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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:
65L99Numerical methods for ODE
References:
[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 · doi:10.1016/j.cnsns.2010.01.038
[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 · doi:10.1016/j.cnsns.2009.08.014
[3]Marinca, V.; Herisanu, N.: Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer, Int commun heat mass transfer 35, 710-715 (2008)
[4]Marinca, V.; Herisanu, N.; Nemes, I.: Optimal homotopy asymptotic method with application to thin film flow, Cent eur J phys 6, 648-653 (2008)
[5]Marinca, V.; Hersanu, N.; Bota, C.; Marinca, B.: An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate, Appl math lett 22, 245-251 (2009) · Zbl 1163.76318 · doi:10.1016/j.aml.2008.03.019
[6]Yabushita, K.; Yamashita, M.; Tsuboi, K.: An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method, J phys A: math theor 40, 8403-8416 (2007)
[7]Liao, S. J.: An explicit, totally analytic approximate solution for Blasius viscous flow problems, Int J nonlinear mech 34, 759-778 (1999)
[8]Liao, S. J.: Beyond perturbation: introduction to homotopy analysis method, (2003)
[9]Liao, S. J.: Notes on the homotopy analysis method: some definitions and theorems, Commun nonlinear sci numer simul 14, 983-997 (2009) · Zbl 1221.65126 · doi:10.1016/j.cnsns.2008.04.013
[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 · doi:10.1016/j.cnsns.2009.09.002