A general approximation for the solution of the one-dimensional nonlinear diffusion equation is presented. It applies to arbitrary soil properties and boundary conditions. The approximation becomes more accurate when the soil-water diffusivity approaches a delta function, yet the result is still very accurate for constant diffusivity suggesting that the present formulation is a reliable one. Three examples are given where the method is applied, for a constant water content at the surface, when a saturated zone exists and for a time-dependent surface flux.
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Parlange, J., Hogarth, W.L., Parlange, M.B. et al. Approximate Analytical Solution of the Nonlinear Diffusion Equation for Arbitrary Boundary Conditions. Transport in Porous Media 30, 45–55 (1998). https://doi.org/10.1023/A:1006508721609
- nonlinear diffusion
- analytic solutions
- exact solutions