Skip to main content
SpringerLink
Log in

The renormalization method based on the Taylor expansion and applications for asymptotic analysis

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Based on the Taylor expansion, we propose a renormalization method (TR method, for simplicity) for asymptotic analysis. The standard renormalization group (RG) method for asymptotic analysis can be derived out from this new method, and hence, the mathematical essence of the RG method is also recovered. The biggest advantage of the proposed method is that the secular terms in perturbation series are automatically eliminated, but in usual perturbation theory, we need more efforts and tricks to eliminate these terms. At the same time, the mathematical foundation of the method is simple, and the logic of the method is very clear. Therefore, it is very easy to use in practice. As application, we obtain the uniform valid asymptotic solutions to some typical problems including vector field, boundary layer, boundary value problems of nonlinear wave equations, the normal form theory and reduction equations of dynamical systems. Further, by combining the topological deformation with the TR method, a modified method, namely the homotopy renormalization method (for simplicity, HTR), is proposed to overcome some weaknesses of the standard RG method and TR method. In this HTR method, since there is a freedom to choose the first-order approximate solution in perturbation expansion, we can improve the global solution. In particular, for those equations not including a small parameter, the HTR method can also be applied. Some concrete applications including multi-solution problems, the forced Duffing equation and the Blasius equation are given.

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

Access this article

Log in via an institution

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. Schwinger, J.: On quantum-electrodynamics and the magnetic moment of the electron. Phys. Rev. 73, 416–417 (1948)

    Article  MATH  Google Scholar 

  2. Schwinger, J.: Quantum electrodynamics. I. A covariant formulation. Phys. Rev. 74, 1439–1461 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  3. Feynman, R.P.: Relativistic cut-off for quantum electrodynamics. Phys. Rev. 74, 1430–1438 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  4. Feynman, R.P.: Space–time approach to non-relativistic quantum mechanics. Rev. Mod. Phys. 20, 367–387 (1948)

    Article  MathSciNet  Google Scholar 

  5. Tomonaga, S.I., Oppenheimer, J.R.: On infinite field reactions in quantum field theory. Phys. Rev. 74, 224–225 (1948)

    Article  MATH  Google Scholar 

  6. Tati, T., Tomonaga, S.: A self-consistent subtraction method in the quantum field theory. I. Prog. Theor. Phys. 3, 391–406 (1948)

    Article  MathSciNet  Google Scholar 

  7. Dyson, F.J.: The electromagnetic shift of energy levels. Phys. Rev. 73, 617–626 (1948)

    Article  MATH  Google Scholar 

  8. Dyson, F.J.: The radiation theories of Tomonaga, Schwinger, and Feynman. Phys. Rev. 75, 486–502 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gell-Mann, M., Low, F.E.: Quantum electrodynamics at small distances. Phys. Rev. 95, 1300–1312 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  10. Wilson, K.G.: Renormalization group and critical phenomena. I. Renormalization group and the Kadanoff scaling picture. Phys. Rev. B 4, 3174–3183 (1971)

    Article  MATH  Google Scholar 

  11. Goldenfeld, N., Martin, O., Oono, Y.: Intermediate asymptotics and renormalization group theory. J. Sci. Comput. 4, 355–372 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  12. Goldenfeld, N., Martin, O., Oono, Y.: Anomalous dimensions and the renormalization group in a nonlinear diffusion process. Phys. Rev. Lett. 64, 1361–1364 (1990)

    Article  Google Scholar 

  13. Goldenfeld, N., Martin, O., Oono, Y.: Asymptotics of Partial Differential Equations and the Renormalisation Group. Asymptotics Beyond All Orders. Springer, New York (1991)

    Google Scholar 

  14. Chen, L.Y., Goldenfeld, N., Oono, Y., Paquette, G.: Selection, stability and renormalization. Phys. A 204, 111–133 (1994)

    Article  Google Scholar 

  15. Chen, L.Y., Goldenfeld, N., Oono, Y.: Renormalization-group theory for the modified porous-medium equation. Phys. Rev. A 44, 6544–6550 (1991)

    Article  Google Scholar 

  16. Chen, L.Y., Goldenfeld, N.: Renormalization-group theory for the propagation of a turbulent burst. Phys. Rev. A 45, 5572–5577 (1992)

    Article  Google Scholar 

  17. Chen, L.Y., Goldenfeld, N., Oono, Y.: Renormalization group theory for global asymptotic analysis. Phys. Rev. Lett. 73, 1311–1315 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chen, L.Y., Goldenfeld, N., Oono, Y.: Renormalization group theory and variational calculations for propagating fronts. Phys. Rev. E 49, 4502–4511 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  19. Paquette, G.C., Chen, L.Y., Goldenfeld, N., Oono, Y.: Structural stability and renormalization group for propagating fronts. Phys. Rev. Lett. 72, 76–79 (1994)

    Article  Google Scholar 

  20. Chen, L.Y., Goldenfeld, N., Oono, Y.: Renormalization group and singular perturbations: multiple scales, boundary layers, and reductive perturbation theory. Phys. Rev. E 54, 376–394 (1996)

    Article  Google Scholar 

  21. Nozaki, K., Oono, Y., Shiwa, Y.: Reductive use of renormalization group. Phys. Rev. E 62, R4501–R4504 (2000)

    Article  MathSciNet  Google Scholar 

  22. Shiwa, Y.: Renormalization-group theoretical reduction of the Swift–Hohenberg model. Phys. Rev. E 63, 016119-7 (2000)

    Article  MathSciNet  Google Scholar 

  23. Shiwa, Y.: Renormalization-group for amplitude equations in cellular pattern formation with and without conservation law. Prog. Theor. Phys. 125, 871–878 (2011)

    Article  MATH  Google Scholar 

  24. Tao, T., Geng, C.: Breather dynamics in the perturbed sine-Gordon equation. Commun. Theor. Phys. 40, 390–392 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. Tao, T., Geng, C., Jian-Wei, L.: Anomalous dimension in the solution of the modified porous medium equation. Commun. Theor. Phys. 37, 741–744 (2002)

    Article  MathSciNet  Google Scholar 

  26. Tao, T., Geng, C., Jian-Wei, L.: Anomalous dimension in the solution of a nonlinear diffusion equation. Commun. Theor. Phys. 36, 617–619 (2001)

    Article  Google Scholar 

  27. Kunihiro, T.: A geometrical formulation of the renormalization group method for global analysis. Prog. Theor. Phys. 94, 503–514 (1995)

    Article  MathSciNet  Google Scholar 

  28. Kunihiro, T.: A geometrical formulation of the renormalization group method for global analysis II: partial differential equations. Jpn. J. Ind. Appl. Math. 14, 51–69 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kunihiro, T.: The renormalization-group method applied to asymptotic analysis of vector fields. Prog. Theor. Phys. 97, 179–200 (1997)

    Article  MathSciNet  Google Scholar 

  30. Ei, S.I., Fujii, K., Kunihiro, T.: Renormalization-group method for reduction of evolution equations; invariant manifolds and envelopes. Ann. Phys. 280, 236–298 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  31. Hatta, Y., Kunihiro, T.: Renormalization group method applied to kinetic equations: roles of initial values and time. Ann. Phys. 298, 24–57 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kunihiro, T.: Renormalization-group resummation of a divergent series of the perturbative wave functions of quantum systems. Prog. Theor. Phys. Suppl. 131, 459–470 (1998)

    Article  Google Scholar 

  33. Kunihiro, T.: Renormalization-group resummation of a divergent series of the perturbative wave functions of the quantum anharmonic oscillator. Phys. Rev. D 57, R2035–R2039 (1998)

    Article  Google Scholar 

  34. Kunihiro, T., Matsukidaira, J.: Dynamical reduction of discrete systems based on the renormalization-group method. Phys. Rev. E 57, 4817–4829 (1998)

    Article  Google Scholar 

  35. Kunihiro, T., Tsumura, K.: Application of the renormalization-group method to the reduction of transport equations. J. Phys. A 39, 8089 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  36. Kuramoto, Y.: On the reduction of evolution equations in extended systems. Prog. Theor. Phys. Suppl. 99, 244–262 (1989)

    Article  MathSciNet  Google Scholar 

  37. Ziane, M.: On a certain renormalization group method. J. Math. Phys. 41, 3290–3299 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  38. Chiba, H.: Extension and unification of singular perturbation methods for ODEs based on the renormalization group method. SIAM J. Appl. Dyn. Syst. 8, 1066–1115 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Chiba, H.: Approximation of center manifolds on the renormalization group method. J. Math. Phys. 49, 102703 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  40. Chiba, H.: Simplified renormalization group method for ordinary differential equations. J. Differ. Equ. 246, 1991–2019 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  41. Boyanovsky, D., De Vega, H.J., Wang, S.Y.: Dynamical renormalization group approach to quantum kinetics in scalar and gauge theories. Phys. Rev. D 61, 065006-34 (2000)

    Google Scholar 

  42. DeVille, R.E.L., Harkin, A., Holzer, M., Kaper, T.J.: Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations. Phys. D 237, 1029–1052 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  43. Gorban, A.N., Karlin, I.V., Zinovyev, A.Y.: Constructive methods of invariant manifolds for kinetic problems. Phys. Rep. 396, 197–403 (2004)

    Article  MathSciNet  Google Scholar 

  44. Kirkinis, E.: Reduction of amplitude equations by the renormalization group approach. Phys. Rev. E 77, 011105-8 (2006)

    MathSciNet  Google Scholar 

  45. Paquette, G.C.: Renormalization group analysis of differential equations subject to slowly modulated perturbations. Phys. A 276, 122–163 (2000)

    Article  Google Scholar 

  46. Polonyi, J.: Lectures on the functional renormalization group method. Cent. Eur. J. Phys. 1, 1–71 (2003)

    Article  Google Scholar 

  47. Boyanovsky, D., De Vega, H.J.: Dynamical renormalization group approach to relaxation in quantum field theory. Ann. Phys. 307, 335–371 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  48. O’Malley, R.E., Kirkinis, E.: Variation of parameters and the renormalization group method. Stud. Appl. Math. 134, 215–232 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  49. Bricmont, J., Kupiainen, A.: Renormalization group and the Ginzburg–Landau equation. Commun. Math. Phys. 150, 193–208 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  50. Yamaguchi, Y.Y., Nambu, Y.: Renormalization group equations and integrability in Hamiltonian systems. Prog. Theor. Phys. 100, 199–204 (1998)

    Article  Google Scholar 

  51. Frasca, M.: Theory of quantum resonance: a renormalization-group approach. Phys. Rev. A 58, 771–774 (1998)

    Article  Google Scholar 

  52. Kirkinis, E.: Secular series and renormalization group for amplitude equations. Phys. Rev. E 78, 032104-4 (2008)

    Article  MathSciNet  Google Scholar 

  53. Shirkov, D.V., Kovalev, V.F.: Bogolyubov renormalization group and symmetry of solution in mathematical physics. Phys. Rep. 352, 219 (2000)

    Article  MATH  Google Scholar 

  54. Verga, A.: Elements of similarity and singularity. In: Peyresq Lectures on Nonlinear Phenomena, vol. 2, pp. 285–318. World Scientific, Singapore (2002)

  55. Palit, A., Datta, D.P.: Comparative study of homotopy analysis and renormalization group methods on Rayleigh and Van der Pol equations. In: Differential Equations and Dynamical Systems, pp. 1–27. Springer-Verlag, Berlin (2014)

  56. Palit, A.: A study of an improved renormalization group method in non linear oscillation. In: Proceedings of ICFM 2015 International Conference on Frontiers in Mathematics 2015 March 26–28, 2015, Gauhati University, Guwahati, Assam, India

  57. Liao, S.J.: Beyond Perturbation: Introduction to the Homotopy Analysis Method. Champan Hall, CRC, Boca Raton (2003)

    Book  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  60. Liu, C., Liu, Y.: Comparison of a general series expansion method and the homotopy analysis method. Mod. Phys. Lett. B 24, 1699–1706 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  61. Liu, C.: The essence of the generalized Newton binomial theorem. Commun. Nonlinear Sci. Numer. Simul. 15, 2766–2768 (2010)

    Article  Google Scholar 

  62. Taniuti, T., Nishihara, K.: Nonlinear Waves. Pitman, Boston (1983)

    MATH  Google Scholar 

  63. Nayfeh, T.A., Nayfeh, A.H., Mook, D.T.: A theoretical and experimental investigation of a three-degree-of-freedom structure. Nonlinear Dyn. 6, 353–374 (1994)

    Article  Google Scholar 

  64. Nayfeh, T.A., Asrar, W., Nayfeh, A.H.: Three-mode interactions in harmonically excited systems with quadratic nonlinearities. Nonlinear Dyn. 3, 385–410 (1992)

    Article  Google Scholar 

  65. Nayfeh, A.H., Nayfeh, J.F., Mook, D.T.: On methods for continuous systems with quadratic and cubic nonlinearities. Nonlinear Dyn. 3, 145–162 (1992)

  66. Nayfeh, A.H.: Perturbation Methods. Wiley, London (2008)

    Google Scholar 

  67. Tsien, S.T.: The Poincare–Lighthill–Kuo method. Adv. Appl. Mech. 4, 281–349 (1956)

    Article  MathSciNet  Google Scholar 

  68. Segal, L.A., Handelman, G.H.: Mathematics Applied to Continuous Medium Mechanics. Macmillan, New York (1977)

    Google Scholar 

  69. Chernkykh, K.F., Aleshkov, Y.Z., Ponyatovsky, V.V., Shamina, V.A.: Introduction in Mechanics of Continuous Medium. Leningrad University Press, Leningrad (1984)

    Google Scholar 

  70. Kauderer, H.: Nichtlineare Mechanik. Springer, Berlin (1958)

    Book  MATH  Google Scholar 

  71. Boertjens, G.J., Van Horssen, W.T.: On mode interactions for a weakly nonlinear beam equation. Nonlinear Dyn. 17, 23–40 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  72. Andrianov, I.V., Danishevs’kyy, V.V.: Asymptotic approach for non-linear periodic vibrations of continuous structures. J. Sound Vib. 249, 465–481 (2002)

    Article  MATH  Google Scholar 

  73. Andrianov, I.V., Awrejcewicz, J.: Analysis of jump phenomena using Pad approximations. J. Sound Vib. 260, 577–588 (2003)

    Article  MATH  Google Scholar 

  74. Andrianov, I.V., Awrejcewicz, J., Danishevs’kyy, V.V., Ivankov, A.O.: Asymptotic Methods in the Theory of Plates with Mixed Boundary Conditions. Wiley, Chichester (2014)

    Book  MATH  Google Scholar 

  75. Stoker, J.J.: Nonlinear Vibrations in Mechanical and Electrical Systems. Interscience Publishers, New York (1950)

    MATH  Google Scholar 

  76. Watson, L.T.: Globally convergent homotopy methods: a tutorial. Appl. Math. Comput. 31, 369–396 (1989)

    MathSciNet  MATH  Google Scholar 

  77. Alexander, J.C., Yorke, J.A.: The homotopy continuation method: numerically implementable topological procedures. Trans. Am. Math. Soc. 242, 271–284 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  78. Cronin, J.: Fixed points and topological degree in nonlinear analysis. Am. Math. Soc. 11 (1995)

  79. Liu, C.: Applications of complete discrimination system for polynomial for classifications of traveling wave solutions to nonlinear differential equations. Comput. Phys. Commun. 181, 317–324 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  80. Blasius, H.: Grenzschichten in flussigkeiten mit kleiner reibung. Z. Math. Phys. 56, 1–37 (1908)

    MATH  Google Scholar 

Download references

Acknowledgements

Thanks to anonymous referees for their valuable comments on the first version of the paper. I would like to thank Prof. Nayfeh for his helpful suggestions. I also appreciate the Editor’s kindly help. Special thanks to the referees of the second version for their detailed comments and suggestions so that I can improve the paper. This project was supported by Scientific Research Fund of Education Department of Heilongjiang Province of China under Grant No.: 12541083.

Author information

Authors and Affiliations

  1. Department of Mathematics, Northeast Petroleum University, Daqing, 163318, China

    Cheng-shi Liu

Authors
  1. Cheng-shi Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cheng-shi Liu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, Cs. The renormalization method based on the Taylor expansion and applications for asymptotic analysis. Nonlinear Dyn 88, 1099–1124 (2017). https://doi.org/10.1007/s11071-016-3298-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-016-3298-8

Keywords

Search

Navigation