Skip to main content
SpringerLink
Log in
Book cover

The Optimal Homotopy Asymptotic Method pp 9–22Cite as

Optimal Homotopy Asymptotic Method

  • Chapter

Abstract

The notion of homotopy is an important part of topology and thus of differential geometry. The homotopy continuation method or shortly speaking homotopy was known as early as in the 1930s. Thus, in 1892, Lyapunov [1] introduced the so-called “artificial small parameters method” considering a linear differential equation with variable coefficients in the form

$$ \frac{du}{dt}=M(t)u $$

with M(t) a time periodic matrix. He replaced this equation with the equation

$$ \frac{du}{dt}=\varepsilon M(t)u $$

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. A.M. Lyapunov, General Problem on Stability of Motion (Taylor and Francis, London, 1992)

    Google Scholar 

  2. T.M. Wu, A new formula of solving nonlinear equations by Adomian decomposition method and homotopy methods. Appl. Math. Comput. 172, 903–907 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. A.P. Morgan, Solving Polynomial Systems using Continuation for Engineering and Scientific Problems (Prentice Hall, Englewood, New York, 1987)

    MATH  Google Scholar 

  4. C.B. Garcia, W.S. Zangwill, Pathways to Solutions. Fixed Points and Equilibrum (Prentice-Hall, Book Company, Englewood Clifts, NY, 1981)

    Google Scholar 

  5. E.L. Allgower, K. Georg, Numerical Continuation Methods, an Introduction (Springer, New York, NY, 1990)

    Book  MATH  Google Scholar 

  6. S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, PhD thesis, Shanghai Jiao Tong University (1992)

    Google Scholar 

  7. S.J. Liao, A Kind of Linearity Invariance Under Homotopy and Some Simple Applications of it in Mechanics, Bericht, no. 520 (Institut fur Schiffbau der Universitaet Hamburg, Germany, 1992)

    Google Scholar 

  8. S.J. Liao, Application of process analysis method to the solution of 2D non-linear progressive gravity waves. J. Ship Res. 36, 30–37 (1992)

    Google Scholar 

  9. S.J. Liao, A kind of approximate solution technique which does not depend upon small parameters (II)—An application in fluid mechanics. Int. J. Nonlinear Mech. 32, 815–822 (1997)

    Article  MATH  Google Scholar 

  10. S.J. Liao, An explicit, totally analytic approximate solution for Blasius viscous flow problems. Int. J. Nonlinear Mech. 34, 759–778 (1999)

    Article  MATH  Google Scholar 

  11. S.J. Liao, Beyond Perturbation-Introduction to the Homotopy Analysis Method (Chapman and Hall, CRC, Boca Raton, FL, 2004)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. S.J. Liao, Homotopy Analysis Method in Nonlinear Differential Equations (Higher Education Press, Springer, Beijing, Berlin, 2012)

    Book  MATH  Google Scholar 

  14. C.-S. Liu, The essence of the homotopy analysis method. Appl. Math. Comput. 216, 1299–1303 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. C.-S. Liu, The essence of the generalized Taylor theorem as the foundation of the homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul. 16(1254–1262) (2011)

    Google Scholar 

  16. S.J. Liao, An optimal homotopy-analysis approach for strongly nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 15, 2003–2016 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  17. V. Marinca, N. Herişanu, I. Nemeş, Optional homotopy asymptotic method with application to thin film flow. Cent. Eur. J. Phys. 6, 648–653 (2008)

    Article  Google Scholar 

  18. V. Marinca, N. Herişanu, Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer. Int. Commun. Heat Mass Transf. 35, 710–715 (2008)

    Article  Google Scholar 

  19. N. Herişanu, V. Marinca, T. Dordea, Gh. Madescu, A new analytical approach to nonlinear vibration of an electrical machine. Proc. Rom. Acad. Ser. A. 9, 229–236 (2008)

    Google Scholar 

  20. V. Marinca, N. Herisanu, Nonlinear Dynamical Systems in Engineering. Some Approximate Approaches (Springer, Berlin-Heidelberg, 2011)

    Book  MATH  Google Scholar 

  21. P. Amore, A. Aranda, Improved Lindstedt-Poincaré method for the solution of nonlinear problems. J. Sound Vib. 283, 1115–1136 (2009)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Politehnica University of Timisoara, Timisoara, Romania

    Vasile Marinca & Nicolae Herisanu

  2. Center for Advanced and Fundamental Technical Research, Romanian Academy, Timisoara Branch, Timisoara, Romania

    Vasile Marinca & Nicolae Herisanu

Authors
  1. Vasile Marinca
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nicolae Herisanu
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Marinca, V., Herisanu, N. (2015). Optimal Homotopy Asymptotic Method. In: The Optimal Homotopy Asymptotic Method. Springer, Cham. https://doi.org/10.1007/978-3-319-15374-2_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-15374-2_2

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-15373-5

  • Online ISBN: 978-3-319-15374-2

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation