On conformable delta fractional calculus on time scales

Volume 16, Issue 3, pp 324--335 http://dx.doi.org/10.22436/jmcs.016.03.03

Publication Date: September 15, 2016 Submission Date: May 15, 2016

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 4317 Downloads
• 7081 Views

Authors

Dafang Zhao - School of Mathematics and Statistics, Hubei Normal University, Huangshi, Hubei 435002, P. R. China. Tongxing Li - School of Informatics, Linyi University, Linyi, Shandong 276005, P. R. China.

Abstract

In this paper, we introduce and investigate the concepts of conformable delta fractional derivative and conformable delta fractional integral on time scales. Basic properties of the theory are proved.

Share and Cite

Keywords

• Conformable delta fractional derivative
• conformable delta fractional integral
• time scale.

MSC

•  26A33
•  26E70

References

• [1] T. Abdeljawad, On conformable fractional calculus , J. Comput. Appl. Math., 279 (2015), 57-66.
  • View Article
  • MathSciNet
  • Google Scholar


• [2] T. Abdeljawad, M. Al Horani, R. Khalil, Conformable fractional semigroups of operators, J. Semigroup Theory Appl., 2015 (2015 ), 9 pages.
  • Google Scholar


• [3] I. Abu Hammad, R. Khalil, Fractional Fourier series with applications, Amer. J. Comput. Appl. Math., 4 (2014), 187-191.
  • View Article
  • Google Scholar


• [4] M. Abu Hammad, R. Khalil, Abel's formula and Wronskian for conformable fractional differential equations , Int. J. Differ. Equ. Appl., 13 (2014), 177-183.
  • View Article
  • Google Scholar
  • MATH


• [5] M. Abu Hammad, R. Khalil , Conformable fractional heat differential equations , Int. J. Pure. Appl. Math., 94 (2014), 215-221.


• [6] M. Abu Hammad, R. Khalil , Legendre fractional differential equation and Legendre fractional polynomials, Int. J. Appl. Math. Res., 3 (2014), 214-219.
  • View Article


• [7] R. P. Agarwal, M. Bohner, T. Li, Oscillatory behavior of second-order half-linear damped dynamic equations, Appl. Math. Comput., 254 (2015), 408-418.
  • View Article
  • MathSciNet
  • Google Scholar


• [8] M. Al Horani, M. Abu Hammad, R. Khalil, Variation of parameters for local fractional nonhomogeneous linear-differential equations, J. Math. Computer Sci., 16 (2016), 147-153.
  • View Article
  • Google Scholar


• [9] H. Batarfi, J. Losada, J. J. Nieto, W. Shammakh, Three-point boundary value problems for conformable fractional differential equations, J. Funct. Spaces, 2015 (2015 ), 6 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] B. Bayour, D. F. M. Torres , Existence of solution to a local fractional nonlinear differential equation, J. Comput. Appl. Math., 312 (2017 ), 127-133
  • View Article
  • MathSciNet
  • Google Scholar


• [11] N. Benkhettou, A. M. C. Brito da Cruz, D. F. M. Torres, A fractional calculus on arbitrary time scales: Fractional differentiation and fractional integration, Signal Process., 107 (2015), 230-237.
  • View Article
  • Google Scholar


• [12] N. Benkhettou, A. M. C. Brito da Cruz, D. F. M. Torres, Nonsymmetric and symmetric fractional calculi on arbitrary nonempty closed sets , Math. Methods Appl. Sci., 39 (2016), 261-279.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] N. Benkhettou, S. Hassani, D. F. M. Torres, A conformable fractional calculus on arbitrary time scales, J. King Saud Univ. Sci., 28 (2016), 93-98.
  • View Article
  • Google Scholar


• [14] M. Bohner, T. Li, Oscillation of second-order p-Laplace dynamic equations with a nonpositive neutral coeficient , Appl. Math. Lett., 37 (2014), 72-76.
  • View Article
  • Google Scholar


• [15] M. J. Bohner, R. R. Mahmoud, S. H. Saker, Discrete, continuous, delta, nabla, and diamond-alpha Opial inequalities, Math. Inequal. Appl., 18 (2015), 923-940.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] M. Bohner, A. Peterson , Dynamic Equations on Time Scales: An Introduction with Application, Birkhäuser, Boston (2001)


• [17] M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston (2003)
  • View Article
  • MathSciNet


• [18] M. Bohner, S. H. Saker, Sneak-out principle on time scales , J. Math. Inequal., 10 (2016), 393-403.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [19] A. Carpinteri, F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Vienna (1997)
  • View Article
  • MathSciNet


• [20] R. Herrmann , Fractional Calculus: An Introduction for Physicists, World Scientific, Singapore (2011)
  • Google Scholar
  • MATH


• [21] S. Hilger, Ein MaBkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph.D. Thesis, Universtät Würzburg (1988)


• [22] R. Khalil, M. Al Horani, D. Anderson , Undetermined coeficients for local fractional differential equations, J. Math. Computer Sci., 16, 140-146. (2016)


• [23] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70.
  • View Article
  • MathSciNet
  • Google Scholar


• [24] T. Li, J. Diblk, A. Domoshnitsky, Yu. V. Rogovchenko, F. Sadyrbaev, Q.-R. Wang, Qualitative analysis of differential, difference equations, and dynamic equations on time scales , Abstr. Appl. Anal., 2015 (2015 ), 3 pages.
  • View Article
  • Google Scholar
  • MATH


• [25] T. Li, S. H. Saker , A note on oscillation criteria for second-order neutral dynamic equations on isolated time scales , Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 4185-4188.
  • View Article
  • MathSciNet
  • Google Scholar


• [26] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations , John Wiley & Sons, Inc., New York (1993)
  • MathSciNet
  • Google Scholar


• [27] K. B. Oldham, J. Spanier , The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Dover Publications, New York-London (2006)


• [28] M. D. Ortigueira, Fractional Calculus for Scientists and Engineers, Springer, Dordrecht (2011)
  • View Article
  • MathSciNet
  • MATH


• [29] A. Peterson, B. Thompson , Henstock-Kurzweil delta and nabla integrals , J. Math. Anal. Appl., 323 (2006), 162-178.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [30] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego (1999)


• [31] J. Sabatier, O. P. Agrawal, J. A. T. Machado , Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht (2007)


• [32] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, CRC Press, Switzerland (1993)
  • Google Scholar