Adil Khurshaid* and Hajra Khurshaid
Volume4-Issue12
Dates: Received: 2023-11-30 | Accepted: 2023-12-21 | Published: 2023-12-22
Pages: 1684-1688
Abstract
Fractional Calculus (FC) has emerged as a valuable tool in various fields. This study explores the historical development of (FC) and examines prominent definitions regarding Fractional Derivatives (FD), such as the Riemann-Liouville, Grunwald-Letnikov, Caputo Fractional Derivative, Katugampula derivatives, Caputo Fractional Derivative, Caputo-Fabrizio Fractional Derivative and as well as Atangana-Baleanu Fractional Derivative. It critically evaluates their strengths, weaknesses and implications on (FD) equations. The findings contribute to establishing a clearer understanding of Fractional Derivatives (FD) and guiding their appropriate use in practical scenarios. It investigates the impact of different definitions on the properties and behaviors of (FD), providing valuable insights for researchers.
FullText HTML
FullText PDF
DOI: 10.37871/jbres1852
Certificate of Publication
Copyright
© 2023 Khurshaid A, et al. Distributed under Creative Commons CC-BY 4.0 
How to cite this article
Khurshaid A, Khurshaid H. Comparative Analysis and Defi nitions of Fractional Derivatives. J Biomed Res Environ Sci. 2023 Dec 22; 4(12): 1684-1688. doi: 10.37871/jbres1852, Article ID: JBRES1852, Available at: https://www.jelsciences.com/articles/ jbres1852.pdf
Subject area(s)
References
- Mainardi F, Gorenflo R. Fractional calculus and special functions. Phys. Univ. Bologna. 2013;1-64.
- Du M, Wang Z, Hu H. Measuring memory with the order of fractional derivative. Scientific Reports. 2013;3(1):1-3.
- De Oliveira EC, Tenreiro Machado JA. A review of definitions for fractional derivatives and integral. Mathematical Problems in Engineering. 2014;1-6.
- Dalir M, Bashour M. Applications of fractional calculus. Applied Mathematical Sciences. 2010;4(21):1021-1032.
- Hartley TT, Lorenzo CF. Dynamics and control of initialized fractional-order systems. Nonlinear Dynamics. 2002;29:201-233.
- You H, Shen Y, Xing H, Yang S. Optimal control and parameters design for the fractional-order vehicle suspension system. Journal of Low Frequency Noise, Vibration and Active Control. 2018;37(3):456-467. doi: 10.1177/0263092317717166.
- Mladenov V, Mastorakis N. Advanced topics on applications of fractional calculus on control problems, system stability and modeling. World Scientific and Engineering Academy and Society. 2014;1-12.
- Miller KG, Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations.1st ed. New York:Wiley; 1993:1-384
- Khaliq A, Mujeeb R. Existence of weak solutions for Ψ-Caputo fractional boundary value problem via variational methods. J. Appl. Anal. Comput. 2021;11:768-1778. doi: 10.11948/20200115.
- Nazeer N, Asjad MI, Azam MK, Akgül A. Study of Results of Katugampola Fractional Derivative and Chebyshev Inequailities. International Journal of Applied and Computational Mathematics. 2022; 8(5):1-16. doi: 10.1007/s40819-022-01426-x.
- Moore EJ, Sirisubtawee S, Koonprasert S. A Caputo–Fabrizio fractional differential equation model for HIV/AIDS with treatment compartment. Advances in Difference Equations. 2019;(1):1-20. doi: 10.1186/s13662-019-2138-9.
- Bas E, Ozarslan R. Real world applications of fractional models by Atangana–Baleanu fractional derivative. Chaos, Solitons & Fractals. 2018; 116:121-125. doi: 10.1016/j.chaos.2018.09.019.
- Atangana A. Fractional Operators and Their Applications. In Elsevier eBooks. 2018;79–112. doi: 10.1016/B978-0-12-809670-3.00005-9.
- Bouzenna FEG, Meftah MT, Difallah M. Application of the Caputo Fabrizio derivative without singular kernel to fractional Schrödinger equations. Pramana. 2020;94(1):1-7. doi: 10.1007/s12043-020-01968-z.
- Atangana A, Koca I. Chaos in a simple nonlinear system with Atangana–Baleanu derivatives with fractional order. Chaos, Solitons & Fractals. 2016; 89:447-454. doi: 10.1016/j.chaos.2016.02.012.
- Soltanpour Moghadam A, Arabameri M, Baleanu D, Barfeie M. Numerical solution of variable fractional order advection‐dispersion equation using Bernoulli wavelet method and new operational matrix of fractional order derivative. Mathematical Methods in the Applied Sciences. 2020;43(7):3936-3953. doi: 10.1002/mma.6164.
