Solution of fractional-order reaction-advection-diffusion equation arising in porous media

In this article, the fractional-order nonlinear reaction-advection-diffusion equation describing contaminant transport in groundwater has been solved using shifted Legendre collocation method. The shifted Legendre polynomial is used to approximate the function. After that the operational matrix for fractional-order derivative in Caputo sense is applied on it. The shifted Legendre collocation points are employed to obtain a system of nonlinear algebraic equations which have been solved using Newton method. The application of the said methodology is demonstrated by applying it to two standard cases. The proposed method is validated by comparing the numerical results with those obtained using exact solutions through error analyses and the results are given in graphical as well as tabular forms. After the validation of its efficiency and effectiveness, the proposed numerical scheme is applied on a mathematical model related to porous media in a fractional order system. The salient feature of this article is the graphical exhibitions of the effects of fractional-order spatial and time derivatives, and also reaction and advection terms on the solution profile for different particular cases.