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RESEARCH ARTICLE

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

Mumtaz Ali1,2 ,  Najeeb Alam Khan1* ,  Muhammad Ayaz1 ,  Nadeem Alam Khan3
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1 Department of Mathematics, University of Karachi, Karachi, Sindh, Pakistan
2 Basic Sciences Department, Balochistan University of Engineering Technology, Khuzdar, Balochistan, Pakistan
3 Department of Computer Sciences, Iqra University, Karachi, Sindh, Pakistan
IJOCTA, 025320140 https://doi.org/10.36922/IJOCTA025320140
Received: 4 August 2025 | Revised: 22 September 2025 | Accepted: 26 September 2025 | Published online: 6 November 2025
© 2025 by the author(s). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
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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

Keywords
Atmospheric model
Atangana-Baleanu-Caputo derivative
Haar wavelet
Operational matrix
Existence and uniqueness
Funding
None.
Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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An International Journal of Optimization and Control: Theories & Applications, Electronic ISSN: 2146-5703 Print ISSN: 2146-0957, Published by AccScience Publishing
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