Skip to main content
SpringerLink
Log in

Application of the concordance method of asymptotic expansions to solving boundary-value problems

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

The concordance method of asymptotic expansions applied for constructing uniform asymptotic expansions of singularly-perturbed partial differential equations and systems is presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. K. I. Babenko (1975) Perturbation theory of stationary flows of a viscous incompressible fluid for small Reynolds numbers Preprint Institute of Applied Mathematics Moscow

    Google Scholar 

  2. B. M. Babich (1973) ArticleTitleOn strict definition of short-wave approximation in the three-dimensional case Zap. Nauchn. Seminarov LOMI 34 23–51

    Google Scholar 

  3. B. M. Babich V. S. Buldyrev (1972) Asymptotic Methods in Problems of Short Wave Diffraction Nauka Moscow

    Google Scholar 

  4. B. M. Babich S. A. Egorov (1973) Solution of the caustic problem by using the methodology of local expansions Problems of Dynamic Theory of Seismic Wave Propagation Nauka Leningrad 4–14

    Google Scholar 

  5. B. M. Babich N. Y. Kirpichnikova (1974) Boundary-Layer Method in Diffraction Problems LGU Leningrad

    Google Scholar 

  6. N. S. Bakhvalov G. P. Panasenko (1984) Average of Processes in Periodic Media. Mathemetical Problem of Composite Materials Nauka Moscow

    Google Scholar 

  7. N. N. Bogolyubov Y. A. Mitropol’skii (1974) Asymptotic Methods in the Theory of Nonlinear Oscillations Nauka Moscow Occurrence Handle0303.34043

    MATH  Google Scholar 

  8. V. F. Butuzov (1973) ArticleTitleAsymptotics of a solution of the equation µ2Δu-k2(x,y)u = f(x, y) in the rectangular domain Differents. Uravn. 9 IssueID9 1654–1660

    Google Scholar 

  9. V. F. Butuzov (1975) ArticleTitleOn the asymptotics of a solution of singularly-perturbed equations of elliptic type in a rectangular domain Differents. Uravn. 11 IssueID6 1030–1041

    Google Scholar 

  10. V. F. Butuzov (1977) ArticleTitleOn construction of boundary-layer functions in certain singularly-perturbed problems of elliptic type Differents. Uravn. 13 IssueID10 1829–1835 Occurrence Handle0374.35006

    MATH  Google Scholar 

  11. V. F. Butuzov (1977) ArticleTitleCorner boundary layer in mixed singularly-perturbed problems for hyperbolic equations Mat. Sb. 104 IssueID3 460–485

    Google Scholar 

  12. V. F. Butuzov A. V. Nesterov (1986) ArticleTitleOn certain singularly-perturbed problems of hyperbolic type with transition layers Differents. Uravn. 22 IssueID10 1739–1744

    Google Scholar 

  13. V. F. Butuzov A. B. Vasil’eva (1988) ArticleTitleOn asymptotic theory of contrast spatial structures Zh. Vychisl. Mat. Mat. Fiz. 28 IssueID3 346–361

    Google Scholar 

  14. M. Dyke ParticleVan (1967) Perturbation Methods in Fluid Mechanics Mir Moscow Occurrence Handle0158.43905

    MATH  Google Scholar 

  15. A. B. Vasil’eva (1976) ArticleTitleOn the development of the theory of ordinary differential equations with small parameter by higher derivatives during 1966-1976 Usp. Mat. Nauk 31 IssueID6 102–122

    Google Scholar 

  16. A. B. Vasil’eva V. M. Volosov (1967) ArticleTitleOn works of A. N. Tikhonov and his students on ordinary differential equations containing a small parameter Usp. Mat. Nauk 22 IssueID2 149–167

    Google Scholar 

  17. A. D. Vasil’eva V. F. Butuzov (1973) Asymptotic Expansions of Solutions of Singularly-Perturbed Equations Nauka Moscow Occurrence Handle0364.34028

    MATH  Google Scholar 

  18. A. B. Vasil’eva V. F. Butuzov (1978) Singularly-Perturbed Equations in Critical Cases MGU Moscow

    Google Scholar 

  19. A. B. Vasil’eva V. F. Butuzov (1990) Asymptotic Methods in the Theory of Singular Perturbations Vysshaya Shkola Moscow Occurrence Handle0747.34033

    MATH  Google Scholar 

  20. A. B. Vasil’eva V. F. Butuzov N. N. Nefedov (1999) ArticleTitleContrast structures in singularly-perturbed problems Fund. Prikl. Mat. 4 IssueID3 799–851 Occurrence Handle0963.34043

    MATH  Google Scholar 

  21. A. B. Vasil’eva M. G. Dmitriev (1982) Singular perturbations in optimal control problems Progress in Science and Technology,Series on Contemporary Problems in Mathematics All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences Moscow 3–77

    Google Scholar 

  22. M. I. Vishik L. A. Lyusternik (1957) ArticleTitleRegular degeneration and the boundary layer for linear differential equations with small parameter Usp. Mat. Nauk 12 IssueID5 3–122 Occurrence Handle0087.29602

    MATH  Google Scholar 

  23. M. I. Vishik L. A. Lyusternik (1960) ArticleTitleSolution of certain problems on perturbations in the case of matrices and self-adjoint and non-self-adjoint differential equations Usp. Mat. Nauk 15 IssueID3 3–78

    Google Scholar 

  24. M. I. Vishik L. A. Lyusternik (1960) ArticleTitleAsymptotic behavior of solutions of linear differential equations with large or rapidly varying coefficients and boundary conditions Usp. Mat. Nauk 15 IssueID4 3–95

    Google Scholar 

  25. R. R. Gadyl’shin (1986) ArticleTitleAsymptotics of an eigenvalue of a singularly-perturbed elliptic problem with small parameter in the boundary condition Differents. Uravn. 22 IssueID4 640–652

    Google Scholar 

  26. R. R. Gadyl’shin (1992) ArticleTitleJoining method of asymptotic expansions in the problem on the acoustic Helmholtz resonator Prikl. Mat. Mekh. 56 IssueID3 412–418

    Google Scholar 

  27. A. R. Danilin (1998) ArticleTitleAsymptotics of bounded controls for a singularly-perturbed elliptic problem in a domain with a small hole Mat. Sb. 189 IssueID11 27–60

    Google Scholar 

  28. A. R. Danilin, “Asymptotics of controls for a singularly-perturbed problem in a rectangle,” Deposited at VINITI (1999).

  29. A. R. Danilin (1999) ArticleTitleAsymptotics of controls for a singularly-perturbed elliptic problem Dokl. Ross. Akad. Nauk 369 IssueID3 305–308 Occurrence Handle1080.49502

    MATH  Google Scholar 

  30. Differential Equations with a Small Parameter [in Russian], UNTs, Akad. Nauk SSSR, Sverdlovsk (1980).

  31. M. G. Dmitriev, “Boundary layer in optimal control problems,” Izv. Akad. Nauk SSSR,Tekh. Kibern., No. 4, 63–69 (1983).

  32. M. G. Dmitriev (1985) ArticleTitleTheory of singular perturbations and certain optimal control problems Differents. Uravn. 21 IssueID10 1693–1698

    Google Scholar 

  33. V. V. Zhikov S. M. Kozlov O. A. Oleinik K. T. Ngoan (1979) ArticleTitleAveraging and G-convergence of differential operators Usp. Mat. Nauk 34 IssueID5 65–133 Occurrence Handle0445.35096

    MATH  Google Scholar 

  34. S. V. Zakharov A. M. Il’in (2001) ArticleTitleFrom a weak discontinuity to the gradient catastrophe Mat. Sb. 192 IssueID10 3–18 Occurrence Handle1028.35015

    MATH  Google Scholar 

  35. A. M. Il’in (1976) ArticleTitleBoundary-value problem for a second-order elliptic operator in a domain with a thin aperture. I. Two-dimensional case Mat. Sb. 99 IssueID4 514–537

    Google Scholar 

  36. A. M. Il’in (1977) ArticleTitleBoundary-value problem for a second-order elliptic equation in a domain with a thin aperture. II. Domain with a small hole Mat. Sb. 103 IssueID3 265–284

    Google Scholar 

  37. A. M. Il’in (1979) On asymptotics of a solution of the boundary-value problem on the half-line for a certain parabolic equation of second order Application of the Concordance Method of Asymptotic Expansions to Boundary-Value Problems for Differential Equations UNTs,Akad. Nauk SSSR Sverdlovsk 81–92

    Google Scholar 

  38. A. M. Il’in, “Study of the asymptotics of a solution of an elliptic boundary-value problem in a domain with a small hole,” Trudy Seminara im. I. G. Petrovskogo, No. 6, 57–82 (1981).

  39. A. M. Il’in (1985) ArticleTitleCauchy problem for a certain quasilinear equation with a small parameter Dokl. Akad. Nauk SSSR 283 IssueID3 530–534

    Google Scholar 

  40. A. M. Il’in (1988) Boundary layer Progress in Science and Technology, Series on Contemporary Problems in Mathematics, Fundamental Directions All-Union Institute for Scientific and Technical Information, USSR Academy of Sciences Moscow 175–214

    Google Scholar 

  41. A. M. Il’in (1989) Concordance of Asymptotic Expansions of Solutions of Boundary-Value Problems Nauka Moscow

    Google Scholar 

  42. A. M. Il’in Y. P. Gor’kov E. F. Lelikova (1975) ArticleTitleAsymptotics of a solution of elliptic equations with small parameter by higher derivatives in a neighborhood of a singular characteristic of the limit equation Trudy Seminara im. I. G. Petrovskogo 1 75–133 Occurrence Handle0332.35026

    MATH  Google Scholar 

  43. A. M. Il’in E. F. Lelikova (1975) ArticleTitleJoining method of asymptotic expansions for the equations εΔu − a(x,y)uy = f(x,y) in a rectangle Mat. Sb. 96 IssueID4 568–583 Occurrence Handle0313.35004

    MATH  Google Scholar 

  44. A. M. Il’in E. F. Lelikova (1982) ArticleTitleAsymptotics of solutions of certain elliptic equations in unbounded domains Mat. Sb. 119 IssueID3 307–324 Occurrence Handle0512.35016

    MATH  Google Scholar 

  45. A. M. Il’in T. N. Nesterova (1978) ArticleTitleAsymptotics of a solution of the Cauchy problem for a certain quasilinear equation with a small parameter Dokl. Akad. Nauk SSSR 240 IssueID1 11–13

    Google Scholar 

  46. A. M. Il’in B. I. Suleimanov (1983) ArticleTitleAsymptotics of the Green function for an elliptic equation of the second order near the boundary of the domain Izv. Akad. Nauk SSSR, Ser. Mat. 47 IssueID6 149–165

    Google Scholar 

  47. A. I. Kalinin (1988) ArticleTitleAlgorithm for asymptotic solution of the quasilinear time-optimal problem Dokl. Akad. Nauk BSSR 32 IssueID3 197–200 Occurrence Handle0850.93223

    MATH  Google Scholar 

  48. A. I. Kalinin (1990) ArticleTitleMethod for asymptotic solution of a singularly-perturbed linear terminal control problem Zh. Vychisl. Mat. Mat. Fiz. 30 IssueID3 366–378 Occurrence Handle0703.49009

    MATH  Google Scholar 

  49. A. I. Kalinin G. A. Romanyuk (1984) Optimization of linear singularly-perturbed systems Constructive Theory of Extremal Problems Universitetskoe Minsk 100–113

    Google Scholar 

  50. L. A. Kalyakin (1979) ArticleTitleConstruction of the asymptotics of a solution of a certain MHD problem with a small parameter. I. Rectilinear flow in a rectangular channel. Superconductive wall orthogonal to the magnetic field Differents. Uravn. 15 IssueID4 668–680

    Google Scholar 

  51. L. A. Kalyakin (1979) ArticleTitleConstruction of the asymptotic of a solution of a certain MHD problem with a small parameter. II. Rectilinear flow in a channel with a rectilinear jut Differents. Uravn. 15 IssueID10 1873–1887

    Google Scholar 

  52. L. A. Kalyakin (1980) Method of joined asymptotic expansions in certain linear MHD problems with singular perturbation Equations with a Small Parameter UNTs Akad. Nauk SSSR Sverdlovsk 16–43

    Google Scholar 

  53. L. A. Kalyakin (1982) ArticleTitleAsymptotics of a solution of a system of two linear MHD equations with a singular perturbation. I. Standard problem in an elliptic layer Differents. Uravn. 18 IssueID10 1724–1738 Occurrence Handle0516.76029

    MATH  Google Scholar 

  54. L. A. Kalyakin (1984) Asymptotic disintegration into simple waves of a solution of a perturbed hyperbolic system of equations Differential Equations with a Small Parameter UNTs Akad. Nauk SSSR Sverdlovsk 36–49

    Google Scholar 

  55. L. A. Kalyakin (1984) ArticleTitleLong-wave asymptotics of a solution of a hyperbolic system of equations Mat. Sb. 124 IssueID5 96–120 Occurrence Handle0566.35066

    MATH  Google Scholar 

  56. L. A. Kalyakin (1985) ArticleTitleLong-wave asymptotics of a solution of the Cauchy problem for a system of equations with a nonlinear perturbation Dokl. Akad. Nauk SSSR 283 IssueID1 18–22

    Google Scholar 

  57. L. A. Kalyakin (1986) ArticleTitleAsymptotic integration of a perturbed hyperbolic system of equations in the class of conditionally periodic functions Trudy Mosk. Mat. Obshch. 49 56–70

    Google Scholar 

  58. V. E. Kapustyan (1992) ArticleTitleAsymptotic bounded control in optimal elliptic problems Avtomatika 3 59–66 Occurrence Handle0789.49015

    MATH  Google Scholar 

  59. V. E. Kapustyan (1992) ArticleTitleAsymptotics of bounded controls in optimal elliptic problems Dokl. Akad. Nauk Ukrainy 2 70–74

    Google Scholar 

  60. V. E. Kapustyan (1992) ArticleTitleAsymptotics of bounded controls in optimal bilinear elliptic problems Dokl. Akad. Nauk Ukrainy 9 35–39

    Google Scholar 

  61. V. E. Kapustyan (1993) ArticleTitleOptimal bisingular elliptic problems with a bounded control Dokl. Akad. Nauk Ukrainy 6 81–85

    Google Scholar 

  62. V. E. Kapustyan (1993) ArticleTitleAsymptotics of controls in optimal singularly-perturbed parabolic problems Dokl. Ross. Akad. Nauk 333 IssueID4 428–431

    Google Scholar 

  63. J. Cole (1972) Perturbation Methods in Applied Mathematics Mir Moscow

    Google Scholar 

  64. N. N. Lebedev (1963) Special Functions and Their Applications Fizmatgiz Moscow-Leningrad

    Google Scholar 

  65. E. F. Lelikova (1976) ArticleTitleOn asymptotics of a solution of an elliptic equation of the second order with a small parameter by higher derivatives Differents. Uravn. 12 IssueID10 1852–1865 Occurrence Handle0338.35006

    MATH  Google Scholar 

  66. E. F. Lelikova (1978) ArticleTitleJoining method of asymptotic expansions for the equations εΔu − au z = f in a parallelepiped Differents. Uravn. 14 IssueID9 1638–1648

    Google Scholar 

  67. J.-L. Lions (1972) Optimal Control of Systems Governed by Partial Differential Equations Mir Moscow

    Google Scholar 

  68. J.-L. Lions (1987) Control of Singular Distributed Systems Mir Moscow

    Google Scholar 

  69. V. G. Maz’ya O. A. Nazarov B. A. Plamenevskii (1981) Asymptotics of Solutions of Elliptic Boundary-Value Problems under Singular Perturbation of a Domain Tbilisskii Universitet Tbilisi Occurrence Handle0462.35001

    MATH  Google Scholar 

  70. V. G. Maz’ya O. A. Nazarov B. A. Plamenevskii (1981) ArticleTitleAsymptotics of solutions of the Dirichlet problem in a domain with a thin tube being cut off Mat. Sb. 116 IssueID2 187–217

    Google Scholar 

  71. V. G. Maz’ya O. A. Nazarov B. A. Plamenevskii (1982) ArticleTitleBoundary-value problem in domains with thin bridges Funkts. Anal. Pril. 175 IssueID2 20–29

    Google Scholar 

  72. V. G. Maz’ya O. A. Nazarov B. A. Plamenevskii (1984) ArticleTitleDirichlet problem in domains with thin bridges Sib. Mat. Zh. Ellipticheskikh Kraevykh Zadach pri Singulyarnom Vozmushchenii Oblasti 25 IssueID2 161–179

    Google Scholar 

  73. V. G. Maz’ya O. A. Nazarov B. A. Plamenevskii (1984) ArticleTitleAsymptotic expansions of eigenvalues of boundary-value problems for the Laplace operator in a domain with small holes Izv. Akad. Nauk SSSR, Ser. Mat. 48 IssueID2 347–371

    Google Scholar 

  74. V. P. Maslov (1965) Perturbation Theory and Asymptotic Methods MGU Moscow

    Google Scholar 

  75. V. P. Maslov (1977) Complex WKB Method in Nonlinear Equations Nauka Moscow Occurrence Handle0449.58001

    MATH  Google Scholar 

  76. V. P. Maslov (1987) Asymptotic Methods for Solution of Pseudodifferential Equations Nauka Moscow

    Google Scholar 

  77. V. P. Maslov (1988) Asymptotic Methods and Perturbation Theory Nauka Moscow Occurrence Handle0653.35002

    MATH  Google Scholar 

  78. V. P. Maslov M. V. Fedoryuk (1976) Quasiclassical Approximation for Equations of Quantum Mechanics Nauka Moscow Occurrence Handle0449.58002

    MATH  Google Scholar 

  79. S. A. Nazarov (1981) ArticleTitleVishik-Lyusternik method for elliptic boundary-value problems in domains with conical points. I. Problem in a cone Sib. Mat. Zh. 22 IssueID4 142–163 Occurrence Handle0479.35032

    MATH  Google Scholar 

  80. S. A. Nazarov (1984) ArticleTitleVishik-Lyusternik method in domains with conical points. II. Problem in a bounded domain Sib. Mat. Zh. 25 IssueID5 132–152

    Google Scholar 

  81. S. A. Nazarov (1990) ArticleTitleAsymptotic solution of variational inequalities for a linear operator with a small parameter by higher derivatives Izv. Akad. Nauk SSSR, Ser. Mat. 54 IssueID4 754–773 Occurrence Handle0704.49016

    MATH  Google Scholar 

  82. S. A. Nazarov (1991) ArticleTitleAsymptotic of a solution of the Dirichlet problem for an equation with rapidly oscillating coefficients in a rectangle Mat. Sb. 182 IssueID5 672–722

    Google Scholar 

  83. S. A. Nazarov (1995) ArticleTitleAsymptotic solutions of a problem with small obstacles Differents. Uravn. 31 IssueID6 1031–1041

    Google Scholar 

  84. S. A. Nazarov B. A. Plamenevskii (1991) Elliptic Problems in Domains with Piecewise-Smooth Boundary Nauka Moscow

    Google Scholar 

  85. A. H. Nayfeh (1976) Perturbation Methods Mir Moscow Occurrence Handle0379.76059

    MATH  Google Scholar 

  86. T. N. Nesterova (1980) On asymptotics of a solution of the Burgers equation in a neighborhood of joining two discontinuity lines Differential Equations with a Small Parameter UNTs Akad. Nauk SSSR Sverdlovsk 66–86

    Google Scholar 

  87. T. N. Nesterova (1983) ArticleTitleJoining method of asymptotic expansions for a solution of a hyperbolic equation with a small parameter Mat. Sb. 120 IssueID4 546–555

    Google Scholar 

  88. V. Y. Novokshenov (1976) ArticleTitleAsymptotic of a solution of a singular integral equation with a small parameter Mat. Sb. 100 IssueID3 455–475 Occurrence Handle0343.45002

    MATH  Google Scholar 

  89. V. Y. Novokshenov (1978) ArticleTitleSingular integral equation with a small parameter on a nite closed interval Mat. Sb. 105 IssueID4 543–573

    Google Scholar 

  90. V. Y. Novokshenov (1980) Asymptotic with respect to a small parameter of a solution of an elliptic pseudodifferential equation in a half-space Differential Equations with a Small Parameter UNTs Akad. Nauk SSSR Sverdlovsk 87–110

    Google Scholar 

  91. I. Proudman J. Pearson (1958) ArticleTitleExpansions at small Reynolds numbers for the flow past a sphere and a circular cylinder Mekhanika 2 IssueID48 3–28

    Google Scholar 

  92. Application of the Concordance Method of Asymptotic Expansion [in Russian], UNTs Akad. Nauk SSSR, Sverdlovsk (1979).

  93. M. I. Rabinovich A. A. Rozenblyum (1972) ArticleTitleOn asymptotic methods of solution of nonlinear partial differential equations Prikl. Mat. Mekh. 36 330–343

    Google Scholar 

  94. M. D. Ramazanov (1978) Problem of flow around a thin wing with an acute trailing edge of nonviscous incompressible fluid Mathematical Analysis and Related Mathematical Problems Nauka Novosibirsk 224–236

    Google Scholar 

  95. Y. S. Soibel’man (1984) ArticleTitleAsymptotics of the capacity of a condenser with plates of an arbitrary form Sib. Mat. Zh. 25 IssueID6 167–181

    Google Scholar 

  96. V. G. Sushko (1985) ArticleTitleOn asymptotic expansions of solutions of a certain parabolic equation with a small parameter Differents. Uravn. 21 IssueID10 1794–1798

    Google Scholar 

  97. A. N. Tikhonov (1948) ArticleTitleOn dependence of solutions of differential equations on a small parameter Mat. Sb. 22 IssueID2 193–204

    Google Scholar 

  98. A. N. Tikhonov (1950) ArticleTitleOn systems of differential equations containing parameters Mat. Sb. 27 IssueID1 147–156 Occurrence Handle0041.42316

    MATH  Google Scholar 

  99. V. A. Trenogin (1970) ArticleTitleDevelopment and application of the Vishik-Lyusternik asymptotic method Usp. Mat. Nauk 25 IssueID4 123–156

    Google Scholar 

  100. M. V. Fedoryuk, “Dirichlet problem for the Laplace operator on the exterior of a thin body of revolution,” Trudy Seminara S. L. Soboleva, No. 1, 113–131 (1980).

  101. M. V. Fedoryuk (1981) ArticleTitleAsymptotic of a solution of the Dirichlet problem for the Laplace and Helmholtz equations in the exterior of a thin cylinder Izv. Akad. Nauk SSSR, Ser. Mat. 45 IssueID1 167–186 Occurrence Handle0477.35034

    MATH  Google Scholar 

  102. M. V. Fedoryuk (1985) ArticleTitleAsymptotics of a solution of the scattering problem on a cylinder with a large perturbation Trudy Mosk. Mat. Obshch. 48 150–162 Occurrence Handle0603.35020

    MATH  Google Scholar 

  103. M. V. Fedoryuk (1983) Asymptotic Methods for Linear Differential Equations Nauka Moscow Occurrence Handle0538.34001

    MATH  Google Scholar 

  104. J. Heading (1965) An Introduction to Phase-Integral Methods Mir Moscow Occurrence Handle0131.08403

    MATH  Google Scholar 

  105. Y. Z. Shaigardanov (1979) ArticleTitleOn asymptotics of a solution of a boundary-value problem for a certain parabolic equation of the fourth order Differents. Uravn. 15 IssueID4 668–680

    Google Scholar 

  106. Y. Z. Shaigardanov (1985) ArticleTitleAsymptotic in parameter of a solution of an elliptic equation of higher order in a neighborhood of the discontinuity line of the limit equation Differents. Uravn. 21 IssueID4 706–715

    Google Scholar 

  107. A. L. Shtaras (1977) ArticleTitleAsymptotic integration of weakly nonlinear partial differential equations Dokl. Akad. Nauk SSSR 237 IssueID3 525–528

    Google Scholar 

  108. A. M. Il’in S. V. Zakharov (2001) ArticleTitleOn the influence of small dissipation on the evolution of weak discontinuities Functional Differential Equations 3 257–271 Occurrence Handle1049.35029

    MATH  Google Scholar 

  109. S. Kaplun (1954) ArticleTitleThe role of coordinate systems in boundary-layer theory Z. Angew. Math. Phys. 5 111–135 Occurrence Handle0055.19004 Occurrence Handle10.1007/BF01600771

    Article  MATH  Google Scholar 

  110. S. Kaplun P. A. Lagerstorm (1957) ArticleTitleAsymptotic expansions of Navier-Stokes solutions for a small Reynolds numbers J. Math. Mech. 6 585–593 Occurrence Handle0080.18501

    MATH  Google Scholar 

  111. P. A. Lagerstorm J. Cole (1955) ArticleTitleExamples illustrating expansion procedures for the Navier-Stokes equations J. Ration. Mech. Anal. 4 817–882 Occurrence Handle0066.19505

    MATH  Google Scholar 

  112. J.-L. Lions (1978) ArticleTitleAsymptotic methods in the optimal control of distributed systems Automatica 14 199–211 Occurrence Handle0377.93035 Occurrence Handle10.1016/0005-1098(78)90085-7

    Article  MATH  Google Scholar 

  113. L. Prandtl, “Über Flussigkeitsbewegung bei sehr kleiner Reibung,” In: Verhandlungen des dritten internationalen Mathematiker Kongresses, Heidelberg-Leipzig (1905), pp. 484–491.

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematic Modelling, Ekaterinburg, Russia

    A. M. Il’in, A. R. Danilin & S. V. Zakharov

Authors
  1. A. M. Il’in
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. R. Danilin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. V. Zakharov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 5, Asymptotic Methods, 2003.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Il’in, A.M., Danilin, A.R. & Zakharov, S.V. Application of the concordance method of asymptotic expansions to solving boundary-value problems. J Math Sci 125, 610–657 (2005). https://doi.org/10.1007/PL00021946

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/PL00021946

Keywords

Search

Navigation