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The essence of the homotopy analysis method. (English) Zbl 1194.34010
Summary: The generalized Taylor expansion including a secret auxiliary parameter \(h\) which can control and adjust the convergence region of the series is the foundation of the homotopy analysis method proposed by Liao. The secret of \(h\) cannot be understood in the frame of the homotopy analysis method. This is a serious shortcoming of Liao’s method. We solve the problem. Through a detailed study of a simple example, we show that the generalized Taylor expansion is just the usual Taylor’s expansion at different point \(t_{1}\). We prove that there is a relationship between \(h\) and \(t_{1}\), which reveals the meaning of \(h\) and the essence of the homotopy analysis method. As an important example, we study the series solution of the Blasius equation. Using the series expansion method at different points, we obtain the same result with Liao’s solution given by the homotopy analysis method.

MSC:
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
34A45 Theoretical approximation of solutions to ordinary differential equations
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[1] S.J. Liao, Proposed homotopy analysis techniques for the solution of nonlinear problems, Ph.D. Dissertation, Shanghai Jiao Tong University, Shanghai, 1992.
[2] Liao, S.J., An approximate solution technique which does not depend upon small parameters: a special example, Int. J. non-linear mech., 30, 371-380, (1995) · Zbl 0837.76073
[3] Liao, S.J., An approximate solution technique which does not depend upon small parameters (part 2): an application in fluid mechanics, Int. J. non-linear mech., 32, 815-822, (1997) · Zbl 1031.76542
[4] Liao, S.J., An explicit, totally analytic approximation of Blasius viscous flow problem, Int. J. non-linear mech., 34, 759-778, (1999) · Zbl 1342.74180
[5] Liao, S.J., A uniformly valid analytic solution of 2D viscous flow past a semi-infinite flat plate, J. fluid mech., 385, 101-128, (1999) · Zbl 0931.76017
[6] Liao, S.J.; Campo, A., Analytic solutions of the temperature distribution in Blasius viscous flow problems, J. fluid mech., 453, 411-425, (2002) · Zbl 1007.76014
[7] Liao, S.J., An explicit analytic solution to the thomas – fermi equation, Appl. math. comput., 144, 495-506, (2003) · Zbl 1034.34005
[8] Liao, S.J., On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet, J. fluid mech., 488, 189-212, (2003) · Zbl 1063.76671
[9] Liao, S.J., On the homotopy analysis method for nonlinear problems, Appl. math. comput., 147, 499-513, (2004) · Zbl 1086.35005
[10] Liao, S.J.; Magyari, E., Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones, Z. angew. math. phys., 57, 777-792, (2006) · Zbl 1101.76056
[11] Liao, S.J.; Tan, Y., A general approach to obtain series solutions of nonlinear differential equations, Stud. appl. math., 119, 297-354, (2007)
[12] Liao, S.J., A general approach to get series solution of non-similarity boundary-layer flows, Commun. nonlinear sci. numer. simulat., 14, 2144-2159, (2009) · Zbl 1221.76068
[13] Liao, S.J., Beyond perturbation: introduction to the homotopy analysis method, (2003), CRC Press LLC Boca Raton
[14] Liao, S.J., On a generalized Taylor theorem: a rational proof of the validity of the so-called homotopy analysis method, J. appl. math. mech., 24, 47-54, (2003)
[15] Blasius, H., Grenzschichten in flussigkeiten mit kleiner reibung, Z. math. phys., 56, 1-37, (1908) · JFM 39.0803.02
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