Solution of One-Dimensional Time Fractional Advection Dispersion Equation by Homotopy Analysis Method

Abstract

This study develops a homotopy analysis method (HAM) for analytically solving a one-dimensional time-fractional advection-dispersion equation (FADE). The HAM is a powerful method for solving nonlinear ordinary and partial differential equations and does not seem to have been employed in hydrology. The advantage of the HAM is that it does not require much information about the boundary conditions of the aquifer domain. The initial condition may be measured for an aquifer, but the boundary conditions do not always have to be specified. The FADE is employed for modeling the fate of contaminants in heterogeneous porous formations subject to an increasing or decreasing source of contamination that is spatially and temporally dependent. Both solute dispersion coefficient and seepage velocity are considered spatially and temporally dependent, exhibiting the heterogeneity of the porous formation. It is found that the contaminant concentration changes with the order of the FADE. This study aids understanding of the physical meaning of parameters involved in velocity and dispersion because the parameters are not linearized. The analytical solution is also compared with the numerical solution obtained by the finite-element method and is validated with field data available in the literature.