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.
Keywords
Hausdorff derivative of a function with respect to a fractal measure structural and fractal derivatives self-similarity time scales hilger derivative of non-integer order
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.
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.
Details
| Primary Language | English |
|---|---|
| Journal Section | Research Article |
| Authors | |
| Submission Date | May 14, 2018 |
| Acceptance Date | November 18, 2018 |
| Publication Date | February 1, 2019 |
| DOI | https://doi.org/10.31801/cfsuasmas.513107 |
| IZ | https://izlik.org/JA54YM56DH |
| Published in Issue | Year 2019 Volume: 68 Issue: 1 |
Cite
Cited By
Linear fractional dynamic equations: Hyers–Ulam stability analysis on time scale
Results in Control and Optimization
https://doi.org/10.1016/j.rico.2023.100347
Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics
