Stability Analysis of Atangana-Baleanu Fractional Delay Differential Equations

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-24-2026-193
Published: Jun 22, 2026
Cite as:
Wisdom Kevin Udogworen, Michael Precious Ineh, Augustine Bwan Panle, Adhir Maharaj, Ndianabasi Peter, Stability Analysis of Atangana-Baleanu Fractional Delay Differential Equations, Int. J. Anal. Appl., 24 (2026), 193.

Main Article Content

Wisdom Kevin Udogworen, Michael Precious Ineh, Augustine Bwan Panle, Adhir Maharaj, Ndianabasi Peter

Abstract

This work investigates the Ulam-Hyers and generalized Ulam-Hyers-Rassias stability of fractional delay differential equations involving the Atangana-Baleanu derivative with Mittag-Leffler kernel. Numerical simulations validate the theoretical results, showing close agreement between exact and approximate solutions and confirming bounded error behaviour. These findings demonstrate the effectiveness of the Atangana-Baleanu fractional delay framework for modelling systems with memory and delay, particularly in epidemiological and immune response applications.

Article Details

References

  1. T. Abdeljawad, {Fractional Operators with Exponential Kernels and a Lyapunov-type inequality}, Adv. in Diff. Eq. 313 (2017). https://doi.org/10.1186/s13662-017-1285-.
  2. T. Abdeljawa, D. Baleanu, {On Fractional Derivatives with exponential Kernel and their discrete versions}, Reports on Mathematics Physics, {80} (2017). https://doi.org/10.1016/S0034-4877(17)30059-9.
  3. T. Abdeljawad, D. Baleanu, {Integration by Parts and its Applications of a New Nonlocal Fractional Derivative with Mittag-Leffler Nonsingular kernel}, The Journal of Nonlinear Science and Applications, {10} (2017). https://doi.org/10.22436/jnsa.010.03.20.
  4. T. Abdeljawad, D. Baleanu, {Discrete fractional differences with nonsingular discrete Mittag-Leffler Kernels}, Adv. in Dif. Eq. 232 (2016). https://doi.org/10.1186/s13662-016-0949-5.
  5. T. Abdeljawad, {A Lyapunov type inequality for fractional operators with non singular Mittag-Leffler kernel}, J. of ineq. and App. {130} (2017). https://doi.org/10.1186/s13660-017-1400-5.
  6. I. Alraddadi, M. P. Ineh, D. K. Igobi, U. Ishtiaq, and I.-L Popa,{Stabilization of fractional hybrid systems with applications in biomedical dynamics}. J. of Math. and Comp. Sci. {41} (2026), 406-420. https://doi.org/10.22436/jmcs.041.03.07.
  7. A. Atangana, and D. Baleanu, {New Fractional Derivatives with NonLocal and Non-singular Kernel: Theory and Application to heat transfer model}, Therm. Sci. {20} (2016), 763-769. https://doi.org/10.2298/TSCI160111018A.
  8. J. Ante, E. Abraham, U. Ebere, W. Udogworen, and C. Akpan, {On the Global Existence of solution and Lyapunov Asymptotic Practical Stability for Nonlinear Impulsive Caputo Fractional Derivative via Comparison Principle}, Scholars journal of physics, Mathematics and Statistics, {11} (2024), 160-172.
  9. J. Atsu, J. E. Ante, C. M. Orazulume, S. O. Essang, R. E. Francis, S. I. Okeke and W. K. Udogworen, {A Lyapunov-based approach to uniform strict stability analysis for nonlinear impulsive Caputo fractional derivatives with weakly singular kernel}, International Journal of Mathematical Analysis and Modelling, {8}, (2025), 198-215. https://tnsmb.org/journal/index.php/ijmam/article/view/221.
  10. M. Caputo and M. Fabrizio, {A New Definition of Fractional Derivative without Singular Kernel}, Progress in Fractional Differentiation and Applications, { 2} (2015),1–7. http://dx.doi.org/10.12785/pfda/010201.
  11. F. Chen. A. Chen, and X. Wang, {On the Solutions for Impulsive Fractional Functional Differential Equations}, Differential Equations and Dynamical Systems, {17} (2009), 379-391. https://doi.org/10.1007/s12591-009-0027-5.
  12. D. Igobi and W. Udogworen, {Results on Existence and Uniqueness of solutions of Fractional Differential Equations of Caputo-Fabrizio type in the sense of Riemann-Liouville}, IAENG International Journal of App. Math. {54} (2024), 1163–1171.
  13. M. P. Ineh, O. O. Marcus, K. N. Inyang, A. Maharaj, and P. A. Nyong, On eventual uniform stability of Caputo fractional hybrid systems with application to library and information systems, Adv. Fixed Point Theory, 16 (2026), Article ID 15. https://doi.org/10.28919/afpt/9826.
  14. M. Ineh, U. Ishtiaq, J. Ante, M. Garayev, and I. Popa, {A robust uniform practical stability approach for Caputo fractional hybrid systems}, AIMS Math. {10} (2025), 7001-7021. https://doi.org/10.3934/math.2025320.
  15. M. Ineh and E. Akpan, {On Lyapunov stability of Caputo Fractional dynamic equations on time scale using vector Lyapunov functions}, Khyyam Journal of Mathematics, {11} (2025), 116-143. https://doi.org/10.22034/kjm.2025.477875.3319.
  16. V. Lakshmikantham, {Theory of Fractional Functional Differential Equation}, Nonlinear Analysis, {69} (2008), 3337-3343.
  17. S. Mohammed, A. Abdeljawad, K. Shah, F. Jarad, {Study of Impulsive problems under Mittag-Leffler power law}, Heliyon6eo5109, {6} (2020), 1-8.
  18. G. Nchama, R. Mariano, and A. Mecias, {Properties of the Caputo-Fabrizio Fractional Derivative}, International Journal of Applied Engineering Research, (2021).
  19. M.I. Syam, and Mohammed Al-Refai, {Fractional differential equations with Atangana-Baleanu fractional derivative: Analysis analysis and applications}, Chaos, Solitons and Fractals: X, {2} (2009).
  20. J. Wang, Y. Zhou, and M. Feckan, {Nonlinear Impulsive Problems for Fractional Differential Equations and Ulam Stability}, Computers and Mathematics with Application, {64} (2012), 3389-3405.
  21. W. Udogworen, D. Igobi, and U. Jim, {Analysis of Atangana-Baleanu-Caputo Impulsive Fractional Delay Differential Equations}, IAENG International Journal of Applied Mathematics, {55} (2025), 3883-3891.
  22. W. Udogworen, D. Igobi, A. Hamarsheh, U. Jim, H. A Nabwey and R. George, {Impulsive fractional delay differential equations with fixed moments and modified Ulam-Hyers-Rassias stability}, {Bound Value Probl.} (2025). https://doi.org/10.1186/s13661-025-02178-5.
  23. W. Udogworen, D. Igobi, A. Hamarsheh, U. Jim, H. A Nabwey and R. George, Analysis of impulsive fractional delay differential equations with Mittag-Leffler kernel and its applications to Hutchinson's model, J. of Math. and Comp. Sci. 42 (2026), 124-142. https://doi.org/10.22436/JMCS.042.01.08.