Fractional conservation of mass

Abstract

The traditional conservation of mass equation is derived using a first-order Taylor series to represent flux change in a control volume, which is valid strictly for cases of linear changes in flux through the control volume. We show that using higher-order Taylor series approximations for the mass flux results in mass conservation equations that are intractable. We then show that a fractional Taylor series has the advantage of being able to exactly represent non-linear flux in a control volume with only two terms, analogous to using a first-order traditional Taylor series. We replace the integer-order Taylor series approximation for flux with the fractional-order Taylor series approximation, and remove the restriction that the flux has to be linear, or piece-wise linear, and remove the restriction that the control volume must be infinitesimal. As long as the flux can be approximated by a power-law function, the fractional-order conservation of mass equation will be exact when the fractional order of differentiation matches the flux power-law. There are two important distinctions between the traditional mass conservation, and its fractional equivalent. The first is that the divergence term in the fractional mass conservation equation is the fractional divergence, and the second is the appearance of a scaling term in the fractional conservation of mass equation that may eliminate scale effects in parameters (e.g., hydraulic conductivity) that should be scale-invariant.