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Comparison of homotopy perturbation method and homotopy analysis method. (English) Zbl 1062.65074
Summary: Comparison of homotopy perturbation method (HPM) and homotopy analysis method is made,revealing that the former is more powerful than the later. Furthermore, the HPM is further developed in this paper by applying the modern perturbation methods.

MSC:
65L05 Numerical methods for initial value problems
34A34 Nonlinear ordinary differential equations and systems, general theory
65H20 Global methods, including homotopy approaches to the numerical solution of nonlinear equations
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[1] Liao, S.J., A second-order approximate analytical solution of a simple pendulum by the process analysis method, ASME J. appl. mech., 59, 970-975, (1992) · Zbl 0769.70017
[2] Liao, S.J., A new kind of nonlinear analytical method based on homotopy technology (1), J. Shanghai mech., 15, 2, 28-33, (1994), (in Chinese)
[3] Liao, S.J., An approximate solution technique not depending on small parameters: a special example, Int. J. non-linear mech., 30, 3, 371-380, (1995) · Zbl 0837.76073
[4] Liao, S.J., A new kind of nonlinear analytical method based on homotopy technology (2), J. Shanghai mech., 16, 3, 129-137, (1995), (in Chinese)
[5] Liao, S.J., A kind of approximate solution technique which does not depend upon small parameters–II: an application in fluid mechanics, Int. J. non-linear mech., 32, 5, 815-822, (1997) · Zbl 1031.76542
[6] Liao, S.J., Homotopy analysis method: A kind of nonlinear analytical technique not depending on small parameters, J. Shanghai mech., 18, 3, 196-200, (1997), (in Chinese)
[7] Liao, S.J., Homotopy analysis method: a new analytic method for nonlinear problems, Appl. math. mech. (English-ed.), 19, 10, 957-962, (1998) · Zbl 1126.34311
[8] S.J. Liao, Homotopy analysis method: a new analytical method for nonlinear problems without small parameters, in: The 3rd International Conference on Nonlinear Mechanics, Shanghai, 1998, pp. 829-833
[9] Liao, S.J., An explicit, totally analytic solution of laminar viscous flow over a semi-infinite flat plate, Commun. nonlinear sci. numer. simulat., 3, 2, 53-57, (1998) · Zbl 0922.34012
[10] Liao, S.J., An explicit, totally analytic approximate solution for Blasius’ viscous flow problem, Int. J. non-linear mech., 34, 759-778, (1999) · Zbl 1342.74180
[11] Liao, S.J., A uniformly valid analytic solution of two dimensional viscous flow over a semi-infinite plat plate, J. fluid mech., 385, 101-128, (1999) · Zbl 0931.76017
[12] Liao, S.J., A non-iterative numerical approach for 2-D viscous flow problems governed by the falkner – skan equation, Int. J. numer. methods fluids, 35, 5, 495-518, (2001) · Zbl 0990.76068
[13] Hillermeier, C., Generalized homotopy approach to multiobjective optimization, Int. J. optim. theory appl., 110, 3, 557-583, (2001) · Zbl 1064.90041
[14] He, J.-H., An approximate solution technique depending upon an artificial parameter, Commun. nonlinear sci. numer. simulat., 3, 2, 92-97, (1998) · Zbl 0921.35009
[15] He, J.-H., Newton-like iteration method for solving algebraic equations, Commun. nonlinear sci. numer. simulat., 3, 2, 106-109, (1998) · Zbl 0918.65034
[16] He, J.-H., Homotopy perturbation technique, Comput. methods appl. mech. engng., 178, 3/4, 257-262, (1999) · Zbl 0956.70017
[17] He, J.-H., A coupling method of homotopy technique and perturbation technique for nonlinear problems, Int. J. nonlinear mech., 35, 1, 37-43, (2000) · Zbl 1068.74618
[18] Mallil, E.; Lahmam, H.; Damil, N.; Potier-Ferry, M., An iterative process based on homotopy and perturbation techniques, Comput. methods appl. mech. engng., 190, 1845-1858, (2000) · Zbl 1004.74079
[19] J.-M. Cadou, N. Moustaghfir, E.H. Mallil, N. Damil, M. Potier-Ferry, Linear iterative solvers based on perturbation techniques, C.R. Acad. Sci. Paris, 2001, T.329, Serie II b: 457-462
[20] Elhage-Hussein, A.; Potier-Ferry, M.; Damil, N., A numerical continuation method based on pade approximants, Int. J. solids struct., 37, 6981-7001, (2000) · Zbl 0980.74021
[21] R.E.L. Mokhtari, J.-M. Cadou, M. Potier-Ferry, Une approche multigrille basee sur les techniques d’homotopie et de perturbation pour la resolution de problems d’elasticite lineare, XV^eme Congres Francais de Mecanique, Nancy, 3-7 Septembre, 2001, pp. 1-6
[22] Jegen, M.D.; Everett, M.E.; Schultz, A., Using homotopy to invert geophysical data, Geophysics, 66, 6, 1749-1760, (2001)
[23] G.-L. Liu, New research directions in singular perturbation theory: artificial parameter approach and inverse-perturbation technique, in: C. Cheng et al. (Ed.), Modern Mathematics and Mechanics (MMM)-VII, Shanghai University Press, Shanghai, 1997, pp. 47-53 (in Chinese)
[24] Damil, N.; Potier-Ferry, M.; Najah, A.; Chari, R.; Lahmam, H., An iterative method based upon pade approximants, Commun. numer. methods engng., 15, 10, 701-708, (1999) · Zbl 0943.65065
[25] Bender, C.M.; Pinsky, K.S.; Simmons, L.M., A new perturbative approach to nonlinear problems, J. math. phys., 30, 7, 1447-1455, (1989) · Zbl 0684.34008
[26] Andrianov, I.; Awrejcewicz, J., Construction of periodic solution to partial differential equations with nonlinear boundary conditions, Int. J. nonlinear sci. numer. simulat., 1, 4, 327-332, (2000) · Zbl 0977.35031
[27] He, J.H., A note on delta – perturbation method, Appl. math. mech., 23, 6, 558-562, (2002)
[28] He, J.H., A review on some new recently developed nonlinear analytical techniques, Int. J. nonlinear sci. numer. simulat., 1, 1, 51-70, (2000) · Zbl 0966.65056
[29] He, J.-H., Modified lindstedt – poincare methods for some strongly nonlinear oscillations. part I: expansion of a constant, Int. J. non-linear mech., 37, 2, 309-314, (2002) · Zbl 1116.34320
[30] He, J.-H., Modified lindstedt – poincare methods for some strongly nonlinear oscillations. part II: a new transformation, Int. J. non-linear mech., 37, 2, 315-320, (2002) · Zbl 1116.34321
[31] He, J.-H., Modified lindstedt – poincare methods for some strongly nonlinear oscillations. part III: double series expansion, Int. J. non-linear sci. numer. simulat., 2, 4, 317-320, (2001) · Zbl 1072.34507
[32] Nayfeh, A.H., Introduction to perturbation techniques, (1981), John Wiley & Sons New York · Zbl 0449.34001
[33] He, J.H., Modified straightforward expansion, Meccanica, 34, 4, 287-289, (1999) · Zbl 1002.70019
[34] Wazwaz, A.-M., A comparison between Adomian decomposition method and Taylor series method in the series solutions, Appl. math. comput., 97, 37-44, (1998) · Zbl 0943.65084
[35] Wang, Z.K.; Gao, T.A., An introduction to homotopy methods, (1990), Chongqing Publishing House, (in Chinese)
[36] He, J.H., Approximate analytical solution of Blasius’s equation, Commun. nonlinear sci. numer. simulat., 3, 4, 206-263, (1998), 4 (1) (1999) 75-78
[37] Howarth, L., On the solution of the laminar boundary layer equation, Proc. R. soc. lond. A, 164, 547-579, (1938) · JFM 64.1452.01
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