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Comparison between the homotopy analysis method and homotopy perturbation method. (English) Zbl 1082.65534
Summary: We show that the so called “homotopy perturbation method” is only a special case of the homotopy analysis method. Both methods are in principle based on Taylor series with respect to an embedding parameter. Besides, both can give very good approximations by means of a few terms, if initial guess and auxiliary linear operator are good enough. The difference is that, “the homotopy perturbation method” had to use a good enough initial guess, but this is not absolutely necessary for the homotopy analysis method. This is mainly because the homotopy analysis method contains the auxiliary parameter \(\hbar\), which provides us with a simple way to adjust and control the convergence region and rate of solution series. So, the homotopy analysis method is more general. Besides, the update of the concept of the “analytical solution” is discussed.

65H20 Global methods, including homotopy approaches to the numerical solution of nonlinear equations
Full Text: DOI
[1] S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems. Ph.D. Thesis, Shanghai Jiao Tong University, 1992.
[2] Liao, S.J., Beyond perturbation: introduction to homotopy analysis method, (2003), Chapman & Hall/CRC Press Boca Raton
[3] Liao, S.J., On the homotopy analysis method for nonlinear problems, Appl. math. comput., 147, 499-513, (2004) · Zbl 1086.35005
[4] Ayub, M.; Rasheed, A.; Hayat, T., Exact flow of a third grade fluid past a porous plate using homotopy analysis method, Int. J. eng. sci., 41, 2091-2103, (2003) · Zbl 1211.76076
[5] Hayat, T.; Khan, M.; Ayub, M., On the explicit analytic solutions of an Oldroyd 6-constant fluid, Int. J. engng. sci., 42, 123-135, (2004) · Zbl 1211.76009
[6] Hayat, T.; Khan, M.; Asghar, S., Homotopy analysis of MHD flows of an Oldroyd 8-constant fluid, Acta mech., 168, 213-232, (2004) · Zbl 1063.76108
[7] Hayat, T.; Khan, M.; Asghar, S., Magnetohydrodynamic flow of an Oldroyd 6-constant fluid, Appl. math. comput., 155, 417-425, (2004) · Zbl 1126.76388
[8] Wu, Y.Y.; Liao, S.J., Solving the one-loop soliton solution of the Vakhnenko equation by means of the homotopy analysis method, Chaos solitons and fractals, 23, 5, 1733-1740, (2004) · Zbl 1069.35060
[9] Wu, W.; Liao, S.J., Solving solitary waves with discontinuity by means of the homotopy analysis method, Chaos, solitons and fractals, 23, 1733-1740, (2004)
[10] S.J. Liao, An analytic solution of unsteady boundary-layer flows caused by an impulsively stretching plate. Communications for Nonlinear Science and Numerical Simulations (in press). · Zbl 1078.76022
[11] 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
[12] He, J.-H., Homotopy perturbation techique, Comput. methods appl. mech. eng., 178, 257-262, (1999)
[13] He, J.-H., Comparison of homotopy perturbation method and homotopy analysis method, Appl. math. comput., 156, 527-539, (2004) · Zbl 1062.65074
[14] Adomian, G., Nonlinear stochastic differential equations, J. math. anal. appli., 55, 441-452, (1976) · Zbl 0351.60053
[15] Lyapunov, A.M., General problem on stability of motion, (1892), Taylor & Francis London, (1992, English translation) · Zbl 0041.32204
[16] Karmishin, A.V.; Zhukov, A.T.; Kolosov, V.G., Methods of dynamics calculation and testing for thin-walled structures, (1990), Mashinostroyenie Moscow, (in Russian)
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