Applied Mathematics and Mechanics

, Volume 22, Issue 3, pp 326–331 | Cite as

A simple fast method in finding particular solutions of some nonlinear PDE

  • Liu Shi-kuo
  • Fu Zun-tao
  • Liu Shi-da
  • Zhao Qiang


The “trial function method” (TFM for short) and a routine way in finding traveling wave solutions to some nonlinear partial differential equations (PDE for short) wer explained. Two types of evolution equations are studied, one is a generalized Burgers or KdV equation, the other is the Fisher equation with special nonlinear forms of its reaction rate term. One can see that this method is simple, fast and allowing further extension.

Key words

trial function method nonlinear PDE shock wave solution solitary wave solution 

CLC numbers

O175 O411 


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  1. [1]
    Kuramoto Y.Chemical Oscillations, Waves and Turbulence [M]. Berlin: Springer-Verlag, 1984.Google Scholar
  2. [2]
    Kuramoto Y. Diffusion-induced chaos in reaction-diffusion systems [J].Prog Theor Phys Suppl, 1978,64(suppl): 346–350.Google Scholar
  3. [3]
    Burgers J M. A mathematical model illustrating the theory of turbulence [J].Adv Appl Mech, 1949,1(2): 171–175.Google Scholar
  4. [4]
    LIU Shi-da, LIU Shi-kuo. KdV-Burgers modeling of turbulence [J].Science in China A, 1992,35(5): 576–577.Google Scholar
  5. [5]
    Kawahara T. Formulation of saturated solutions in a nonlinear dispersive system with instability and dissipation [J].Phys Rev Lett, 1983,51(5): 381–387.CrossRefGoogle Scholar
  6. [6]
    Kwok W, Chow. A class of exact, periodic solutions of nonlinear envelope equations [J].J Math Phys, 199536(8): 4125–4137.MathSciNetCrossRefGoogle Scholar
  7. [7]
    Kudryashov N A. Exact solutions of the generalized Kuramoto-Sivashinsky equation [J].Phys, Lett A, 1990,147(5,6): 287–291.MathSciNetCrossRefGoogle Scholar
  8. [8]
    WANG Ming-liang, ZHOU Yu-bin, LI Zhi-bin. Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics [J].Phys Lett A, 1996,216(1): 67–75.CrossRefGoogle Scholar
  9. [9]
    KONG De-xing, HU Hai-rong. Geometric approach for finding exact solutions to nonlinear partial differential equations [J].Phys Lett A, 1998,246(1,2): 105–112.MathSciNetGoogle Scholar
  10. [10]
    XIONG Shu-lin. A kind of analytical solutions of Burgers-KdV equation [J].Kexue Tongbao (Chinese Science Bulletin), 1989,34(1): 26–29. (in Chinese)MathSciNetGoogle Scholar
  11. [11]
    LIU Shi-da, LIU Shi-kuo. Heteroclinic orbit on the KdV-Burgers equation and Fisher equation [J].Commun Theor Phys, 1991,16(4): 497–500.Google Scholar
  12. [12]
    LIU Shi-da, LIU Shi-kuo. The traveling wave solutions of KdV-Burgers-Kuramoto equation [J].Progress in Natural Science, 1999,9(10): 912–918. (in Chinese)Google Scholar
  13. [13]
    Benney D J. Long nonlinear waves in fluid flow [J].J Math and Phys, 1966,45(1): 52–60.zbMATHMathSciNetGoogle Scholar

Copyright information

© Editorial Committee of Applied Mathematics and Mechanics 1980

Authors and Affiliations

  • Liu Shi-kuo
    • 1
  • Fu Zun-tao
    • 1
    • 2
  • Liu Shi-da
    • 1
    • 2
  • Zhao Qiang
    • 1
  1. 1.Department of GeophysicsPeking UniversityBeijingP R China
  2. 2.SKLTRPeking UniversityBeijingP R China

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