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A note on the homotopy analysis method. (English) Zbl 1195.65104
Summary: The present work is devoted to using an analytic approach, namely the homotopy analysis method, to obtain convergent series solutions of strongly nonlinear problems. On the basis of the homotopy derivative concept described in [J. Liao, Sci. Sin. Ser. A 25, 983–992 (1982; Zbl 0496.90037)], a theorem is proved here which generalizes some lemmas and theorems provided in [Liao, loc. cit.; and A. Molabahrami and F. Khani, Nonlinear Anal., Real World Appl. 10, No. 2, 589–600 (2009; Zbl 1167.35483)]. Significant applicability of the theorem obtained here in some practical situations is demonstrated.
MSC:
65L99Numerical methods for ODE
References:
[1]S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. Thesis, Shanghai Jiao Tong University, 1992.
[2]Liao, S. J.: Beyond perturbation: introduction to homotopy analysis method, (2003)
[3]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
[4]Molabahrami, A.; Khani, F.: The homotopy analysis method to solve the Burgers–Huxley equation, Nonlinear anal. Ser. B: RWA 10, 14-21 (2007)
[5]Cheng, Y.; Liao, S. J.: On the explicit, purely analytic solution of von Kármán swirling viscous flow, Commun. nonlinear sci. Numer. simul. 47, 75-85 (2006)
[6]Turkyilmazoglu, M.: A homotopy treatment of some boundary layer flows, Int. J. Nonlinear sci. Numer. simul. 10, 885-889 (2009)
[7]Jacobsen, J.: The Liouville–bratu–Gelfand problem for radial operators, J. differential equations 184, 283-298 (2002) · Zbl 1015.34013 · doi:10.1006/jdeq.2001.4151
[8]Belendez, A.; Hernandez, A.; Belendez, T.; Marquez, A.; Neipp, C.: An improved heuristic approximation for the period of a nonlinear pendulum; linear analysis of a classical nonlinear problem, Internat. J. Non-linear mech. 8, 329-334 (2007)
[9]Liao, S. J.: Series solution of nonlinear eigenvalue problems by means of the homotopy analysis method, Nonlinear anal. Ser. B: RWA 10, 10-26 (2008)
[10]Wang, J.; Chen, J. -K.; Liao, S.: An explicit solution of the large deformation of a cantilever beam under point load at the free tip, J. comput. Appl. math. 212, 320-330 (2008) · Zbl 1128.74026 · doi:10.1016/j.cam.2006.12.009
[11]Abbasbandy, S.; Tan, Y.; Liao, S. J.: Newton-homotopy analysis method for nonlinear equations, Appl. math. Comput. 188, 1794-1800 (2007) · Zbl 1119.65032 · doi:10.1016/j.amc.2006.11.136
[12]Li, S.; Liao, S. J.: An analytic approach to solve multiple solutions of a strongly nonlinear problem, Appl. math. Comput. 169, 854-865 (2005) · Zbl 1151.35354 · doi:10.1016/j.amc.2004.09.066
[13]Gelfand, I. M.: Some problems in the theory of quasi-linear equations, Amer. math. Soc. transl. Ser. 2 29, 295-381 (1963)
[14]Xu, H.; Liao, S.; Pop, I.: Series solution of unsteady boundary layer flows of non-Newtonian fluids near a forward stagnation point, J. non-Newton fluid mech. 139, 31-43 (2006) · Zbl 1195.76070 · doi:10.1016/j.jnnfm.2006.06.003
[15]Cheng, J.; Liao, S.; Mohapatra, R. N.; Vajravelu, K.: Series solutions of nano boundary layer flows by means of the homotopy analysis method, J. math. Anal. appl. 343, 233-245 (2008) · Zbl 1135.76016 · doi:10.1016/j.jmaa.2008.01.050