Research Article
BibTex RIS Cite
Year 2016, Volume: 4 Issue: 1, 31 - 39, 15.04.2016
https://doi.org/10.36753/mathenot.421356

Abstract

References

  • [1] Butzer, P.L., Kilbas, A. A. and Trujillo, J. J., Mellin transform analysis and integration by parts for Hadamard-type fractional integrals, J. Math. Anal. Appl. 270 (2002), no. 1, 1-15.
  • [2] Butzer, P.L., Kilbas, A. A. and Trujillo, J. J., Fractional calculus in the Mellin setting and Hadamard-type fractional integrals, J. Math. Anal. Appl. 269 (2002), no. 1, 1-27.
  • [3] Butzer, P.L., Kilbas, A. A. and Trujillo, J. J., Stirling functions of the second kind in the setting of difference and fractional calculus, Numer. Funct. Anal. Optim. 24 (2003), no. 7-8, 673-711.
  • [4] Hadamard, J., Essai sur l’etude des fonctions donnees par leur developpment de Taylor, J. Pure Appl. Math. 4 (1892), no. 8, 101-186.
  • [5] Jarad, F., Abdeljawad, T. and Baleanu, D., Caputo-type modification of the Hadamard fractional derivatives, Advances in Difference Equations August 2012 (2012), 2012–142.
  • [6] Kilbas, A.A., Hadamard-type fractional calculus, J. Korean Math. Soc. 38 (2001), no. 6, 1191- 1204.
  • [7] Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J., Theory and applications of fractional differential equations. North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.
  • [8] Kilbas, A.A. and Titioura, A.A., Nonlinear differential equations with Marchaud-Hadamardtype fractional derivative in the weighted space of summable functions, Math. Model. Anal. 12 (2007), no. 3, 343-356.
  • [9] Pooseh, S., Almeida, R. and Torres, D. F. M., Expansion formulas in terms of integer-order derivatives for the Hadamard fractional integral and derivative, Numer. Funct. Anal. Optim. 33 (2012), no. 3, 301–319.
  • [10] Qassim, M. D., Furati, K. M. and Tatar, N.E., On a Differential Equation Involving Hilfer- Hadamard Fractional Derivative, Abstract and Applied Analysis vol. 2012 (2012), Article ID 391062, 17 pages, doi:10.1155/2012/391062.
  • [11] Qian, D., Gong, Z. and Li, C., A generalized Gronwall inequality and its application to fractional differential equations with Hadamard derivatives, 3rd Conference on Nonlinear Science and Complexity (NSC10), Cankaya University, Ankara, Turkey, 28–31 July, 2010.

A Discretization of the Hadamard fractional derivative

Year 2016, Volume: 4 Issue: 1, 31 - 39, 15.04.2016
https://doi.org/10.36753/mathenot.421356

Abstract

We present a new discretization for the Hadamard fractional derivative, that simplifies

the computations. We then apply the method to solve a fractional differential equation

and a fractional variational problem with dependence on the Hadamard fractional

derivative.

References

  • [1] Butzer, P.L., Kilbas, A. A. and Trujillo, J. J., Mellin transform analysis and integration by parts for Hadamard-type fractional integrals, J. Math. Anal. Appl. 270 (2002), no. 1, 1-15.
  • [2] Butzer, P.L., Kilbas, A. A. and Trujillo, J. J., Fractional calculus in the Mellin setting and Hadamard-type fractional integrals, J. Math. Anal. Appl. 269 (2002), no. 1, 1-27.
  • [3] Butzer, P.L., Kilbas, A. A. and Trujillo, J. J., Stirling functions of the second kind in the setting of difference and fractional calculus, Numer. Funct. Anal. Optim. 24 (2003), no. 7-8, 673-711.
  • [4] Hadamard, J., Essai sur l’etude des fonctions donnees par leur developpment de Taylor, J. Pure Appl. Math. 4 (1892), no. 8, 101-186.
  • [5] Jarad, F., Abdeljawad, T. and Baleanu, D., Caputo-type modification of the Hadamard fractional derivatives, Advances in Difference Equations August 2012 (2012), 2012–142.
  • [6] Kilbas, A.A., Hadamard-type fractional calculus, J. Korean Math. Soc. 38 (2001), no. 6, 1191- 1204.
  • [7] Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J., Theory and applications of fractional differential equations. North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.
  • [8] Kilbas, A.A. and Titioura, A.A., Nonlinear differential equations with Marchaud-Hadamardtype fractional derivative in the weighted space of summable functions, Math. Model. Anal. 12 (2007), no. 3, 343-356.
  • [9] Pooseh, S., Almeida, R. and Torres, D. F. M., Expansion formulas in terms of integer-order derivatives for the Hadamard fractional integral and derivative, Numer. Funct. Anal. Optim. 33 (2012), no. 3, 301–319.
  • [10] Qassim, M. D., Furati, K. M. and Tatar, N.E., On a Differential Equation Involving Hilfer- Hadamard Fractional Derivative, Abstract and Applied Analysis vol. 2012 (2012), Article ID 391062, 17 pages, doi:10.1155/2012/391062.
  • [11] Qian, D., Gong, Z. and Li, C., A generalized Gronwall inequality and its application to fractional differential equations with Hadamard derivatives, 3rd Conference on Nonlinear Science and Complexity (NSC10), Cankaya University, Ankara, Turkey, 28–31 July, 2010.
There are 11 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Ricardo Almeida

Nuno R. O. Bastos

Publication Date April 15, 2016
Submission Date September 25, 2015
Published in Issue Year 2016 Volume: 4 Issue: 1

Cite

APA Almeida, R., & Bastos, N. R. O. (2016). A Discretization of the Hadamard fractional derivative. Mathematical Sciences and Applications E-Notes, 4(1), 31-39. https://doi.org/10.36753/mathenot.421356
AMA Almeida R, Bastos NRO. A Discretization of the Hadamard fractional derivative. Math. Sci. Appl. E-Notes. April 2016;4(1):31-39. doi:10.36753/mathenot.421356
Chicago Almeida, Ricardo, and Nuno R. O. Bastos. “A Discretization of the Hadamard Fractional Derivative”. Mathematical Sciences and Applications E-Notes 4, no. 1 (April 2016): 31-39. https://doi.org/10.36753/mathenot.421356.
EndNote Almeida R, Bastos NRO (April 1, 2016) A Discretization of the Hadamard fractional derivative. Mathematical Sciences and Applications E-Notes 4 1 31–39.
IEEE R. Almeida and N. R. O. Bastos, “A Discretization of the Hadamard fractional derivative”, Math. Sci. Appl. E-Notes, vol. 4, no. 1, pp. 31–39, 2016, doi: 10.36753/mathenot.421356.
ISNAD Almeida, Ricardo - Bastos, Nuno R. O. “A Discretization of the Hadamard Fractional Derivative”. Mathematical Sciences and Applications E-Notes 4/1 (April 2016), 31-39. https://doi.org/10.36753/mathenot.421356.
JAMA Almeida R, Bastos NRO. A Discretization of the Hadamard fractional derivative. Math. Sci. Appl. E-Notes. 2016;4:31–39.
MLA Almeida, Ricardo and Nuno R. O. Bastos. “A Discretization of the Hadamard Fractional Derivative”. Mathematical Sciences and Applications E-Notes, vol. 4, no. 1, 2016, pp. 31-39, doi:10.36753/mathenot.421356.
Vancouver Almeida R, Bastos NRO. A Discretization of the Hadamard fractional derivative. Math. Sci. Appl. E-Notes. 2016;4(1):31-9.

20477

The published articles in MSAEN are licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.