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A Boundary-layer Flow with an Infinite Number of Solutions

  • Shijun Liao

Abstract

In this chapter, the Mathematica package BVPh (version 1.0) based on the homotopy analysis method (HAM) is used to gain exponentially and algebraically decaying solutions of a nonlinear boundary-value equation in an infinite interval. Especially, an infinite number of algebraically decaying solutions were found for the first time by means of the HAM, which illustrate the originality and validity of the HAM for nonlinear boundary-value problems.

Keywords

Initial Guess Computer Algebra System Decay Solution Auxiliary Linear Operator Obtain Series Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Banks, W.H.H.: Similarity solutions of the boundary-layer equations for a stretching wall. Journal de Mecanique theorique et appliquee. 2, 375–392 (1983).zbMATHGoogle Scholar
  2. Li, Y.J., Nohara, B.T., Liao, S.J.: Series solutions of coupled Van der Pol equation by means of homotopy analysis method. J. Mathematical Physics 51, 063517 (2010). doi:10.1063/1.3445770.MathSciNetCrossRefGoogle Scholar
  3. Liao, S.J.: The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems. PhD dissertation, Shanghai Jiao Tong University (1992).Google Scholar
  4. Liao, S.J.: A kind of approximate solution technique which does not depend upon small parameters (II) — An application in fluid mechanics. Int. J. Nonlin. Mech. 32, 815–822 (1997).zbMATHCrossRefGoogle Scholar
  5. Liao, S.J.: An explicit, totally analytic approximation of Blasius viscous flow problems. Int. J. Nonlin. Mech. 34, 759–778 (1999a).zbMATHCrossRefGoogle Scholar
  6. Liao, S.J.: A uniformly valid analytic solution of 2D viscous flow past a semi-infinite flat plate. J. Fluid Mech. 385, 101–128 (1999b).MathSciNetzbMATHCrossRefGoogle Scholar
  7. Liao, S.J.: On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet. J. Fluid Mech. 488, 189–212 (2003a).MathSciNetzbMATHCrossRefGoogle Scholar
  8. Liao, S.J.: Beyond Perturbation — Introduction to the Homotopy Analysis Method. Chapman & Hall/ CRC Press, Boca Raton (2003b).CrossRefGoogle Scholar
  9. Liao, S.J.: On the homotopy analysis method for nonlinear problems. Appl. Math. Comput. 147, 499–513 (2004).MathSciNetzbMATHCrossRefGoogle Scholar
  10. Liao, S.J.: A new branch of solutions of boundary-layer flows over an impermeable stretched plate. Int. J. Heat Mass Tran. 48, 2529–2539 (2005).zbMATHCrossRefGoogle Scholar
  11. Liao, S.J.: Series solutions of unsteady boundary-layer flows over a stretching flat plate. Stud. Appl. Math. 117, 2529–2539 (2006).CrossRefGoogle Scholar
  12. Liao, S.J.: Notes on the homotopy analysis method — Some definitions and theorems. Commun. Nonlinear Sci. Numer. Simulat. 14, 983–997 (2009).zbMATHCrossRefGoogle Scholar
  13. Liao, S.J.: On the relationship between the homotopy analysis method and Euler transform. Commun. Nonlinear Sci. Numer. Simulat. 15, 1421–1431 (2010a). doi:10.1016/j.cnsns.2009.06.008.zbMATHCrossRefGoogle Scholar
  14. Liao, S.J.: An optimal homotopy-analysis approach for strongly nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simulat. 15, 2003–2016 (2010b).zbMATHCrossRefGoogle Scholar
  15. Liao, S.J., Campo, A.: Analytic solutions of the temperature distribution in Blasius viscous flow problems. J. Fluid Mech. 453, 411–425 (2002).MathSciNetzbMATHCrossRefGoogle Scholar
  16. Liao, S.J., Pop, I.: Explicit analytic solution for similarity boundary layer equations. Int. J. Heat and Mass Transfer. 47, 75–85 (2004).CrossRefGoogle Scholar
  17. Liao, S.J., Magyari, E.: Exponentially decaying boundary layers as limiting cases of families of algebraically decaying ones. Z. angew. Math. Phys. 57, 777–792 (2006).MathSciNetzbMATHCrossRefGoogle Scholar
  18. Liao, S.J., Tan, Y.: A general approach to obtain series solutions of nonlinear differential equations. Stud. Appl. Math. 119, 297–355 (2007).MathSciNetCrossRefGoogle Scholar
  19. Magyari, E., Pop, I., Keller, B.: New analytical solutions of a well known boundary value problem in fluid mechanics. Fluid Dyn. Res. 33, 313–317 (2003).MathSciNetzbMATHCrossRefGoogle Scholar
  20. Trefethen, L.N.: Computing numerically with functions instead of numbers. Math. in Comp. Sci. 1, 9–19 (2007).MathSciNetzbMATHCrossRefGoogle Scholar
  21. Xu, H., 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. Physics of Fluids. 22, 053601 (2010). doi:10.1063/1.3392770.CrossRefGoogle Scholar

Copyright information

© Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Shijun Liao
    • 1
  1. 1.Shanghai Jiao Tong UniversityShanghaiChina

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