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A note on some recent results of the conformable fractional derivative

Yıl 2019, Cilt: 3 Sayı: 1, 11 - 17, 31.03.2019
https://doi.org/10.31197/atnaa.482525

Öz

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.

Kaynakça

  • \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).
Yıl 2019, Cilt: 3 Sayı: 1, 11 - 17, 31.03.2019
https://doi.org/10.31197/atnaa.482525

Öz

Kaynakça

  • \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).
Toplam 21 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Articles
Yazarlar

O. Taghipour Birgani Bu kişi benim

Sumit Chandok

Nebojsa Dedovic

Stojan Radenovic

Yayımlanma Tarihi 31 Mart 2019
Yayımlandığı Sayı Yıl 2019 Cilt: 3 Sayı: 1

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