On perturbation methods involving expansions in terms of a parameter
Author:
Richard Bellman
Journal:
Quart. Appl. Math. 13 (1955), 195-200
MSC:
Primary 34.0X
DOI:
https://doi.org/10.1090/qam/70792
MathSciNet review:
70792
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Abstract: It is shown by means of some examples from the theories of linear algebraic equations, linear integral equations and nonlinear differential equations that the effectiveness of the method of expanding a solution in a power series in terms of a parameter may in many cases be greatly increased by expanding in terms of a suitably chosen function of the parameter. This is particularly the case when the physical setting of the problem allows only positive values of the parameter to enter.
- Richard Bellman, A note on the summability of formal solutions of linear integral equations, Duke Math. J. 17 (1950), 53–55. MR 32916
- Solomon Lefschetz, Lectures on Differential Equations, Annals of Mathematics Studies, No. 14, Princeton Univeristy Press, Princeton, N. J.; Oxford University Press, London, 1946. MR 0016788
- M. J. Lighthill, A technique for rendering approximate solutions to physical problems uniformly valid, Philos. Mag. (7) 40 (1949), 1179–1201. MR 33941
- J. Shohat, On van der Pol’s and non-linear differential equations, J. Appl. Phys. 15 (1944), 568–574. MR 10322
- J. J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems, Interscience Publishers, Inc., New York, N.Y., 1950. MR 0034932
R. Bellman, On the summability of solutions of linear integral equations, Duke Math. J. 17, 53-5 (1950)
S. Lefschetz, Lectures on differential equations, Princeton University Press, 1946
J. Lighthill, A technique for rendering approximate solutions to physical problems uniformly valid, Phil. Mag. 40, 1179-1201 (1949)
J. Shohat, On Van der Pol’s and related nonlinear differential equations, J. Appl. Physics 15, 568-574 (1944)
J. J. Stoker, Nonlinear vibrations, Interscience, New York, 1950
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Article copyright:
© Copyright 1955
American Mathematical Society