Fractional derivatives in advection-dispersion equations: A comparative study

Abstract

Fractional derivatives capture the heterogeneity of aquifer mediums better than ordinary derivatives; however, due to the global nonacceptance of any general definition, their perfect applications are still questionable. As an effective alternative to the Riemann-Liouville fractional derivative (R-L FD), the present work modeled the fractional advection–dispersion equation (FADE) for spatially varying transport parameters by employing the conformable fractional derivative (CFD) and improved Riemann-Liouville-type conformable fractional derivative (IR-L CFD). In the proposed model, the first order spatial, time derivatives are replaced with the CFD, the IR-L CFD of order $0<\delta ⩽1$. Also, $D\propto v^n;1⩽n⩽2$ ($D$-dispersion, $v$-seepage velocity) is considered for capturing the maximum possible orders of the medium heterogeneity. The FADE is solved by homotopy analysis method (HAM) in the presence of spatially increasing non-linear source of contamination. Initially, the groundwater is considered uniformly polluted.

The performances of the IR-L CFD and CFD are compared using experimental data available in the literature; the findings demonstrate that both CFD and IRLCFD may be used successfully to estimate the fate of pollutants in heterogeneous media; however, the IR-L CFD approximates the R-L FD more accurately than the CFD. Also, the accuracy of the results yielded by the FADE involving the CFD and the IR-L CFD notably relies on the nature of the solute transport parameters. The obtained solutions can be used to verify numerical solutions. The presented model can get solutions of FADEs with rigorous transport conditions that are not solvable otherwise.