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Some issues on HPM and HAM methods: a convergence scheme. (English) Zbl 1219.65083
Summary: The homotopy method for the solution of nonlinear equations is revisited in the present study. An analytic method is proposed for determining the valid region of convergence of control parameter of the homotopy series, as an alternative to the classical way of adjusting the region through graphical analysis. Illustrative examples are presented to exhibit a vivid comparison between the homotopy perturbation method (HPM) and the homotopy analysis method (HAM). For special choices of the initial guesses it is shown that the convergence-control parameter does not cover the HPM. In such cases, blindly using the HPM yields a non convergence series to the sought solution. In addition to this, HPM is shown not always to generate a continuous family of solutions in terms of the homotopy parameter. By the convergence-control parameter this can however be prevented to occur in the HAM.

##### MSC:
 65L99 Numerical methods for ordinary differential equations
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##### References:
  Mickens, R.E., Oscillations in planar dynamic systems, (1996), World Scientific Singapore · Zbl 0840.34001  He, J.H., Variational iteration method—some recent results and new interpretations, J. comput. appl. math., 207, 3-17, (2007) · Zbl 1119.65049  Inc, M.; Cavlak, E., On numerical solutions of a new coupled mkdv system by using the Adomian decomposition method and he’s variational iteration methos, Phys. scr., 78, 045008, (2008) · Zbl 1158.35417  He, J.H.; Wu, Xu-H., Exp-function method for nonlinear wave equations, Chaos solitons fractals, 30, 700-705, (1999)  Belendez, A.; Hernandez, A.; Belendez, T.; Marquez, A.; Neipp, C., Analytic approximations for the period of a simple pendulum, European J. phys., 27, 539-551, (2006)  S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. Thesis, Shanghai Jiao Tong University, 1992.  He, J.H., An approximate solution technique depending on an artificial parameter: a special example, Commun. nonlinear sci. numer. simul., 3, 92-96, (1998) · Zbl 0921.35009  Liao, S.J., Beyond perturbation: introduction to homotopy analysis method, (2003), Chapman & Hall/CRC  Liao, S.J., An explicit, totally analytic approximation of blasius’ viscous flow problems, Internat. J. non-linear mech., 34, 759-778, (1999) · Zbl 1342.74180  Liao, S.J.; Tan, Y., A general approach to obtain series solutions of nonlinear differential equations, Stud. appl. math., 119, 297-354, (2007)  Sajid, M.; Hayat, T., Comparison of ham and hpm methods in nonlinear heat conduction and convection equations, Nonlinear anal. RWA, 9, 2296-2301, (2008) · Zbl 1156.76436  Abbasbandy, S., The application of the homotopy analysis method to nonlinear equations arising in heat transfer, Phys. lett. A, 360, 109-113, (2006) · Zbl 1236.80010  VanGorder, R.A.; Vajravelu, K., Analytic and numerical solutions to the lane – emden equation, Phys. lett. A, 372, 6060-6065, (2008) · Zbl 1223.85004  Liang, S.; Jeffrey, D.J., Comparison of homotopy analysis method and homotopy perturbation method through an evolution equation, Commun. nonlinear sci. numer. simul., 14, 4057-4064, (2009) · Zbl 1221.65281  Crane, L., Flow past a stretching plate, Z. angew. math. phys., 21, 645-647, (1970)  Magyari, E.; Keller, B., Exact solutions for self-similar boundary-layer flows induced by permeable stretching walls, Eur. J. mech. B fluids, 19, 109-122, (2000) · Zbl 0976.76021  Magyari, E.; Pop, I.; Keller, B., Exact dual solutions occurring in Darcy mixed convection flow, Int. J. heat mass transfer, 44, 4563-4566, (2001) · Zbl 1068.76548  Benton, Edward R., Some new exact, viscous, nonsteady solutions of burgers’ equation, Phys. fluids, 9, 1247-1248, (1966) · Zbl 0148.22306
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