Analytic solutions are presented for a nonlinear diffusion-convection model describing constant rate rainfall infiltration in uniform soils and other porous materials. The model is based on the Darcy-Buckingham approach to unsaturated water flow and assumes simple functional forms for the soil water diffusivity D(O) and hydraulic conductivity K(O) which depend on a single free parameter C and readily measured soil hydraulic properties. These D(O) and K(O) yield physically reasonable analytic moisture characteristics. The relation between this model and other models which give analytic solutions is explored. As C--> co, the model reduces to the weakly nonlinear Burgers' equation, which has been applied in certain field situations. At the other end of the range as C-> 1, the model approaches a Green-Ampt-like model. A wide range of realistic soil hydraulic properties is encompassed by varying the C parameter. The general features of the analytic solutions are illustrated for selected C values. Gradual and steep wetting profiles develop during rainfall, aspects seen in the laboratory and field. In addition, the time-dependent surface water content and surface water pressure potential are presented explicitly. A simple traveling wave approximation is given which agrees closely with the exact solution at comparatively early infiltration times. 1. INTRODUCTION Over 20 years have passed since Rubin and Steinhardt [1963] used numerical techniques to illuminate the general features of rainfall infiltration. The search for analytical or quasi-analytical solutions has, however, continued unabated. This sustained quest is no mere dilettantism but reflects continuing needs for such solutions. These are principally fourfold; they provide general physical insight into the infiltration process. Additionally, they form the bases for rational approximations and simplifications in terms of readily measured soil water properties. They also furnish bench marks for the validation and improvement of numerical schemes. Finally, they are useful for testing various inverse techniques used in the estimation of soil hydraulic properties. We introduce here a nonlinear model of constant rate rainfall which shows considerable versatility in meeting these and related requirements. Adopting the Darcy-Buckingham approach, we seek solutions to the highly nonlinear, one-dimensional Fokker-Planck diffusion-convection equation subject to flux boundary conditions, which is taken to describe rainfall infiltration in uniform soils [Philip, 1969]. Braester [1973], in an attempt to solve this problem, linearized the flow equation. The linear model has several severe limitations [Philip, 1957a; Parlange, 1976], the most notable being that the linear convection term does not permit the development of a traveling wave solution at large infiltration times. This problem does not arise in the exactly solvable Burgers' equation with its weakly nonlinear convection term [Philip, 1973, 1974']. The correct form of Knight's solution of Burgers' equation for the cons...
Nonclassical symmetry solutions of physically relevant partial differential equations are considered via the reduction methods of Bluman and Cole and Clarkson and Kruskal. Consistency conditions will be provided to show that, if satisfied, these two methods are equivalent in the sense that they lead to the same symmetry solutions. The Boussinesq equation and Burgers’ equation are used as illustrative examples. Exact solutions, one of which is new, will be presented for Burgers’ equation obtained from the Bluman and Cole method, yet not obtainable by Clarkson and Kruskal’s method.
The paper starts by giving a motivation for this research and justifying the considered stochastic diffusion models for cosmic microwave background radiation studies. Then it derives the exact solution in terms of a series expansion to a hyperbolic diffusion equation on the unit sphere. The Cauchy problem with random initial conditions is studied. All assumptions are stated in terms of the angular power spectrum of the initial conditions. An approximation to the solution is given and analysed by finitely truncating the series expansion. The upper bounds for the convergence rates of the approximation errors are derived. Smoothness properties of the solution and its approximation are investigated. It is demonstrated that the sample Hölder continuity of these spherical fields is related to the decay of the angular power spectrum. Numerical studies of approximations to the solution and applications to cosmic microwave background data are presented to illustrate the theoretical results.
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