Existence and UML-stability results for fractional Duffing equation with sequential Caputo-Riemann-Liouville derivatives
*Houas MohamedCorresponding authorm.houas.st@univ-dbkm.dzAffiliation 1Department of MathematicsFaculty of Sciences of Matter and Computer ScienceKhemis Miliana UniversityKhemis Miliana, AlgeriaAffiliation 2Laboratory FIMAKhemis Miliana UniversityKhemis Miliana, Algeria0000-0001-6256-0511View full profile →
* Corresponding author · click or hover a name for details
- Received:
- 01 Jul 2024
- Published Online:
- 09 May 2026
- Article type:
- Research Article
- Language:
- EN
- Article no.:
- JIM-2137
- Pages:
- 1–14
Abstract
Keywords
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References
[1] M. I. Abbas, “Existence and uniqueness of Mittag-Leffler-Ulam stable solution for fractional integrodifferential equations with nonlocal initial conditions,” European Journal of Pure and Applied Mathematics, vol. 8, no. 4, pp. 478-498 (2015).
[2] A. H. Ganie, M. Houas, and M. M. AlBaidani, “Solvability and Mittag-Leffler-Ulam stability for fractional Duffing problem with three sequential fractional derivatives,” Mathematical Methods in the Applied Sciences, vol. 47, no. 4, pp. 1807-1822 (2024).
[3] M. Bezziou, I. Jebril, and Z. Dahmani, “A new nonlinear Duffing system with sequential fractional derivatives,” Chaos, Solitons & Fractals, vol. 151, Art. no. 111247, pp. 1-7 (2021).
[4] H. Chen and Y. Li, “Rate of decay of stable periodic solutions of Duffing equations,” Journal of Differential Equations, vol. 236, pp. 493-503 (2007).
[5] S. Chandrasekhar, “An introduction to the study of stellar structure,” Ciel et Terre, vol. 55, pp. 412-415 (1939).
[6] J. Deng and L. Ma, “Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations,” Applied Mathematics Letters, vol. 23, no. 6, pp. 676-680 (2010).
[7] G. Duffing, Forced Oscillations with Variable Natural Frequency and Their Technical Significance. Braunschweig, Germany: Vieweg (1918).
[8] C. L. Ejikeme, M. O. Oyesanya, D. F. Agbebaku, and M. B. Okofu, “Solution to nonlinear Duffing oscillator with fractional derivatives using Homotopy Analysis Method (HAM),” Global Journal of Pure and Applied Mathematics, vol. 14, no. 10, pp. 1363-1383 (2018).
[9] A. Granas and J. Dugundji, Fixed Point Theory. New York, NY, USA: Springer-Verlag (2003).
[10] Y. Gouari, Z. Dahmani, and I. Jebril, “Application of fractional calculus on a new differential problem of Duffing type,” Advances in Mathematics: Scientific Journal, vol. 9, pp. 10989-11002 (2020).
[11] M. Houas and M. E. Samei, “Existence and stability of solutions for linear and nonlinear damping of q-fractional Duffing-Rayleigh problem,” Mediterranean Journal of Mathematics, vol. 20, no. 3, Art. no. 148, pp. 1-28 (2023).
[12] M. Houas, “Solvability and stability for fractional differential equations involving two Riemann-Liouville fractional orders,” Journal of Interdisciplinary Mathematics, vol. 26, no. 8, pp. 1699-1715 (2023).
[13] M. Houas and M. E. Samei, “Existence and Mittag-Leffler-Ulam stability results for Duffing type problem involving sequential fractional derivatives,” International Journal of Applied and Computational Mathematics, vol. 8, no. 4, Art. no. 185, pp. 1-24 (2022).
[14] M. Houas and M. Bezziou, “Existence and stability results for fractional differential equations with two Caputo fractional derivatives,” Facta Universitatis, Series: Mathematics and Informatics, vol. 34, no. 2, pp. 341-357 (2019).
[15] M. Houas, M. E. Samei, S. S. Santra, and J. Alzabut, “On a Duffing-type oscillator differential equation on the transition to chaos with fractional q-derivatives,” Journal of Inequalities and Applications, vol. 2024, Art. no. 12, pp. 1-28 (2024).
[16] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations. Amsterdam, The Netherlands: Elsevier (2006).
[17] V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis, vol. 69, pp. 2677-2682 (2008).
[18] K. D. Kucche and S. T. Sutar, “Analysis of nonlinear fractional differential equations involving Atangana-Baleanu-Caputo derivative,” Chaos, Solitons & Fractals, vol. 143, Art. no. 110556, pp. 1-9 (2021).
[19] S. Y. Lin, “Generalized Gronwall inequalities and their applications to fractional differential equations,” Journal of Inequalities and Applications, vol. 2013, Art. no. 549, pp. 1-9 (2013).
[20] A. C. Lazer and P. J. McKenna, “On the existence of stable periodic solutions of differential equations of Duffing type,” Proceedings of the American Mathematical Society, vol. 110, pp. 125-133 (1990).
[21] J. Niu, R. Liu, Y. Shen, and S. Yang, “Chaos detection of Duffing system with fractional order derivative by Melnikov method,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 29, pp. 123-126 (2019).
[22] A. U. K. Niazi, J. Wei, M. U. Rehman, and P. Denghao, “Ulam-Hyers-Mittag-Leffler stability for nonlinear fractional neutral differential equations,” Sbornik: Mathematics, vol. 209, no. 9, pp. 1337-1350 (2018).
[23] P. Pirmohabbati, A. H. Refahi Sheikhani, H. Saberi Najafi, and A. Abdolahzadeh Ziabari, “Numerical solution of full fractional Duffing equations with cubic-quintic-heptic nonlinearities,” AIMS Mathematics, vol. 5, no. 2, pp. 1621-1641 (2020).
[24] P. Pirmohabbati, A. H. R. Sheikhani, H. S. Najafi, and A. A. Ziabari, “Numerical solution of strongly nonlinear full fractional Duffing equation,” Journal of Interdisciplinary Mathematics, vol. 23, no. 8, pp. 1531-1551 (2020).
[25] L. Podlubny, Fractional Differential Equations. New York, NY, USA: Academic Press (1999).
[26] J. Sunday, “The Duffing oscillator: Applications and computational simulations,” Asian Research Journal of Mathematics, vol. 2, no. 3, pp. 1-13 (2017).
[27] K. Tablennehas and Z. Dahmani, “A three sequential fractional differential problem of Duffing type,” Applied Mathematics E-Notes, vol. 21, pp. 587-598 (2021).
[28] J. Wang and Y. Zhou, “Mittag-Leffler-Ulam stabilities of fractional evolution equations,” Applied Mathematics Letters, vol. 25, pp. 723-728 (2012).
[29] J. Wang and Y. Zhang, “Ulam-Hyers-Mittag-Leffler stability of fractional-order delay differential equations,” Optimization, vol. 63, no. 8, pp. 1181-1190 (2014).
