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

What is all about

Source code of the manuscript entitled "A study on partial dynamic equation on time scales involving derivatives of polynomials" along with Mathematica programs in order to verify results.

How to use Mathematica package

• Open the package file AStudyOnDinamicEquationsPackage.m in Wolfram Mathematica, I use version 13.0
• Execute the package using Shift+Enter
• Open the notebook file AStudyOnDynamicEquationsNotebook.nb
• Execute the line: Needs["AStudyOnDynamicEquations"]
• Execute the line: mainTheorem[m_] := Expand[timeScaleDerivativeX[m, t, sigma[t]] + timeScaleDerivativeB[m, t, t]]
• Continue executing according to the guideline below

Reproduce the results using Mathematica

Few examples of the outcomes of the manuscript and how to reproduce them

Example for Z x Z time scale.

To reproduce example 4.2 proceed as follows with Mathematica:

• Set sigma[x_] := x + 1 in Mathematica package and execute definition.
• Execute timeScaleDerivativeX[1, x, b] which produces -3 b + 3 b^2.
• Execute Expand[timeScaleDerivativeX[1, t, sigma[t]]] which produces 3 t + 3 t^2.
• Execute timeScaleDerivativeB[1, x, b] which produces 1 - 6 b^2 + 6 b x.
• Execute timeScaleDerivativeB[1, t, t] which produces 1.
• Execute mainTheorem[1] which produces 1 + 3 t + 3 t^2.

Example for R x R time scale.

To reproduce example 4.7 proceed as follows with Mathematica:

• Set sigma[x_] := x + Globaldx` in Mathematica package and execute definition.
• Execute timeScaleDerivativeX[1, x, b] which produces -3 b + 3 b^2.
• Execute Limit[Expand[timeScaleDerivativeB[1, x, b]], dx -> 0] which produces 6 b - 6 b^2 - 3 x + 6 b x.
• Execute timeScaleDerivativeX[1, t, t] which produces -3 t + 3 t^2.
• Execute Limit[Expand[timeScaleDerivativeB[1, t, t]], dx -> 0] which produces 3t.
• Execute Limit[mainTheorem[1], dx -> 0] which produces 3t^2.