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On the asymptotic validity of perturbation methods for hyperbolic differential equations

Part of the Lecture Notes in Mathematics book series (LNM,volume 711)

Abstract

In this work a justification for the following formal perturbation methods is presented:

  1. (i)

    Luke's procedure for equations of the type φtt−φss=εf(φ).

  2. (ii)

    The Chikwendu-Kevorkian perturbation method for equations of the type φtt−φss=εf(φt, φs).

  3. (iii)

    The Krylov-Bogoljubov-Mitropolski-Montgomery-Tidman approach for equations of the type φtt−φss+p2φ=εf(φ, φt, φs).

The justification implies a proof of the validity of the asymptotic approximations for the initial value problem: φ=φ(t,s), t⩾0, −∞<s<∞, φ(0,s)=ρ(s), φt(0,s)=μ(s).

As a mathematical tool the theory of integral inequalities is used.

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References

  1. W. Walter, Differential and integral inequalities. Springer-Verlag, Berlin, 1970.

    CrossRef  Google Scholar 

  2. D. Montgomery, D.A. Tidman, Secular and nonsecular behaviour for the cold plasma equations. The Physics of Fluids, vol. 7 no. 2, 1964.

    Google Scholar 

  3. J.C. Luke, A perturbation method for nonlinear dispersive wave problems. Proceedings of the Royal Society of London, Ser. A, vol. 292, 1966.

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  4. S.C. Chikwendu, J. Kevorkian, A perturbation method for hyperbolic equations with small nonlinearities. SIAM J. Appl. Math., vol. 22 no. 2, 1972.

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  5. W. Eckhaus, Formal approximations and singular perturbations. SIAM Review, vol. 19, no. 4, October 1977.

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  6. W. Eckhaus, New approach to the asymptotic theory of nonlinear oscillations and wave-propagation. J. Math. An. and Appl., vol. 49 no. 4, 1975.

    Google Scholar 

  7. A. Nayfeh, Perturbation methods. John Wiley, New York, 1973.

    MATH  Google Scholar 

  8. A.H.P. van der Burgh, Studies in the asymptotic theory of non-linear resonance. Ph.D. Thesis, Delft, 1974.

    Google Scholar 

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© 1979 Springer-Verlag

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van der Burgh, A.H.P. (1979). On the asymptotic validity of perturbation methods for hyperbolic differential equations. In: Verhulst, F. (eds) Asymptotic Analysis. Lecture Notes in Mathematics, vol 711. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0062956

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  • DOI: https://doi.org/10.1007/BFb0062956

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-09245-2

  • Online ISBN: 978-3-540-35332-4

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