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**On the homotopy analysis method for nonlinear problems.**
*(English)*
Zbl 1086.35005

Summary: A powerful, easy-to-use analytic tool for nonlinear problems in general, namely the homotopy analysis method, is further improved and systematically described through a typical nonlinear problem, i.e., the algebraically decaying viscous boundary layer flow due to a moving sheet. Two rules, the rule of solution expression and the rule of coefficient ergodicity, are proposed, which play important roles in the frame of the homotopy analysis method and simplify its applications in science and engineering. An explicit analytic solution is given for the first time, with recursive formulas for coefficients. This analytic solution agrees well with numerical results and can be regarded as a definition of the solution of the considered nonlinear problem.

### MSC:

35A25 | Other special methods applied to PDEs |

76M25 | Other numerical methods (fluid mechanics) (MSC2010) |

76D10 | Boundary-layer theory, separation and reattachment, higher-order effects |

Full Text:
DOI

### References:

[1] | Cole, J. D., Perturbation Methods in Applied Mathematics (1968), Blaisdell Publishing Company: Blaisdell Publishing Company Waltham Massachusetts · Zbl 0162.12602 |

[2] | Ali Hasan Nayfeh, Perturbation Methods (2000), John Wiley and Sons: John Wiley and Sons New York · Zbl 0995.35001 |

[3] | Lyapunov, A. M., General Problem on Stability of Motion (English translation) (1992), Taylor and Francis: Taylor and Francis London, (Original work published 1892) · Zbl 0786.70001 |

[4] | Karmishin, A. V.; Zhukov, A. I.; Kolosov, V. G., Methods of Dynamics Calculation and Testing for Thin-Walled Structures (1990), Mashinostroyenie: Mashinostroyenie Moscow, (in Russian) |

[5] | Adomian, G., Solving Frontier Problems of Physics: The Decomposition Method (1994), Kluwer Academic Publishers: Kluwer Academic Publishers Boston and London · Zbl 0802.65122 |

[6] | S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, PhD thesis, Shanghai Jiao Tong University, 1992; S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, PhD thesis, Shanghai Jiao Tong University, 1992 |

[7] | Liao, S. J., An explicit, totally analytic approximation of Blasius viscous flow problems, Int. J. Non-Linear Mech., 34, 4, 759-778 (1999) · Zbl 1342.74180 |

[8] | Liao, S. J., A simple way to enlarge the convergence region of perturbation approximations, Int. J. Non-linear Dynam., 19, 2, 93-110 (1999) · Zbl 0949.70003 |

[9] | 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 |

[10] | 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 |

[11] | Liao, S. J., An analytic approximation of the drag coefficient for the viscous flow past a sphere, Int. J. Non-Linear Mech., 37, 1-18 (2002) · Zbl 1116.76335 |

[12] | Liao, S. J., An explict analytic solution to the Thomas-Fermi equation, Appl. Math. Comput., 144, 433-444 (2003) |

[13] | S.J. Liao, K.F. Cheung, Analytic solution for nonlinear progressive waves in deep water, J. Engrg. Math., in press; S.J. Liao, K.F. Cheung, Analytic solution for nonlinear progressive waves in deep water, J. Engrg. Math., in press · Zbl 1112.76316 |

[14] | Kuiken, H. K., On boundary layers in fluid mechanics that decay algebraically along stretches of wall that are not vanishingly small, IMA J. Appl. Math., 27, 387-405 (1981) · Zbl 0472.76045 |

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