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An optimal homotopy-analysis approach for strongly nonlinear differential equations. (English) Zbl 1222.65088
Summary: An optimal homotopy-analysis approach is described by means of the nonlinear Blasius equation as an example. This optimal approach contains at most three convergence-control parameters and is computationally rather efficient. A new kind of averaged residual error is defined, which can be used to find the optimal convergence-control parameters much more efficiently. It is found that all optimal homotopy-analysis approaches greatly accelerate the convergence of series solution. And the optimal approaches with one or two unknown convergence-control parameters are strongly suggested. This optimal approach has general meanings and can be used to get fast convergent series solutions of different types of equations with strong nonlinearity.

MSC:
65L99Numerical methods for ODE
34A25Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.)
34A45Theoretical approximation of solutions of ODE
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[1] Lindstedt, A.: Über die integration einer für die strörungstheorie wichtigen differentialgleichung, Astron nach 103, 211-220 (1882) · Zbl 14.0926.02 · doi:10.1002/asna.18821031404
[2] Cole, J. D.: Perturbation methods in applied mathematics, (1968) · Zbl 0162.12602
[3] Von Dyke, M.: Perturbation methods in fluid mechanics, (1975) · Zbl 0329.76002
[4] Murdock, J. A.: Perturbations: theory and methods, (1991) · Zbl 0810.34047
[5] Kevorkian, J.; Cole, J. D.: Multiple scales and singular perturbation methods, Applied mathematical sciences 114 (1995) · Zbl 0846.34001
[6] Nayfeh, A. H.: Perturbation methods, (2000) · Zbl 0995.35001
[7] Lyapunov, A. M.: General problem on stability of motion, (1992) · Zbl 0786.70001
[8] Karmishin, AV, Zhukov, AT, Kolosov, VG. Methods of dynamics calculation and testing for thin-walled structures. Mashinostroyenie, Moscow; 1990 [in Russian].
[9] Awrejcewicz, J.; Andrianov, I. V.; Manevitch, L. I.: Asymptotic approaches in nonlinear dynamics, (1998) · Zbl 0910.70001
[10] Adomian, G.: Nonlinear stochastic differential equations, J math anal appl 55, 441-452 (1976) · Zbl 0351.60053 · doi:10.1016/0022-247X(76)90174-8
[11] Adomian, G.; Adomian, G. E.: A global method for solution of complex systems, Math model 5, 521-568 (1984) · Zbl 0556.93005 · doi:10.1016/0270-0255(84)90004-6
[12] Cherruault, Y.: Convergence of Adomian’s method, Kyberneters 8, No. 2, 31-38 (1988) · Zbl 0697.65051
[13] 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 · doi:10.1016/0898-1221(91)90220-X
[14] Adomian, G.: Solving frontier problems of physics: the decomposition method, (1994) · Zbl 0802.65122
[15] Rach, R.: A new definition of Adomian polymonial, Kybernetes 37, 910-955 (2008) · Zbl 1176.33023 · doi:10.1108/03684920810884342
[16] Liao, SJ. The proposed homotopy analysis technique for the solution of nonlinear problems. PhD thesis, Shanghai Jiao Tong University; 1992.
[17] Hilton, P. J.: An introduction to homotopy theory, (1953) · Zbl 0051.40302
[18] Sen, S.: Topology and geometry for physicists, (1983) · Zbl 0529.53001
[19] Liao, S. J.: An explicit, totally analytic approximation of Blasius viscous flow problems, Int J non-linear mech 34, No. 4, 759-778 (1999) · Zbl 05137896
[20] Liao, S. J.: Beyond perturbation: introduction to the homotopy analysis method, (2003)
[21] Liao, S. J.; Tan, Y.: A general approach to obtain series solutions of nonlinear differential equations, Stud appl math 119, 297-355 (2007)
[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 · doi:10.1016/S0020-7462(96)00101-1
[23] Liang, S. X.; Jeffrey, D. J.: Comparison of homotopy analysis method and homotopy perturbation method through an evaluation equation, Commun nonlinear sci numer simul 14, 4057-4064 (2009) · Zbl 1221.65281 · doi:10.1016/j.cnsns.2009.02.016
[24] Liao, S. J.: On the relationship between the homotopy analysis method and Euler transform, Commun nonlinear sci numer simul. 15, 1421-1431 (2010) · Zbl 1221.65206 · doi:10.1016/j.cnsns.2009.06.008
[25] 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 · doi:10.1016/j.cnsns.2008.04.013
[26] Liao, S. J.: On the homotopy analysis method for nonlinear problems, Appl math comput 147, 499-513 (2004) · Zbl 1086.35005 · doi:10.1016/S0096-3003(02)00790-7
[27] Liao, SJ. Series solution of deformation of a beam with arbitrary cross section under an axial load. ANZIAM J. online. · Zbl 1196.34027 · doi:10.1017/S1446181109000339
[28] Liao, S. J.: Series solutions of unsteady boundary-layer flows over a stretching flat plate, Stud appl math 117, No. 3, 2529-2539 (2006) · Zbl 1145.76352 · doi:10.1111/j.1467-9590.2006.00354.x
[29] Liao, S. J.: A general approach to get series solution of non-similarity boundary layer flows, Commun nonlinear sci numer simul 14, 2144-2159 (2009) · Zbl 1221.76068 · doi:10.1016/j.cnsns.2008.06.013
[30] Yang, C.; Liao, S. J.: On the explicit, purely analytic solution of von kármán swirling viscous flow, Commun nonlinear sci numer simul 11, No. 1, 83-139 (2006) · Zbl 1075.35059 · doi:10.1016/j.cnsns.2004.05.006
[31] He, J. H.: Homotopy perturbation technique, Comput methods appl mech eng 178, 257-262 (1999) · Zbl 0956.70017
[32] Sajid, M.; Hayat, T.: Comparison of HAM and HPM methods for nonlinear heat conduction and convection equations, Nonlinear anal: real world appl 9, 2296-2301 (2008) · Zbl 1156.76436 · doi:10.1016/j.nonrwa.2007.08.007
[33] 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
[34] Van Gorder, R. A.; Vajravelu, K.: Analytic and numerical solutions to the Lane-Emden equation, Phys lett A 372, 6060-6065 (2008) · Zbl 1223.85004 · doi:10.1016/j.physleta.2008.08.002
[35] Molabahrami, A.; Khani, F.: The homotopy analysis method to solve the Burgers -- Huxley equation, Nonlinear anal B: real world appl 10, 589-600 (2009) · Zbl 1167.35483 · doi:10.1016/j.nonrwa.2007.10.014
[36] 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 simul 14, 473-488 (2009)
[37] Yabushita, K.; Yamashita, M.; 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 05178236
[38] 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 · doi:10.1017/S1446181100010087 · http://www.austms.org.au/Publ/ANZIAM/V47P4/2378.html
[39] Zhu, S. P.: An exact and explicit solution for the valuation of American put options, Quant finan 6, 229-242 (2006) · Zbl 1136.91468 · doi:10.1080/14697680600699811
[40] Zou, L.; Zong, Z.; Wang, Z.; He, L.: Solving the discrete KdV equation with homotopy analysis method, Phys lett A 370, 287-294 (2007) · Zbl 1209.65122 · doi:10.1016/j.physleta.2007.05.068
[41] Allan, F. M.: Derivation of the Adomian decomposition method using the homotopy analysis method, Appl math comput 190, 6-14 (2007) · Zbl 1125.65063 · doi:10.1016/j.amc.2006.12.074
[42] Song, H.; Tao, L.: Homotopy analysis of 1D unsteady, nonlinear groundwater flow through porous media, J coastal res 50, 292-295 (2007)
[43] Abbasbandy, S.: Solitary wave equations to the Kuramoto -- Sivashinsky equation by means of the homotopy analysis method, Nonlinear dyn 52, 35-40 (2008) · Zbl 1173.35646 · doi:10.1007/s11071-007-9255-9
[44] Akyildiz, F. T.; Vajravelu, K.: Magnetohydrodynamic flow of a viscoelastic fluid, Phys lett A 372, 3380-3384 (2008) · Zbl 1220.76073 · doi:10.1016/j.physleta.2008.01.073
[45] Abbasbandy, S.; Parkes, E. J.: Solitary smooth hump solutions of the Camassa -- Holm equation by means of the homotopy analysis method, Chaos solitons fract 36, 581-591 (2008) · Zbl 1139.76013 · doi:10.1016/j.chaos.2007.10.034
[46] Abbas, Z.; Wang, Y.; Hayat, T.; Oberlack, M.: Hydromagnetic flow in a viscoelastic fluid due to the oscillatory stretching surface, Int J nonlinear mech 43, 783-793 (2008) · Zbl 1203.76169 · doi:10.1016/j.ijnonlinmec.2008.04.009
[47] Wu, Y.; Cheung, K. F.: Explicit solution to the exact Riemann problems and application in nonlinear shallow water equations, Int J numer methods fluids 57, 1649-1668 (2008) · Zbl 1210.76033 · doi:10.1002/fld.1696
[48] Wu, Y. Y.; Cheung, K. F.: Homotopy solution for nonlinear differential equations in wave propagation problems, Wave motion 46, 1-14 (2009) · Zbl 1231.65138 · doi:10.1016/j.wavemoti.2008.07.002
[49] Liu, Y. P.; Li, Z. B.: The homotopy analysis method for approximating the solution of the modified Korteweg-de Vries equation, Chaos solitons fract 39, 1-8 (2009) · Zbl 1197.65166 · doi:10.1016/j.chaos.2007.01.148
[50] Zhao, J, Wong, HY. A closed-form solution to American options under general diffusion processes. Quant Finan. online. · Zbl 1278.91171
[51] Liao, S. J.: A new branch of solutions of boundary-layer flows over an impermeable stretched plate, Int J heat mass transfer 48, No. 12, 2529-2539 (2005) · Zbl 1189.76142 · doi:10.1016/j.ijheatmasstransfer.2005.01.005
[52] Liao, S. J.; Magyari, E.: Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones, Zamp 57, No. 5, 777-792 (2006) · Zbl 1101.76056 · doi:10.1007/s00033-006-0061-x
[53] 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 · doi:10.1016/j.ijnonlinmec.2007.03.007
[54] Marinca, V.; Herisanu, N.; Nemes, I.: Optimal homotopy asymptotic method with application to thin film flow, Central eur J phys 6, 648-653 (2008)
[55] Marinca, V.; Herisanu, N.: Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer, Int commun heat mass transfer 35, 710-715 (2008)
[56] Marinca, V.; Herisanu, N.: An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate, Appl math lett 22, 245-251 (2009) · Zbl 1163.76318 · doi:10.1016/j.aml.2008.03.019
[57] Niu, Z., Wang, C. A one-step optimal homotopy analysis method for nonlinear differential equations. Commun Nonlinear Sci Numer Simul. Online. · Zbl 1222.65093 · doi:10.1016/j.cnsns.2009.12.026