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A study on partial dynamic equation on time scales involving derivatives of polynomials

Petro Kolosov

Abstract

Let $P(m,b,x)$ be a $2m+1$-degree polynomial in $x,b$. Let be a two-dimensional timescale $Λ^2 = \mathbb{T}_1 \times \mathbb{T}_2 = \{t=(x, b) \colon \; x\in\mathbb{T}_1, \; b\in\mathbb{T}_2 \}$ such that $\mathbb{T}_1 = \mathbb{T}_2$. In this manuscript we derive and discuss an identity that connects the timescale derivative of odd-power polynomial with partial derivatives of polynomial $P(m,b,x)$ evaluated in particular points. For every $t\in\mathbb{T}_1$ and $(x,b) \in Λ^2$ \[ \frac{Δx^{2m+1}}{Δx}(t) = \frac{\partial P(m,b,x)}{Δx} (m, σ(t), t) + \frac{\partial P(m,b,x)}{Δb} (m, t, t) \] such that $σ(t) > t$ is forward jump operator. In addition, we discuss various derivative operators in context of partial cases of above equation, we show finite difference, classical derivative, $q-$derivative, $q-$power derivative on behalf of it.

A study on partial dynamic equation on time scales involving derivatives of polynomials

Abstract

Let be a -degree polynomial in . Let be a two-dimensional timescale such that . In this manuscript we derive and discuss an identity that connects the timescale derivative of odd-power polynomial with partial derivatives of polynomial evaluated in particular points. For every and such that is forward jump operator. In addition, we discuss various derivative operators in context of partial cases of above equation, we show finite difference, classical derivative, derivative, power derivative on behalf of it.

Paper Structure

This paper contains 10 sections, 12 theorems, 46 equations.

Table of Contents

  1. Definitions
  2. Introduction
  3. Main results
  4. Discussion and examples
  5. Time scale of integers $\mathbb{T} = \mathbb{Z} \times \mathbb{Z}$
  6. Time scale of real numbers $\mathbb{T} = \mathbb{R} \times \mathbb{R}$
  7. Quantum time scale $\mathbb{T} = q^\mathbb{R} \times q^\mathbb{R}$
  8. Quantum power time scale $\mathbb{T} = \mathbb{R}^q \times \mathbb{R}^q$
  9. Pure quantum power time scale $\mathbb{T} = q^{\mathbb{R}^n} \times q^{\mathbb{R}^n}$
  10. Addendum 1: Mathematica scripts

Key Result

Theorem 3.1

Let $P(m,b,x)$ be a $2m+1$-degree polynomial in $x,b$. Let be a two-dimensional timescale $\Lambda^2 = \mathbb{T}_1 \times \mathbb{T}_2 = \{t=(x, b) \colon \; x\in\mathbb{T}_1, \; b\in\mathbb{T}_2 \}$ such that $\mathbb{T}_1 = \mathbb{T}_2$. For every $t\in\mathbb{T}_1$ and $(x,b) \in \Lambda^2$ where

Theorems & Definitions (22)

  • Theorem 3.1
  • Corollary 4.1
  • Example 4.2
  • Example 4.3
  • Corollary 4.4
  • Corollary 4.5
  • Corollary 4.6
  • Example 4.7
  • Example 4.8
  • Corollary 4.9
  • ...and 12 more