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Mat. Sb. (N.S.), 1975, Volume 96(138), Number 4, Pages 568–583 (Mi msb3409)  

This article is cited in 30 scientific papers (total in 32 papers)

A method of joining asymptotic expansions for the equation $\varepsilon\Delta u-a(x,y)u_y=f(x,y)$ in a rectangle

A. M. Il'in, E. F. Lelikova

Abstract: In the closed rectangle ($0\leqslant x\leqslant l_1$, $0\leqslant y\leqslant l_2$) as $\varepsilon\to0$ an asymptotic expansion of the solution of the Dirichlet problem for the equation indicated in the title is constructed and proved. Near the corners $(0,0)$ and $(l_1,0)$ the method of joining two different asymptotic expansions is employed.
Bibliography: 9 titles.

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English version:
Mathematics of the USSR-Sbornik, 1975, 25:4, 533–548

Bibliographic databases:

UDC: 517.9
MSC: Primary 35J25, 35B40; Secondary 41A60
Received: 12.07.1974

Citation: A. M. Il'in, E. F. Lelikova, “A method of joining asymptotic expansions for the equation $\varepsilon\Delta u-a(x,y)u_y=f(x,y)$ in a rectangle”, Mat. Sb. (N.S.), 96(138):4 (1975), 568–583; Math. USSR-Sb., 25:4 (1975), 533–548

Citation in format AMSBIB
\by A.~M.~Il'in, E.~F.~Lelikova
\paper A~method of joining asymptotic expansions for the equation $\varepsilon\Delta u-a(x,y)u_y=f(x,y)$ in a~rectangle
\jour Mat. Sb. (N.S.)
\yr 1975
\vol 96(138)
\issue 4
\pages 568--583
\jour Math. USSR-Sb.
\yr 1975
\vol 25
\issue 4
\pages 533--548

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    This publication is cited in the following articles:
    1. A. M. Il'in, “A boundary value problem for the elliptic equation of second order in a domain with a narrow slit. 1. The two-dimensional case”, Math. USSR-Sb., 28:4 (1976), 459–480  mathnet  crossref  mathscinet  zmath  isi
    2. V. F. Butuzov, “The angular boundary layer in mixed singularly perturbed problems for hyperbolic equations”, Math. USSR-Sb., 33:3 (1977), 403–425  mathnet  crossref  mathscinet  zmath  isi
    3. Lube G., Tobiska L., “Singularly Perturbed Elliptic Problems of Arbitrary Order with Change of Type”, Math. Nachr., 98 (1980), 123–134  crossref  mathscinet  zmath  isi
    4. Nazarov S., “The Superpower Boundary-Layer in the Problem of the Strained Plates Bend”, no. 1, 1980, 72–80  mathscinet  zmath  isi
    5. Slutski A., “On the Asymptotics of Solutions of Degenerative Elliptic-Equations with a Small Parameter at Higher Derivatives”, no. 3, 1981, 59–64  zmath  isi
    6. Kalyakin L., “Asymptotic-Expansion of a Solution of a System of 2 Linear Mhd Equations with a Singular Perturbation .1. a Standard Problem in an Elliptic Layer”, Differ. Equ., 18:10 (1982), 1238–1249  mathscinet  isi
    7. T. N. Nesterova, “The method of matching asymptotic expansions for the solution of a hyperbolic equation with a small parameter”, Math. USSR-Sb., 48:2 (1984), 541–550  mathnet  crossref  mathscinet  zmath
    8. Lelikova E., “The Asymptotic-Expansion of a Solution of an Elliptic Equation Containing a Small Parameter in a Domain with a Conical Point”, Differ. Equ., 19:2 (1983), 231–243  mathscinet  zmath  isi
    9. Tobiska L., “Discretization Processes for the Solution of Singularly Perturbed Boundary-Value-Problems”, Z. Angew. Math. Mech., 63:2 (1983), 115–123  crossref  mathscinet  zmath  isi
    10. L. A. Kalyakin, “Long wave asymptotics of asolution of a hyperbolic system of equations”, Math. USSR-Sb., 52:1 (1985), 91–114  mathnet  crossref  mathscinet  zmath
    11. Ilin A., “Investigation of Singularly Perturbed Boundary-Value-Problems by the Asymptotic-Expansion-Consistency Method”, Differ. Equ., 21:10 (1985), 1192–1196  mathscinet  zmath  isi
    12. Kaliakin L., “Long-Wave Asymptotics of the Cauchy-Problem Solution for a System of Equations with Nonlinear Perturbation”, 283, no. 1, 1985, 18–22  mathscinet  isi
    13. Bardos C., Cea V., Grisvard P., “Error-Estimates Related to Singular Perturbations of 1st-Order Equations and Systems”, Comput. Math. Appl., 13:9-11 (1987), 801–829  crossref  mathscinet  zmath  isi
    14. Kaliakin L., “Matching Method for the Problem of Asymptotic Decomposition of Plane-Wave Packet in a Dispersive Medium”, 301, no. 5, 1988, 1048–1052  mathscinet  isi
    15. L. A. Kalyakin, “Long wave asymptotics. Integrable equations as asymptotic limits of non-linear systems”, Russian Math. Surveys, 44:1 (1989), 3–42  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    16. V. V. Varlamov, A. V. Nesterov, “Asymptotic representation of the solution of the problem of the propagation of acoustic waves in a non-uniform compressible relaxing medium”, U.S.S.R. Comput. Math. Math. Phys., 30:3 (1990), 47–55  mathnet  crossref  mathscinet  zmath
    17. S. A. Nazarov, “Asymptotics of the solution of the Dirichlet problem for an equation with rapidly oscillating coefficients in a rectangle”, Math. USSR-Sb., 73:1 (1992), 79–110  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    18. B. I. Berezin, V. G. Sushko, “Small-parameter asymptotic expansions of the solution of a degenerate problem”, U.S.S.R. Comput. Math. Math. Phys., 31:9 (1991), 53–57  mathnet  mathscinet  zmath  isi
    19. S. A. Nazarov, “Asymptotic of the solution of a boundary value problem in a thin cylinder with nonsmooth lateral surface”, Russian Acad. Sci. Izv. Math., 42:1 (1994), 183–217  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    20. V. M. Babich, L. A. Kalyakin, M. D. Ramazanov, N. Kh. Rozov, “Arlen Mikhailovich Il'in (on the occasion of the 70th anniversary)”, Proc. Steklov Inst. Math. (Suppl.), 2003no. , suppl. 1, S1–S7  mathnet  mathscinet  zmath  elib
    21. E. F. Lelikova, “On the asymptotics of a solution of a second order elliptic equation with small parameter at a higher derivative”, Proc. Steklov Inst. Math. (Suppl.), 2003no. , suppl. 1, S129–S143  mathnet  mathscinet  zmath  elib
    22. Danilin, AR, “Asymptotic behaviour of solutions of a singular elliptic system in a rectangle”, Sbornik Mathematics, 194:1–2 (2003), 31  mathnet  crossref  mathscinet  zmath  isi  elib
    23. Kopteva N., “How Accurate Is the Streamline-Diffusion Fem Inside Characteristic (Boundary and Interior) Layers?”, Comput. Meth. Appl. Mech. Eng., 193:45-47 (2004), 4875–4889  crossref  mathscinet  zmath  isi
    24. A. M. Il'in, E. F. Lelikova, “On asymptotic approximations of solutions of an equation with a small parameter”, St. Petersburg Math. J., 22:6 (2011), 927–939  mathnet  crossref  mathscinet  zmath  isi
    25. E. F. Lelikova, “The asymptotics of the solution of an equation with a small parameter in a domain with angular points”, Sb. Math., 201:10 (2010), 1495–1510  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    26. “Arlen Mikhailovich Ilin (k vosmidesyatiletiyu so dnya rozhdeniya)”, Ufimsk. matem. zhurn., 4:2 (2012), 3–12  mathnet  mathscinet
    27. Lelikova E.F., “Asymptotic Behavior of the Solution of a Small-Parameter Equation in a Domain with a Conical Point on the Boundary”, Dokl. Math., 85:1 (2012), 33–39  crossref  mathscinet  zmath  isi
    28. Lelikova E.F., “Ob asimptotike resheniya uravneniya s malym parametrom v oblasti s konicheskoi tochkoi na granitse”, Doklady Akademii nauk, 442:2 (2012), 166–166  mathscinet  zmath  elib
    29. D. A. Tursunov, U. Z. Erkebaev, “Asimptotika resheniya bisingulyarno vozmuschennoi zadachi Dirikhle v koltse s kvadratichnym rostom na granitse”, Vestn. Yuzhno-Ur. un-ta. Ser. Matem. Mekh. Fiz., 8:2 (2016), 52–61  mathnet  crossref  elib
    30. A. R. Danilin, S. V. Zakharov, O. O. Kovrizhnykh, E. F. Lelikova, I. V. Pershin, O. Yu. Khachai, “Ekaterinburgskoe nasledie Arlena Mikhailovicha Ilina”, Tr. IMM UrO RAN, 23, no. 2, 2017, 42–66  mathnet  crossref  elib
    31. S. F. Dolbeeva, V. N. Pavlenko, S. V. Matveev, O. N. Dementev, A. V. Melnikov, E. A. Sbrodova, A. A. Solovev, V. I. Ukhobotov, V. E. Fedorov, E. A. Fominykh, A. A. Ershov, “Arlen Mikhailovich Ilin. 85 let so dnya rozhdeniya”, Chelyab. fiz.-matem. zhurn., 2:1 (2017), 5–9  mathnet  elib
    32. Danilin A.R., “Asymptotics of the Solution of a Singular Optimal Distributed Control Problem With Essential Constraints in a Convex Domain”, Differ. Equ., 56:2 (2020), 251–263  crossref  isi
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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