Novel improved fractional operators and their scientific applications

Abd-Allah Hyder,M. A. Barakat

doi:10.1186/s13662-021-03547-x

Abd-Allah Hyder, M. A. Barakat

Open Access

https://doi.org/10.1186/s13662-021-03547-x

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Abstract

The motivation of this research is to introduce some new fractional operators called “the improved fractional (IF) operators”. The originality of these fractional operators comes from the fact that they repeat the method on general forms of conformable integration and differentiation rather than on the traditional ones. Hence the convolution kernels correlating with the IF operators are served in conformable abstract forms. This extends the scientific application scope of their fractional calculus. Also, some results are acquired to guarantee that the IF operators have advantages analogous to the familiar fractional integral and differential operators. To unveil the inverse and composition properties of the IF operators, certain function spaces with their characterizations are presented and analyzed. Moreover, it is remarkable that the IF operators generalize some fractional and conformable operators proposed in abundant preceding works. As scientific applications, the resistor–capacitor electrical circuits are analyzed under some IF operators. In the case of constant and periodic sources, this results in novel voltage forms. In addition, the overall influence of the IF operators on voltage behavior is graphically simulated for certain selected fractional and conformable parameter values. From the standpoint of computation, the usage of new IF operators is not limited to electrical circuits; they could also be useful in solving scientific or engineering problems.

Highlights

The traditional notions of derivatives and integrals are extended from integer orders to positive real or complex orders in fractional calculus [2, 4, 5, 29, 48, 53]
We show that the left improved fractional integral (LIFI), right improved fractional integral (RIFI), left improved fractional derivative (LIFD), and right improved fractional derivative (RIFD) are inclusive and cover many fractional and conformable operators proposed in multiple past works
2.2 Some advantages of the LIFI, RIFI, LIFD, and RIFD We provide some results to guarantee that the LIFI, RIFI, LIFD, and RIFD have advantages analogous to those of the familiar fractional integral and differential operators

Summary

Introduction

The traditional notions of derivatives and integrals are extended from integer orders to positive real or complex orders in fractional calculus [2, 4, 5, 29, 48, 53]. Remark 2 The LIFI and RIFI include some fractional integral operators defined in many previous works. (11) coincide with the left and right fractional integrals defined by Jarad et al. (2) If ω(θ, ρ) = θ1–ρ , μ = ν = 0, the LIFI and RIFI in (10) and (11) coincide with the generalized fractional left and right integrals defined by Katugampola in [33], respectively.

Results

Conclusion