A New Truncated M-Fractional Derivative Type Unifying Some Fractional Derivative Types with Classical Properties

Sousa, J. Vanterler da C.; Oliveira, E. Capelas de

doi:10.28924/2291-8639-16-2018-83

Research Article

A New Truncated M-Fractional Derivative Type Unifying Some Fractional Derivative Types with Classical Properties

J. Vanterler da C. Sousa¹*
E. Capelas de Oliveira²

unicamp
Unicamp

* Corresponding author

Abstract

We introduce a truncated $M$-fractional derivative type for $\alpha$-differentiable functions that generalizes four other fractional derivatives types recently introduced by Khalil et al., Katugampola and Sousa et al., the so-called conformable fractional derivative, alternative fractional derivative, generalized alternative fractional derivative and $M$-fractional derivative, respectively. We denote this new differential operator by $_{i}\mathscr{D}_{M}^{\alpha,\beta }$, where the parameter $\alpha$, associated with the order of the derivative is such that $ 0 <\alpha<1 $, $\beta>0$ and $ M $ is the notation to designate that the function to be derived involves the truncated Mittag-Leffler function with one parameter.

The definition of this truncated $M$-fractional derivative type satisfies the properties of the integer-order calculus. We also present, the respective fractional integral from which emerges, as a natural consequence, the result, which can be interpreted as an inverse property. Finally, we obtain the analytical solution of the $M$-fractional heat equation and present a graphical analysis.

How to Cite

J. V. d. C. Sousa, E. C. d. Oliveira, A New Truncated M-Fractional Derivative Type Unifying Some Fractional Derivative Types with Classical Properties, Int. J. Anal. Appl. 16 (2018), 83–96. https://doi.org/10.28924/2291-8639-16-2018-83

BibTeX

@article{Sousa2018,
  author  = {Sousa, J. Vanterler da C. and Oliveira, E. Capelas de},
  title   = {A New Truncated M-Fractional Derivative Type Unifying Some Fractional Derivative Types with Classical Properties},
  journal = {Int. J. Anal. Appl.},
  volume  = {16},
  year    = {2018},
  pages   = {83--96},
  doi     = {10.28924/2291-8639-16-2018-83}
}

References

[1] R. Herrmann, Fractional calculus: An Introduction for Physicists, World Scientific Publishing Company, Singapore, 2011.
[2] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, Elsevier, Amsterdam, Vol. 207, 2006.
[3] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, San Diego, Vol. 198, 1999.
[4] E. Capelas de Oliveira, J. A. Tenreiro Machado, A review of definitions for fractional derivatives and integral, Math. Probl. Eng., 2014, (2014) (238459).
[5] U. N. Katugampola, New fractional integral unifying six existing fractional integrals, arxiv.org/abs/1612.08596, (2016).
[6] R. Figueiredo Camargo, E. Capelas de Oliveira, Fractional Calculus (In Portuguese), Editora Livraria da F ´isica, SËœ ao Paulo, 2015.
[7] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math, 264, (2014) 65-70.
[8] U. N. Katugampola, A new fractional derivative with classical properties, arXiv:1410.6535v2, (2014).
[9] J. Vanterler da C. Sousa, E. Capelas de Oliveira, M-fractional derivative with classical properties, Submitted, (2017).
[10] R. Gorenflo, A. A. Kilbas, F. Mainardi, S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer, Berlin, 2014.
[11] A. A. Kilbas, H. M.Srivastava, J. J. Trujillo, Theory and Applications of the Fractional Differential Equations, Vol. 204, Elsevier, Amsterdam, 2006.
[12] M. Rafikov, J. M. Balthazar, Optimal pest control problem in population dynamics, Computational & Applied Mathematics 24 (1) (2005) 65-81.
[13] O. S. Iyiola, E. R. Nwaeze, Some new results on the new conformable fractional calculus with application using d'Alembert approach, Progr. Fract. Differ. Appl. 2, (2016) 115-122.
[14] Y. Cenesiz, A. Kurt, The solutions of time and space conformable fractional heat equations with conformable fourier transform, Acta Univ. Sapientiae, Math. 7 (2) (2015) 130-140.