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Structural derivatives on time scales

Year 2019, Volume: 68 Issue: 1, 1186 - 1196, 01.02.2019
https://doi.org/10.31801/cfsuasmas.513107

Abstract

We introduce the notion of structural derivative on time scales. The new operator of differentiation unifies the concepts of fractal and fractional order derivative and is motivated by lack of classical differentiability of some self-similar functions. Some properties of the new operator are proved and illustrated with examples.

References

  • Atangana, A., Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system, Chaos Solitons Fractals 102 (2017), 396--406.
  • Bayour, B. and Torres, D. F. M., Complex-valued fractional derivatives on time scales, in Differential and difference equations with applications, 79--87, Springer Proc. Math. Stat., 164, Springer, 2016.
  • Bohner, M. and Peterson, A. Dynamic equations on time scales, Birkhäuser Boston, Boston, MA, 2001.
  • Bohner, M. and Peterson, A., Advances in dynamic equations on time scales, Birkhäuser Boston, Boston, MA, 2003.
  • Chen, W., Time-space fabric underlying anomalous diffusion, Chaos Solitons Fractals 28 (2006), no. 4, 923--929.
  • Chen, W., Liang, Y.-J. and Hei, X.-D., Local structural derivative and its applications, Chin. J. Solid Mech. 37 (2016), no. 5, 456--460.
  • Karci, A. and Karadogan, A., Fractional order derivative and relationship between derivative and complex functions, Math. Sci. Appl. E-Notes 2 (2014), no. 1, 44--54.
  • Strunin, D. V. and Suslov, S. A. Phenomenological approach to 3D spinning combustion waves: numerical experiments with a rectangular rod, Int. J. Self Prop. High Temp. Synth. 14 (2005), no. 1, 33--39.
  • Tarasov, V. E., Fractional hydrodynamic equations for fractal media, Ann. Physics 318 (2005), no. 2, 286--307.
  • Weberszpil, J., Lazo, M. J. and Helayel-Neto, J. A., On a connection between a class of q-deformed algebras and the Hausdorff derivative in a medium with fractal metric, Physica A 436 (2015), 399--404.
  • Weberszpil, J. and Sotolongo-Costa, O., Structural derivative model for tissue radiation response, J. Adv. Phys. 13 (2017), no. 4, 4779--4785.
  • Wio, H. S., Escudero, C., Revelli, J. A., Deza, R. R. and de la Lama, M. S., Recent developments on the Kardar-Parisi-Zhang surface-growth equation, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 369 (2011), no. 1935, 396--411.
  • Yablonskiy, D. A., Bretthorst, G. L. and Ackerman, J. J. H., Statistical model for diffusion attenuated MR signal, Mag. Res. Medicine 50 (2003), no. 4, 664--669.
Year 2019, Volume: 68 Issue: 1, 1186 - 1196, 01.02.2019
https://doi.org/10.31801/cfsuasmas.513107

Abstract

References

  • Atangana, A., Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system, Chaos Solitons Fractals 102 (2017), 396--406.
  • Bayour, B. and Torres, D. F. M., Complex-valued fractional derivatives on time scales, in Differential and difference equations with applications, 79--87, Springer Proc. Math. Stat., 164, Springer, 2016.
  • Bohner, M. and Peterson, A. Dynamic equations on time scales, Birkhäuser Boston, Boston, MA, 2001.
  • Bohner, M. and Peterson, A., Advances in dynamic equations on time scales, Birkhäuser Boston, Boston, MA, 2003.
  • Chen, W., Time-space fabric underlying anomalous diffusion, Chaos Solitons Fractals 28 (2006), no. 4, 923--929.
  • Chen, W., Liang, Y.-J. and Hei, X.-D., Local structural derivative and its applications, Chin. J. Solid Mech. 37 (2016), no. 5, 456--460.
  • Karci, A. and Karadogan, A., Fractional order derivative and relationship between derivative and complex functions, Math. Sci. Appl. E-Notes 2 (2014), no. 1, 44--54.
  • Strunin, D. V. and Suslov, S. A. Phenomenological approach to 3D spinning combustion waves: numerical experiments with a rectangular rod, Int. J. Self Prop. High Temp. Synth. 14 (2005), no. 1, 33--39.
  • Tarasov, V. E., Fractional hydrodynamic equations for fractal media, Ann. Physics 318 (2005), no. 2, 286--307.
  • Weberszpil, J., Lazo, M. J. and Helayel-Neto, J. A., On a connection between a class of q-deformed algebras and the Hausdorff derivative in a medium with fractal metric, Physica A 436 (2015), 399--404.
  • Weberszpil, J. and Sotolongo-Costa, O., Structural derivative model for tissue radiation response, J. Adv. Phys. 13 (2017), no. 4, 4779--4785.
  • Wio, H. S., Escudero, C., Revelli, J. A., Deza, R. R. and de la Lama, M. S., Recent developments on the Kardar-Parisi-Zhang surface-growth equation, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 369 (2011), no. 1935, 396--411.
  • Yablonskiy, D. A., Bretthorst, G. L. and Ackerman, J. J. H., Statistical model for diffusion attenuated MR signal, Mag. Res. Medicine 50 (2003), no. 4, 664--669.
There are 13 citations in total.

Details

Primary Language English
Journal Section Review Articles
Authors

Benaoumeur Bayour This is me 0000-0003-3504-4655

Delfim F. M. Torres 0000-0001-8641-2505

Publication Date February 1, 2019
Submission Date May 14, 2018
Acceptance Date November 18, 2018
Published in Issue Year 2019 Volume: 68 Issue: 1

Cite

APA Bayour, B., & Torres, D. F. M. (2019). Structural derivatives on time scales. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(1), 1186-1196. https://doi.org/10.31801/cfsuasmas.513107
AMA Bayour B, Torres DFM. Structural derivatives on time scales. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. February 2019;68(1):1186-1196. doi:10.31801/cfsuasmas.513107
Chicago Bayour, Benaoumeur, and Delfim F. M. Torres. “Structural Derivatives on Time Scales”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68, no. 1 (February 2019): 1186-96. https://doi.org/10.31801/cfsuasmas.513107.
EndNote Bayour B, Torres DFM (February 1, 2019) Structural derivatives on time scales. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68 1 1186–1196.
IEEE B. Bayour and D. F. M. Torres, “Structural derivatives on time scales”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 68, no. 1, pp. 1186–1196, 2019, doi: 10.31801/cfsuasmas.513107.
ISNAD Bayour, Benaoumeur - Torres, Delfim F. M. “Structural Derivatives on Time Scales”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68/1 (February 2019), 1186-1196. https://doi.org/10.31801/cfsuasmas.513107.
JAMA Bayour B, Torres DFM. Structural derivatives on time scales. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68:1186–1196.
MLA Bayour, Benaoumeur and Delfim F. M. Torres. “Structural Derivatives on Time Scales”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 68, no. 1, 2019, pp. 1186-9, doi:10.31801/cfsuasmas.513107.
Vancouver Bayour B, Torres DFM. Structural derivatives on time scales. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68(1):1186-9.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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