On the relationship between the homotopy analysis method and Euler transform.

*(English)*Zbl 1221.65206Summary: A new transform, namely the homotopy transform, is defined for the first time. Then, it is proved that the famous Euler transform is only a special case of the so-called homotopy transform which depends upon one non-zero auxiliary parameter \(\hbar\) and two convergent series. In the frame of the homotopy analysis method, a general analytic approach for highly nonlinear differential equations, the so-called homotopy transform is obtained by means of a simple example. This fact indicates that the Euler transform is equivalent to the homotopy analysis method in some special cases. On one side, this explains why the convergence of the series solution given by the homotopy analysis method can be guaranteed. On the other side, it also shows that the homotopy analysis method is more general and thus more powerful than the Euler transform.

##### MSC:

65L99 | Numerical methods for ordinary differential equations |

34E10 | Perturbations, asymptotics of solutions to ordinary differential equations |

34A45 | Theoretical approximation of solutions to ordinary differential equations |

41A58 | Series expansions (e.g., Taylor, Lidstone series, but not Fourier series) |

40G05 | Cesàro, Euler, Nörlund and Hausdorff methods |

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\textit{S. Liao}, Commun. Nonlinear Sci. Numer. Simul. 15, No. 6, 1421--1431 (2010; Zbl 1221.65206)

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##### References:

[1] | Nayfeh, A.H., Perturbation methods, (1973), John Wiley & Sons New York |

[2] | Von Dyke, M., Perturbation methods in fluid mechanics, (1975), The Parabolic Press Stanford, California |

[3] | Hinch, E.J., Perturbation methods, Cambridge texts in applied mathematics, (1991), Cambridge University Press Cambridge |

[4] | Murdock, J.A., Perturbations: theory and methods, (1991), John Wiley & Sons New York · Zbl 0810.34047 |

[5] | Kahn, P.B.; Zarmi, Y., Nonlinear dynamics: exploration through normal forms, (1997), John Wiley & Sons Int. New York |

[6] | Nayfeh, A.H., Perturbation methods, (2000), John Wiley & Sons New York |

[7] | Lyapunov AM. General problem on stability of motion [Trans.]. London: Taylor & Francis; 1992 [Original work published 1892]. |

[8] | Adomian, G., Nonlinear stochastic differential equations, J math anal appl, 55, 441-452, (1976) · Zbl 0351.60053 |

[9] | Adomian, G., A review of the decomposition method and some recent results for nonlinear equations, Comp math appl, 21, 101-127, (1991) · Zbl 0732.35003 |

[10] | Karmishin AV, Zhukov AT, Kolosov VG. Methods of dynamics calculation and testing for thin-walled structures. Moscow: Mashinostroyenie; 1990 [in Russian]. |

[11] | Awrejcewicz, J.; Andrianov, I.V.; Manevitch, L.I., Asymptotic approaches in nonlinear dynamics, (1998), Springer-Verlag Berlin · Zbl 0910.70001 |

[12] | Liao SJ. The proposed homotopy analysis technique for the solution of nonlinear problems. Ph.D. Thesis, Shanghai Jiao Tong University; 1992. |

[13] | Eynde, R.V., Historical evolution of the concept homotopic paths, Arch hist exact sci, 45, 2, 128-188, (1992) · Zbl 0766.01015 |

[14] | Liao, S.J., A kind of approximate solution technique which does not depend upon small parameters (II): an application in fluid mechanics, Int J nonlinear mech, 32, 815-822, (1997) · Zbl 1031.76542 |

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

[16] | Liao, S.J.; Cheung, K.F., Homotopy analysis of nonlinear progressive waves in deep water, J eng math, 45, 2, 105-116, (2003) · Zbl 1112.76316 |

[17] | Liao, S.J., Beyond perturbation: introduction to the homotopy analysis method, (2003), Chapman & Hall/CRC Press Boca Raton |

[18] | Liao, S.J., On the homotopy analysis method for nonlinear problems, Appl math comput, 147, 499-513, (2004) · Zbl 1086.35005 |

[19] | Liao, S.J.; Tan, Y., A general approach to obtain series solutions of nonlinear differential equations, Stud appl math, 119, 297-355, (2007) |

[20] | Liao, S.J., Notes on the homotopy analysis method: some definitions and theorems, Commun nonlinear sci numer simul, 14, 983-997, (2009) · Zbl 1221.65126 |

[21] | 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 |

[22] | Allan, F.M.; Syam, M.I., On the analytic solutions of the non-homogenous Blasius problem, J comput appl math, (2005) · Zbl 1071.65108 |

[23] | Hayat, T.; Khan, M., Homotopy solutions for a generalized second grade fluid past a porous plate, Nonlinear dyn, 42, 395-405, (2005) · Zbl 1094.76005 |

[24] | Zhu, S.P., An exact and explicit solution for the valuation of American put options, Quant finance, 6, 229-242, (2006) · Zbl 1136.91468 |

[25] | 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 |

[26] | Wang, Z.; Zou, L.; Zhang, H., Applying homotopy analysis method for solving differential-difference equation, Phys lett A, 369, 77-84, (2007) · Zbl 1209.65119 |

[27] | Tao, L.; Song, H.; Chakrabarti, S., Nonlinear progressive waves in water of finite depth – an analytic approximation, Coastal eng, 54, 825-834, (2007) |

[28] | Haya, T.; Khan, M.; Sajid, M.; Asghar, S., Rotating flow of a third grade fluid in a porous space with Hall current, Nonlinear dyn, 49, 83-91, (2007) · Zbl 1181.76149 |

[29] | Abbasbandy, S., Soliton solutions for the 5th-order KdV equation with the homotopy analysis method, Nonlinear dyn, 51, 83-87, (2008) · Zbl 1170.76317 |

[30] | Liao, S.J., A new branch of solutions of boundary-layer flows over an impermeable stretched plate, Int J heat mass transfer, 48, 12, 2529-2539, (2005) · Zbl 1189.76142 |

[31] | Liao, S.J.; Magyari, E., Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones, Zamp, 57, 5, 777-792, (2006) · Zbl 1101.76056 |

[32] | Agnew, R.P., Euler transformations, J math, 66, 313-338, (1944) · Zbl 0060.16004 |

[33] | Adams, E.P.; Hippisley, C.R.L., Smithsonian mathematical formulae tables and table of elliptic functions, (1922), Smithsonian Institute Washington |

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