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

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.