Perturbation Methods and Computer Algebra in Mechanics

  • R. H. Rand
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 343)


Perturbation methods are a collection of diverse schemes for obtaining approximate analytic solutions to problems which involve a small parameter ϵ, usually in the form of a power series in ϵ In these notes we will discuss computer algebra treatments of regular perturbations (section 2), and of three singular perturbation methods (composite expansions, section 4, averaging, section 5 and two variable expansion method, section 8).


Hopf Bifurcation Perturbation Method Computer Algebra Transition Curve Jacobian Elliptic Function 
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  1. 1.
    C.M. Bender and S.A. Orszag, “Advanced Mathematical Methods for Scientists and Engineers”, McGraw-Hill (1978)MATHGoogle Scholar
  2. 2.
    P. Byrd and M. Friedman, “Handbook of Elliptic Integrals for Engineers and Physicists”, Springer (1954)MATHGoogle Scholar
  3. 3.
    V.T. Coppola and R.H. Rand, ”Averaging Using Elliptic Functions: Approximation of Limit Cycles”, Acta Mechanica, 81:125–142 (1990)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    V.T. Coppola and R.H. Rand, “MACSYMA Program to Implement Averaging Using Elliptic Functions”, in Computer Aided Proofs in Analysis, eds. K.R. Meyer and D.S. Schmidt, pp.71–89, Springer (1991)CrossRefGoogle Scholar
  5. 5.
    J. Guckenheimer and P. Holmes, “Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields”, Springer (1983)CrossRefMATHGoogle Scholar
  6. 6.
    P.B. Kahn and Y. Zarmi, “Minimal Normal Forms in Harmonic Oscillations with Small Nonlinear Perturbations”, Physica D, 54:65–74 (1991)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    A. Nayfeh, “Perturbation Methods”, Viley (1973)MATHGoogle Scholar
  8. 8.
    R.H. Rand, “Computer Algebra in Applied Mathematics: An Introduction to MACSYMA”, Pitman (1984)MATHGoogle Scholar
  9. 9.
    R.H. Rand, “The Use of Symbolic Computation in Perturbation Analysis”, in Symbolic Computation in Fluid Mechanics and Heat Transfer, ed. H.H. Bau et al., ASME publication HTD-105/AMD-97, pp.41–45 (1988)Google Scholar
  10. 10.
    R.H. Rand, “Using Computer Algebra to Handle Elliptic Functions in the Method of Averaging”, in Symbolic Computations and Their Impact on Mechanics, eds. A.K. Noor, I. Elishakoff, G. Hulbert, pp.311–326, Amer.Soc.Mech.Eng., PVP-Vol.205, (1990)Google Scholar
  11. 11.
    R.H. Rand and D. Armbruster, “Perturbation Methods, Bifurcation Theory and Computer Algebra”, Springer (1987)CrossRefMATHGoogle Scholar
  12. 12.
    J.A. Sanders and F. Verhulst, “Averaging Methods in Nonlinear Dynamical Systems”, Springer (1985)CrossRefMATHGoogle Scholar
  13. 13.
    J.J. Stoker, “Nonlinear Vibrations”, Interscience (1950)MATHGoogle Scholar
  14. 14.
    M. Van Dyke, “Analysis and Improvement of Perturbation Series”, Q.J.Mech.Appl.Math., 27:423–450 (1974)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Wien 1994

Authors and Affiliations

  • R. H. Rand
    • 1
  1. 1.Cornell UniversityIthacaUSA

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