Analysis of Cauchy problems for variable-order derivatives with Mittag-Leffler kernel

Year 2024, , 64 - 78, 31.12.2024

İlknur Koca

https://doi.org/10.53391/mmnsa.1544150

Abstract

In this paper, the Cauchy problem for variable-order fractional differential equations incorporating the Mittag-Leffler kernel is explored. The variable-order derivative is modeled as a bounded function that adapts to the underlying dynamics of the system. The existence of a solution by utilizing a fixed-point theorem along with an iterative series that converges to the precise solution is established. The uniqueness of the solution is guaranteed by enforcing conditions like generalized Lipschitz continuity and linear growth conditions. This study contributes to the broader understanding of fractional calculus and its applications in complex systems where classical models are insufficient.

Keywords

Cauchy problems, Mittag-Leffler kernel, existence and uniqueness, variable-order derivative

References

• [1] Atangana, A. Fractional Operators with Constant and Variable Order with Application to Geohydrology. Academic Press: United Kingdom, (2017).
• [2] Goufo, E.F.D. and Atangana, A. Dynamics of traveling waves of variable order hyperbolic Liouville equation: Regulation and control. Discrete & Continuous Dynamical Systems-Series S, 13(3), (2020).
• [3] Khan, H., Alzabut, J., Gulzar, H., Tunç, O. and Pinelas, S. On system of variable order nonlinear p-Laplacian fractional differential equations with biological application. Mathematics, 11(8), 1913, (2023).
• [4] Zhuang, P., Liu, F., Anh, V. and Turner, I. Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM Journal on Numerical Analysis, 47(3), 1760-1781, (2009).
• [5] Patnaik, S., Hollkamp, J.P. and Semperlotti, F. Applications of variable-order fractional operators: a review. Proceedings of the Royal Society A, 476(2234), 20190498, (2020).
• [6] Uçar, E., Uçar, S., Evirgen, F. and Özdemir, N. A fractional SAIDR model in the frame of Atangana–Baleanu derivative. Fractal and Fractional, 5(2), 32, (2021).
• [7] Uçar, E., Uçar, S., Evirgen, F. and Özdemir, N. Investigation of E-cigarette smoking model with mittag-leffler kernel. Foundations of Computing and Decision Sciences, 46(1), 97-109, (2021).
• [8] Venkatesh, A., Manivel, M. and Baranidharan, B. Numerical study of a new time-fractional Mpox model using Caputo fractional derivatives. Physica Scripta, 99(2), 025226, (2024).
• [9] Alkahtani, B.S.T. and Koca I. Fractional stochastic sır model. Results in Physics, 24, 104124, (2021).
• [10] Koca, I. Efficient numerical approach for solving fractional partial differential equations with non-singular kernel derivatives. Chaos, Solitons & Fractals, 116, 278-286, (2018).
• [11] Atangana, A. and Baleanu, D. New fractional derivatives with nonlocal and nonsingular kernel: theory and application to heat transfer model. Thermal Science, 20(2), 763-769, (2016).
• [12] Iwa, L.L., Omame, A. and Inyama, S.C. A fractional-order model of COVID-19 and Malaria co-infection. Bulletin of Biomathematics, 2(2), 133-161, (2024).
• [13] Din, A. and Abidin, M.Z. Analysis of fractional-order vaccinated Hepatitis-B epidemic model with Mittag-Leffler kernels. Mathematical Modelling and Numerical Simulation with Applications, 2(2), 59-72, (2022).
• [14] Sulaiman, T.A., Yavuz, M., Bulut, H. and Baskonus, H.M. Investigation of the fractional coupled viscous Burgers’ equation involving Mittag-Leffler kernel. Physica A: Statistical Mechanics and its Applications, 527, 121126, (2019).
• [15] Atangana, A. and Koca, I. Fractional Differential and Integral Operators with Respect to a Function: Theory, Methods and Applications. Springer: Singapore, (2025).
• [16] Atangana, A. Extension of rate of change concept: from local to nonlocal operators with applications. Results in Physics, 19, 103515, (2020).
• [17] Umarov, S. and Steinberg, S. Variable order differential equations with piecewise constant order-function and diffusion with changing modes. Zeitschrift für Analysis und ihre Anwendungen, 28(4), 431-450, (2009).