Numerical Approximation of the Fractional Model of Atmospheric Dynamics of CO2 Using the Gegenbauer Wavelet Collocation Method

Abstract

This work provides an understanding of the fractional order nonlinear mathematical model that describes the dynamic variation of carbon dioxide ($C{O}_2)$ gas atmospheric concentration. The impact of changes in the human population $R\left(\xi \right)$ and forest biomass $T\left(\xi \right)$ on the dynamics of $C{O}_2$ gas concentration in the atmosphere is also highlighted in this work. It investigates the model's solution by applying an effective wavelet method known as the Gegenbauer wavelet collocation method (GWCM). The considered model is a nonlinear coupled system of fractional ordinary differential equations (SFODEs). The operational matrices of integration are constructed using the Gegenbauer wavelets. Processing is accelerated by utilizing the properties of the Gegenbauer wavelet expansions and the operational matrix of integration to convert nonlinear SFODEs into a system of algebraic equations. The Newton-iterative strategy is used to solve this system of algebraic equations to determine the unknown coefficients and to arrive at an approximate solution. Numerical illustration demonstrates the method's effectiveness and accuracy. In addition to demonstrating the effectiveness of the applied strategy, the results show that it is appropriate for high approximations of the atmospheric $C{O}_2$ gas concentration fractional model solution. All the numerical computations are carried out with the help of Mathematica software.