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First Integrals of the Diffusion Equation; An Extension of the Fujita Solutions¹

ABSTRACT

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.

¹ Contribution of the School of Australian Environmental Studies, Griffith University, Brisbane, Queensland, 4111. Australia.

² Professor of Applied Mathematics and Senior Lecturer, Griffith University, respectively.

³ Professor of Engineering and Applied Science, Yale University, New Haven, Conn.

Received for publication August 14, 1979. Accepted for publication April 28, 1980.

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