On the fractional Fourier transforms with respect to functions and its applications

The aim of this study is to introduce the fractional Fourier transform in the framework of Φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varPhi $$\end{document}-fractional calculus, which is called the generalized fractional Fourier transform. This proposed Fourier transform offers a more comprehensive form that encompasses both the classical Fourier transform and the fractional Fourier transform. We have established the connection between the fractional Fourier transform and differential and integral operators and investigated the inverse operator and certain properties of this transform method. Furthermore, we have determined the convolution for the fractional Fourier transform with respect to functions and some operational properties of convolution. Finally, we employ the generalized fractional Fourier transform to solve partial and ordinary differential equations of fractional order.