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Journal of Environmental Accounting and Management 13(3) (2025) 281--288 | DOI:10.5890/JEAM.2025.09.004

Mohd Khalid$^{1}$, Subhash Alha$^{1}$, Ali Akgül$^{2,3,4,5\dagger}$

$^{1}$ Department of Mathematics, Maulana Azad National Urdu University, Gachibowli, Hyderabad-500032, India

$^2$ Department of Electronics and Communication Engineering, Saveetha School of Engineering, SIMATS, Chennai, Tamilnadu, India

$^3$ Siirt University, Art and Science Faculty, Department of Mathematics, 56100 Siirt, Turkey

$^4$ Department of Computer Engineering, Biruni University, 34010 Topkapı, Istanbul, Turkey

$^5$ Near East University, Mathematics Research Center, Department of Mathematics, Near East Boulevard, PC: 99138, Nicosia /Mersin 10 – Turkey

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Abstract

In this paper, new hybrid fractional operators are introduced, which are generated by replacing classical derivatives with proportional derivatives. To be more precise, we combine regularised Hilfer-Prabhakar (RHP) and two-parameter Atangana-Baleanu (SABC) fractional derivatives with constant proportional derivatives. This paper provides an in-depth analysis of the implications and uses of these hybrid operators as they relate to differential equations with constant proportional derivatives.

References

1. [1]	Farman, M., Akgül, A., Sultan, M., Riaz, S., Asif, H., Agarwal, P., and Hassani, M.K. (2024), Numerical study and dynamics analysis of diabetes mellitus with co-infection of COVID-19 virus by using fractal fractional operator, Scientific Reports, 14(1), 16489.

2. [2]	Zehra, A., Jamil, S., Farman, M., and Nisar, K.S. (2024), Modeling and analysis of Hepatitis B dynamics with vaccination and treatment with novel fractional derivative, Plos One, 19(7), 0307388.

3. [3]	Farman, M., Shehzad, A., Nisar, K.S., Hincal, E., and Akgul, A. (2024), A mathematical fractal-fractional model to control tuberculosis prevalence with sensitivity, stability, and simulation under feasible circumstances, Computers in Biology and Medicine, 178, 108756.

4. [4]	Baleanu, D., Jajarmi, A., Sajjadi, S.S., and Mozyrska, D. (2019), A new fractional model and optimal control of a tumor-immune surveillance with non-singular derivative operator, Chaos: An Interdisciplinary Journal of Nonlinear Science, 29(8), 083127.

5. [5]	Baleanu, D., Shekari, P., Torkzadeh, L., Ranjbar, H., Jajarmi, A., and Nouri, K. (2023), Stability analysis and system properties of Nipah virus transmission: A fractional calculus case study. Chaos, Solitons \& Fractals, 166, 112990.

6. [6]	Defterli, O., Baleanu, D., Jajarmi, A., Sajjadi, S.S., Alshaikh, N., and Asad, J.H. (2022), Fractional treatment: an accelerated mass-spring system, Romanian Reports in Physics, 74(4), 122.

7. [7]	Baleanu, D., Arshad, S., Jajarmi, A., Shokat, W., Ghassabzade, F.A., and Wali, M. (2023), Dynamical behaviours and stability analysis of a generalized fractional model with a real case study, Journal of Advanced Research, 48, 157-173.

8. [8]	Atangana, A. and Baleanu, D. (2016), New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Thermal Science, 20(2), 762-769.

9. [9]	Caputo, M. and Fabrizio, M. (2015), A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation \& Applications, 1(2), 73-85.

10. [10]	Khalil, R., Al Horani, M., Yousef, A., and Sababheh, M. (2014), A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264, 65-70.

11. [11]	Hilfer, R. and Luchko, Y. (2019), Desiderata for fractional derivatives and integrals, Mathematics, 7(2), 149.

12. [12]	Anderson, D.R. and Ulness, D.J. (2015), Newly defined conformable derivatives, Advances in Dynamical Systems and Applications, 10(2), 109-137.

13. [13]	Baleanu, D., Fernandez, A., and Akgül, A. (2020), On a fractional operator combining proportional and classical differintegrals, Mathematics, 8(3), 360.

14. [14]	Karatas Akgül, E., Akgül, A., and Baleanu, D. (2020), Laplace transform method for economic models with constant proportional Caputo derivative, Fractal and Fractional, 4(3), 30.

15. [15]	Akgül, A. and Baleanu, D. (2021), Analysis and applications of the proportional Caputo derivative, Advances in Difference Equations, 2021(136), 1-12.

16. [16]	Akgül, A. (2022), Some fractional derivatives with different kernels, International Journal of Applied and Computational Mathematics, 8(4), 183.

17. [17]	Garra, R., Gorenflo, R., Polito, F., and Tomovski, Ž. (2014), Hilfer–Prabhakar derivatives and some applications, Applied Mathematics and Computation, 242, 576-589.

18. [18]	Chinchole, S.M. and Bhadane, A. (2022), A new definition of fractional derivatives with Mittag-Leffler kernel of two parameters, Communications in Mathematics and Applications, 13(1), 19.

19. [19]	Khalid, M. and Akgül, A. (2024), Fractional frontier: Navigating Cauchy-type equations with formable and Fourier transformations, Contemporary Mathematics, 2693-2708.

20. [20]	Khalid, M., Mallah, I.A., Alha, S., and Akgul, A. (2024), Exploring the Elzaki transform: Unveiling solutions to reaction-diffusion equations with generalized composite fractional derivatives, Contemporary Mathematics, 1426-1438.

21. [21]	Khalid, M. and Alha, S. (2023), New generalized integral transform on Hilfer–Prabhakar fractional derivatives and its applications, International Journal of Dynamics and Control, 12(1), 24-31.

22. [22]	Kexue, L. and Jigen, P. (2011), Laplace transform and fractional differential equations, Applied Mathematics Letters, 24(12), 2019-2023.

23. [23]	Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006), Theory and Applications of Fractional Differential Equations, Elsevier, 204.