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The Separability of the Diffusion Equation with Drift Term
Author(s) -
Seeger A.
Publication year - 1970
Publication title -
physica status solidi (b)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.51
H-Index - 109
eISSN - 1521-3951
pISSN - 0370-1972
DOI - 10.1002/pssb.19700410205
Subject(s) - diffusion equation , scalar (mathematics) , diffusion , term (time) , mathematical analysis , constant (computer programming) , convection–diffusion equation , mathematics , factorization , degenerate energy levels , mathematical physics , physics , quantum mechanics , geometry , algorithm , computer science , programming language , economy , economics , service (business)
The paper investigates the diffusion equation with drift term, ∂ C /∂ t = div ( D grad C + + μ C grad U ), in particular the question of its separability in three dimensions. ( D is the diffusion coefficient, μ the mobility, both assumed to be constant scalar quantities; U denotes the potential giving rise to the drift.) The transformation to a new dependent variable f = C exp (μ U /2 D ) transforms the equation into a Schrödinger equation. Using this it is shown that, for appropriate choices of the potentials U , the diffusion‐drift equation is separable with modulation factor in the ellipsoidal coordinates (and its degenerate special cases) but in no other coordinate systems. The conditions to which U has to be subjected in order to lead to separability are stated and shown to have a close relationship to the factorization of the solutions of the original equation.