Generalized Fick's law and fractional ADE for pollution transport in a river: Detailed derivation

The fractional advection-dispersion equation (ADE) is a generalization of the classical ADE in which the second-order derivative is replaced with a fractional-order derivative. While the fractional ADE is analyzed as a stochastic process in the Fourier and Laplace space so far, in this study a fractional ADE for describing solute transport in rivers is derived in detail with a finite difference scheme in the real space. In contrast to the classical ADE, the fractional ADE is expected to be able to provide solutions that resemble the highly skewed and heavy-tailed time-concentration distribution curves of water pollutants observed in rivers.