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Reply to “Comments on ‘A one-step optimal homotopy analysis method for nonlinear differential equations”’. (English) Zbl 1222.65093
Summary: V. Marinca and N. Herişanu [ibid. 15, No. 11, 3735–3739 (2010; Zbl 1222.65089)] made some comments on our paper [ibid. 15, No. 8, 2026–2036 (2010; Zbl 1222.65091)] and pointed out “some fundamental mistakes and misinterpretations along with a false conclusion”. Unfortunately, Marinca’s comments are wrong. Here, we further reveal the essence of Marinca’s approach, and point out the reason why their method is indeed time-consuming: their method is nothing more than a traditional method in approximation theory. Numerical results for a given example and related MATHEMATICA code are given to support our view-points.

MSC:
65L99 Numerical methods for ordinary differential equations
Software:
Mathematica
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[1] Marinca, V.; Herisanu, N., Comments on “A one-step optimal homotopy analysis method for nonlinear differential equations”, Commun. nonlinear sci numer simul, 15, 3735-3739, (2010) · Zbl 1222.65089
[2] Niu, Z.; Wang, C., A one-step optional homotopy analysis method for nonlinear differential equations, Commun nonlinear sci numer simul, 15, 2026-2036, (2010) · Zbl 1222.65091
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[10] Liao, S.J., An optimal homotopy-analysis approach for strongly nonlinear differential equations, Commun nonlinear sci numer simul, 15, 2003-2016, (2010) · Zbl 1222.65088
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