AIMS Mathematics, 2020, 5(3): 2126-2142. doi: 10.3934/math.2020141.

Research article

Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

On stability of a class of second alpha-order fractal differential equations

1 Department of Mathematics, Faculty of Sciences, Van Yuzuncu Yil University, 65080, Turkey
2 Young Researchers and Elite Club, Urmia Branch, Islamic Azad University, Urmia, Iran

In this paper, we give a review of fractal calculus which is an expansion of standard calculus. Fractal calculus is applied for functions that are not differentiable or integrable on totally disconnected fractal sets such as middle-μ Cantor sets. Analogues of the Lyapunov functions and their features are given for asymptotic behaviors of fractal differential equations. The stability of fractal differentials in the sense of Lyapunov is defined. For the suggested fractal differential equations, sufficient conditions for the stability and uniform boundedness and convergence of the solutions are presented and proved. We present examples and graphs for more details of the results.
  Figure/Table
  Supplementary
  Article Metrics

Keywords fractal calculus; staircase function; Cantor-like sets; fractal stability; fractal convergence

Citation: Cemil Tunç, Alireza Khalili Golmankhaneh. On stability of a class of second alpha-order fractal differential equations. AIMS Mathematics, 2020, 5(3): 2126-2142. doi: 10.3934/math.2020141

References

  • 1. B. B. Mandelbrot, The Fractal Geometry of Nature, New York: WH freeman, 1983.
  • 2. M. F. Barnsley, Fractals Everywhere, Academic press, 2014.
  • 3. C. Cattani, Fractals and hidden symmetries in DNA, Math. Probl. Eng., 2010, 2010.
  • 4. S. Kumari, R. Chugh, J. Cao, et al. Multi fractals of generalized multivalued iterated function systems in b-metric spaces with applications, Math., 7 (2019), 967.
  • 5. M. S. Tavazoei, M. Haeri, An optimization algorithm based on chaotic behavior and fractal nature, J. Comput. Appl. Math., 206 (2007), 1070-1081.    
  • 6. E. Nakaguchi, M. Efendiev, On a new dimension estimate of the global attractor for chemotaxisgrowth systems, Osaka J. Math., 45 (2008), 273-281.
  • 7. M. Efendiev, S. Zelik, Finite- and infinite-dimensional attractors for porous media equations, P. Lond. Math. Soc., 96 (2008), 51-77.    
  • 8. I. Chitescu, H. Georgescu, R.Miculescu, Approximation of infinite dimensional fractals generated by integral equations, J. Comput. Appl. Math., 234 (2010), 1417-1425.    
  • 9. S. Ree, L. E. Reichl, Fractal analysis of chaotic classical scattering in a cut-circle billiard with two openings, Phys. Rev. E, 65 (2002), 055205.
  • 10. E. de Amo, I. Chitescu, M. D. Carrillo, et al. A new approximation procedure for fractals, J. Comput. Appl. Math., 151 (2003), 355-370.    
  • 11. L. Nottale, J. Schneider, Fractals and nonstandard analysis, J. Math. Phys., 25 (1984), 1296-1300.    
  • 12. M. Czachor, Waves along fractal coastlines: From fractal arithmetic to wave equations, Acta Phys. Pol. B, 50 (2019), 813-831.    
  • 13. M. N. Célérier, L. Nottale, Quantum-classical transition in scale relativity, J. Phys. A: Math. Gen., 37 (2004), 931.
  • 14. K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley Sons, 2004.
  • 15. K. Falconer, Techniques in Fractal Geometry, Chichester (W. Sx.): Wiley, 1997.
  • 16. G. Edgar, Measure, Topology, and Fractal Geometry, Springer Science Business Media, 2007.
  • 17. J. Kigami, Analysis on Fractals, Cambridge University Press, 2001.
  • 18. F. H. Stillinger, Axiomatic Basis for Spaces with Noninteger Dimension, J. Math. Phys., 18 (1977), 1224-1234.    
  • 19. V. E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer Science Business Media, 2011.
  • 20. M. Zubair, M. J. Mughal, Q. A. Naqvi, Electromagnetic Fields and Waves in Fractional Dimensional Space, Springer Science Business Media, 2012.
  • 21. W. Chen Y. Liang, New methodologies in fractional and fractal derivatives modeling, Chaos Soliton. Fract., 102 (2017), 72-77.    
  • 22. Y. Liang, Q. Y. Allen, W. Chen, et al. A fractal derivative model for the characterization of anomalous diffusion in magnetic resonance imaging, Commun. Nonlinear Sci., 39 (2016), 529-537.    
  • 23. H. Richard, Fractional Calculus: an Introduction for Physicists, World Scientific, 2014.
  • 24. S. Das, Functional Fractional Calculus, Springer Science Business Media, 2011.
  • 25. C. Song, S. Fei, J. Cao, et al. Robust synchronization of fractional-order uncertain chaotic systems based on output feedback sliding mode control, Math., 7 (2019), 599.
  • 26. G. Rajchakit, A. Pratap, R. Raja, et al. Hybrid control scheme for projective lag synchronization of Riemann-Liouville sense fractional order memristive BAM neuralnetworks with mixed delays, Math., 7 (2019), 759.
  • 27. Y. Wei, L. Yin, X. Long, The coupling integrable couplings of the generalized coupled Burgers equation hierarchy and its Hamiltonian structure, Adv. Differ. Equ., 2019 (2019), 1-17.    
  • 28. X. Li, Z. Liu, J. Li, et al. Existence and controllability for nonlinear fractional control systems with damping in Hilbert spaces, Acta Math. Sci., 39 (2019), 229-242.    
  • 29. X. Li, Y. Li, Z. Liu, et al. Sensitivity analysis for optimal control problems described by nonlinear fractional evolution inclusions, Fract. Calc. Appl. Anal., 21 (2018), 1439-1470.    
  • 30. K. M. Kolwankar, A. D. Gangal, Fractional differentiability of nowhere differentiable functions and dimensions, Chaos, 6 (1996), 505-513.    
  • 31. K. M. Kolwankar, A. D. Gangal, Local fractional Fokker-Planck equation, Phys. Rev. Lett., 80 (1998), 214.
  • 32. A. Parvate, A. D. Gangal, Calculus on fractal subsets of real line-I: Formulation, Fract., 17 (2009), 53-148.    
  • 33. A. Parvate, A. D. Gangal, Calculus on fractal subsets of real line-II: Conjugacy with ordinary calculus, Fract., 19 (2011), 271-290.    
  • 34. A. Parvate, S. Satin, A. D. Gangal, Calculus on fractal curves in Rn, Fract., 19 (2011), 15-27.    
  • 35. A. K. Golmankhaneh, A. Fernandez, A. K. Golmankhaneh, et al. Diffusion on middle-ξ Cantor sets, Entropy, 20 (2018), 1-13.
  • 36. A. K. Golmankhaneh, A. S. Balankin, Sub-and super-diffusion on Cantor sets: Beyond the paradox, Phys. Lett. A, 382 (2018), 960-967.    
  • 37. A. K. Golmankhaneh, U. Branch, Quantum mechanics on the fractal time-space, In: Proceedings of the 8th International Conference on Fractional Differentiation and its Applications, 2016, 677-687.
  • 38. A. K. Golmankhaneh, C. Tunç, Sumudu transform in fractal calculus, Appl. Math. Comput., 350 (2019), 386-401.
  • 39. A. K. Golmankhaneh, C. Tunç, On the Lipschitz condition in the fractal calculus, Chaos, Soliton Fract., 95 (2017), 140-147.    
  • 40. C. Tunç, S. A. Mohammed, On the asymptotic analysis of bounded solutions to nonlinear differential equations of second order, Adv. Differ. Equ., 2019 (2019), 1-19.    
  • 41. G. C. Wu, D. Baleanu, W. H. Luo, Lyapunov functions for Riemann-Liouville-like fractional difference equations, Appl. Math. Comput., 314 (2017), 228-236.
  • 42. M. Rivero, S. V. Rogosin, J. A. T. Machado, et al. Stability of fractional order systems, Math. Probl. Eng., 2013 (2013).
  • 43. R. Agarwal, S. Hristova, D. O'Regan, A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations, Fract. Calc. Appl. Anal., 19 (2016), 290-318.
  • 44. A. Q. M. Khaliq, X. Liang, K. M. Furati, A fourth-order implicit-explicit scheme for the space fractional nonlinear Schrödinger equations, Numer. Algorithms, 75 (2016), 147-172.
  • 45. C. Tunç, Uniformly stability and boundedness of solutions of second order nonlinear delay differential equations, Appl. Comput. Math., 10 (2011), 449-462.
  • 46. C. Tunç, E. Tunç, On the asymptotic behavior of solutions of certain second-order differential equations, J. Franklin I., 344 (2007), 391-398.    
  • 47. T. Yoshizawa, Stability Theory by Liapunov's Second Method, 1966.
  • 48. T. Hara, On the asymptotic behavior of the solutions of some third and fourth order nonautonomous differential equations, Publ. Res. I. Math. Sci., 9 (1974), 649-673.
  • 49. R. DiMartino, W. Urbina, On Cantor-Like Sets and Cantor-Lebesgue Singular Functions, arXiv preprint arXiv:1403.6554, 2014.
  • 50. G. Braden, Fractal Time: The Secret of 2012 and a New World Age, Hay House Inc, 2010.

 

Reader Comments

your name: *   your email: *  

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved