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, nf, the order of spatial fractional derivatives, ρ, and the steady-state infiltration rate during the overland flow, As. The analyses of the data show that the mean spatial order of fractional derivatives is ρ=1.25, the mean flow pattern parameter m=1.50, and the mean fractional roughness coefficient is nf=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.

中文翻译：

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陆上流的随机分数运动波动方程：HPM 解决方案和应用


本文提出了对陆上流的随机分数运动波动方程 （rfKWE） 分析的新发现。rfKWE 具有时间和空间分数阶导数的阶数，粗糙度参数、有效降雨强度和入渗速率为随机变量。新的解是借助一种称为同伦扰动法 （HPM） 的数值方法推导出来的，并提出了不同情况下的近似解。这些解是用来自实验室模拟降雨的陆上流槽的数据来评估的。结果表明，在渗透表面上，由分数阶导数的时间顺序表示的陆上流动的时间非局域性随着时间的推移而减少，而如果存在陆上流动，则由分数阶导数的空间顺序表示的空间非局域性继续存在。结果表明，广泛使用的单位放电-高度关系是 rfKWE 解的特例。演示了确定分数粗糙度系数 nf、空间分数导数阶数 ρ 和陆上流动期间稳态渗透率 As 的程序。数据分析表明，分数阶导数的平均空间阶数为 ρ=1.25，平均流型参数 m=1.50，平均分数阶粗糙度系数为 nf=0.002，小于常规粗糙度系数 n=0.108。利用这些参数的平均值及其标准差，进行了模拟以演示这些方法的使用，这也是经典 KWE 和 rfKWE 模型的比较。