Applied Mathematics and Mechanics

, Volume 33, Issue 12, pp 1569–1582 | Cite as

Solving shock wave with discontinuity by enhanced differential transform method (EDTM)

  • Li Zou (邹 丽)Email author
  • Zhen Wang (王 振)
  • Zhi Zong (宗 智)
  • Dong-yang Zou (邹东阳)
  • Shuo Zhang (张 朔)


An enhanced differential transform method (EDTM), which introduces the Padé technique into the standard differential transform method (DTM), is proposed. The enhanced method is applied to the analytic treatment of the shock wave. It accelerates the convergence of the series solution and provides an exact power series solution.

Key words

enhanced differential transform method (EDTM) shock wave Padé technique Burgers equation 

Chinese Library Classification


2010 Mathematics Subject Classification

34E05 35C20 35R05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Nayfeh, A. H. Introduction to Perturbation Techniques, John Wiley & Sons, New York (1981)zbMATHGoogle Scholar
  2. [2]
    Nayfeh, A. H. Problems in Perturbation, John Wiley & Sons, New York (1985)zbMATHGoogle Scholar
  3. [3]
    Lyapunov, A. M. General Problem on Stability of Motion, Taylor & Francis, London (1992)Google Scholar
  4. [4]
    Adomian, G. Nonlinear stochastic differential equations. Journal of Mathematical Analysis and Applications, 55, 441–452 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Adomian, G. and Adomian, G. E. A global method for solution of complex systems. Mathematical Model, 5, 521–568 (1984)Google Scholar
  6. [6]
    Adomian, G. Solving Frontier Problems of Physics: the Decomposition Method, Kluwer Academic Publishers, Boston/London (1994)zbMATHGoogle Scholar
  7. [7]
    Liao, S. J. Homotopy analysis method: a new analytic method for nonlinear problems. Applied Mathematics and Mechanics (English Edition), 19(10), 957–962 (1998) DOI 10.1007/BF02457955MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Liao, S. J. Beyond Perturbation: Introduction to the Homotopy Analysis Method (in Chinese), Science Press, Beijing (2007)Google Scholar
  9. [9]
    Lu, D. Q. Interaction of viscous waves with a free surface. Applied Mathematics and Mechanics (English Edition), 25(6), 647–655 (2004) DOI 10.1007/BF02438207MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    Lu, D. Q., Dai, S. Q., and Zhang, B. S. Hamiltonian formulation of nonlinear water waves in a two-fluid system. Applied Mathematics and Mechanics (English Edition), 20(4), 343–349 (1999) DOI 10.1007/BF02458559MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    Zhou, J. K. Differential Transform and Its Applications for Electrical Circuits (in Chinese), Huazhong University Press, Wuhan (1986)Google Scholar
  12. [12]
    Ravi-Kanth, A. S. V. and Aruna, K. Differential transform method for solving the linear and nonlinear Klein-Gordon equation. Computer Physics Communications, 180(5), 708–711 (2009)MathSciNetCrossRefGoogle Scholar
  13. [13]
    Chen, C. K. and Ho, S. H. Solving partial differential equations by two-differential transform method. Applied Mathematics and Computation, 106, 171–179 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    Jang, M. J., Chen, C. L., and Liu, Y. C. Two-dimensional differential transformation method for partial differential equation. Applied Mathematics and Computation, 121, 261–270 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    Abdel-Halim Hassan, I. H. Different applications for the differential transformation in the differential equations. Applied Mathematics and Computation, 129, 183–201 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    Ayaz, F. On the two-dimensional differential transform method. Applied Mathematics and Computation, 143, 361–374 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    Ayaz, F. Solutions of the system of differential equations by differential transform method. Applied Mathematics and Computation, 147, 547–567 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    Wang, Z., Zou, L., and Zhang, H. Q. Applying homotopy analysis method for solving differential-difference equation. Physics Letters A, 369, 77–84 (2007)zbMATHCrossRefGoogle Scholar
  19. [19]
    Zou, L., Zong, Z., Wang, Z., and He, L. Solving the discrete KdV equation with homotopy analysis method. Physics Letters A, 370, 287–294 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    Adbel-Halim Hassan, I. H. Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems. Chaos, Solitons & Fractals, 36(1), 53–65 (2008)MathSciNetCrossRefGoogle Scholar
  21. [21]
    Kangalgil, F. and Ayaz, F. Solitary wave solutions for the KdV and mKdV equations by differential transform method. Chaos, Solitons & Fractals, 41(1), 464–472 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    Cole, J. D. On a quasi-linear parabolic equation occurring in aerodynamics. Quarterly of Applied Mathematics, 9, 225–236 (1951)MathSciNetzbMATHGoogle Scholar
  23. [23]
    Bateman, H. Some recent researches on the motion of fluids. Monthly Weather Review, 43, 163–170 (1915)CrossRefGoogle Scholar
  24. [24]
    Burgers, J. M. A mathematical model illustrating the theory of turbulence. Advances in Applied Mechanics, 1, 171–199 (1948)MathSciNetCrossRefGoogle Scholar
  25. [25]
    Zhang, X. H., Ouyang, J., and Zhang, L. Element-free characteristic Galerkin method for Burgers equation. Engineering Analysis with Boundary Elements, 33, 356–362 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    Kutluay, S., Esen, A., and Dag, I. Numerical solutions of the Burgers equation by the least-squares quadratic B-spline finite element method. Journal of Computational and Applied Mathematics, 167(1), 21–33 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  27. [27]
    Whitham, G. B. Linear and Nonlinear Waves, John Wiley & Sons, New York (1974)zbMATHGoogle Scholar
  28. [28]
    Rosenau, P. and Hyman, J. M. Compactons: solitons with finite wavelength. Physical Review Letters, 75, 564–567 (1993)CrossRefGoogle Scholar
  29. [29]
    Tian, L. X. and Yin, J. L. Shock-peakon and shock-compacton solutions for K(p, q) equation by variational iteration method. Journal of Mathematical Analysis and Applications, 207, 46–52 (2007)MathSciNetzbMATHGoogle Scholar
  30. [30]
    Camassa, R. and Holm, D. An integrable shallow water equation with peaked solitons. Physical Review Letters, 71, 1661–1664 (1993)MathSciNetCrossRefGoogle Scholar

Copyright information

© Shanghai University and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Li Zou (邹 丽)
    • 1
    • 2
    Email author
  • Zhen Wang (王 振)
    • 3
  • Zhi Zong (宗 智)
    • 1
    • 4
  • Dong-yang Zou (邹东阳)
    • 1
    • 2
  • Shuo Zhang (张 朔)
    • 1
    • 2
  1. 1.State Key Laboratory of Structure Analysis for Industrial EquipmentDalianLiaoning Province, P. R. China
  2. 2.School of Aeronautics and AstronauticsDalian University of TechnologyDalianLiaoning Province, Liaoning Province, P. R. China
  3. 3.School of MathematicsDalian University of TechnologyDalianLiaoning Province, P. R. China
  4. 4.School of Naval ArchitectureDalian University of TechnologyDalianLiaoning Province, P. R. China

Personalised recommendations