Abstract
In the present paper, we have considered three methods with which to control the error in the homotopy analysis of elliptic differential equations and related boundary value problems, namely, control of residual errors, minimization of error functionals, and optimal homotopy selection through appropriate choice of auxiliary function H(x). After outlining the methods in general, we consider three applications. First, we apply the method of minimized residual error in order to determine optimal values of the convergence control parameter to obtain solutions exhibiting central symmetry for the Yamabe equation in three or more spatial dimensions. Secondly, we apply the method of minimizing error functionals in order to obtain optimal values of the convergnce control parameter for the homotopy analysis solutions to the Brinkman–Forchheimer equation. Finally, we carefully selected the auxiliary function H(x) in order to obtain an optimal homotopy solution for Liouville’s equation.
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Liao, S.J.: On the Proposed Homotopy Analysis Techniques for Nonlinear Problems and its Application. Ph.D. dissertation. Shanghai Jiao Tong University (1992)
Liao, S.J.: Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman & Hall/CRC Press, Boca Raton (2003)
Liao, S.J.: An explicit, totally analytic approximation of Blasius viscous flow problems. Internat. J. Non-Linear Mech. 34, 759–778 (1999)
Liao, S.J.: On the homotopy analysis method for nonlinear problems. Appl. Math. Comput. 147, 499–513 (2004)
Liao, S.J., Tan, Y.: A general approach to obtain series solutions of nonlinear differential equations. Stud. Appl. Math. 119, 297–354 (2007)
Liao, S.J.: Notes on the homotopy analysis method: some definitions and theorems. Commun. Nonlinear Sci. Numer. Simul. 14, 983–997 (2009)
Van Gorder, R.A., Vajravelu, K.: 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. Commun. Nonlinear Sci. Numer. Simul. 14, 4078–4089 (2009)
Liao, S.: An optimal homotopy-analysis approach for strongly nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 15, 2315–2332 (2010)
Abbasbandy, S.: The application of homotopy analysis method to nonlinear equations arising in heat transfer. Phys. Lett. A 360, 109–113 (2006)
Abbasbandy, S.: Homotopy analysis method for heat radiation equations. Int. Comm. Heat Mass Transfer 34, 380–387 (2007)
Liao, S.J., Su, J., Chwang, A.T.: Series solutions for a nonlinear model of combined convective and radiative cooling of a spherical body. Int. J. Heat Mass Transfer 49, 2437–2445 (2006)
Liao, S.J., Campo, A.: Analytic solutions of the temperature distribution in Blasius viscous flow problems. J. Fluid Mech. 453, 411–425 (2002)
Liao, S.J.: An explicit, totally analytic approximation of Blasius viscous flow problems. Internat. J. Non-Linear Mech. 34, 759–778 (1999)
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)
Liao, S.J.: On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet. J. Fluid Mech. 488, 189–212 (2003)
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. 210, 189–196 (2009)
Hang, X., Lin, Z.L., Liao, S.J., Wu, J.Z., Majdalani, J.: Homotopy based solutions of the Navier-Stokes equations for a porous channel with orthogonally moving walls. Phys. Fluids 22, 053601 (2010)
Sajid, M., Hayat, T., Asghar, S.: Comparison between the HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt. Nonlinear Dynam. 50, 27–35 (2007)
Hayat, T., Sajid, M.: On analytic solution for thin film flow of a fourth grade fluid down a vertical cylinder. Phys. Lett. A 361, 316–322 (2007)
Turkyilmazoglu, M.: Purely analytic solutions of the compressible boundary layer flow due to a porous rotating disk with heat transfer. Phys. Fluids 21, 106104 (2009)
Abbasbandy, S., Zakaria, F.S.: Soliton solutions for the fifth-order KdV equation with the homotopy analysis method. Nonlinear Dynam. 51, 83–87 (2008)
Wu, W., Liao, S.J.: Solving solitary waves with discontinuity by means of the homotopy analysis method. Chaos, Solitons & Fractals 26, 177–185 (2005)
Sweet, E., Van Gorder, R.A.: Analytical solutions to a generalized Drinfel’d - Sokolov equation related to DSSH and KdV6. Appl. Math. Comput. 216, 2783–2791 (2010)
Wu, Y., Wang, C., Liao, S.J.: Solving the one-loop soliton solution of the Vakhnenko equation by means of the homotopy analysis method. Chaos, Solitons & Fractals 23, 1733–1740 (2005)
Cheng, J., Liao, S.J., 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)
Van Gorder, R.A., Sweet, E., Vajravelu, K.: Nano boundary layers over stretching surfaces. Commun. Nonlinear Sci. Numer. Simul. 15, 1494–1500 (2010)
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)
Bataineh, A.S., Noorani, M.S.M., Hashim, I.: Homotopy analysis method for singular IVPs of Emden-Fowler type. Commun. Nonlinear Sci. Numer. Simul. 14, 1121–1131 (2009)
Van Gorder, R.A., Vajravelu, K.: Analytic and numerical solutions to the Lane-Emden equation. Phys. Lett. A 372, 6060–6065 (2008)
Liao, S.: A new analytic algorithm of Lane-Emden type equations. Appl. Math. Comput. 142, 1–16 (2003)
Andersson, L.: Chruściel, P.T., Friedrich, H.: On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein’s field equations. Comm. Math. Phys. 149, 587–612 (1992)
Brendle, S.: Blow-up phenomena for the Yamabe equation. J. Amer. Math. Soc. 21, 951–980 (2008)
Alice Chang, S.Y., Han, Z.C., Yang, P.: Classification of singular radial solutions to the [sigma] k Yamabe equation on annular domains. J. Differential Equations 216, 482–501 (2005)
Van Gorder, R.A.: Analytical solutions to a quasilinear differential equation related to the Lane-Emden equation of the second kind. Celest. Mech. Dynam. Astron. 109, 137–145 (2011)
Van Gorder, R.A.: An elegant perturbation solution for the Lane-Emden equation of the second kind. New Astron. 16, 65–67 (2011)
Van Gorder, R.A.: Exact first integrals for a Lane–Emden equation of the second kind modeling a thermal explosion in a rectangular slab. New Astron. 16, 492–497 (2011)
Momoniat, E., Harley, C.: Approximate implicit solution of a Lane–Emden equation. New Astron. 11, 520–526 (2006)
Wazwaz, A.M.: A new algorithm for solving differential equations of Lane–Emden type. Appl. Math. Comp. 118, 287–310 (2001)
Chandrasekhar, S.: An Introduction to the Study of Stellar Structure, p. 90. Dover, New York (1967)
Hooman, K.: A perturbation solution for forced convection in a porous-saturated duct. J. Comput. Appl. Math. 211, 57–66 (2008)
Hooman, K., Merrikh, A.A.: Analytical solution of forced convection in a duct of rectangular cross section saturated by a porous medium. Heat Transfer 128, 596–600 (2006)
Van Gorder, R.A., Vajravelu, K., Akyildiz, F.T.; Solutions to the Brinkman-Forchheimer momentum equation for a unidirectional flow over a rectangular domain. Int. J. Fluid Mech. Res. 36, 552–565 (2009)
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Van Gorder, R.A. Control of error in the homotopy analysis of semi-linear elliptic boundary value problems. Numer Algor 61, 613–629 (2012). https://doi.org/10.1007/s11075-012-9554-1
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DOI: https://doi.org/10.1007/s11075-012-9554-1