Computational Mathematics and Modeling

, Volume 19, Issue 3, pp 271–281 | Cite as

Application of the dual operator method to an equation describing the behavior of a boundary function

Article

Abstract

The dual operator is an analogue of the conjugate operator in linear theory. In this study the dual operator is applied to a second-order differential equation describing the behavior of the zero-order boundary function in the boundary function method used to derive the asymptotic solution of the singularly perturbed integro-differential plasma-sheath equation. This approach produces is a three-point difference scheme. The results of a numerical solution of the Cauchy problem are reported.

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References

  1. 1.
    G. A. Emmert, R. M. Wieland, A. T. Mense, and J. N. Davidson, “Electric sheath and presheath in a collisionless, finite ion temperature plasma,” Phys. Fluids, 23, No. 4, 803–812 (1980).MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    D. S. Filippychev, “Boundary function method for asymptotic solution of the plasma-sheath equation,” Prikl. Mat. Informatika, No. 19, 21–40 (2004).Google Scholar
  3. 3.
    A. B. Vasil’eva and V. F. Butuzov, Asymptotic Expansions of Solutions of Singularly Perturbed Equations [in Russian], Nauka, Moscow (1973).Google Scholar
  4. 4.
    A. B. Vasil’eva and V. F. Butuzov, Singularly Perturbed Equations in Critical Cases [in Russian], Izd. MGU, Moscow (1978).Google Scholar
  5. 5.
    A. B. Vasil’eva and V. F. Butuzov, Asymptotic Methods in Singular Perturbation Theory [in Russian], Vysshaya Shkola, Moscow (1990).Google Scholar
  6. 6.
    D. S. Filippychev, “Numerical solution of the differential equation describing the behavior of the zero-order boundary function,” Prikl. Mat. Informatika, No. 23, 24–35 (2006).Google Scholar
  7. 7.
    D. G. Cacuci, R. B. Perez, and V. Protopopescu, “Duals and propagators, A canonical formalism for nonlinear equations,” J. Math. Phys., 29, No. 2, 335–361 (1988).CrossRefMathSciNetGoogle Scholar
  8. 8.
    D. G. Cacuci and O. A. Karakashian, “Benchmarking the propagator method for nonlinear systems: A Burgers-Korteweg-de Vries equation,” J. Comput. Phys., 89, No. 1, 63–79 (1990).MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    D. S. Filippychev, “Application of the dual operator formalism to obtain the zero-order boundary function for the plasma-sheath equation,” Prikl. Matem. Informatika, No. 22, 76–90 (2005).Google Scholar
  10. 10.
    D. S. Filippychev, “Numerical solution of the differential equation for a boundary function,” Vestnik MGU, Ser. 15, Vychisl. Matem. Kibern., No. 1, 10–14 (2006).Google Scholar
  11. 11.
    D. S. Filippychev, “Simulation of the plasma-sheath equation on a condensing grid,” Prikl. Matem. Informatika, No. 13, 35–54 (2003).Google Scholar
  12. 12.
    D. S. Filippychev, “Simulation of the plasma-sheath equation,” Vestnik MGU, Ser. 15, Vychisl. Matem. Kibernetika, No. 4, 32–39 (2004).Google Scholar
  13. 13.
    D. S. Filippychev, “Boundary-function equation and its numerical solution,” Prikl. Matem. Informatika, No. 24, 24–34 (2007).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.Faculty of Computational Mathematics and CyberneticsMoscow State UniversityMoscowRussia

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