f}(t,s)=f(t-s+t_0)$
+
|$f(t-s+t_0)$
 
|-
 
|-
 
|[[Multiples_of_integers | $h\mathbb{Z}$]]
 
|[[Multiples_of_integers | $h\mathbb{Z}$]]
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|
 
|
 
|}
 
|}
 +
</center>
  
 
=See also=
 
=See also=
[[Convolution]]<br />
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[[Unilateral convolution]] <br />
 +
[[Unilateral Laplace transform]]<br />
 +
 
 +
=References=
 +
*{{PaperReference|The convolution on time scales|2007|Martin Bohner|author2=Gusein Sh. Guseinov|prev=|next=}}: Definition 2.1

Latest revision as of 14:51, 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 $f$).

Properties

Delta integral of certain shift of f is delta integral of f
Delta partial derivative of shift along diagonal

Examples

Time Scales Shift
$\mathbb{T}$ $\hat{f}(t,s)=$
$\mathbb{R}$ $f(t-s)$
$\mathbb{Z}$ $f(t-s+t_0)$
$h\mathbb{Z}$
$\mathbb{Z}^2$
$\overline{q^{\mathbb{Z}}}, q > 1$
$\overline{q^{\mathbb{Z}}}, q < 1$
$\mathbb{H}$

See also

Unilateral convolution
Unilateral Laplace transform

References