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On the selection of auxiliary functions, operators, and convergence control parameters in the application of the homotopy analysis method to nonlinear differential equations: a general approach. (English) Zbl 1221.65208
Summary: The homotopy analysis method of Liao has been useful in obtaining analytical solutions to various nonlinear differential equations. In this method, one has great freedom to select auxiliary functions, operators, and parameters in order to ensure the convergence of the approximate solutions and to increase both the rate and region of convergence. We discuss in this paper the selection of the initial approximation, auxiliary linear operator, auxiliary function, and convergence control parameter in the application of the homotopy analysis method, in a fairly general setting. Further, we discuss various convergence requirements on solutions.
MSC:
65L99Numerical methods for ODE
References:
[1]Liao, S. J.: Beyond perturbation: introduction to the homotopy analysis method, (2003)
[2]Liao SJ. On the proposed homotopy analysis techniques for nonlinear problems and its application. Ph.D. dissertation, Shanghai Jiao Tong University; 1992.
[3]Liao, S. J.: An explicit totally analytic approximation of Blasius viscous flow problems, Int J nonlinear mech 34, 759-778 (1999)
[4]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
[5]Liao, S. J.; Tan, Y.: A general approach to obtain series solutions of nonlinear differential equations, Stud appl math 119, 297-355 (2007)
[6]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
[7]Akyildiz, F. T.; Vajravelu, K.; Mohapatra, R. N.; Sweet, E.; Van Gorder, R. A.: Implicit differential equation arising in the steady flow of a sisko fluid, Appl math comput (2009)
[8]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
[9]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
[10]Abbanbandy, S.: Soliton solutions for the Fitzhugh – Nagumo equation with the homotopy analysis method, Appl math model 32, 2706-2714 (2008) · Zbl 1167.35395 · doi:10.1016/j.apm.2007.09.019
[11]Abbanbandy, S.: Homotopy analysis method for the Kawahara equation, Nonlinear anal real world appl (2008)
[12]Abbanbandy, S.: Homotopy analysis method for heat radiation equations, Int commun heat mass transfer 34, 380-387 (2007)
[13]Abbanbandy, S.: The application of homotopy analysis method to solve a generalized Hirota – satsuma coupled KdV equation, Phys lett A 361, 478-483 (2007)
[14]Sajiad, M.; Hayat, T.: Comparison of HAM and HPM methods in 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
[15]Sajiad, M.; Hayat, T.: Comparison of HAM and HPM solutions in heat radiation equations, Int commun heat mass transfer (2008)
[16]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 · doi:10.1016/S0020-7225(03)00207-6
[17]Chen, Y. M.; Liu, J. K.: Exact A study of homotopy analysis method for limit cycle of van der Pol equation, Commun nonlinear sci numer simul 14, 1816-1821 (2009) · Zbl 1221.65198 · doi:10.1016/j.cnsns.2008.07.010
[18]Ziabakhsh, Z.; Domairry, G.: Analytic solution of natural convection flow of a non-Newtonian fluid between two vertical flat plates using homotopy analysis method, Commun nonlinear sci numer simul 14, 1868-1880 (2009)
[19]Jafari, H.; Seifi, S.: Solving a system of nonlinear fractional partial differential equations using homotopy analysis method, Commun nonlinear sci numer simul 14, 1962-1969 (2009) · Zbl 1221.35439 · doi:10.1016/j.cnsns.2008.06.019
[20]Jafari, H.; Seifi, S.: Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation, Commun nonlinear sci numer simul 14, 2006-2012 (2009) · Zbl 1221.65278 · doi:10.1016/j.cnsns.2008.05.008
[21]Alomari, A. K.; Noorani, M. S. M.; Nazar, R.: Adaptation of homotopy analysis method for the numeric-analytic solution of Chen system, Commun nonlinear sci numer simul 14, 2336-2346 (2009) · Zbl 1221.65192 · doi:10.1016/j.cnsns.2008.06.011
[22]Alomari, A. K.; Noorani, M. S. M.; Nazar, R.: Explicit series solutions of some linear and nonlinear Schrödinger equations via the homotopy analysis method, Commun nonlinear sci numer simul 14, 1196-1207 (2009) · Zbl 1221.35389 · doi:10.1016/j.cnsns.2008.01.008
[23]Alizadeh-Pahlavan, A.; Sadeghy, K.: On the use of homotopy analysis method for solving unsteady MHD flow of Maxwellian fluids above impulsively stretching sheets, Commun nonlinear sci numer simul 14, 1355-1365 (2009) · Zbl 1221.76213 · doi:10.1016/j.cnsns.2008.03.001
[24]Molabahrami, A.; Khani, F.: The homotopy analysis method to solve the Burgers – Huxley equation, Nonlinear anal real world appl 10, 589-600 (2009) · Zbl 1167.35483 · doi:10.1016/j.nonrwa.2007.10.014
[25]Hashim, I.; Abdulaziz, O.; Momani, S.: Homotopy analysis method for fractional ivps, Commun nonlinear sci numer simul 14, 674-684 (2009) · Zbl 1221.65277 · doi:10.1016/j.cnsns.2007.09.014
[26]Domairry, G.; Fazeli, M.: Homotopy analysis method to determine the fin efficiency of convective straight fins with temperature-dependent thermal conductivity, Commun nonlinear sci numer simul 14, 489-499 (2009)
[27]Domairry, G.; Mohsenzadeh, A.; Famouri, M.: The application of homotopy analysis method to solve nonlinear differential equation governing Jeffrey-Hamel flow, Commun nonlinear sci numer simul 14, 85-95 (2009) · Zbl 1221.76056 · doi:10.1016/j.cnsns.2007.07.009
[28]Cheng J, Liao S. On the interaction of deep water waves and exponential shear currents. Z Angew Math Phys. doi:10.1007/s00033-008-7050-1.
[29]Abbasbandy, S.; Zakaria, F. Samadian: Soliton solutions for the fifth-order KdV equation with the homotopy analysis method, Nonlinear dyn 51, 83-87 (2008) · Zbl 1170.76317 · doi:10.1007/s11071-006-9193-y
[30]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
[31]Sajid, M.; Awais, M.; Nadeem, S.; Hayat, T.: The influence of slip condition on thin film flow of a fourth grade fluid by the homotopy analysis method, Comput math appl 56, 2019-2026 (2008) · Zbl 1165.76312 · doi:10.1016/j.camwa.2008.04.022
[32]Bataineh, A. Sami; Noorani, M. S. M.; Hashim, I.: Approximate analytical solutions of systems of pdes by homotopy analysis method, Comput math appl 55, 2913-2923 (2008) · Zbl 1142.65423 · doi:10.1016/j.camwa.2007.11.022
[33]Song, Lina; Zhang, Hongqing: Application of homotopy analysis method to fractional KdV-Burgers – Kuramoto equation, Phys lett A 367, 88-94 (2007) · Zbl 1209.65115 · doi:10.1016/j.physleta.2007.02.083
[34]Wang, Chun; Pop, Ioan: Analysis of the flow of a power-law fluid film on an unsteady stretching surface by means of homotopy analysis method, J non-Newtonian fluid mech 138, 161-172 (2006) · Zbl 1195.76132 · doi:10.1016/j.jnnfm.2006.05.011
[35]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
[36]Rashidi MM, Ganji DD, Dinarvand S. Approximate traveling wave solutions of coupled Whitham – Broer – Kaup shallow water equations by homotopy analysis method. Diff Equat Nonlinear Mech. doi:10.1155/2008/243459. · Zbl 1180.35431 · doi:10.1155/2008/243459
[37]Bataineh, A. Sami; Noorani, M. S. M.; Hashim, I.: Approximate solutions of singular two-point BVPs by modified homotopy analysis method, Phys lett A 372, 4062-4066 (2008) · Zbl 1220.34026 · doi:10.1016/j.physleta.2008.03.026
[38]Wang, Z.; Zou, L.; Zhang, H.: Solitary solution of discrete mkdv equation by homotopy analysis method, Commun theor phys (Beijing, China) 49, 1373-1378 (2008)
[39]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 math theor 40, 8403-8416 (2007)
[40]Elwakila, E. A. E.; Abdou, M. A.: New applications of the homotopy analysis method, Z naturforsch 63a, 385-392 (2008)
[41]Inc, M.: On numerical solution of Burgers equation by homotopy analysis method, Phys lett A 372, 356-360 (2008) · Zbl 1217.76019 · doi:10.1016/j.physleta.2007.07.057
[42]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 · doi:10.1016/j.physleta.2007.04.070
[43]Wang, C.; Wu, Y.; Wu, W.: Solving the nonlinear periodic wave problems with the homotopy analysis method, Wave motion 41, 329-337 (2005) · Zbl 1189.35293 · doi:10.1016/j.wavemoti.2004.08.002