Asymptotic solutions of partial differential equations with damping and delay
Authors:
G. N. Bojadziev and R. W. Lardner
Journal:
Quart. Appl. Math. 33 (1975), 205-214
MSC:
Primary 35R10; Secondary 35C99
DOI:
https://doi.org/10.1090/qam/447854
MathSciNet review:
447854
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Abstract: Certain hyperbolic partial differential equations with small nonlinearities involving significant damping as well as time delay are investigated. The Krylov—Bogoliubov—Mitropolskii method is extended and applied. An application is given to longitudinal vibrations of a nonlinear elastic rod.
Yu. A. Mitropolskii and B. I. Moseenkov, Lectures on the application of asymptotic methods to the solution of equations with partial derivatives (in Russian), Ac. of Sci. Ukr. SSR, Kiev, 1968
V. I. Fodchuk, Application of the averaging method to the differential-difference equations of the neutral type, in Proceedings of the fourth conference on nonlinear oscillations, Prague (1968), 161–165
- G. N. Boyadžiev and R. W. Lardner, Monofrequent oscillations in mechanical systems governed by second order hyperbolic differential equations with small non-linearities, Internat. J. Non-Linear Mech. 8 (1973), 289–302 (English, with French, German and Russian summaries). MR 334623, DOI https://doi.org/10.1016/0020-7462%2873%2990019-X
Z. Osinski, Longitudinal, torsional and bending vibrations of a uniform bar with nonlinear internal friction and relaxation, in Nonlinear vibration problems, P. Ac. of Sci. Warsaw 4, 159–166 (1962)
I. P. Popov and N. P. Paltov, Approximate methods for analyzing nonlinear automatic systems (in Russian), Gos. Izdat. Fiz.-Math. Lit., Moscow, 1960
- Kenneth L. Cooke, Differential-difference equations, Internat. Sympos. Nonlinear Differential Equations and Nonlinear Mechanics, Academic Press, New York, 1963, pp. 155–171. MR 0146481
Yu. A. Mitropolskii and B. I. Moseenkov, Lectures on the application of asymptotic methods to the solution of equations with partial derivatives (in Russian), Ac. of Sci. Ukr. SSR, Kiev, 1968
V. I. Fodchuk, Application of the averaging method to the differential-difference equations of the neutral type, in Proceedings of the fourth conference on nonlinear oscillations, Prague (1968), 161–165
G. N. Bojadziev and R. W. Lardner, Monofrequent oscillations in mechanical systems governed by second order hyperbolic differential equations with small non-linearities, Int. J. Non-Linear Mechanics 8, 289–302 (1973)
Z. Osinski, Longitudinal, torsional and bending vibrations of a uniform bar with nonlinear internal friction and relaxation, in Nonlinear vibration problems, P. Ac. of Sci. Warsaw 4, 159–166 (1962)
I. P. Popov and N. P. Paltov, Approximate methods for analyzing nonlinear automatic systems (in Russian), Gos. Izdat. Fiz.-Math. Lit., Moscow, 1960
R. E. Bellman and K. L. Cooke, Differential-difference equations, Academic Press, New York, 1963
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Article copyright:
© Copyright 1975
American Mathematical Society
