Computational approaches to time scale dynamic equations using the Elzaki transform

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 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 T = R, difference equations for T = N₀, and q-difference equations for T = qN₀, where qN₀ = ᵗ | t ∈ N₀ ~for~ q >1\ which hold significant relevance in quantum theory. The proposed transform can be applied to various nonstandard timescales, including T = hN₀, T = N²₀, and T = Tₙ, which represent the space of harmonic numbers. Numerous examples and applications illustrate the efficacy of the ET on timescales in addressing dynamic equations.