Dynamics Of The Atmospheric Fractional Carbon-Dioxide Gas Model With Deep Neural Network

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.