Mathematical modeling for adsorption process of dye removal nonlinear equation using power law and exponentially decaying kernels

In this research work, a new time-invariant nonlinear mathematical model in fractional (non-integer) order settings has been proposed under three most frequently employed strategies of the classical Caputo, the Caputo–Fabrizio, and the Atangana–Baleanu–Caputo with the fractional parameter χ , where 0 < χ ≤ 1. The model consists of a nonlinear autonomous transport equation used to study the adsorption process in order to get rid of the synthetic dyeing substances from the wastewater effluents. Such substances are used at large scale by various industries to color their products with the textile and chemical industries being at the top. The non-integer-order model suggested in the present study depicts the past behavior of the concentration of the solution on the basis of having information of the initial concentration present in the dye. Being nonlinear, it carries the possibility to have no exact solution. However, the Lipchitz condition shows the existence and uniqueness of the underlying model’s solution in non-integer-order settings. From a numerical simulation viewpoint, three numerical techniques having first order convergence have been employed to illustrate the numerical results obtained.