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Control of error in the homotopy analysis of semi-linear elliptic boundary value problems

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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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References

  1. Liao, S.J.: On the Proposed Homotopy Analysis Techniques for Nonlinear Problems and its Application. Ph.D. dissertation. Shanghai Jiao Tong University (1992)

  2. Liao, S.J.: Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman & Hall/CRC Press, Boca Raton (2003)

    Book  Google Scholar 

  3. Liao, S.J.: An explicit, totally analytic approximation of Blasius viscous flow problems. Internat. J. Non-Linear Mech. 34, 759–778 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Liao, S.J.: On the homotopy analysis method for nonlinear problems. Appl. Math. Comput. 147, 499–513 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Liao, S.J., Tan, Y.: A general approach to obtain series solutions of nonlinear differential equations. Stud. Appl. Math. 119, 297–354 (2007)

    Article  MathSciNet  Google Scholar 

  6. Liao, S.J.: Notes on the homotopy analysis method: some definitions and theorems. Commun. Nonlinear Sci. Numer. Simul. 14, 983–997 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. 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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Liao, S.: An optimal homotopy-analysis approach for strongly nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 15, 2315–2332 (2010)

    Google Scholar 

  9. Abbasbandy, S.: The application of homotopy analysis method to nonlinear equations arising in heat transfer. Phys. Lett. A 360, 109–113 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Abbasbandy, S.: Homotopy analysis method for heat radiation equations. Int. Comm. Heat Mass Transfer 34, 380–387 (2007)

    Article  MathSciNet  Google Scholar 

  11. 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)

    Article  MATH  Google Scholar 

  12. Liao, S.J., Campo, A.: Analytic solutions of the temperature distribution in Blasius viscous flow problems. J. Fluid Mech. 453, 411–425 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Liao, S.J.: An explicit, totally analytic approximation of Blasius viscous flow problems. Internat. J. Non-Linear Mech. 34, 759–778 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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)

    Article  MathSciNet  MATH  Google Scholar 

  15. 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)

    Article  MathSciNet  MATH  Google Scholar 

  16. 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)

    Article  MathSciNet  MATH  Google Scholar 

  17. 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)

    Article  Google Scholar 

  18. 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)

    Article  MathSciNet  MATH  Google Scholar 

  19. 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)

    Article  MATH  Google Scholar 

  20. 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)

    Article  Google Scholar 

  21. Abbasbandy, S., Zakaria, F.S.: Soliton solutions for the fifth-order KdV equation with the homotopy analysis method. Nonlinear Dynam. 51, 83–87 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Wu, W., Liao, S.J.: Solving solitary waves with discontinuity by means of the homotopy analysis method. Chaos, Solitons & Fractals 26, 177–185 (2005)

    Article  MATH  Google Scholar 

  23. 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)

    Article  MathSciNet  MATH  Google Scholar 

  24. 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)

    MathSciNet  MATH  Google Scholar 

  25. 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)

    Article  MathSciNet  MATH  Google Scholar 

  26. Van Gorder, R.A., Sweet, E., Vajravelu, K.: Nano boundary layers over stretching surfaces. Commun. Nonlinear Sci. Numer. Simul. 15, 1494–1500 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. 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)

    Article  MATH  Google Scholar 

  28. 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)

    Article  MathSciNet  MATH  Google Scholar 

  29. Van Gorder, R.A., Vajravelu, K.: Analytic and numerical solutions to the Lane-Emden equation. Phys. Lett. A 372, 6060–6065 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. Liao, S.: A new analytic algorithm of Lane-Emden type equations. Appl. Math. Comput. 142, 1–16 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  31. 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)

    Article  MathSciNet  MATH  Google Scholar 

  32. Brendle, S.: Blow-up phenomena for the Yamabe equation. J. Amer. Math. Soc. 21, 951–980 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  33. 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)

    Article  MathSciNet  MATH  Google Scholar 

  34. 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)

    Article  Google Scholar 

  35. Van Gorder, R.A.: An elegant perturbation solution for the Lane-Emden equation of the second kind. New Astron. 16, 65–67 (2011)

    Article  Google Scholar 

  36. 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)

    Article  Google Scholar 

  37. Momoniat, E., Harley, C.: Approximate implicit solution of a Lane–Emden equation. New Astron. 11, 520–526 (2006)

    Article  Google Scholar 

  38. Wazwaz, A.M.: A new algorithm for solving differential equations of Lane–Emden type. Appl. Math. Comp. 118, 287–310 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  39. Chandrasekhar, S.: An Introduction to the Study of Stellar Structure, p. 90. Dover, New York (1967)

    Google Scholar 

  40. Hooman, K.: A perturbation solution for forced convection in a porous-saturated duct. J. Comput. Appl. Math. 211, 57–66 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  41. 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)

    Article  Google Scholar 

  42. 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)

    Article  Google Scholar 

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  1. Department of Mathematics, University of Central Florida, Orlando, FL, 32816-1364, USA

    Robert A. Van Gorder

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  1. Robert A. Van Gorder
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Correspondence to Robert A. Van Gorder.

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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

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