[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Discontinuity, Nonlinearity and Complexity
ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com

Oscillatory Criteria for Some non Conformable Differential Equation with Damping

Discontinuity, Nonlinearity, and Complexity 10(3) (2021) 461--469 | DOI:10.5890/DNC.2021.09.009

Juan E. N '{a}poles Valdes

UNNE, FaCENA, Av. Libertad 5450, (3400) Corrientes, Argentina and UTN, FRRE, French 414, (3500) Resistencia, Chaco, Argentina

Download Full Text PDF

 

Abstract

In this paper we present some criteria on the oscillation of solutions of a Non Conformable Differential Equations of $\alpha+\alpha$ order, under natural considerations. The local derivative considered was defined by the author in a previous work and a change of variables is used to transform the generalized differential equation into an ordinary differential equation of second order and using a Generalized Riccatti Transformation, together with known integration techniques, we obtain the desired results.

References

  1. [1]  Kilbas, A., Srivastava, H., and Trujillo, J. (2006), Theory and Applications of Fractional Differential Equations, in Math. Studies, North-Holland, New York.
  2. [2]  Miller, K.S. (1993), An Introduction to Fractional Calculus and Fractional Differential Equations, J. Wiley and Sons, New York.
  3. [3]  Oldham, K. and Spanier, J. (1974), The Fractional Calculus. Theory and Applications of Differentiation and Integration of Arbitrary Order, Academic Press, USA.
  4. [4]  Podlubny, I. (1999), Fractional Differential Equations, Academic Press, USA.
  5. [5]  Aguila Camacho, N., Duarte Mermoud, M.A., and Gallegos, J.A. (2014), Lyapunov functions for fractional order systems, Commun Nonlinear Sci Numer Simulat, {\bf19}, 2951-2957.
  6. [6]  Aguila Camacho, N. and Duarte Mermoud, M.A. (2016), Boundedness of the solutions for certain classes of fractional differential equations with application to adaptive systems, ISA Transactions, {\bf 60}, 82--88.
  7. [7]  Baleanu, D., Mustafa, O.G., and Agarwal, R.P. (2010), Asymptotically linear solutions for some linear fractional differential equations, Abstract Appl. Anal., Article ID 865139.
  8. [8]  Baranowski, J., Zagorowska, M., Bauer, W., Dziwinski, T., and Piatek, P. (2015), Applications of Direct Lyapunov Method in Caputo Non-Integer Order Systems, Electronika IR Electrotechnika, ISSN 1392-1215, {\b 21}(2).
  9. [9]  Burton, T.A. (2011), Fractional differential equations and Lyapunov functionals, Nonlinear Anal., {\bf 74}, 5648-5662.
  10. [10]  Chen, D., Zhang, R., Liu, X., and Ma, X. (2014), Fractional order Lyapunov stability theorem and its applications in synchronization of complex dynamical networks, Commun Nonlinear Sci Numer Simulat, {\bf 19}, 4105-4121.
  11. [11]  Delavari, H., Baleanu, D., and Sadati, J. (2012), Stability analysys of Caputo fractional order nonlinear systems revisited, Nonlinear Dyn, {\bf 67}, 2433-2439.
  12. [12]  Deng, W. (2010), Smoothness and stability of the solutions for nonlinear fractional differential equations, Nonlinear Anal., {\bf 72}, 1768-1777.
  13. [13]  Duarte Mermoud, M.A., Aguila Camacho, N., Gallegos, J.A., and Castro Linares, R. (2015), Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems, Commun Nonlinear Sci Numer Simulat, {\bf 22}, 650-659.
  14. [14]  Feng, Q. and Meng, F. (2013), Interval oscillation criteria for a class of nonlinear fractional differential equations, WSEAS Transactions on Mathematics, {\bf 12}(5), 564-571.
  15. [15]  Hu, J.B., Lu, G.P., Zhang, S.B., and Zhao, L.D. (2015), Lyapunov stability theorem about fractional system without and with delay, Commun Nonlinear Sci Numer Simulat, {\bf 20}, 905-913.
  16. [16]  Lakshmikantham, V., Leela, S., and Sambandham, M. (2008), Lyapunov theory for fractional differential equations, Commun. Appl. Anal., {\bf 12}, 365-376.
  17. [17]  Li, Y., Chen, Y.Q., and Podlubny, I. (2010), Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag Leffler stability, Comput. Math. Appl., {\bf 59}, 1810-1821.
  18. [18]  Trigeassou, J.C., Maamri, N., Sabatier, J., and Oustaloup, A. (2011), A Lyapunov approach to the stability of fractional differential equations, Signal Process, {\bf 91}, 437-445.
  19. [19]  Vargas De Le\{o}n, C. (2015), Volterra-type Lyapunov functions for fractional-order epidemic systems, Commun Nonlinear Sci Numer Simulat {\bf 24}, 75-85.
  20. [20]  Guzman, P.M. and N\apoles Valdes, J.E. (2019), A note on the oscillatory character of some non conformable generalized Li\enard system, Advanced Mathematical Models $\&$ Applications {\bf 4}, No.2, pp.127-133
  21. [21]  N\{a}poles V., J.E., On the oscillatory character of some non conformable fractional differential equation, submited.
  22. [22]  Khalil, R., Al Horani, M., Yousef, A., and Sababheh, M. (2014), A new definition of fractional derivative, J. Comput. Appl. Math., {\bf 264}, 65-70.
  23. [23]  Abdeljawad, T. (2015), On conformable fractional calculus. J. Comput. Appl. Math., {\bf 279}, 57-66.
  24. [24]  Katugampola, U.N. (2014), A new fractional derivative with classical properties. arXiv:1410.6535
  25. [25]  Guzman, P.M., Langton, G., Lugo, L.M., Medina, J., and N\{a}poles Vald\es, J.E. (2018), A new definition of a fractional derivative of local type, J. Math. Anal., {\bf 9}(2), 88-98.
  26. [26]  N\{a}poles V., J.E., Guzman, P.M., and Lugo, L.M. (2018), Some New Results on Nonconformable Fractional Calculus, Advances in Dynamical Systems and Applications, {\bf 13}, Number 2, pp. 167-175.
  27. [27] Guzman, P.M., Lugo, L.M., and N\{a}poles Vald\es, J.E. (2020), On the stability of solutions of fractional non conformable differential equations, Stud. Univ. Babes-Bolyai Math., 65(4), 495-502.
  28. [28]  He, J.E., Elagan, S.K., and Li, Z.B. (2012), Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus, Phys. Lett. A, 376, 257-259.
  29. [29]  Li, Z.B. (2010), An extended fractional complex transform, Int. J. Nonlinear Sci. Numer. Simul., {\bf 11}, 0335-0337.
  30. [30]  Li, Z.B. and He, J.H. (2011), Application of the fractional complex transform to fractional differential equations, Nonlinear Sci. Lett. A, {\bf 2}, 121-126.
  31. [31]  Yang, X.J. (2012), The Zero-mass Renormalization Group Differential Equations and Limit Cycles in Non-smooth Initial Value Problems, Prespacetime Journal, July, {\bf 3}, Issue 9, 913-923.
  32. [32]  Liu, C.S. (2015), Counterexamples on Jumaries two basic fractional calculus formulae, Commun Nonlinear Sci Numer Simulat, {\bf 22}, No.1-3, 92-94. DOI: 10.1016/j.cnsns.2014.07.022
  33. [33]  Bayram, M., Adiguzel, H., and Secer, A. (2016), Oscillation criteria for nonlinear fractional differential equation with damping term, Open Phys.; {\bf 14}, 119-128
  34. [34]  Bayram, M., Secer, A., and Adiguzel, H. (2017), On the oscillation of fractional order nonlinear differential equations, Sakarya \"{Universitesi Fen Bilimleri Enstit\"{u}s\"{u} Dergisi}, {\bf 21}(6), 1512-1523.
  35. [35]  Bekir, A., G\"{u}ner, O., and Cevikel, A.C. (2013), Fractional Complex Transform and exp-Function Methods for Fractional Differential Equations, Abstract and Applied Analysis, Article ID 426462, 8 pages http://dx.doi.org/10.1155/2013/426462
  36. [36]  Feng, Q. (2004), Oscillatory criteria for two fractional differential equations, WSEAS Transactions on Mathematics, Volume {\bf 13}, 800-810.
  37. [37]  Feng, Q. and Meng, F. (2013), Oscillation of solutions to nonlinear forced fractional differential equations, Electronic J. of Differential Equations Vol. 2013, {\bf 169}, 1-10.
  38. [38]  Ganesan, V., and Kumar, M.S. (2016), Oscillation theorems for fractional order neutral defferential equations, International J. of Math. Sci. Engg. Appls. (IJMSEA) ISSN 0973-9424, {\bf 10} No. III (December), 23-37.
  39. [39]  Li, Z.B. and He, J.H. (2010), Fractional complex transform for fractional differential equations, Math. Comput. Appl. {\bf 15}, 970-973.
  40. [40]  Mahdy, A.M.S. and Marai, G.M.A. (2018), Fractional Complex Transform for Solving the Fractional Differential Equations, Global Journal of Pure and Applied Mathematics, {\bf 14}, Number 1, pp. 17-37
  41. [41]  Qin, H. and Zheng, B. (2013), Oscillation of a Class of Fractional Differential Equations with Damping Term, The ScientificWorld Journal, Article ID {\bf 685621}, 9 pages http://dx.doi.org/10.1155/2013/685621
  42. [42]  Zayed, E.M.E., Amer, Y.A., and Shohib, R.M.A. (2016), The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics, Journal of the Association of Arab Universities for Basic and Applied Sciences {\bf 19}, 59-69.

Copyright © 2011-2023 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates