Some new properties and applications of a fractional Fourier transform

Abstract

In this paper, we deal with the fractional Fourier transform in the form introduced a little while ago by the first named author and his coauthors. This transform is closely connected with the Fractional Calculus operators and has been already employed for solving of both the fractional diffusion equation and the fractional Schrödinger equation. In this paper, we continue the investigation of the fractional Fourier transform, and in particular prove some new operational relations for a linear combination of the left- and righthand sided fractional derivatives. As an application of the obtained results, we provide a schema for solving the fractional differential equations with both leftand right-hand sided fractional derivatives without and with delays and give some examples of realization of our method for several fractional differential equations.