Notes on the homotopy analysis method: some definitions and theorems.

*(English)*Zbl 1221.65126Summary: We describe, very briefly, the basic ideas and current developments of the homotopy analysis method, an analytic approach to get convergent series solutions of strongly nonlinear problems, which recently attracts interests of more and more researchers. Definitions of some new concepts such as the homotopy-derivative, the convergence-control parameter and so on, are given to redescribe the method more rigorously. Some lemmas and theorems about the homotopy-derivative and the deformation equation are proved. Besides, a few open questions are discussed, and a hypothesis is put forward for future studies.

##### MSC:

65H99 | Nonlinear algebraic or transcendental equations |

35A25 | Other special methods applied to PDEs |

35C10 | Series solutions to PDEs |

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\textit{S. Liao}, Commun. Nonlinear Sci. Numer. Simul. 14, No. 4, 983--997 (2009; Zbl 1221.65126)

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##### References:

[1] | Krylov, N.; Bogoliubov, N.N., Introduction to nonlinear mechanics, (1947), Princeton University Press Princeton (NJ) · Zbl 0063.03382 |

[2] | Bogoliubov, N.N.; Mitropolsky, Y.A., Asymptotic methods in the theory of nonlinear oscillations, (1961), Gordon and Breach New York |

[3] | Cole, J.D., Perturbation methods in applied mathematics, (1968), Blaisdell Publishing Company Waltham (MA) · Zbl 0162.12602 |

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

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

[6] | Mickens, R.E., An introduction to nonlinear oscillations, (1981), Cambridge University Press Cambridge · Zbl 0459.34002 |

[7] | Nayfeh, A.H., Introduction to perturbation techniques, (1981), John Wiley & Sons New York · Zbl 0449.34001 |

[8] | Nayfeh, A.H., Problems in perturbation, (1985), John Wiley & Sons New York |

[9] | Lagerstrom, P.A., Matched asymptotic expansions: ideas and techniques of applied mathematical sciences, vol. 76, (1988), Springer-Verlag New York · Zbl 0666.34064 |

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

[11] | Hinch, E.J., Perturbation methods, Cambridge texts in applied mathematics, (1991), Cambridge University Press Cambridge · Zbl 0746.34001 |

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

[13] | Lyapunov AM (1892) General problem on stability of motion. Taylor & Francis, London; 1992 [English translation]. |

[14] | Karmishin, A.V.; Zhukov, A.T.; Kolosov, V.G., Methods of dynamics calculation and testing for thin-walled structures, (1990), Mashinostroyenie Moscow, [in Russian] |

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

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

[17] | Rach, R., On the Adomian method and comparisons with picard’s method, J math anal appl, 10, 139-159, (1984) |

[18] | Adomian, G.; Adomian, G.E., A global method for solution of complex systems, Math model, 5, 521-568, (1984) · Zbl 0556.93005 |

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

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

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

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

[23] | 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 |

[24] | 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 |

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

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

[27] | Liao, S.J., Beyond perturbation: a review on the basic ideas of the homotopy analysis method and its applications, Adv mech, 38, 1, 1-34, (2008), [in Chinese] |

[28] | Hayat, T.; Javed, T.; Sajid, M., Analytic solution for rotating flow and heat transfer analysis of a third-grade fluid, Acta mech, 191, 219-229, (2007) · Zbl 1117.76069 |

[29] | Hayat, 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 |

[30] | Hayat, T.; Sajid, M., On analytic solution for thin film flow of a forth grade fluid down a vertical cylinder, Phys lett A, 361, 316-322, (2007) · Zbl 1170.76307 |

[31] | Hayat, T.; Sajid, M., Analytic solution for axisymmetric flow and heat transfer of a second grade fluid past a stretching sheet, Int J heat mass transf, 50, 75-84, (2007) · Zbl 1104.80006 |

[32] | Hayat, T.; Abbas, Z.; Sajid, M.; Asghar, S., The influence of thermal radiation on MHD flow of a second grade fluid, Int J heat mass transf, 50, 931-941, (2007) · Zbl 1124.80325 |

[33] | Hayat, T.; Sajid, M., Homotopy analysis of MHD boundary layer flow of an upper-convected Maxwell fluid, Int J eng sci, 45, 393-401, (2007) · Zbl 1213.76137 |

[34] | Hayat, T.; Ahmed, N.; Sajid, M.; Asghar, S., On the MHD flow of a second grade fluid in a porous channel, Comp math appl, 54, 407-414, (2007) · Zbl 1123.76072 |

[35] | Hayat, T.; Khan, M.; Ayub, M., The effect of the slip condition on flows of an Oldroyd 6-constant fluid, J comput appl, 202, 402-413, (2007) · Zbl 1147.76550 |

[36] | Sajid M, Siddiqui, A, Hayat, T. Wire coating analysis using MHD Oldroyd 8-constant fluid. Int J Eng Sci 2007;45:381-92. |

[37] | Sajid, M.; Hayat, T.; Asghar, S., Non-similar analytic solution for MHD flow and heat transfer in a third-order fluid over a stretching sheet, Int J heat mass transf, 50, 1723-1736, (2007) · Zbl 1140.76042 |

[38] | Sajid, M.; Hayat, T.; Asghar, S., Non-similar solution for the axisymmetric flow of a third-grade fluid over radially stretching sheet, Acta mech, 189, 193-205, (2007) · Zbl 1117.76006 |

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

[40] | Abbasbandy, S., The application of the homotopy analysis method to solve a generalized hirota – satsuma coupled KdV equation, Phys lett A, 361, 478-483, (2007) · Zbl 1273.65156 |

[41] | LiuYP, Li ZB. The homotopy analysis method for approximating the solution of the modified Korteweg-de Vries equation. Chaos, Solitons and Fractals. [online]. |

[42] | Zou L, Zong Z, Wang Z, He L. Solving the discrete KdV equation with homotopy analysis method. Phys. Lett. A. · Zbl 1209.65122 |

[43] | Song, L.; Zhang, H.Q., Application of homotopy analysis method to fractional kdv – burgers – kuramoto equation, Phys lett A, 367, 88-94, (2007) · Zbl 1209.65115 |

[44] | 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 |

[45] | Abbasbandy, S., Homotopy analysis method for heat radiation equations, Int commun heat mass transf, 34, 380-387, (2007) |

[46] | Sajid M, Hayat T. Comparison of HAM and HPM methods for nonlinear heat conduction and convection equations. Nonlinear Anal: Real World Appl, doi:10.1016/j.nonrwa.2007.08.007 [online]. · Zbl 1156.76436 |

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

[48] | Zhu, S.P., A closed-form analytical solution for the valuation of convertible bonds with constant dividend yield, Anziam J, 47, 477-494, (2006) · Zbl 1147.91336 |

[49] | Wu Y, Cheung KF. Explicit solution to the exact Riemann problems and application in nonlinear shallow water equations. Int J Numer Meth Fluids, doi:10.1002/fld.1696 [online]. |

[50] | Yamashita, M.; Yabushita, K.; Tsuboi, K., An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method, J phys A, 40, 8403-8416, (2007) · Zbl 1331.70041 |

[51] | Bouremel, Y., Explicit series solution for the glauert-jet problem by means of the homotopy analysis method, Commun nonlinear sci numer simulat, 12, 5, 714-724, (2007) · Zbl 1115.76065 |

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

[53] | Song, H.; Tao, L., Homotopy analysis of 1D unsteady, nonlinear groundwater flow through porous media, J coastal res, 50, 292-295, (2007) |

[54] | Molabahrami A, Khani F. The homotopy analysis method to solve the Burgers-Huxley equation. Nonlinear Anal B: Real World Appl, doi:10.1016/j.nonrwa.2007.10.014 [online]. · Zbl 1167.35483 |

[55] | Bataineh, A.S.; Noorani, M.S.M.; Hashim, I., Solutions of time-dependent emden – fowler type equations by homotopy analysis method, Phys lett A, 371, 72-82, (2007) · Zbl 1209.65104 |

[56] | 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 |

[57] | Mustafa Inc. On exact solution of Laplace equation with Dirichlet and Neumann boundary conditions by the homotopy analysis method. Phys Lett A 2007;365:412-15. · Zbl 1203.65275 |

[58] | Cai WH. Nonlinear Dynamics of thermal-hydraulic networks. PhD thesis, University of Notre Dame; 2006. |

[59] | Song, Y.; Zheng, L.C.; Zhang, X.X., On the homotopy analysis method for solving the boundary layer flow problem over a stretching surface with suction and injection, J univ sci technol Beijing, 28, 782-784, (2006), [in Chinese] |

[60] | 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 |

[61] | Liao, S.J., A new branch of solutions of boundary-layer flows over a permeable stretching plate, Int J non-linear mech, 42, 819-830, (2007) · Zbl 1200.76046 |

[62] | Sen, S., Topology and geometry for physicists, (1983), Academic Press Florida · Zbl 0529.53001 |

[63] | Poincaré, H., Second complément à l’analysis situs, Proc London math soc, 32, 1, 277-308, (1900) · JFM 31.0477.10 |

[64] | Alizadeh-Pahlavan A, Aliakbar V, Vakili-Farahani F, Sadeghy, K. MHD flows of UCM fluids above porous stretching sheets using two-auxiliary-parameter homotopy analysis method. Commun Nonlinear Sci Numer Simulat, doi:10.1016/j.cnsns.2007.09.011 [online]. |

[65] | Marinca V, Herisanu N, Nemes I. A modified homotopy analysis method with application to thin film flow of a fourth grade fluid down a vertical cylinder. Central Eur J Phys [online]. |

[66] | Bataineh AS, Noorani MSM, Hashim I. On a new reliable modification of the homotopy analysis method. Commun Nonlinear Sci Numer Simulat, doi:10.1016/j.cnsns.2007.10.007 [online]. |

[67] | 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 |

[68] | He, J.H., Newton-like iteration method for solving algebraic equations, Commun nonlinear sci numer simulat, 3, 106-109, (1998) · Zbl 0918.65034 |

[69] | Sajid, M.; Hayat, T.; Asghar, S., Comparison between the HAM and HPM solutions of tin film flows of non-Newtonian fluids on a moving belt, Nonlinear dyn, 50, 27-35, (2007) · Zbl 1181.76031 |

[70] | Liao, S.J.; Chwang, A.T., General boundary element method for nonlinear problems, Int J numer meth fluids, 23, 467-483, (1996) · Zbl 0863.76037 |

[71] | Liao, S.J., General boundary element method for nonlinear heat transfer problems governed by hyperbolic heat conduction equation, Comput mech, 20, 5, 397-406, (1997) · Zbl 0890.65121 |

[72] | Liao, S.J., On the general boundary element method and its further generalization, Int J numer meth fluids, 31, 627-655, (1999) · Zbl 0954.76060 |

[73] | Liao, S.J.; Chwang, A.T., General boundary element method for unsteady nonlinear heat transfer problems, Int J numer heat transfer (part B), 35, 2, 225-242, (1999) |

[74] | Zhao, X.Y.; Liao, S.J., A short note on the general boundary element method for viscous flows with high Reynolds number, Int J numer meth fluids, 42, 349-359, (2003) · Zbl 1055.76037 |

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