## 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:

 65L99 Numerical methods for ordinary differential equations
Full Text:

### References:

  Liao, S.J., Beyond perturbation: introduction to the homotopy analysis method, (2003), Chapman & Hall/CRC Press Boca Raton  Liao SJ. On the proposed homotopy analysis techniques for nonlinear problems and its application. Ph.D. dissertation, Shanghai Jiao Tong University; 1992.  Liao, S.J., An explicit totally analytic approximation of blasius’ viscous flow problems, Int J nonlinear mech, 34, 759-778, (1999) · Zbl 1342.74180  Liao, S.J., On the homotopy analysis method for nonlinear problems, Appl math comput, 147, 499-513, (2004) · Zbl 1086.35005  Liao, S.J.; Tan, Y., A general approach to obtain series solutions of nonlinear differential equations, Stud appl math, 119, 297-355, (2007)  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  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), online · Zbl 1160.76002  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  Akyildiz, F.T.; Vajravelu, K., Magnetohydrodynamic flow of a viscoelastic fluid, Phys lett A, 372, 3380-3384, (2008) · Zbl 1220.76073  Abbanbandy, S., Soliton solutions for the fitzhugh – nagumo equation with the homotopy analysis method, Appl math model, 32, 2706-2714, (2008) · Zbl 1167.35395  Abbanbandy, S., Homotopy analysis method for the Kawahara equation, Nonlinear anal real world appl, (2008), online  Abbanbandy, S., Homotopy analysis method for heat radiation equations, Int commun heat mass transfer, 34, 380-387, (2007)  Abbanbandy, S., The application of homotopy analysis method to solve a generalized hirota – satsuma coupled KdV equation, Phys lett A, 361, 478-483, (2007) · Zbl 1273.65156  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  Sajiad, M.; Hayat, T., Comparison of HAM and HPM solutions in heat radiation equations, Int commun heat mass transfer, (2008), online  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  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  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)  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  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  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  Alomari, A.K.; Noorani, M.S.M.; Nazar, R., Explicit series solutions of some linear and nonlinear schrodinger equations via the homotopy analysis method, Commun nonlinear sci numer simul, 14, 1196-1207, (2009) · Zbl 1221.35389  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  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  Hashim, I.; Abdulaziz, O.; Momani, S., Homotopy analysis method for fractional ivps, Commun nonlinear sci numer simul, 14, 674-684, (2009) · Zbl 1221.65277  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)  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  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. · Zbl 1173.76007  Abbasbandy, S.; Samadian Zakaria, F., Soliton solutions for the fifth-order KdV equation with the homotopy analysis method, Nonlinear dyn, 51, 83-87, (2008) · Zbl 1170.76317  Allan, F.M., Derivation of the Adomian decomposition method using the homotopy analysis method, Appl math comput, 190, 6-14, (2007) · Zbl 1125.65063  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  Sami Bataineh, A.; 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  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  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  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  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.  Sami Bataineh, A.; 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  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) · Zbl 1392.34080  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) · Zbl 1331.70041  Elwakila, E.A.E.; Abdou, M.A., New applications of the homotopy analysis method, Z naturforsch, 63a, 385-392, (2008)  Inc, M., On numerical solution of burgers’ equation by homotopy analysis method, Phys lett A, 372, 356-360, (2008) · Zbl 1217.76019  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  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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.