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Homotopy perturbation technique. (English) Zbl 0956.70017
Summary: The homotopy perturbation technique does not depend upon a small parameter in the equation. By the homotopy technique in topology, a homotopy can be constructed with an imbedding parameter \(p\in [0,1]\), which is considered as a “small parameter”. Here we give some examples, and demonstrate that the approximations obtained by the proposed method are uniformly vaild not only for small parameters, but also for very large parameters.

MSC:
70K60 General perturbation schemes for nonlinear problems in mechanics
34A45 Theoretical approximation of solutions to ordinary differential equations
34E10 Perturbations, asymptotics of solutions to ordinary differential equations
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[1] Liao, S.J., An approximate solution technique not depending on small parameters: a special example, Int. J. non-linear mechanics, 30, 3, 371-380, (1995) · Zbl 0837.76073
[2] Liao, S.J., Boundary element method for general nonlinear differential operators, Engineering analysis with boundary element, 20, 2, 91-99, (1997)
[3] A.H. Nayfeh, Introduction to Perturbation Techniques, Wiley, New York, 1981 · Zbl 0449.34001
[4] C.C. Lin, Mathematics Applied to Deterministic Problems in Natural Sciences, Macmillan, New York, 1974
[5] Y.B. Wang et al., An Introduction to Perturbation Techniques (in Chinese), Shanghai Jiaotong University Press, 1986
[6] He, J.H., A new approach to nonlinear partial differential equations, communications in nonlinear science and numerical simulation, 2, 4, 230-235, (1997)
[7] J.H. He, Nonlinear oscillation with fractional derivative and its approximation, International Conference on Vibration Engineering ’98, 1998, Dalian, China
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