Fractional Governing Equations of Diffusion Wave and Kinematic Wave Open-Channel Flow in Fractional Time-Space. I. Development of the Equations

Abstract

In this study the fractional governing equations for diffusion wave and kinematic wave approximations to unsteady open-channel flow in prismatic channels in fractional time-space were developed. The governing fractional equations were developed from the mass and motion conservation equations in order to provide a physical basis to these equations. A fractional form of the resistance formula for open-channel flow was also developed. Detailed dimensional analyses of the derived equations were then performed in order to ensure dimensional consistency of the derivations. It is shown that these fractional equations of unsteady open-channel flow are fundamentally nonlocal in terms of nonlocal fluxes. The derived fractional governing equations of diffusion wave and kinematic wave open-channel flow can accommodate both the long-memory nonlocal behavior of open-channel flow as well as its local, finite memory behavior, as is numerically demonstrated in the accompanying paper by the authors.