Theoretical treatment and implementation of the SCM included Appell-Changhee polynomials for the fractional delayed carbon absorption-emission model

Abstract

Some researchers have started to perform in-depth studies on carbon dioxide emissions phenomena because global warming has caused immense damage. In this work, as a generalization of these studies, we will investigate and describe the fractional delayed Carbon absorption -emission model which consists of two fractional (Caputo sense) differential equations. From this point, we study the behavior of the solution for this model by using an approximate technique based on Appell -type Changhee polynomials (ACPs). We present an approximation of the fractional-order derivative by using ACPs. We implement the spectral collocation method (SCM) and use the properties of the ACPs to convert the provided model into a set of algebraic equations. A special focus is placed on the examination of the existence and stability of the equilibrium point for the model according to the characteristic roots with the help of the Hopf bifurcation technique. We verify the effectiveness and efficiency of the presented numerical scheme by comparing its results to those of the fourth-order Runge-Kutta method (RKM4). The results confirm that the current method is a straightforward and efficient tool for simulating the solutions to such models.