Random fractional partial differential equations and solutions for water movement in soils: Theory and applications

Abstract

AbstractThis paper analyses a set of random fractional partial differential equations (rfPDEs) for water movement in soils. The rfPDEs for both rigid and swelling soils are solved for both a random flux boundary condition (BC), and random concentration BC. Solutions from a random flux BC are presented for the large‐time and small‐time situations with the large‐time solution as a very simple method for determining the flux through the surface of the soil. The equation of cumulative infiltration is presented with random parameters of the rfPDE subject to a random concentration BC. The simulations using the results of the rfPDE for the two types of BCs yielded encouraging and stable results based on two sets of field data: the first set of the data was measurements at a single site while the second set was from 26 measurements in a small catchment. The results suggest that the presented procedures are very useful methods for the interpolation, extrapolation, and prediction of hydrological variables and parameters such as water content, hydraulic conductivity or the flux through the surface of the soil. The methodologies presented in this paper are able to reveal and reproduce the realistic hydrological processes in nature which are often stochastic and random.