Papers › A study on partial dynamic equation on time scales involving derivatives of polynomials
A study on partial dynamic equation on time scales involving derivatives of polynomials
Petro Kolosov
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Let P(m,b,x) be a 2m+1-degree polynomial in x,b. Let be a two-dimensional timescale Λ² = 𝕋₁ ×𝕋₂ = {t=(x, b) x∈𝕋₁, b∈𝕋₂ } such that 𝕋₁ = 𝕋₂. 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∈𝕋₁ and (x,b) ∈Λ² (Δx²ᵐ⁺¹)/Δx(t) = (∂P(m,b,x))/Δx (m, σ(t), t) + (∂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.
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