Notes on the q-Analogues of the Natural Transforms and Some Further Applications

TL;DR

This paper explores q-analogues of the natural transform, extending classical integral transforms to q-calculus, and applies these to special functions, convolutions, and derivatives, with potential applications in solving fluid flow problems.

Contribution

It introduces q-analogues of the natural transform and derives new formulas for special functions within q-calculus, expanding the mathematical toolkit for applied analysis.

Findings

Derived q-analogues of exponential, trigonometric, and hyperbolic functions

Established convolution and differentiation properties in q-calculus

Extended natural transform applications to special functions

Abstract

As an extension to the Laplace and Sumudu transforms the classical Natural transform was proposed to solve certain fluid flow problems. In this paper, we investigate q-analogues of the q-Natural transform of some special functions. We derive the q-analogues of the q-integral transform and further apply to some general special functions such as : the exponential functions, the q-trigonometric functions, the q-hyperbolic functions and the Heaviside Function. Some further results involving convolutions and differentiations are also obtained.