Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



On asymptotic approximations of solutions of an equation with a small parameter

Authors: A. M. Il′in and E. F. Lelikova
Translated by: B. M. Bekker
Original publication: Algebra i Analiz, tom 22 (2010), nomer 6.
Journal: St. Petersburg Math. J. 22 (2011), 927-939
MSC (2010): Primary 35J47
Published electronically: August 19, 2011
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A second order elliptic equation with a small parameter at one of the highest order derivatives is considered in a three-dimensional domain. The limiting equation is a collection of two-dimensional elliptic equations in two-dimensional domains depending on one parameter. By the method of matching of asymptotic expansions, a uniform asymptotic approximation of the solution of a boundary-value problem is constructed and justified up to an arbitrary power of a small parameter.

References [Enhancements On Off] (What's this?)

  • 1. Norman Levinson, The first boundary value problem for 𝜖Δ𝑢+𝐴(𝑥,𝑦)𝑢ₓ+𝐵(𝑥,𝑦)𝑢_{𝑦}+𝐶(𝑥,𝑦)𝑢=𝐷(𝑥,𝑦) for small 𝜖, Ann. of Math. (2) 51 (1950), 428–445. MR 0033433,
  • 2. O. A. Oleĭnik, On equations of elliptic type with a small parameter in the highest derivatives, Mat. Sbornik N.S. 31(73) (1952), 104–117 (Russian). MR 0052012
  • 3. S. L. Kamenomostskaya, The first boundary problem for equations of elliptic type with a small parameter with the highest derivatives, Izv. Akad. Nauk SSSR. Ser. Mat. 19 (1955), 345–360 (Russian). MR 0074667
  • 4. M. I. Višik and L. A. Lyusternik, Regular degeneration and boundary layer for linear differential equations with small parameter, Uspehi Mat. Nauk (N.S.) 12 (1957), no. 5(77), 3–122 (Russian). MR 0096041
  • 5. V. A. Trenogin, The development and applications of the Ljusternik-Višik asymptotic method, Uspehi Mat. Nauk 25 (1970), no. 4 (154), 123–156 (Russian). MR 0274921
  • 6. Saul Kaplun and P. A. Lagerstrom, Asymptotic expansions of Navier-Stokes solutions for small Reynolds numbers, J. Math. Mech. 6 (1957), 585–593. MR 0091693
  • 7. Milton Van Dyke, Perturbation methods in fluid mechanics, Applied Mathematics and Mechanics, Vol. 8, Academic Press, New York-London, 1964. MR 0176702
  • 8. V. M. Babich and N. Ya. Kirpichnikova, \cyr Metod pogranichnogo sloya v zadachakh difraktsii, Izdat. Leningrad. Univ., Leningrad, 1974 (Russian). MR 0481585
    V. M. Babič and N. Y. Kirpičnikova, The boundary-layer method in diffraction problems, Springer Series in Electrophysics, vol. 3, Springer-Verlag, Berlin-New York, 1979. Translated from the Russian and with an introduction by Edward F. Kuester. MR 555574
  • 9. A. M. Il′in and E. F. Lelikova, The method of matching asymptotic expansions for the equation 𝜖Δ𝑢-𝑎(𝑥,𝑦)𝑢_{𝑦}=𝑓(𝑥,𝑦) in a rectangle, Mat. Sb. (N.S.) 96(138) (1975), no. 4, 568–583, 645 (Russian). MR 0382824
  • 10. A. M. Il′in, \cyr Soglasovanie asimptoticheskikh razlozheniĭ resheniĭ kraevykh zadach, “Nauka”, Moscow, 1989 (Russian). With an English summary. MR 1007834
    A. M. Il′in, Matching of asymptotic expansions of solutions of boundary value problems, Translations of Mathematical Monographs, vol. 102, American Mathematical Society, Providence, RI, 1992. Translated from the Russian by V. Minachin [V. V. Minakhin]. MR 1182791
  • 11. E. F. Lelikova, On the asymptotic behavior of a solution of a second-order elliptic equation with small parameter at a highest derivative, Trudy Inst. Mat. Mekh. Ural. Otdel. Ross. Akad. Nauk 9 (2003), no. 1, 107-119. (Russian)
  • 12. E. F. Lelikova, On the asymptotic behavior of the solution of an equation with a small parameter, Dokl. Akad. Nauk 429 (2009), no. 4, 447–450 (Russian); English transl., Dokl. Math. 80 (2009), no. 3, 852–855. MR 2649174,
  • 13. -, On the asymptotic behavior of the solution of a second-order elliptic equation with small parameter at a highest derivative, Trudy Moskov. Mat. Obshch. 71 (2010), 162-199; English transl., Trans. Mosc. Math. Soc. 71 (2010), 141-174.
  • 14. O. A. Ladyženskaja and N. N. Ural′ceva, \cyr Lineĭnye i kvazilineĭnye uravneniya èllipticheskogo tipa, Izdat. “Nauka”, Moscow, 1964 (Russian). MR 0211073
    Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968. MR 0244627
  • 15. Lipman Bers, Fritz John, and Martin Schechter, Partial differential equations, American Mathematical Society, Providence, R.I., 1979. With supplements by Lars Gȧrding and A. N. Milgram; With a preface by A. S. Householder; Reprint of the 1964 original; Lectures in Applied Mathematics, 3A. MR 598466

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 35J47

Retrieve articles in all journals with MSC (2010): 35J47

Additional Information

A. M. Il′in
Affiliation: Chelyabinsk State University, Ul. Brat′yev Kashirinyh 129, Chelyabinsk 454001, Russia

E. F. Lelikova
Affiliation: Institute of Mathematics and Mechanics, of Ural Branch of Russian Academy of Science, Ul. Sof′i Kovalevskoi 16, Ekaterinburg 620990, Russia

Keywords: Asymptotic, boundary value problem, small parameter, matching of asymptotic expansions
Received by editor(s): June 11, 2010
Published electronically: August 19, 2011
Additional Notes: Supported by RFBR (grants nos. 08-01-00260, 09-01-00530), by a grant NSh-6249.2010.1, and by a grant FTsP 02.740.110612
Dedicated: Dedicated to our teacher Vasiliĭ Mikhaĭlovich Babich
Article copyright: © Copyright 2011 American Mathematical Society