On Piecewise Fractional Differential Operator with Application to Biological Model of Prey Predator

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-23-2025-301
Published: Nov 28, 2025
Cite as:
Muhammad Nisarul Haq, Muhammad Sarwar, Kamaleldin Abodayeh, Saowaluck Chasreechai, Thanin Sitthiwirattham, On Piecewise Fractional Differential Operator with Application to Biological Model of Prey Predator, Int. J. Anal. Appl., 23 (2025), 301.

Main Article Content

Muhammad Nisarul Haq, Muhammad Sarwar, Kamaleldin Abodayeh, Saowaluck Chasreechai, Thanin Sitthiwirattham

Abstract

In this work, using the concept of piecewise fractional order differential operators and fixed point theory, existence results for the general Cauchy type dynamical system are studied. Further, Ulam-Hyers (U.H) stability and generalized U.H stability results are also investigated for the considered systems under the Caputo-Fabrizio type piecewise derivative. As an application of the considered system, biological prey-predator model, under piecewise fractional order differential operators having non-singular kernel are also studied.

Article Details

References

  1. H.M. Srivastava, Y. Ling, G. Bao, Some Distortion Inequalities Associated with the Fractional Derivatives of Analytic and Univalent Functions, J. Inequal. Pure Appl. Math. 2 (2021), 23.
  2. M. Darus, R.W. Ibrahim, Radius Estimates of a Subclass of Univalent Functions, Mat. Vesnik 63 (2011), 55–58. https://eudml.org/doc/253329.
  3. C.A. Monje, Y. Chen, B.M. Vinagre, D. Xue, V. Feliu, Fractional-Order Systems and Controls: Fundamentals and Applications, Springer, 2010. https://doi.org/10.1007/978-1-84996-335-0.
  4. R. Caponetto, G. Dongola, L. Fortuna, I. Petras, Fractional Order Systems: Modeling and Control Applications, World Scientific, 2010.
  5. G. Mani, A.J. Gnanaprakasam, L. Guran, R. George, Z.D. Mitrović, Some Results in Fuzzy B-Metric Space with B-Triangular Property and Applications to Fredholm Integral Equations and Dynamic Programming, Mathematics 11 (2023), 4101. https://doi.org/10.3390/math11194101.
  6. G. Mani, A. Gnanaprakasam, O. Ege, A. Aloqaily, N. Mlaiki, Fixed Point Results in $C^star$-Algebra-Valued Partial B-Metric Spaces with Related Application, Mathematics 11 (2023), 1158. https://doi.org/10.3390/math11051158.
  7. S. Westerlund, Dead Matter Has Memory!, Phys. Scr. 43 (1991), 174–179. https://doi.org/10.1088/0031-8949/43/2/011.
  8. M. Caputo, F. Mainardi, A New Dissipation Model Based on Memory Mechanism, Pure Appl. Geophys. 91 (1971), 134–147. https://doi.org/10.1007/bf00879562.
  9. W. Wyss, The Fractional Diffusion Equation, J. Math. Phys. 27 (1986), 2782–2785. https://doi.org/10.1063/1.527251.
  10. P. Agarwal, J.J. Nieto, M. Luo, Extended Riemann-Liouville Type Fractional Derivative Operator with Applications, Open Math. 15 (2017), 1667–1681. https://doi.org/10.1515/math-2017-0137.
  11. M. Caputo, M. Fabrizio, A new Definition of Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl. 1 (2015), 73–85.
  12. J. Losada, J.J. Nieto, Properties of a New Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl. 1 (2015), 87–92.
  13. J.T. Machado, V. Kiryakova, F. Mainardi, Recent History of Fractional Calculus, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), 1140–1153. https://doi.org/10.1016/j.cnsns.2010.05.027.
  14. L. Byszewski, Strong Maximum Principles for Parabolic Nonlinear Problems with Nonlocal Inequalities Together with Arbitrary Functionals, J. Math. Anal. Appl. 156 (1991), 457–470. https://doi.org/10.1016/0022-247x(91)90409-s.
  15. L. Byszewski, V. Lakshmikantham, Theorem About the Existence and Uniqueness of a Solution of a Nonlocal Abstract Cauchy Problem in a Banach Space, Appl. Anal. 40 (1991), 11–19. https://doi.org/10.1080/00036819008839989.
  16. G. Wu, D. Zeng, D. Baleanu, Fractional Impulsive Differential Equations: Exact Solutions, Integral Equations and Short Memory Case, Fract. Calc. Appl. Anal. 22 (2019), 180–192. https://doi.org/10.1515/fca-2019-0012.
  17. G. Wu, M. Luo, L. Huang, S. Banerjee, Short Memory Fractional Differential Equations for New Memristor and Neural Network Design, Nonlinear Dyn. 100 (2020), 3611–3623. https://doi.org/10.1007/s11071-020-05572-z.
  18. A. Atangana, S. İğret Araz, New Concept in Calculus: Piecewise Differential and Integral Operators, Chaos, Solitons Fractals 145 (2021), 110638. https://doi.org/10.1016/j.chaos.2020.110638.
  19. G. Mani, S. Haque, A.J. Gnanaprakasam, O. Ege, N. Mlaiki, The Study of Bicomplex-Valued Controlled Metric Spaces with Applications to Fractional Differential Equations, Mathematics 11 (2023), 2742. https://doi.org/10.3390/math11122742.
  20. K. Shah, T. Abdeljawad, B. Abdalla, M.S. Abualrub, Utilizing Fixed Point Approach to Investigate Piecewise Equations with Non-Singular Type Derivative, AIMS Math. 7 (2022), 14614–14630. https://doi.org/10.3934/math.2022804.
  21. K. Shah, T. Abdeljawad, A. Ali, Mathematical Analysis of the Cauchy Type Dynamical System Under Piecewise Equations with Caputo Fractional Derivative, Chaos Solitons Fractals 161 (2022), 112356. https://doi.org/10.1016/j.chaos.2022.112356.
  22. M.A. Alqudah, T. Abdeljawad, Eiman, K. Shah, F. Jarad, et al., Existence Theory and Approximate Solution to Prey–Predator Coupled System Involving Nonsingular Kernel Type Derivative, Adv. Differ. Equ. 2020 (2020), 520. https://doi.org/10.1186/s13662-020-02970-w.
  23. S. Muhammad, O.J. Algahtani, S. Saifullah, A. Ali, Theoretical and Numerical Aspects of the Malaria Transmission Model with Piecewise Technique, AIMS Math. 8 (2023), 28353–28375. https://doi.org/10.3934/math.20231451.