The two-variable technique for singular partial differential problems and its justification
Author:
M. Bouthier
Journal:
Quart. Appl. Math. 38 (1980), 263-276
MSC:
Primary 35B25
DOI:
https://doi.org/10.1090/qam/592195
MathSciNet review:
592195
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Abstract: Dirichlet problems with a small parameter in factor of the highest derivative are considered for bounded domains. A two-variable technique is formalized in order to carry out the study of the main boundary layer. A “secular “ hypothesis is made, and a unique and uniformly valid asymptotic expansion is obtained. However, it is shown that the “secular” hypothesis may be weakened and this yields a whole set of expansions. Then the asymptotic validity of each expansion in the set can be proven for second-order operators by means of an extension of a theorem due to Eckhaus and Jager.
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Comstock, Singular perturbation of elliptic equation, SIAM jl. Appl. Math. 1971, 20, 491–502
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Erdelyi, Two variable expansions for singular perturbations, J. Inst. Math. Applic., 4, 113–119 (1968)
Hormander, Linear partial differential operators, Springer-Verlag, Berlin, 1963, 89
Kaplun, Fluid mechanics and singular perturbations, Academic Press, New York, 1967
A. Lagerstrom and R. G. Casten, Basic concepts underlying singular perturbation techniques, SIAM Rev. 14, 63–120 (1972)
Levinson, The first boundary value problem, Ann. Math. 51, 428–445 (1950)
L. Lions, Perturbations singulières dans les problémes aux limites et en controle optimal, Springer-Verlag, Berlin, 1973
H. Nayfeh, Perturbation methods, Springer Verlag, New York, 1973
Protter and H. Weinberger, Maximum principles in differential equations, Prentice-Hall, London (1967)
L. Reiss, On multivariable asymptotic expansions, SIAM Rev. 13, 189–196 (1971)
R. Smith, The multivariable method in singular perturbation analysis, SIAM Rev. 17, 221–273 (1975)
Van Dyke, Perturbation methods in fluid mechanics. Academic Press, New York, 1964
B. Whitham, Two-timing variational principle and waves, J. Fluid Mech. 44, 373–395 (1970)
J. Wollkind, Singular perturbation techniques: a comparison of the method of the matched asymptotic expansions with that of multiple scales, SIAM Rev. 19, 502–516 (1977)
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© Copyright 1980
American Mathematical Society
