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Additional Techniques

  • Kuppalapalle Vajravelu
  • Robert A. van Gorder

Abstract

In the previous chapters, we have discussed the general idea of the method of homotopy analysis. We have also discussed some ways of ensuring that the obtained series solutions converge appropriately.Now, we shall discuss some additional techniques that might be useful for those applying the method of homotopy analysis to nonlinear boundary or initial value problems. As the more advanced methods are special made for certain types of problems, not all methods discussed here will be relevant for simpler models. However, for more complicated problems, some of the method discussed here may be useful to the reader.

Keywords

Residual Error Homotopy Analysis Method Additional Technique Perturbation Solution Compliant Substrate 
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. [1]
    S.J. Liao, An optimal homotopy-analysis approach for strongly nonlinear differential equations, Communications in Nonlinear Science and Numerical Simulation, 15 (2010) 2003.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    A. Föppl, Vorlesungen über technische Mechanik, 5th edn., p.132, B.G. Teubner, Leipzig, Germany, 1907.zbMATHGoogle Scholar
  3. [3]
    T. von Kármán, Festigkeitsproblem im Maschinenbau, Encyk. D. Math. Wiss., IV (1910) 311.Google Scholar
  4. [4]
    E. Cerda and L. Mahadevan, Geometry and Physics of Wrinkling, Phys. Rev. Lett., 90 (2003) 074302.CrossRefGoogle Scholar
  5. [5]
    L.D. Landau and E.M. Lifshitz, Theory of Elasticity, 3rd edn., ISBN 075062633X. Elsevier Science, January 1984.Google Scholar
  6. [6]
    X. Chen and J.W. Hutchinson, Herringbone buckling patterns of compressed thin films on compliant substrates, ASME Journal of Applied Mechanics, 71 (2004) 597.zbMATHCrossRefGoogle Scholar
  7. [7]
    Z.Y. Huang, W. Hong and Z. Suo, Nonlinear analyses of wrinkles in a film bonded to a compliant substrate, Journal of the Mechanics and Physics of Solids, 53 (2005) 2101.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    B. Audoly, Mode-dependent toughness and the delamination of compressed thin films, Journal of the Mechanics and Physics of Solids, 48 (2000) 2101.Google Scholar
  9. [9]
    J.W. Hutchinson and Z. Suo, Mixed-mode cracking in layered materials, Advances in Applied Mechanics, 29 (1992) 63.zbMATHCrossRefGoogle Scholar
  10. [10]
    J.-S. Wang and A.G. Evans, Measurement and analysis of buckling and buckle propagation in compressed oxide layers on superalloy substrates, Acta Materialia, 46 (1998) 4993.CrossRefGoogle Scholar
  11. [11]
    M.W. Moon, K.R. Lee, K.H. Oh and J.W. Hutchinson, Buckle delamination on patterned substrates, Acta Materialia, 52 (2004) 3151.CrossRefGoogle Scholar
  12. [12]
    B. Audoly and A. Boudaoud, Buckling of a stiff film bound to a compliant substrate—Part I: Formulation, linear stability of cylindrical patterns, secondary bifurcations, Journal of the Mechanics and Physics of Solids, 56 (2008) 2401.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    B. Audoly and A. Boudaoud, Buckling of a stiff film bound to a compliant substrate—Part II: A global scenario for the formation of herringbone pattern, Journal of the Mechanics and Physics of Solids, 56 (2008) 2422.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    R.A. Van Gorder, Analytical method for the construction of solutions to the Foppl-von Karman equations governing deflections of a thin flat plate, Int. J. Non-Linear Mech., 47 (2012) 1.CrossRefGoogle Scholar
  15. [15]
    E. Sweet and R.A. Van Gorder, Exponential-type solutions to a generalized Drinfel'd-Sokolov equation, Physica Scripta, 82 (2010) 03500.CrossRefGoogle Scholar
  16. [16]
    S. Abbasbandy, E. Shivanian and K. Vajravelu, Mathematical properties of ħ-curve in the frame work of the homotopy analysis method, Communications in Nonlinear Science and Numerical Simulation, 16 (2011) 4268.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    S. Abbasbandy and E. Shivanian, Predictor homotopy analysis method and its application to some nonlinear problems, Communications in Nonlinear Science and Numerical Simulation, 16 (2011) 2456.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    S.J. Liao, An optimal homotopy-analysis approach for strongly nonlinear differential equations, Communications in Nonlinear Science and Numerical Simulation, 15 (2010) 2003.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    S.J. Liao, On the homotopy multiple-variable method and its applications in the interactions of nonlinear gravity waves, Communications in Nonlinear Science and Numerical Simulation, 16 (2011) 1274.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    O.M. Phillips, On the dynamics of unsteady gravity waves of finite amplitude Part 1: The elementary interactions, Journal of Fluid Mechanics, 9 (1960) 193.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  • Kuppalapalle Vajravelu
    • 1
  • Robert A. van Gorder
    • 1
  1. 1.Department of MathematicsUniversity of Central FloridaOrlandoUSA

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