On a ratio theorem for a class of nonlinear boundary value problems
Authors:
C. Rogers and A. J. Bracken
Journal:
Quart. Appl. Math. 44 (1987), 639-648
MSC:
Primary 35Q20; Secondary 35L60
DOI:
https://doi.org/10.1090/qam/872816
MathSciNet review:
872816
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Abstract: A generalization of Ussing’s flux ratio theorem is established as a consequence of a symmetry property of a Green’s function. Bäcklund transformations are then used to construct ratio theorems for certain nonlinear boundary value problems.
- Arnold Sommerfeld, Partial Differential Equations in Physics, Academic Press, Inc., New York, N. Y., 1949. Translated by Ernst G. Straus. MR 0029463
H. H. Ussing, K. Eskesen, and J. Lim, The flux-ratio transients as a tool for separating transport paths, in Epithelial Ion and Water Transport, pp. 257–264 (A. D. C. Macknight and J. P. Leader, eds.), Raven Press, New York, 1981
H. H. Ussing, Interpretation of tracer fluxes, in Membrane Transport in Biology, Vol. 1, pp. 115–140 (G. Giebisch, D. C. Tosteson, and H. H. Ussing, eds.), Springer-Verlag, New York, 1978
O. Sten-Knudsen and H. H. Ussing, The flux-ratio equation under nonstationary conditions, J. Membr. Biol. 63, 223–242 (1981)
- L. Bass and A. J. Bracken, The flux-ratio equation under nonstationary boundary conditions, Math. Biosci. 66 (1983), no. 1, 87–92. MR 718591, DOI https://doi.org/10.1016/0025-5564%2883%2990078-0
L. Bass and D. S. McAnally, The ratio of nonstationary tracer fluxes into and out of a hollow circular cylinder, J. Membr. Biol. In press
- C. Rogers and P. L. Sachdev, The Burgers hierarchy: on nonlinear initial and boundary value problems, Nuovo Cimento B (11) 83 (1984), no. 2, 127–134 (English, with Italian and Russian summaries). MR 772826, DOI https://doi.org/10.1007/BF02721585
- J. G. Kingston and C. Rogers, Reciprocal Bäcklund transformations of conservation laws, Phys. Lett. A 92 (1982), no. 6, 261–264. MR 683534, DOI https://doi.org/10.1016/0375-9601%2882%2990081-0
- A. S. Fokas and Y. C. Yortsos, On the exactly solvable equation$\ S_{t}=[(\beta S+\gamma )^{-2}S_{x}]_{x}+\alpha (\beta S+\gamma )^{-2}S_{x}$ occurring in two-phase flow in porous media, SIAM J. Appl. Math. 42 (1982), no. 2, 318–332. MR 650227, DOI https://doi.org/10.1137/0142025
- C. Rogers, M. P. Stallybrass, and D. L. Clements, On two-phase filtration under gravity and with boundary infiltration: application of a Bäcklund transformation, Nonlinear Anal. 7 (1983), no. 7, 785–799. MR 707086, DOI https://doi.org/10.1016/0362-546X%2883%2990034-2
A. Sommerfeld, Partial Differential Equations in Physics, pp. 50–51, Academic Press, New York, 1964
H. H. Ussing, K. Eskesen, and J. Lim, The flux-ratio transients as a tool for separating transport paths, in Epithelial Ion and Water Transport, pp. 257–264 (A. D. C. Macknight and J. P. Leader, eds.), Raven Press, New York, 1981
H. H. Ussing, Interpretation of tracer fluxes, in Membrane Transport in Biology, Vol. 1, pp. 115–140 (G. Giebisch, D. C. Tosteson, and H. H. Ussing, eds.), Springer-Verlag, New York, 1978
O. Sten-Knudsen and H. H. Ussing, The flux-ratio equation under nonstationary conditions, J. Membr. Biol. 63, 223–242 (1981)
L. Bass and A. J. Bracken, The flux-ratio equation under nonstationary boundary conditions, Math. Biosci. 66, 87–92 (1983)
L. Bass and D. S. McAnally, The ratio of nonstationary tracer fluxes into and out of a hollow circular cylinder, J. Membr. Biol. In press
C. Rogers and P. L. Sachdev, The Burgers hierarchy: on nonlinear initial and boundary value problems, Nuovo Cimento. In press
J. G. Kingston and C. Rogers, Reciprocal Bäcklund transformations of conservation laws, Phys. Lett. 92A, 261–264 (1982)
A. S. Fokas and Y. C. Yortsos, On the exactly soluble equations ${S_t} = {\left [ {{{\left ( {\beta S + \gamma } \right )}^{ - 2}}{S_x}} \right ]_x} + \alpha {\left ( {\beta S + \gamma } \right )^{ - 2}}{S_x}$ occurring in two phase flow in porous media, Soc. Ind. Appl. Math. J. Appl. 42, 318–332 (1982)
C. Rogers, M. P. Stallybrass, and D. L. Clements, On two phase filtration under gravity and with boundary infiltration: application of a Bäcklund transformation, Nonlinear Anal. Methods Appl. 7, 785–799 (1983)
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Article copyright:
© Copyright 1987
American Mathematical Society
