Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

On the method of directly defining inverse mapping for nonlinear differential equations

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

In scientific computing, it is time-consuming to calculate an inverse operator \(\mathcal {A}^{-1}\) of a differential equation \(\mathcal {A}\phi = f\), especially when \(\mathcal {A}\) is a highly nonlinear operator. In this paper, based on the homotopy analysis method (HAM), a new approach, namely the method of directly defining inverse mapping (MDDiM), is proposed to gain analytic approximations of nonlinear differential equations. In other words, one can solve a nonlinear differential equation \(\mathcal {A}\phi = f\) by means of directly defining an inverse mapping \(\mathcal J\), i.e. without calculating any inverse operators. Here, the inverse mapping \(\mathcal {J}\) is even unnecessary to be explicitly expressed in a differential form, since “mapping” is a more general concept than “differential operator”. To guide how to directly define an inverse mapping \(\mathcal {J}\), some rules are provided. Besides, a convergence theorem is proved, which guarantees that a convergent series solution given by the MDDiM must be a solution of problems under consideration. In addition, three nonlinear differential equations are used to illustrate the validity and potential of the MDDiM, and especially the great freedom and large flexibility of directly defining inverse mappings for various types of nonlinear problems. The method of directly defining inverse mapping (MDDiM) might open a completely new, more general way to solve nonlinear problems in science and engineering, which is fundamentally different from traditional methods.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Nayfeh, A.H.: Perturbation Methods. Wiley (2000)

  2. Liao, S.: Proposed Homotopy Analysis Techniques for the Solution of Nonlinear Problem, Ph.D. thesis, Shanghai Jiao Tong University (1992)

  3. Liao, S.: Beyond Perturbation: Introduction to the Homotopy Analysis Method. CRC Press (2003)

  4. Liao, S.: Homotopy Analysis Method in Nonlinear Differential Equations. Springer and Higher Education Press (2012)

  5. Liao, S. (ed.): Advances in the Homotopy Analysis Method. World Scientific Publisher (2013)

  6. Kuppalapalle, V., Van Gorder, R.A.: Nonlinear Flow Phenomena and Homotopy Analysis: Fluid Flow and Heat Transfer. Springer-Verlag, New York (2013)

  7. Adomian, G.: Solving Frontier Problems of Physics: The Decomposition Method. Kluwer Academic Publishers (1994)

  8. He, J.: Homotopy perturbation technique. Comput. Meth. Appl. Mech. Eng. 178, 257–262 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. He, J.: Variational iteration method - a kind of non-linear analytical technique: Some examples. Int. J. Non-Linear Mech. 34, 699–708 (1999)

    Article  MATH  Google Scholar 

  10. 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 Dyn. 50, 27–35 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Van Gorder, R.A.: The variational iteration method is a special case of the homotopy analysis method. Applied Mathematics Letters 45, 81–85 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Abbasbandy, S.: The application of the homotopy analysis method to solve a generalized Hirota-Satsuma coupled KdV equation. Phys. Lett. A 361, 478–483 (2007)

    Article  MATH  Google Scholar 

  13. Song, H., Tao, L.: Homotopy analysis of 1D unsteady, nonlinear groundwater flow through porous media. J. Coastal Res. 50, 292–295 (2007)

    Google Scholar 

  14. Wu, Y.Y., Cheung, K.F.: Homotopy solution for nonlinear differential equations in wave propagation problems. Wave Motion 46, 1–14 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Van Gorder, R.A., Vajravelu, K.: On the selection of auxiliary functions, operators, and convergence-control parameters in the applications of the homotopy analysis method to nonlinear differential equation: A general approach. Commun. Nonlinear Sci. Numer. Simul. 14, 4078–4089 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mastroberardino, A.: Homotopy analysis method applied to electrohydrodynamic flow. Commun. Nonlinear Sci. Numer. Simulat. 16, 2730–2736 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Motsa, S.S., Sibanda, P., Auad, F.G., Shateyi, S.: A new spectral homotopy analysis method for the MHD Jeffery-Hamel problem. Comput. Fluids 39, 1219–1225 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Liang, S., Liu, S.: An open problem on the optimality of an asymptotic solution to Duffing’s nonlinear oscillation problem. Commun. Nonlinear Sci. Numer. Simul. 19, 4189–4195 (2014)

    Article  MathSciNet  Google Scholar 

  19. Liao, S., Cheung, K.F.: Homotopy analysis of nonlinear progressive waves in deep water. J. Eng. Math. 45, 105–116 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  21. Cheng, J., Zhu, S., Liao, S.: An explicit series approximation to the optimal exercise boundary of american put options. Commun. Nonlinear Sci. Numer. Simul. 15, 1148–1158 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Turkyilmazoglu, M.: Purely analytic solutions of the compressible boundary layer flow due to a porous rotating disk with heat transfer. Phys. Fluids 21, 106014 (2009)

    MATH  Google Scholar 

  23. Liao, S.: On the homotopy multiple-variable method and its applications in the interactions of nonlinear gravity waves. Commun. Nonlinear Sci. Numer. Simul. 16, 1274–1303 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Xu, D., Lin, Z., Liao, S., Stiassnie, M.: On the steady-state fully resonant progressive waves in water of finite depth. J. Fluid Mech. 710, 379–418 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Liu, Z., Liao, S.: Steady-state resonance of multiple wave interactions in deep water. J. Fluid Mech. 742, 664–700 (2014)

    Article  MathSciNet  Google Scholar 

  26. Liu, Z., Xu, D., Li, J., Peng, T., Alsaedi, A., Liao, S.: On the existence of steady-state resonant waves in experiments. J. Fluid Mech. 763, 1–23 (2015)

    Article  Google Scholar 

  27. Liao, S., Xu, D., Liu, Z.: On the discovery of steady-state resonant water waves. In: Tobisch, E. (ed.) New Approaches to Nonlinear Waves. Lecture Notes in Physics, chap. 3, vol. 908, pp. 43–82. Springer (2015)

  28. Liao, S., Tan, Y.: A general approach to obtain series solutions of nonlinear differential equations. Stud. Appl. Math. 119(4), 297–354 (2007)

    Article  MathSciNet  Google Scholar 

  29. Liao, S.: Notes on the homotopy analysis method: Some definitions and theorems. Commun. Nonlinear Sci. Numer. Simul. 14(4), 983–997 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China

    Shijun Liao & Yinlong Zhao

  2. Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration (CISSE), Shanghai, 200240, China

    Shijun Liao

  3. MOE Key Laboratory in Scientific and Engineering Computing, Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, China

    Shijun Liao

Authors
  1. Shijun Liao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yinlong Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shijun Liao.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liao, S., Zhao, Y. On the method of directly defining inverse mapping for nonlinear differential equations. Numer Algor 72, 989–1020 (2016). https://doi.org/10.1007/s11075-015-0077-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-015-0077-4

Keywords

Search

Navigation