## Iteration perturbation method for strongly nonlinear oscillations.(English)Zbl 1015.70019

Summary: We propose a perturbation technique coupling with iteration method, yielding a tool for an analytical solution of nonlinear equations of the form $$u''+u+\varepsilon f(u,u')=0$$, $$u(0)=A$$, $$u'(0)=0$$. The results obtained are valid not only for weakly nonlinear problems, but also for strongly nonlinear problems. Furthermore, the approximate solutions are valid for the whole solution domain, and even the first-step iteration leads to high accuracy. Some examples are given to illustrate the effectiveness of the method.

### MSC:

 70K60 General perturbation schemes for nonlinear problems in mechanics
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### References:

 [1] Acton, J.R., Solving Equations With Physical Understanding (1985) [2] Andrianov, I., International Journal of Nonlinear Sciences and Numerical Simulation 1 (4) pp 327– (2000) · Zbl 0977.35031 [3] He, J.H., Communications in Nonlinear Science and Numerical Simulation 4 (2) pp 109– (1999) · Zbl 0928.34013 [4] He, J.H., Mechanics and Practice 21 (1) pp 17– (1999) [5] He, J.H., Computer Methods in Applied Mechanics and Engineering 178 pp 257– (1999) · Zbl 0956.70017 [6] He, J.H., Meccanica 34 (3) pp 287– (1999) [7] He, J.H., Communications in Nonlinear Science and Numerical Simulation 4 (1) pp 78– (1999) · Zbl 0932.34057 [8] He, J.H., Communications in Nonlinear Science and Numerical Simulation 4 (1) pp 81– (1999) · Zbl 0932.34058 [9] He, J.H., International Journal of Non-Linear Mechanics 35 pp 37– (2000) · Zbl 1068.74618 [10] He, J.H., Meccanica 35 pp 299– (2000) · Zbl 0986.70016 [11] He, J.H., Journal of Sound and Vibration 229 (5) pp 1257– (2000) · Zbl 1235.70139 [12] He, J.H., International Journal of Nonlinear Sciences and Numerical Simulation 1 (1) pp 51– (2000) [13] Howarth, L., Proceedings of the Royal Society of London 164 pp 547– (1938) · JFM 64.1452.01 [14] Liao, S.J., International Journal of Non-Linear Mechanics 30 (3) pp 371– (1995) · Zbl 0837.76073 [15] Liao, S.J., Journal Fluid Mechanics 385 pp 101– (1999) · Zbl 0931.76017 [16] Nayfeh, A.H., Introduction to Perturbation Techniques (1981) · Zbl 0449.34001 [17] Nayfeh, A.H., Problems in Perturbation (1985) · Zbl 0573.34001
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