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

• Published in: Hydrological processes Vol. 37; no. 3
• Main Author: Su, Ninghu
• Format: Journal Article
• Language: English
• Published: Hoboken, USA John Wiley & Sons, Inc 01.03.2023; Wiley Subscription Services, Inc
• Subjects: Boundary conditions; Differential equations; equations; Fluctuations; Hydraulic conductivity; Hydrologic processes; Hydrology; Interpolation; Moisture content; Parameters; Partial differential equations; prediction; random flux boundary; random fractional partial differential equations (rfPDEs); random soil parameters; Soil; Soil surfaces; Soil swelling; Soil water movement; Soils; Water content; water movement; Watersheds
• ISSN: 0885-6087; 1099-1085
• DOI: 10.1002/hyp.14844

This 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. Computed random fluxes into and out of the soil, r$$ r $$, from the measured water content, θ$$ \theta $$, and hydraulic conductivity, K0$$ {K}_0 $$. The random influx is computed as an inverse problem using the methods based on random fractional partial differential equations presented in this paper. The methods have been applied to data from single site and a catchment.