A note on some recent results of the conformable fractional derivative

Year 2019, Volume: 3 Issue: 1, 11 - 17, 31.03.2019

O. Taghipour Birgani Sumit Chandok , Nebojsa Dedovic , Stojan Radenovic

https://doi.org/10.31197/atnaa.482525

Cited By: 13

Abstract

In this note, we discuss, improve and complement some recent results of the conformable fractional derivative introduced and established by Katugampola

[arxiv:1410.6535v1] and Khalil et al. [J. Comput. Appl. Math. 264(2014) 65-70]. Among other things we show that each function $f$ defined on $(a,b)$, $a>0$ has a conformable fractional derivative (CFD) if and only if it has a classical first derivative. At the end of the paper, we prove the Rolle's, Cauchy, Lagrange's and Darboux's theorem in the context of Conformable Fractional Derivatives.

Keywords

Conformable fractional derivative; fractional derivative; fractional integral; Riemann-Liouvelle definition; Caputo definition; fractional differential equations

References

• \bibitem{AC} F. B. Adda and J. Cresson, \emph{Fractional differentialequations and the Schr\"{o}dinger equation,} App. Math. Comput.\textbf{161}(2005) 323-345
• \bibitem{AMA} B. Ahmad, M. M. Matar and R. P. Agarwal, \emph{Existenceresults for fractional differential equations of arbitrary order withnonlocal integral boundary conditions,} Boundary Value Problem (2015)2015:220
• \bibitem{Alm} R. Almeida, \emph{What is the best fractional derivative tofit data?} to appear
• \bibitem{TAb} T. Abdeljawad, \emph{On conformable fractional calculus,} J.Comput. Appl. Math. \textbf{729} (2015) 57-66.
• \bibitem{AiZ} J. Alzabut, T. Abdeljawad, \emph{A generalized discretefractional Gronwall inequality and its application on the uniqueness ofsolutions for nonlinear delayed fractional difference system}, to appear
• \bibitem{BKMG} L. B. Budhia, P. Kumam, J. M. Moreno and D. Gopal, \emph{%Extensions of almost-F and F-Suzuki contractions with graph and someapplications to fractional calculus,} Fixed Point Theory Appl. (2016) 2016:2
• \bibitem{Die} K. Diethelm and Neville J. Ford, \emph{Analysis of FractionalDifferential Equations,} J. Math. Anal. Appl. Volume 265, Issue 2, 15January 2002, Pages 229-248
• \bibitem{KaT} U. N. Katugampola, \emph{A new fractional derivative withclassical properties}, arXiv:1410.6535v1 [math.CA] 24Oct2014
• \bibitem{KHYS} R. Khalil, A. Al Horani, A. Yousef, M. Sabadheh, \emph{A newdefinition of fractional derivative,} J. Comput. Appl. Math. 264 (2014) 65-70
• \bibitem{KiL} A. Kilbas, H. Srivistava, J. Trujillo, \emph{Theory andApplications of Fractional Differential Equations, in: Math. Studies.,}North-Holand, New York, 2006
• \bibitem{LV} V. Lakshmikantham, A. S. Vatsala, \emph{Basic theory offractional differential equations,} Nonlinear Anal. \textbf{69} (2008)2677-2682
• \bibitem{LGS} H. Lakzian, D. Gopal and W. Sintunavarat, \emph{New fixedpoint results for mappings of contractive type with an application tononlinear fractional differential equations,} J. Fixed Point Theory Appl.DOI 10.1007/s11784-015-0275-7
• \bibitem{LoV} A. Loverro, \emph{Fractioanal Calculus: History, Definitionsand Applications for the Engineer, }Department of Aerospace and MechanicalEngineering, University of Notre Dame, Notre Dame, IN 46556, U.S.A
• \bibitem{Mi} K. S. Miler, \emph{An Introduction to Fractional Calculus andFractional Differential Equations,} J. Wiley and Sons, New Yorkl, 1993
• \bibitem{OlD} K. Oldham, J. Spanier, \emph{The Fractional Calculus, Theoryand Applications of Differentiation and Integration of Arbitrary Order,}Academic Press, USA, 1974
• \bibitem{OM} E. C. de Oliveira and J. A. T. Machado, \emph{A review ofdefinitions for fractional derivatives and integral,} Mathematical Problemsin Engineering, Volume 2014, Article ID 238459, 6 pages
• \bibitem{OTM} M. D. Ortigueira, J. A. T. Machado, \emph{What is afractional derivative?} Journal of Computational Physics \textbf{293} (2015)4-13
• \bibitem{Po} I. Podlubny, \emph{Fractional Differential Equations,}Academic Press, USA, 1999
• \bibitem{MR} M. Rahimy, \emph{Applications of Fractional DifferentialEquations}, Applied Mathematical Science, Vol. \textbf{4}, 2010, no. 50,2453-2461
• \bibitem{Su} C.M. Su, J. P. Sun, Y. H. Zhao, \emph{Existence and uniquenessof solutions for BVP of nonlinear fractional differential equation}, toappear
• \bibitem{Zh} S. Zhang, \emph{The existence of a positive solution for anonlinear fractional differential equation,} Journal of Math. Anal. Appl.\textbf{252}, 804-812 (2000).