Two new numerical approaches for the fractional distribution of the model of a system of lakes via modified hat and quasi-hat functions

In this article, two numerical approaches are presented to solve a system of three fractional differential equations that express the pollution of lakes.In our recent study, a new class of hat functions, called quasi-hat functions (QHFs), are constructed.The proposed approaches utilize modified hat functions (MHFs) and quasi-hat functions (QHFs).Fractional-order operational of MHFs and QHFs are used to build algorithms that transform the main problem into a system of six equations with six unknowns and three equations with three unknowns, respectively.Absolute errors of obtained approximate solutions and convergence analysis of the utilized approach will be studied.Finally, three examples are provided to illustrate the capabilities of these algorithms.The pollution monitoring results are reported in some tables and figures for different values of $\alpha$.