New Generalızed Fourıer Transforms and theır Applıcatıons to Ordınary, Partıal and Fractıonal Dıfferentıal Equatıons

Abstract

This article presents new generalized definitions of Fourier, Fourier sine, Fourier cosine, inverse Fourier, inverse Fourier sine and inverse Fourier cosine transforms, which encompass various studies on the generalized Fourier transforms in the existing literature. We also give some fundamental properties such that linearity, shifting, differentiability and convolution. Moreover, the solutions to the ordinary electric current differential equation and the fractional motion differential equation are obtained through the use of the generalized Fourier and inverse Fourier transforms. Subsequently, the solutions to the partial diffusion differential equation are obtained through the use of the generalized Fourier sine, inverse Fourier sine, Fourier cosine, and inverse Fourier cosine transforms. Furthermore, we illustrate the relations of the new generalized Fourier transforms with other the generalized Fourier transforms available in the literature. Finally, we provide tables of the new generalized Fourier transforms, and then graphs of the approximate behaviours of the solution of the ordinary electric current differential equation.