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| Format: | Preprint |
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2016
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| Online Access: | https://arxiv.org/abs/1608.00801 |
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| _version_ | 1866910608247488512 |
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| author | Kolosov, Petro |
| author_facet | Kolosov, Petro |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1608_00801 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | A study on partial dynamic equation on time scales involving derivatives of polynomials Kolosov, Petro Classical Analysis and ODEs Analysis of PDEs 26E70, 05A30 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. |
| title | A study on partial dynamic equation on time scales involving derivatives of polynomials |
| topic | Classical Analysis and ODEs Analysis of PDEs 26E70, 05A30 |
| url | https://arxiv.org/abs/1608.00801 |