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Variables‐Separable Solution of the Horizontal Flow Equation with Nonconstant Diffusivity
Author(s) -
Swartzendruber Dale
Publication year - 1966
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/sssaj1966.03615995003000010010x
Subject(s) - separation of variables , mathematics , mathematical analysis , thermal diffusivity , constant (computer programming) , boundary value problem , diffusion , transformation (genetics) , diffusion equation , partial differential equation , product (mathematics) , flow (mathematics) , boundary (topology) , boltzmann constant , separable space , boltzmann equation , thermodynamics , physics , chemistry , geometry , biochemistry , economy , computer science , gene , programming language , economics , service (business)
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.