DYNAMICS OF THE ATMOSPHERIC FRACTIONAL CARBON-DIOXIDE GAS MODEL WITH DEEP NEURAL NETWORK

Tariq Mahmood (), Muhammad Arfan (), Yahya Almalki, Ibrahim Mekawy () and Mohamed Abdalla
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Tariq Mahmood: School of Mathematics and Statistics, Northwestern Polytechnical University, Xian, Shaanxi 710129, P. R. China
Muhammad Arfan: Department of Mathematics, University of Malakand, Chakadara Dir (L), Khyber Pakhtunkhwa, Pakistan
Yahya Almalki: Department of Mathematics, Faculty of Science, King Khalid University, Abha 61471, Saudi Arabia
Ibrahim Mekawy: Department of Management Information Systems, College of Business and Economics, Qassim University, Buraydah 51452, Saudi Arabia
Mohamed Abdalla: Department of Mathematics, Faculty of Science, South Valley University, Qena 83523, Egypt

FRACTALS (fractals), 2025, vol. 33, issue 08, 1-16

Abstract: This work deals with the novel approach of fractional piecewise differential and integral operator. Atmospheric dynamics of carbon dioxide (CO2) gas has been investigated under piecewise operator with fractional Caputo and Atangana–Baleanu operators, respectively, and the whole interval is split into two sub-intervals. The model under consideration is composed of nonlinear differential equations which represent the dynamics of the human population and forest biomass in the atmosphere in relation to the CO2 gas concentration. Various fixed-point theorems are applied to determine whether the model mentioned above has a unique solution. For the numerical solution of this study, we propose the piecewise fractional Adams–Bashforth scheme and examined the model in terms of fractional orders to illustrate and verify the efficacy of the proposed strategy. All classes of the said system are tested for graphical representations. The first interval is plotted for Caputo while the second interval is shown for the Atangana–Baleanu fractional operator. From the numerical simulation, we find the crossover behavior of the atmospheric dynamics of CO2 gas in human life for its controlling and for the future predication. To obtain a better performance of the considered model, we apply the Levenberg–Marquardt technique and consider epochs 1000. In order to precisely train the data, we tested the approximate solution and absolute error, regression, and error-histograms by using the obtained figures with a neural network.

Keywords: Fractional Calculus; Carbon Dioxide Gas Model; Theoretical Analysis; Piecewise Approach; Nonlinear Equations; Numerical Simulations; Neural Network (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1142/S0218348X25401395

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