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

Abstract

Let $P(m,b,x)$ be a $2m+1$-degree polynomial in $x,b$. Let be a two-dimensional timescale ${\Lambda }^2={\mathbb{T}}_1\times {\mathbb{T}}_2=\{t=(x,b):x\in {\mathbb{T}}_1,b\in {\mathbb{T}}_2\}$ 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 $(x,b)\in {\Lambda }^2$ \[ \frac{Δx^{2m+1}}{Δx}(t) = \frac{\partial P(m,b,x)}{Δx} (m, σ(t), t) + \frac{\partial P(m,b,x)}{Δb} (m, t, t) \] such that $\sigma (t)>t$ 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.

16 pages, the manuscript source code is available at https://github.com/kolosovpetro/AStudyOnDynamicEquations

References in corpus (1)

• A Truly Conformable Calculus on Time Scales