Space and Time Fractional Governing Equations of Unsteady Overland Flow

Abstract

Combining fractional continuity and motion equations, the space and time fractional governing equations of unsteady overland flow were derived. The kinematic and diffusion wave approximations were obtained from the space and time fractional continuity and motion equations of the overland flow process. When the fractional powers of space and time derivatives go to 1, the fractional governing equations become the conventional governing equations of unsteady overland flow, and the conventional equations can be obtained as the special cases of the proposed fractional governing equations. Similar to findings of the fractional open channel flow process reported previously, the numerical example herein demonstrated that as the fractional powers of the space and time derivatives decrease from 1, overland flows have longer durations, and both the occurrence time and magnitude of the peak flows decrease. The proposed space and time fractional unsteady overland flow equations may allow modeling anomalous hydrographs by taking into account nonlocality in time and space, and may provide further insights into nonlocal transport in hillslopes reported in the literature.