Random fractional kinematic wave equations of overland flow: The HPM solutions and applications

This paper presents new findings from analyses of a random fractional kinematic wave equation (rfKWE) for overland flow. The rfKWE is featured with orders of temporal and spatial fractional derivatives and with the roughness parameter, the effective rainfall intensity and infiltration rate as random variables. The new solutions are derived with the aid of a numerical method named the homotopy perturbation method (HPM) and approximate solutions are presented for different situations. The solutions are evaluated with data from overland flow flumes with simulated rainfall in the laboratory. The results suggest that on an infiltrating surface the temporal nonlocality of overland flow represented by the temporal order of fractional derivatives diminishes over time while the spatial nonlocality manifested by the spatial order of fractional derivatives continue if there is overland flow. It shows that the widely used unit discharge-height relationship is a special case of the solution of the rfKWE. Procedures are demonstrated for determining the fractional roughness coefficient, $n_f$, the order of spatial fractional derivatives, $\rho$, and the steady-state infiltration rate during the overland flow, $A_s$. The analyses of the data show that the mean spatial order of fractional derivatives is $\rho =1.25$, the mean flow pattern parameter $m=1.50$, and the mean fractional roughness coefficient is $n_f=0.002$ which is smaller than the conventional roughness coefficient, $n=0.108$. With these average values of the parameters and their standard deviations, simulations were performed to demonstrate the use of the methods, which is also a comparison of the classic KWE and rfKWE models.