Poincare-Lighthill and linear-time-scales methods for linear perturbation problems
Authors:
A. Klimas, F. X. Murphy Jr. and G. Sandri
Journal:
Quart. Appl. Math. 31 (1973), 237-243
DOI:
https://doi.org/10.1090/qam/99703
MathSciNet review:
QAM99703
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Abstract: For a class of multidimensional linear perturbation problems of considerable significance in applications, the Poincaré—Lighthill technique is shown to give first-order expansion terms which grow unbounded relative to the leading term (secular behavior), while the method of linear time scales leads to well-behaved expansion terms. A solvable example is introduced for comparison with the exact solution.
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G. V. Ramanathan and G. Sandri, J. Math. Phys. 10, 1973 (1969)
M. J. Lighthill, Phil. Mag. 40, 1179 (1949)
M. J. Lighthill, A. für Flug. 9, 267 (1961)
H. S. Tsien, The Poincaré-Lighthill-Kuo method in advanced applied mechanics 4, Academic Press, Inc., N. Y., 1956, p. 281
C. Comstock, U. S. Naval Postgraduate School Report No. NPS-53ZK71071A, 20 July 1971
G. Sandri, Ann Phys. (NY) 24, 332, 380 (1963)
E. Frieman, J. Math. Phys. 4, 410 (1963)
J. Cole and J. Kevorkian, in Nonlinear differential equations and nonlinear mechanics, J. P. LaSalle and S. Lefshetz, eds., Academic Press, Inc., N. Y., 1963
A. Klimas, R. V. Ramnath and G. Sandri, J. Math. Anal. Appl. 32, 482 (1970). It is interesting to compare the quantities $G,B,\Gamma$ of [8] with the quantities ${\tilde y_w},{\tilde y_s},\phi$ of [16]. We have ${\tilde y_w} = G,{\tilde y_s} = B,\phi = \Gamma$.
N. Krylov and N. Bogolubov, Introduction to nonlinear mechanics, Princeton University Press, Princeton, N. J., 1947
N. Bogolubov and Y. Mitropolsky, Asymptotic methods in the theory of nonlinear oscillations, Gordon and Breach, N. Y., 1961
G. F. Carrier, Boundary layer problems in applied mathematics, CPAM 7, 11 (1954); in Advances in applied mechanics III, Academic Press, N. Y., 1953
S. Kaplun, Fluid mechanics and singular perturbations, Academic Press, N. Y., 1967
M. D. Van Dyke, Perturbation methods in fluid Mechanics, Academic Press, Inc., N. Y., 1964
J. Cole, Perturbation methods in applied mathematics, Blaisdell, 1967
R. Bellman, Perturbation techniques in mathematics, physics and engineering, Holt, Rinehart and Winston, San Francisco, 1964
W. Wasow, J. Rat. Mech. Anal. 4, 751 (1955)
P. D. Usher, Coordinate stretching and interface location, II, a new PL expansion, J. Comp. Phys. 3, 29 (1968)
P. D. Usher, Quart. Appl. Math. 28, 463 (1971)
M. F. Pritulo, On the determination of uniformly accurate solutions of differential equations by the method of perturbation of coordinates, P.M.M. 26, 444 (1962)
F. X. Murphy, Jr. and G. Sandri, Lectures in nonequliibrium statistical mechanics, C.U.N. Y.mimeograph (1971) and Not. Am. Math. Soc. 18, 928 (1971)
G. V. Ramanathan and G. Sandri, J. Math. Phys. 10, 1973 (1969)
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Article copyright:
© Copyright 1973
American Mathematical Society
