Abstract
This article aims to develop fractional order convolution theory to bring forth innovative methods for generating fractional Fourier transforms by having recourse to solutions for fractional difference equations. It is evident that fractional difference operators are used to formulate for finding the solutions of problems of distinct physical phenomena. While executing the fractional Fourier transforms, a new technique describing the mechanism of interaction between fractional difference equations and fractional differential equations will be introduced as h tends to zero. Moreover, by employing the theory of discrete fractional Fourier transform of fractional calculus, the modeling techniques will be improved, which would help to construct advanced equipments based on fractional transforms technology using fractional Fourier decomposition method. Numerical examples with graphs are verified and generated by MATLAB.
Highlights
Miller and Ross [27], Oldham and Spanier [29], and Podlubny [31] have developed continuous fractional calculus
Discrete delta fractional calculus has been developed by Atici and Eloe [12, 13, 15], Goodrich [21,22,23], and Holm [24]
The integral transforms, like Mellin, Laplace, Fourier, were applied to obtain the solution of differential equations. These transforms made effectively possible to change a signal in the time domain into that in the frequency s-domain in the field of Digital Signal Processing (DSP) [34]
Summary
Miller and Ross [27], Oldham and Spanier [29], and Podlubny [31] have developed continuous fractional calculus. For recent developments in the theory of discrete fractional calculus, applications of Mittag-Leffler function and fractional integral inequalities, we refer to [4,5,6, 8, 9, 16, 19, 25, 26, 32]. The integral transforms, like Mellin, Laplace, Fourier, were applied to obtain the solution of differential equations. These transforms made effectively possible to change a signal in the time domain into that in the frequency s-domain in the field of Digital Signal Processing (DSP) [34]. The more recent applications of fractional Fourier transform in Xray models and simulations are developed in [28, 30]. In [33], the forward complex DFT, Baleanu et al Advances in Difference Equations (2019) 2019:212 written in polar form, is given by
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