General information
Latest issue
Impact factor
License agreement
Submit a manuscript

Search papers
Search references

Latest issue
Current issues
Archive issues
What is RSS

Uspekhi Mat. Nauk:

Personal entry:
Save password
Forgotten password?

Uspekhi Mat. Nauk, 1963, Volume 18, Issue 3(111), Pages 15–86 (Mi umn6350)  

This article is cited in 87 scientific papers (total in 88 papers)

Asymptotic behaviour of solutions to certain problems involving non-linear differential equations containing a small parameter multiplying the highest derivatives

A. B. Vasil'eva

Full text: PDF file (7409 kB)

English version:
Russian Mathematical Surveys, 1963, 18:3, 13–84

Bibliographic databases:

MSC: 34E05, 35B40, 34A34, 35F25, 35B10, 35L20
Received: 06.07.1962

Citation: A. B. Vasil'eva, “Asymptotic behaviour of solutions to certain problems involving non-linear differential equations containing a small parameter multiplying the highest derivatives”, Uspekhi Mat. Nauk, 18:3(111) (1963), 15–86; Russian Math. Surveys, 18:3 (1963), 13–84

Citation in format AMSBIB
\by A.~B.~Vasil'eva
\paper Asymptotic behaviour of solutions to certain problems involving non-linear differential equations containing a~small parameter multiplying the highest derivatives
\jour Uspekhi Mat. Nauk
\yr 1963
\vol 18
\issue 3(111)
\pages 15--86
\jour Russian Math. Surveys
\yr 1963
\vol 18
\issue 3
\pages 13--84

Linking options:

    SHARE: FaceBook Twitter Livejournal

    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Katsuhiko Sekine, “Initial value problem for elementary and composite particles”, Nuclear Physics, 76:3 (1966), 513  crossref
    2. A. B. Vasil'eva, V. M. Volosov, “The work of Tikhonov and his pupils on ordinary differential equations containing a small parameter”, Russian Math. Surveys, 22:2 (1967), 124–142  mathnet  crossref  mathscinet  zmath
    3. F.G. Heineken, H.M. Tsuchiya, R. Aris, “On the mathematical status of the pseudo-steady state hypothesis of biochemical kinetics”, Mathematical Biosciences, 1:1 (1967), 95  crossref
    4. D. B. MacMillan, “Asymptotic Methods for Systems of Differential Equations in which some Variables have very Short Response Times”, SIAM J Appl Math, 16:4 (1968), 704  crossref  mathscinet
    5. S. A. Lomov, “The construction of asymptotic solutions of certain problems involving parameters”, Math. USSR-Izv., 2:4 (1968), 849–877  mathnet  crossref  mathscinet  zmath
    6. Rutherford Aris, “A note on mechanism and memory in the kinetics of biochemical reactions”, Mathematical Biosciences, 3 (1968), 421  crossref
    7. Robert E O'Malley, “Topics in singular perturbations”, Advances in Mathematics, 2:4 (1968), 365  crossref
    8. R. E. O’Malley, Jr, “On a Boundary Value Problem for a Nonlinear Differential Equation with a Small Parameter”, SIAM J Appl Math, 17:3 (1969), 569  crossref  mathscinet  zmath
    9. K. Kuen Tam, “A Note on the Asymptotic Solution of the Flow Between Two Oppositely Rotating Infinite Plane Disks”, SIAM J Appl Math, 17:6 (1969), 1305  crossref  zmath
    10. P.V Kokotović, J.B Cruz, “An approximation theorem for linear optimal regulators”, Journal of Mathematical Analysis and Applications, 27:2 (1969), 249  crossref
    11. Ralph Nossal, Barry Ninham, “Kinetic equations for surface adsorption”, Biopolymers (Biospectroscopy), 9:1 (1970), 103  crossref
    12. V. A. Trenogin, “The development and applications of the asymptotic method of Lyusternik and Vishik”, Russian Math. Surveys, 25:4 (1970), 119–156  mathnet  crossref  mathscinet  zmath
    13. R.E O'Malley, “Singular perturbations of a boundary value problem for a system of nonlinear differential equations”, Journal of Differential Equations, 8:3 (1970), 431  crossref
    14. W.R. Perkins, P.V. Kokotović, “Deterministic parameter estimation for near-optimum feedback control”, Automatica, 7:4 (1971), 439  crossref
    15. Y.K. Kao, S.G. Bankoff, “Perturbation analysis of recycle reactors”, Chemical Engineering Science, 26:2 (1971), 189  crossref
    16. S. A. Lomov, “The method of perturbations for singular problems”, Math. USSR-Izv., 6:3 (1972), 631–648  mathnet  crossref  mathscinet  zmath
    17. W.H. Ray, M. Marek, S. Elnashaie, “The effect of heat and mass dispersion on tubular reactor performance”, Chemical Engineering Science, 27:8 (1972), 1527  crossref
    18. D.R. Schneider, N.R. Amundson, R. Aris, “On a mechanism for autocatalysis”, Chemical Engineering Science, 27:5 (1972), 895  crossref
    19. Linn Zien, “An Upper Bound for the Singular Parameter in a Stable, Singularly Perturbed System”, Journal of the Franklin Institute, 295:5 (1973), 373  crossref
    20. Y. K. KAO, S. G. BANKOFF, “OPTIMAL DESIGN OF A CHEMICAL REACTOR WITH FAST AND SLOW REACTIONS”, Chemical Engineering Communications, 1:3 (1974), 141  crossref
    21. R.E O'Malley, “Boundary layer methods for certain nonlinear singularly perturbed optimal control problems”, Journal of Mathematical Analysis and Applications, 45:2 (1974), 468  crossref
    22. Koichi Asatani, “Suboptimal control of fixed-end-point minimum energy problem via singular perturbation theory”, Journal of Mathematical Analysis and Applications, 45:3 (1974), 684  crossref
    23. Ferdinand Verhulst, “Asymptotic expansions in the perturbed two-body problem with application to systems with variable mass”, Celestial Mech, 11:1 (1975), 95  crossref  mathscinet  zmath
    24. Peddapullaiah Sannuti, “Asymptotic Expansions of Singularly Perturbed Quasi-Linear Optimal Systems”, SIAM J Control, 13:3 (1975), 572  crossref  mathscinet  zmath
    25. H.G. Othmer, “On the temporal characteristics of a model for the Zhabotinskii-Belousov reaction”, Mathematical Biosciences, 24:3-4 (1975), 205  crossref
    26. M Balachandra, “Some new basic results for singularly perturbed ordinary differential equations”, Journal of Mathematical Analysis and Applications, 49:2 (1975), 302  crossref
    27. F. A. Howes, “Effective Characterization of the Asymptotic Behavior of Solutions of Singularly Perturbed Boundary Value Problems”, SIAM J Appl Math, 30:2 (1976), 296  crossref  mathscinet  zmath
    28. P.V. Kokotovic, R.E. O'Malley, P. Sannuti, “Singular perturbations and order reduction in control theory — An overview”, Automatica, 12:2 (1976), 123  crossref
    29. Duc-Long Nguyen, K. J. Parbhakar, “Adiabatic approximation used in nonlinear theory of backscattering”, Phys Fluids, 20:5 (1977), 864  crossref
    30. B. Mitchell, “Taylor series methods for the solution of the point reactor kinetics equations”, Annals of Nuclear Energy, 4:4-5 (1977), 169  crossref
    31. F. A. Howes, “The Asymptotic Solution of a Class of Singularly Perturbed Nonlinear Boundary Value Problems via Differential Inequalities”, SIAM J Math Anal, 9:2 (1978), 215  crossref  mathscinet  zmath
    32. F. A. Howes, “Singularly Perturbed Nonlinear Boundary Value Problems with Turning Points. II”, SIAM J Math Anal, 9:2 (1978), 250  crossref  mathscinet  zmath
    33. F.A. Howes, “An asymptotic theory for a class of nonlinear Robin problems”, Journal of Differential Equations, 30:2 (1978), 192  crossref
    34. D. S. NAIDU, P. K. RAJAGOPALAN, “Application of Vasileva's singular perturbation method to a problem in ecology”, International Journal of Systems Science, 10:7 (1979), 761  crossref
    35. F.A. Howes, “Singularly perturbed superquadratic boundary value problems”, Nonlinear Analysis: Theory, Methods & Applications, 3:2 (1979), 175  crossref
    36. P. De Mottoni, A. Tesei, “A Comparative Analysis of Some Semi-Linear Parabolic Systems Modelling the Dynamics of Fast Nuclear Reactors”, Z angew Math Mech, 60:11 (1980), 615  crossref  mathscinet  zmath
    37. F.A. Howes, “Some singularly perturbated superquadratic boundary value problems whose solutions exhibit boundary and shock layer behavior”, Nonlinear Analysis: Theory, Methods & Applications, 4:4 (1980), 683  crossref
    38. R. SUEBAYYAN, K. MUHAMMAD ZAKARIAH, K. R. PALANISAMY, “Optimal control of singularly perturbed linear systems : single term Walsh series approach”, International Journal of Systems Science, 13:12 (1982), 1339  crossref
    39. D.S. Naidu, S. Sen, “Singular perturbation method for the transient analysis of a transformer”, Electric Power Systems Research, 5:4 (1982), 307  crossref
    40. G.S. Ladde, D.D. Šiljak, “Multiparameter singular perturbations of linear systems with multiple time scales”, Automatica, 19:4 (1983), 385  crossref
    41. MORISHIGE KIMURA, “On the matrix Riccati equation for a singularly perturbed linear discrete control system”, International Journal of Control, 38:5 (1983), 959  crossref
    42. Wlodzimierz Klonowski, “Simplifying principles for chemical and enzyme reaction kinetics”, Biophysical Chemistry, 18:2 (1983), 73  crossref
    43. Petar V. Kokotović, “Applications of Singular Perturbation Techniques to Control Problems”, SIAM Rev, 26:4 (1984), 501  crossref  mathscinet  zmath  isi
    44. F HOWES, “Boundary layer behavior in perturbed second-order systems”, Journal of Mathematical Analysis and Applications, 104:2 (1984), 467  crossref
    45. Walter G. Kelley, “Boundary and interior layers for singularly pertubed boundary value problems”, Nonlinear Analysis: Theory, Methods & Applications, 8:12 (1984), 1455  crossref
    46. Flaviano Battelli, Claudio Lazzari, “On the pseudo-steady-state approximation and Tikhonov theorem for general enzyme systems”, Mathematical Biosciences, 75:2 (1985), 229  crossref
    47. G.S. Ladde, S.G. Rajalakshmi, “Diagonalization and stability of multi-time-scale singularly perturbed linear systems”, Applied Mathematics and Computation, 16:2 (1985), 115  crossref
    48. F BATTELLI, C LAZZARI, “Dichotomies and stability in singularly perturbed systems of O.D.E”, Nonlinear Analysis, 11:2 (1987), 259  crossref
    49. Flaviano Battelli, Claudio Lazzari, “Asymptotic expansions obtained by a center manifold theorem”, Annali di Matematica, 151:1 (1988), 369  crossref  mathscinet  zmath  isi
    50. G.S. Ladde, S.G. Rajalakshmi, “Singular perturbations of linear systems with multiparameters and multiple time scales”, Journal of Mathematical Analysis and Applications, 129:2 (1988), 457  crossref
    51. PAUL M. SHARKEY, “Composite control of non-linear singularly perturbed systems: meeting a design objective”, International Journal of Control, 50:6 (1989), 2173  crossref
    52. John S Jeffries, Donald R Smith, “A green function approach for a singularly perturbed vector boundary-value problem”, Advances in Applied Mathematics, 10:1 (1989), 1  crossref
    53. Akira Ukawa, “Lattice QCD simulations beyond the quenched approximation”, Nuclear Physics B - Proceedings Supplements, 10:1 (1989), 66  crossref
    54. “On the method of matched asymptotic expansions”, Journal of Guidance, Control, and Dynamics, 12:2 (1989), 277  crossref
    55. Xiao-Biao Lin, “Heteroclinic bifurcation and singularly perturbed boundary value problems”, Journal of Differential Equations, 84:2 (1990), 319  crossref
    56. “Computational singular perturbation method for dynamic systems”, Journal of Guidance, Control, and Dynamics, 14:3 (1991), 661  crossref
    57. Ognyan Christov, “On the Kolmogorov’s condition for a special case of the Kirchhoff top”, J Math Phys (N Y ), 35:7 (1994), 3448  crossref  mathscinet  zmath
    58. A. M. Samoilenko, “N. N. Bogolyubov and non-linear mechanics”, Russian Math. Surveys, 49:5 (1994), 109–154  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    59. R. Sh. Islamov, “Analytic investigation of the glow discharge anode region in elevated-pressure electronegative gases”, J Appl Phys, 79:11 (1996), 8253  crossref  adsnasa  isi  elib
    60. Alexander Vigodner, “Limits of Singularly Perturbed Control Problems with Statistical Dynamics of Fast Motions”, SIAM J. Control Optim, 35:1 (1997), 1  crossref
    61. Simon J. Fraser, “Double perturbation series in the differential equations of enzyme kinetics”, J Chem Phys, 109:2 (1998), 411  crossref  isi
    62. Marc R. Roussel, Simon J. Fraser, “Invariant manifold methods for metabolic model reduction”, Chaos, 11:1 (2001), 196  crossref  zmath  isi
    63. “Singular Perturbations and Time Scales in Guidance and Control of Aerospace Systems: A Survey”, Journal of Guidance, Control, and Dynamics, 24:6 (2001), 1057  crossref
    64. E. Fiorani, G. Giachetta, G. Sardanashvily, “Geometric quantization of time-dependent completely integrable Hamiltonian systems”, J Math Phys (N Y ), 43:10 (2002), 5013  crossref  mathscinet  zmath  adsnasa  isi  elib
    65. Walter Kelley, “Perturbation problems with quadratic dependence on the first derivative”, Nonlinear Analysis: Theory, Methods & Applications, 51:3 (2002), 469  crossref
    66. Alessandra Celletti, Claude Froeschlé, Elena Lega, “Frequency analysis of the stability of asteroids in the framework of the restricted three-body problem”, Celest Mech, 90:3-4 (2004), 245  crossref  mathscinet  zmath
    67. Erich Bohl, Ivo Marek, “Input-output systems in biology and chemistry and a class of mathematical models describing them”, Appl Math, 50:3 (2005), 219  crossref  mathscinet  zmath
    68. John Guckenheimer, Martin Wechselberger, Lai-Sang Young, “Chaotic attractors of relaxation oscillators”, Nonlinearity, 19:3 (2006), 701  crossref  mathscinet  zmath  isi
    69. D. Subbaram Naidu, “Singular perturbations and hysteresis. Michael P. Mortell, Robert E. O'Malley, Alexei Pokrovskii and Vladimir Sobolev, Society of Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005. No. of pages: xiv+344. Price: $66.00. ISBN 0-089871-597-0”, Int J Robust Nonlinear Control, 17:12 (2007), 1155  crossref  mathscinet
    70. Ferdinand Verhulst, “Singular perturbation methods for slow–fast dynamics”, Nonlinear Dyn, 50:4 (2007), 747  crossref  mathscinet  isi
    71. Stepan P. Sosnitskii, “ON THE STABILITY OF TRIANGULAR LAGRANGIAN POINTS IN THE RESTRICTED THREE-BODY PROBLEM”, Astronomical Journal, 135:1 (2008), 187  crossref  adsnasa  isi
    72. Tulga Ersal, Hosam K. Fathy, D. Geoff Rideout, Loucas S. Louca, Jeffrey L. Stein, “A Review of Proper Modeling Techniques”, J Dyn Sys Meas Control, 130:6 (2008), 061008  crossref  isi
    73. Derkunova E.A., “Singulyarno vozmuschennoe uravnenie v chastnykh proizvodnykh pervogo poryadka v sluchae smeny ustoichivosti”, Vestn. Yuzhno-Uralskogo gos. un-ta. Ser.: Matematicheskoe modelirovanie i programmirovanie, 2009, no. 37, 57–64  elib
    74. Georgios Lukes-Gerakopoulos, Theocharis A. Apostolatos, George Contopoulos, “Observable signature of a background deviating from the Kerr metric”, Phys Rev D, 81:12 (2010), 124005  crossref  elib
    75. Daniel Cobb, “Eigenvalue Conditions for Convergence of Singularly Perturbed Matrix Exponential Functions”, SIAM J Control Optim, 48:7 (2010), 4327  crossref
    76. A. Celletti, S. Di Ruzza, C. Lhotka, L. Stefanelli, “Nearly-integrable dissipative systems and celestial mechanics”, Eur Phys J Spec Top, 186:1 (2010), 33  crossref
    77. T. Sh. Kal'menov, A. Sh. Shaldanbaev, “On a recurrence method for solving a singularly perturbed Cauchy problem for a second order equation”, Siberian Adv. Math., 21:4 (2011), 274–281  mathnet  crossref  mathscinet  elib
    78. Anshu Siddarth, John Valasek, “Kinetic State Tracking for a Class of Singularly Perturbed Systems”, J. Guidance, 34:3 (2011), 734  crossref
    79. Lena Noethen, Sebastian Walcher, “Tikhonov's theorem and quasi-steady state”, DCDS-B, 16:3 (2011), 945  crossref
    80. Constantinos I. Siettos, “Equation-Free multiscale computational analysis of individual-based epidemic dynamics on networks”, Applied Mathematics and Computation, 2011  crossref
    81. Alexandra Goeke, Christian Schilli, Sebastian Walcher, Eva Zerz, “Computing quasi-steady state reductions”, J Math Chem, 2012  crossref
    82. Alessandra Celletti, “Some Aspects of Conservative and Dissipative KAM Theorems”, Milan J. Math, 2012  crossref
    83. Dimitris A. Goussis, “The role of slow system dynamics in predicting the degeneracy of slow invariant manifolds: The case of vdP relaxation–oscillations”, Physica D: Nonlinear Phenomena, 2013  crossref
    84. M. Hilairet, M. Ghanes, O. Béthoux, V. Tanasa, J-P. Barbot, “A passivity-based controller for coordination of converters in a fuel cell system”, Control Engineering Practice, 21:8 (2013), 1097  crossref
    85. Peter Barmettler, Davide Fioretto, Vladimir Gritsev, “Non-equilibrium dynamics of Gaudin models”, EPL, 104:1 (2013), 10004  crossref
    86. D.T.. Maris, D.A.. Goussis, “The “hidden” dynamics of the Rössler attractor”, Physica D: Nonlinear Phenomena, 295-296 (2015), 66  crossref
    87. Alessandra Celletti, Giuseppe Pucacco, Danilo Stella, “Lissajous and Halo Orbits in the Restricted Three-Body Problem”, J Nonlinear Sci, 2015  crossref
    88. V. F. Butuzov, A. I. Bychkov, “Nachalno-kraevaya zadacha dlya singulyarno vozmuschennogo parabolicheskogo uravneniya v sluchayakh dvukratnogo i trekhkratnogo kornya vyrozhdennogo uravneniya”, Chebyshevskii sb., 16:4 (2015), 41–76  mathnet  elib
  • Успехи математических наук Russian Mathematical Surveys
    Number of views:
    This page:1992
    Full text:437
    First page:2

    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019