Mathematics and Computational Sciences

Computational approaches to time scale dynamic equations using the Elzaki transform

Document Type : Original Article

Authors

Shivaji Tarate 1
Kishor Kshirsagar 1
Vasant Nikam 2
Tarig Elzaki 3

1 Department of Mathematics, New Arts, Commerce and Science College, Ahilyanagar, India

2 Department of Mathematics, Loknete Vyankatrao Hiray Arts, Science and Commerce College, Nashik, India

3 Department of Mathematics, College of Sciences and Arts, University of Jeddah, Alkamel, Saudi Arabia

https://doi.org/10.30511/mcs.2026.2072721.1484
Abstract
Integral transform methods are effective techniques for addressing a range of dynamic equations characterized by initial or boundary value conditions, frequently expressed in the form of integral equations. This article presents the ET on an arbitrary timescale $\mathbb{T}$ as a new integral transform for addressing specific problems. The ET on timescales appears absent from the existing literature. This new approach primarily unifies discrete and continuous analysis, allowing for the treatment of differential, difference, and $q$-difference equations within a singular framework. This study's results pertain to ordinary differential equations for $\mathbb{T} = \mathbb{R}$, difference equations for $\mathbb{T} = \mathbb{N}_0$, and $q$-difference equations for $\mathbb{T} = q^{\mathbb{N}_0}$, where $q^{\mathbb{N}_0} = \{q^t | t \in \mathbb{N}_0 ~\text{for}~ q >1\} $ which hold significant relevance in quantum theory. The proposed transform can be applied to various nonstandard timescales, including $\mathbb{T} = h\mathbb{N}_0$, $\mathbb{T} = \mathbb{N}^{2}_{0}$, and $\mathbb{T} = \mathbb{T}_n$, which represent the space of harmonic numbers. Numerous examples and applications illustrate the efficacy of the ET on timescales in addressing dynamic equations.

Keywords

Elzaki Transform
Dynamic Equation
Time-scale
Subjects
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Mathematics and Computational Sciences
Volume 7, Issue 2
Spring 2026
Pages 156-174

  • Receive Date 25 September 2025
  • Revise Date 08 May 2026
  • Accept Date 08 May 2026

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