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Variables-Separable Solution of the Horizontal Flow Equation with Nonconstant Diffusivity¹

ABSTRACT

Solution of the nonlinear diffusion-type equation is approached by both an additive form and a product form of separation of variables. The problem considered is the special case of one-dimensional water absorption by a horizontal, semi-infinite column of uniform soil initially at a uniform water content, and which from time zero on has a higher constant water content applied and maintained at one end. The additive separation of variables implies boundary conditions for which the water content depends on distance and time. On this basis, it is shown that an equation proposed in the literature, and arrived at by a transformation equivalent to a special case of additive variables-separation, is not a solution to the problem.

However, the problem is solved by using a product form of separation of variables, which reduces to the equivalent of the Boltzmann transformation by virtue of one boundary condition. Hence, in the light of this approach, the Boltzmann transformation appears as a consequence of the governing partial differential equation and one boundary condition, rather than as an additional assumption in need of separate justification.

¹ Joint contribution from the Department of Agronomy, Purdue University Agr. Exp. Sta. (Journal Paper No. 2555), Lafayette, Ind., and the US Salinity Laboratory, Soil and Water Conservation Research Division, ARS, USDA, Riverside, Calif., in cooperation with the 17 Western States and Hawaii. The work was done by the author during sabbatical leave at the US Salinity Laboratory; helpful discussions with Drs. W. R. Gardner and S. L. Rawlins are acknowledged with thanks.

² Professor of Soils.

Received for publication May 27, 1965. Accepted for publication July 13, 1965.