Application of a fast Fibonacci wavelet method for the fractional cryosphere model

Year 2025, Volume: 5 Issue: 2, 257 - 279, 30.06.2025

Mulualem Aychluh

https://doi.org/10.53391/mmnsa.1529457

Abstract

This paper presents an efficient numerical approach for approximating the fractional-order cryosphere model using a Caputo-Fabrizio derivative with a non-singular exponential decay kernel. We develop a Fibonacci wavelet-based collocation method to solve the system, transforming the governing equations into nonlinear algebraic equations via operational matrix integration. The resulting system is solved using the Newton-Raphson method. We establish the existence and uniqueness of solutions through the Banach fixed-point theorem and Picard operator theory, and provide a comprehensive convergence analysis of the proposed scheme. Numerical simulations demonstrate the method's effectiveness, with results validated against fourth- and seventh-order fractional Runge-Kutta (RK4 and RK7) methods. The novelty of this work lies in the first application of the Fibonacci wavelet and collocation technique to the cryosphere model, and we represent the model’s nonlinear equations using Caputo-Fabrizio fractional derivative. The MATLAB R2016a generated results highlight the method's precision in capturing oscillatory behaviors and sensitivity to fractional order variations $(0<\wp<1)$, offering improved climate dynamics modeling through modified memory effect representation.

Keywords

Cryosphere model , Fibonacci wavelet method , Caputo-Fabrizio fractional derivative , collocation method

References

• [1] Aychluh, M. Nonlinear analysis of the fractional Lorenz-84 model with a Rabotnov exponential kernel law. Journal of Applied Mathematics and Computing, 71(2), 2231-2260, (2025).
• [2] Marshall, S.J. The Cryosphere. Princeton University Press: New Jersey, (2011).
• [3] Deng, K., Azorin-Molina, C., Yang, S., Hu, C., Zhang, G., Minola, L. and Chen, D. Changes of Southern Hemisphere westerlies in the future warming climate. Atmospheric Research, 270, 106040, (2022).
• [4] Habenom, H., Aychluh, M., Suthar, D.L., Al-Mdallal, Q. and Purohit, S.D. Modeling and analysis on the transmission of covid-19 Pandemic in Ethiopia. Alexandria Engineering Journal, 61(7), 5323-5342, (2022).
• [5] Aychluh, M., Purohit, S.D., Agarwal, P. and Suthar, D.L. Atangana–Baleanu derivative-based fractional model of COVID-19 dynamics in Ethiopia. Applied Mathematics in Science and Engineering, 30(1), 635-660, (2022).
• [6] Diethelm, K. The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type (Vol. 2004). Springer: Berlin, (2010).
• [7] Caputo, M. and Fabrizio, M. A new definition of fractional derivative without singular kernel. Progress in Fractional Differentiation & Applications, 1(2), 73-85, (2015).
• [8] Chakraborty, A., Veeresha, P., Ciancio, A., Baskonus, H.M. and Alsulami, M. The effect of climate change on the dynamics of a modified surface energy balance-mass balance model of Cryosphere under the frame of a non-local operator. Results in Physics, 54, 107031, (2023).
• [9] Nicolis, C. Long-term climatic transitions and stochastic resonance. Journal of Statistical Physics, 70, 3–14, (1993).
• [10] Aychluh, M., Suthar, D.L. and Purohit, S.D. Analysis of the nonlinear Fitzhugh–Nagumo equation and its derivative based on the Rabotnov fractional exponential function. Partial Differential Equations in Applied Mathematics, 11, 100764, (2024).
• [11] Aychluh, M. and Ayalew, M. The fractional power series method for solving the nonlinear Kuramoto-Sivashinsky equation. International Journal of Applied and Computational Mathematics, 11, 29, (2025).
• [12] Keshavarz, E., Ordokhani, Y. and Razzaghi, M. Bernoulli wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations. Applied Mathematical Modelling, 38(24), 6038-6051, (2014).
• [13] Ayalew, M., Ayalew, M. and Aychluh, M. Numerical approximation of space-fractional diffusion equation using Laguerre spectral collocation method. International Journal of Mathematics for Industry, 1-17, (2024).
• [14] Heydari, M.H., Hooshmandasl, M.R. and Ghaini, F.M. A new approach of the Chebyshev wavelets method for partial differential equations with boundary conditions of the telegraph type. Applied Mathematical Modelling, 38(5-6), 1597-1606, (2014).
• [15] Suthar, D.L., Habenom, H. and Aychluh, M. Effect of vaccination on the transmission dynamics of COVID-19 in Ethiopia. Results in Physics, 32, 105022, (2022).
• [16] Ur Rehman, M. and Saeed, U. Gegenbauer wavelets operational matrix method for fractional differential equations. Journal of the Korean Mathematical Society, 52(5), 1069-1096, (2015).
• [17] Iqbal, M.A., Saeed, U. and Mohyud-Din, S.T. Modified Laguerre wavelets method for delay differential equations of fractional-order. Egyptian Journal of Basic and Applied Sciences, 2(1), 50-54, (2015).
• [18] Venkatesh, S.G., Ayyaswamy, S.K. and Balachandar, S.R. Legendre Wavelets based approximation method for solving advection problems. Ain Shams Engineering Journal, 4(4), 925-932, (2013).
• [19] Srinivasa, K. and Mundewadi, R.A. Wavelets approach for the solution of nonlinear variable delay differential equations. International Journal of Mathematics and Computer in Engineering, 1(2), 139-148, (2023).
• [20] Mulimani, M. and Srinivasa, K. A new approach to the Benjamin-Bona-Mahony equation via ultraspherical wavelets collocation method. International Journal of Mathematics and Computer in Engineering, 2(2), 179-192, (2024).
• [21] Golberg, M. and Chen, C. Discrete Projection Methods for Integral Equations. Computational Mechanics: Boston, (1996).
• [22] Shiralashetti, S.C. and Lamani, L. Fibonacci wavelet based numerical method for the solution of nonlinear Stratonovich Volterra integral equations. Scientific African, 10, e00594, (2020).
• [23] Sabermahani, S., Ordokhani, Y. and Yousefi, S.A. Fibonacci wavelets and their applications for solving two classes of time-varying delay problems. Optimal Control Applications and Methods, 41(2), 395–416, (2020).
• [24] Sabermahani, S. and Ordokhani, Y. Fibonacci wavelets and Galerkin method to investigate fractional optimal control problems with bibliometric analysis. Journal of Vibration and Control, 27(15-16), 1778-1792, (2021).
• [25] Srivastava, H.M., Shah, F.A. and Nayied, N.A. Fibonacci wavelet method for the solution of the non-linear Hunter–Saxton equation. Applied Sciences, 12(15), 7738, (2022).
• [26] Shah, F.A., Irfan, M., Nisar, K.S., Matoog, R.T. and Mahmoud, E.E. Fibonacci wavelet method for solving time-fractional telegraph equations with Dirichlet boundary conditions. Results in Physics, 24, 104123, (2021).
• [27] Vivek, Kumar, M. and Mishra, S.N. A fast Fibonacci wavelet-based numerical algorithm for the solution of HIV-infected CD4+ T cells model. The European Physical Journal Plus, 138, 458, (2023).
• [28] Chakraborty, A. and Veeresha, P. Effects of global warming, time delay and chaos control on the dynamics of a chaotic atmospheric propagation model within the frame of Caputo fractional operator. Communications in Nonlinear Science and Numerical Simulation, 128, 107657 (2024).
• [29] Kumbinarasaiah, S. and Mulimani, M. Fibonacci wavelets-based numerical method for solving fractional order (1+1)-dimensional dispersive partial differential equation. International Journal of Dynamics and Control, 11, 2232–2255, (2023).
• [30] Moore, E.J., Sirisubtawee, S. and Koonprasert, S. A Caputo–Fabrizio fractional differential equation model for HIV/AIDS with treatment compartment. Advances in Difference Equations, 2019(200), 1-20, (2019).
• [31] Losada, J. and Nieto, J.J. Properties of a new fractional derivative without singular kernel. Progress in Fractional Differentiation and Applications, 1(2), 87–92, (2015).
• [32] Kumbinarasaiah, S. Hermite wavelets approach for the multi-term fractional differential equations. Journal of Interdisciplinary Mathematics, 24(5), 1241-1262, (2021).