Existence and uniqueness analysis of a fractional atmospheric system using Haar-based operational matrices

Abstract

Understanding and accurately modeling the dynamics of climate-related processes is essential for predicting and mitigating the effects of global warming. This study introduces a fractional order atmospheric model that simultaneously captures the interactions among three key variables: permafrost thaw, atmospheric temperature, and greenhouse gas concentration. The model was formulated using the Atangana–Baleanu–Caputo fractional derivative, allowing for the inclusion of memory effects that are critical in climate dynamics. To solve the resulting nonlinear fractional differential equations, we constructed an operational matrix of the Atangana–Baleanu fractional integral operator based on Haar wavelets. Using Haar series expansions and operational matrices, the system was transformed into an objective function. This objective function was then minimized using differential evolution optimization to determine unknown Haar coefficients. The proposed method was validated against traditional numerical and predictor–corrector methods, with theoretical analysis confirming existence, uniqueness, and a provable upper bound for the approximation error. Numerical experiments under various parameter settings demonstrated the high accuracy, efficiency, and flexibility of the method. These results highlight the potential of fractional order modeling as a powerful framework for analyzing complex environmental systems and improving climate prediction models.

Keywords

Atmospheric model / Atangana-Baleanu-Caputo derivative / Haar wavelet / Operational matrix / Existence and uniqueness

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