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First Integrals of the Diffusion Equation; An Extension of the Fujita Solutions
Author(s) -
Parlange J.Y.,
Braddock R. D.,
Chu B. T.
Publication year - 1980
Publication title -
soil science society of america journal
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.836
H-Index - 168
eISSN - 1435-0661
pISSN - 0361-5995
DOI - 10.2136/sssaj1980.03615995004400050005x
Subject(s) - thermal diffusivity , sorptivity , fujita scale , diffusion , mathematics , integral equation , mathematical analysis , dissipative system , constant (computer programming) , extension (predicate logic) , power law , power function , function (biology) , thermodynamics , physics , statistics , computer science , compressive strength , evolutionary biology , biology , meteorology , programming language
In general, solutions of the nonlinear diffusion equation have to be obtained by numerical or analytical iterative integration. It is shown here that if the diffusivity has a dependence on the water content which obeys a power law, then iterations can be avoided, and the solution obtained at once. This unique case results from the existence of a first integral of the diffusion equation. Previously known analytical results, i.e., the constant diffusivity solution, the delta‐function solution, and the Fujita solutions belong to that general class. For this general class of solutions it is also possible to show that the sorptivity has a dependence on the surface water content which obeys a power law. This represents the only known case when such an analytical relationship exists. This relationship is used to discuss the representation of the square of the sorptivity as an integral of the diffusivity, when the latter has an arbitrary dependence on the water content.