Numerical Solution of the Time Fractional Reaction-advection-diffusion Equation in Porous Media

In this work, we obtained the numerical solution for the system of nonlinear time-fractional order advection-reaction-diffusion equation using the homotopy perturbation method using Laplace transform method with fractional order derivatives in Liouville-Caputo sense. The solution obtained is very useful and significant to analyze many physical phenomenons. The present technique demonstrates the coupling of homotopy perturbation method and the Laplace transform technique using He's polynomials, which can be applied to numerous coupled systems of nonlinear fractional models to find the approximate numerical solutions. The salient features of the present work is the graphical presentations of the numerical solution of the concerned nonlinear coupled equation for several particular cases and showcasing the effect of reaction terms on the nature of solute concentration of the considered mathematical model for different particular cases. To validate the reliability, efficiency and accuracy of the proposed efficient scheme, a comparison of numerical solutions and exact solution are reported for Burgers' coupled equations and other particular cases of concerned nonlinear coupled systems. Here we find high consistency and compatibility between exact and numerical solution to a high accuracy. Presentation of absolute errors for given examples are reported in tabulated and graphical forms that ensure the convergence rate of the numerical scheme.