The $\psi$-$\omega$-Mellin transform

TL;DR

This paper introduces a generalized Mellin transform within weighted fractional calculus, establishing connections with Laplace and Fourier transforms, and providing new properties, inversion formulas, and practical applications.

Contribution

It proposes a novel $oldsymbol{\psi ext{-}\omega ext{-}Mellin}$ transform and related operators, extending integral transform theory in weighted fractional calculus with respect to functions.

Findings

Developed a new $oldsymbol{\psi ext{-}\omega ext{-}Mellin}$ transform.

Established properties, inversion, and convolution theorems for the new operators.

Presented a practical application demonstrating the transform's utility.

Abstract

This manuscript introduces a generalization of the Mellin integral transform within the framework of weighted fractional calculus with respect to an increasing function. The proposed transform is much more suitable for working with fractional operators that involve a weight and are defined with respect to a function. This work also explores the connections between the Laplace and Fourier integral transforms in the same context. To achieve this, a new formulation of the weighted Fourier integral transform with respect to a function is presented, along with a new version of the bilateral Laplace transform. We study some of the properties of these new operators, obtain an inversion formula and a convolution theorem, and also present a practical application as an illustrative example.