Comparison of the matched asymptotic expansions method and the two-variable technique
Author:
M. Bouthier
Journal:
Quart. Appl. Math. 41 (1984), 407-422
MSC:
Primary 76D10; Secondary 35C20, 35J40
DOI:
https://doi.org/10.1090/qam/724052
MathSciNet review:
724052
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Abstract: Either the matched asymptotic expansions method or the two-variable technique are available for treating boundary layer problems. A comparison of the two methods is achieved on dealing with elliptic boundary value problems. The two-variable technique is proved to be slightly more powerful than the matched expansions method. Nevertheless it fails to determine a closed class of approximate solutions. Such a class, which involves the results of both the asymptotic methods is set out with help of an asymptotic equivalence theorem.
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M. Bouthier, Développements asymptotiques à deux types d’échelles et perturbation singulière de problèmes aux limites elliptiques, Paris, Thèse d’Etat, Université Paris 6, 1977
M. Bouthier, The two-variable technique for singular partial differential problems and its justification, Quart. Appl. Math. 38, 263–276 (1980)
G. Comstock, Singular perturbation of elliptic equations, SIAM J. Appl. Math. 20, 491–502 (1971)
W. Eckhaus, Asymptotic analysis of singular perturbations, North-Holland, Amsterdam, 1979
A. Erdelyi, Two-variable expansions for singular perturbations, J. Inst. Math. Applic. 4, 113–119 (1968)
A. Friedman, Partial differential equations, Holt Rinchart and Winston, New York, 1969
S. Kaplun, Fluid mechanics and singular perturbations, Academic Press, New York, 1967
J. Kevorkian and J. D. Cole, Perturbations methods, Applied Mathematics, Springer-Verlag, New York, 1981
K. Kuen Tam, On the asymptotic solution of the Orr-Sommerfeld equation by the method of multiple scales. J. Fluid Mech. 34, 145–158 (1968)
P. A. Lagerstrom and R. G. Casten, Basic concepts underlying singular perturbation techniques, SIAM Review 14, 63–120 (1972)
L. E. Levine and E. S. Lubot, Optimal approximate solutions and time scales, SIAM J. Appl. Math. 29, 439–448
A. H. Nayfeh, Perturbation methods, Wiley, New York, 1973
E. L. Reiss, On multivariable asymptotic expansions, SIAM Review 13, 189–196 (1971)
S. Rosenblat, Asymptotically equivalent singular perturbation problems, Studies in Applied Mathematics, 55, 249–280 (1976)
J. W. Searl, Expansions for singular perturbations, J. Inst. Maths Appl., 8, 131–138 (1971)
D. R. Smith, The multivariable method in singular perturbation analysis, SIAM Review 17, 221–273 (1975)
A. Van Harten, On an elliptic singular perturbation problem, in Ordinary and Partial Differential Equations, Lecture Notes in Math., Springer-Verlag, New York, 485–494, 1976
D. J. Wollkind, Singular perturbation techniques: a comparison of the method of matched asymptotic expansions with that of multiple scales, SIAM Review 19, 502–516 (1977)
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© Copyright 1984
American Mathematical Society