On the Simultaneous Use of Asymptotic and Numerical Methods to Solve Nonlinear Two Point Problems with Boundary and Interior Layers

  • Robert E. O’MalleyJr.
Part of the Progress in Scientific Computing book series (PSC, volume 5)


The purpose of this paper is to provide a broad-brush survey concerning boundary value problems for certain systems of nonlinear singularly perturbed ordinary differential equations. The aim is to emphasize important and difficult open problems needing much more study, in terms of both mathematical and numerical analysis and computational experiments. The presentation will, regrettably, be removed from both direct applications and substantial achievements. Gradually, we will, however, become more specific and ultimately will discuss some currently tractible problems.


Singular Perturbation Shock Layer Singular Perturbation Problem Singular Perturbation Theory Decay Solution 
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  1. [1]
    L. Abrahamsson and S. Osher (1982), “Monotone difference schemes for singular perturbation problems,” SIAM J. Num. Anal. 19, 979–992.Google Scholar
  2. [2]
    U. Ascher and R. Weiss (1983), “Collocation for singular perturbations I: First order systems with constant coefficients,” SIAM J. Num. Anal. 20, 537–557.Google Scholar
  3. [3]
    D. L. Brown (1982), Solution Adaptive Mesh Procedures for the Numerical Solution of Singular Perturbation Problems Ph.D. thesis, California Institute of Technology, Pasadena.Google Scholar
  4. [4]
    J. -L. Callot (1981),, Bifurcations du Portrait de Phase pour des Equations Différentielles Linéaires du Second Ordre ayant pour Type l’ Equation d’ Hermite, Thèse, Université Louis Pasteur, Strasbourg.Google Scholar
  5. [5]
    R. C. Y. Chin, G. W. Hedstrom, and F. A. Howes (1984), A Survey of Analytical and Numerical Methods for Multiple Scale Problems preprint, Lawrence Livermore Laboratory.Google Scholar
  6. [6]
    J. H. Chow (editor) (1982), Time-Scale Modeling of Dynamic Networks with Applications to Power Systems Lecture Notes in Control and Information Sciences 46, Springer-Verlag, Berlin.Google Scholar
  7. [7]
    E. A. Coddington and N. Levinson (1952), “A boundary value problem for a nonlinear differential equation with a small parameter,” Proc. Amer. Math. Soc. 3, 73–81.Google Scholar
  8. [8]
    G. Dahlquist (1969), A numerical method for some ordinary differential equations with large Lipschitz constants Information Processing 68 (A. J. H. Morrell, editor), North-Holland, Amsterdam, 183–186.Google Scholar
  9. [9]
    G. Dahlquist (1984), On Transformations of Graded Matrices with Applications to Stiff ODE’s preprint, Royal Institute of Technology, Stockholm.Google Scholar
  10. [10]
    F. R. deHoog and R. M. M. Mattheij (1983), On Dichotomy and Well-Conditioning in BVP Report, Catholic University, Nijmegen.Google Scholar
  11. [11]
    F. Diener (1981), Méthode du Plan d’Observabilité Thèse, Université Louis Pasteur, Strasbourg.Google Scholar
  12. [12]
    M. Diener (1981), Étude Générique des Canards These, Université Louis Pasteur, Strasbourg.Google Scholar
  13. [13]
    M. Diener (1984), “The canard unchained or how fast/ slow dynamical systems bifurcate,” Math. Intelligencer 6, no. 3, 38–49.Google Scholar
  14. [14]
    W. Eckhaus (1983), “Relaxation oscillations, including a standard chase on French ducks,” Lecture Notes in Math. 985, Springer Verlag, Heidelberg, 449–494.Google Scholar
  15. [15]
    N. Fenichel (1979), “Geometric singular perturbation theory for ordinary differential equations,” J. Differential Equations 31, 53–98.Google Scholar
  16. [16]
    P. C. Fife (1976), “Boundary and interior transition layer phenomena for pairs of second order differential equations,” J. Math. Anal. Appl. 54, 66–93.Google Scholar
  17. [17]
    P. C. Fife (1979), Mathematical Aspects of Reacting and Diffusing Systems Lecture Notes in Biomathematics 28, Springer Verlag, Berlin.Google Scholar
  18. [19]
    J. E. Flaherty and R. E. O’Malley, Jr. (1977), “The numerical solution of boundary value problems for stiff differential equations,” Math. Comp. 31, 66–93.Google Scholar
  19. [20]
    J. E. Flaherty and R. E. O’Malley, Jr. (1984), “Numerical methods for stiff systems of two-point boundary value problems,” SIAM J. Sci. Stat. Comp.Google Scholar
  20. [21]
    P. Habets (1982), Singular Perturbations in Nonlinear Systems and Optimal Control report, Université Catholique de Louvain.Google Scholar
  21. [23]
    B. D. Hassard (1980), “Computation of invariant manifolds,” New Approaches to Nonlinear Problems in Dynamics P. J. Holmes (editor), SIAM, Philadelphia, 27–42.Google Scholar
  22. [24]
    P. W. Hemker (1977), A Numerical Study of Stiff Two-Point Boundary Problems thesis, Mathematisch Centrum, Amsterdam.Google Scholar
  23. [25]
    P. W. Hemker (1982), Numerical Aspects of Singular Perturbation Problems report, Mathematisch Centrum, Amsterdam.Google Scholar
  24. [26]
    F. Hoppensteadt (1971), “Properties of solutions of ordinary differential equations with a small parameter,” Comm. Pure Appl. Math. 24, 807–840.Google Scholar
  25. [27]
    F. A. Howes (1978), “Boundary and interior layer behavior and their interaction,” Memoirs Amer. Math. Soc. 203.Google Scholar
  26. [29]
    F. A. Howes (1983), “Shock layer behavior in perturbed second-order systems,” Proceedings, Berkeley Conference on Control and Fluid Dynamics to appear.Google Scholar
  27. [31]
    F. A. Howes (1984), “Asymptotic structures in nonlinear dissipative and dispersive systems,” Physica D to appear.Google Scholar
  28. [32]
    F. A. Howes (1984), “Multi-dimensional reactionconvection-diffusion equations,” Proceedings, Dundee Conference on Differential Equations to appear.Google Scholar
  29. [33]
    F. A. Howes and R. E. O’Malley, Jr. (1980), “Singular perturbations of semilinear second-order systems,” Lecture Notes in Math. 827, Springer-Verlag, Berlin, 131–150.Google Scholar
  30. [34]
    A. M. Il’in (1969), “Differencing scheme for a differential equation with a small parameter affecting the highest derivatives,” Math Notes 6, 596–602.Google Scholar
  31. [36]
    N. Kopell (1983), “Invariant manifolds and the initialization problem for some atmospheric equations,” preprint, Northeastern University.Google Scholar
  32. [38]
    H. -O. Kreiss and N. Nichols (1975), Numerical Methods for Singular Perburbation Problems report, Uppsala University.Google Scholar
  33. [39]
    H. -O. Kreiss, N. K. Nichols, and D. L. Brown (1983), Numerical Methods for Stiff Two-Point Boundary Value Problems preprint, California Institute of Technology.Google Scholar
  34. [40]
    H. L. Kurland (1984), Singularly Perturbed Systems and the Morse-Conley Index preprint, Boston University.Google Scholar
  35. [41]
    M. Lentini, M. R. Osborne, and R. D. Russell (1983), The Close Relationships Between Methods for Solving Two-Point Boundary Value Problems preprint, Simon Fraser University.Google Scholar
  36. [42]
    J. J. Levin (1957), “The asymptotic behavior of the stable initial manifold of a system of nonlinear differential equations,” Trans. Amer. Math. Soc. 85, 357–368.Google Scholar
  37. [43]
    J. J. Levin and N. Levinson (1954), “Singular perturbations of nonlinear systems of differential equations and an associated boundary layer equation,” J. Rational Mech. Anal. 3, 247–270.Google Scholar
  38. [44]
    J. Lorenz (1981), Nonlinear Singular Perturbation Problems and the Engquist-Osher Difference Scheme Report, Catholic University, Nijmegen.Google Scholar
  39. [45]
    J. Lorenz (1982), “Nonlinear boundary value problems with turning points and properties of difference schemes,” Lecture Notes in Math. 942, Springer Verlag, Berlin, 150–169.Google Scholar
  40. [46]
    J. Lorenz (1983), Stability and Monotonicity Properties of Stiff Quasilinear Boundary Problems preprint.Google Scholar
  41. [47]
    J. Lorenz (1983), Analysis of Difference Schemes for a Stationary Shock Problem preprint,Universitat Trier.Google Scholar
  42. [48]
    R. Lutz and M. Goze (1981), Non-Standard Analysis Lecture Notes in Math. 881, Springer-Verlag, Heidelberg.Google Scholar
  43. [49]
    R. Lutz and T. Sari (1982), “Application of nonstandard analysis to boundary value problems in singular perturbation theory,” Lecture Notes in Math. 942, Springer-Verlag, Berlin, 113–135.Google Scholar
  44. [50]
    A. Majda (1983), “The stability of multi-dimensional shock fronts,” Memoirs Amer. Math. Soc. 41, no. 275.Google Scholar
  45. [51]
    Mao Zu-fan (1982), Partitioning a Stiff Ordinary Differential Equation by a Scaling Technique report, Royal Institute of Technology, Stockholm.Google Scholar
  46. [52]
    V. P. Maslov and G. A. Omel’yanov (1981), “Asymptotic soliton-form solutions of equations with small dispersion,” Russian Math. Surveys 36:3, 73–149.Google Scholar
  47. [53]
    R. M. M. Mattheij (1984), Decouplinq and Stability of Algorithms for Boundary Value Problems preprint, Catholic University, Nijmegen.Google Scholar
  48. [54]
    R. M. M. Mattheij and R. E. O’Malley, Jr. (1984), “On solving boundary value problems for multi-scale systems using asymptotic approximations and multiple shooting,” BIT, to appear.Google Scholar
  49. [55]
    W. L. Miranker (1981), Numerical Methods for Stiff Equations Reidel, Dordrecht.Google Scholar
  50. [56]
    K. Nipp (1980), An Algorithmic Approach to Singular Perturbation Problems in Ordinary Differential Equations with an Application to the BelousovZhabotinskii Reaction dissertation, Eidgen. Tech. Hochschule, Zurich.Google Scholar
  51. [57]
    R. E. O’Malley, Jr. (1974), Introduction to Singular Perturbations Academic Press, New York.Google Scholar
  52. [58]
    R. E. O’Malley, Jr. (1980), “On multiple solutions of singularly perturbed systems in the conditionally stable case,” Singular Perturbations and Asymptotics (R. E. Meyer and S. V. Parter, editors), Academic Press, New York, 87–108.Google Scholar
  53. [59]
    R. E. O’Malley, Jr. (1983), “Shock and transition layers for singularly perturbed second-order vector systems,” SIAM J. Appl. Math. 43, 935943.Google Scholar
  54. [60]
    S. Osher (1981), “Nonlinear singular perturbation problems and one-sided difference schemes,” SIAM J. Num. Anal. 18, 129–144.Google Scholar
  55. [63]
    G. Peponides, P. V. Kokotovic, and J. H. Chow (1982), “Singular perturbations and time scales in nonlinear models of power systems,” IEEE Trans. Circuits and Systems 29, 758–767.Google Scholar
  56. [64]
    A. Saberi and H. Khalil (1984), “Quadratic-type Liapunov functions for singularly perturbed systems,” IEEE Trans. Automatic Control 29, 542–550.Google Scholar
  57. [65]
    V. R. Saksena, J. O’Reilly, and P. V. Kokotovic (1984), “Singular perturbations and time-scale methods in control theory: Survey 1976–1983,” Automatica 20, 273–293.Google Scholar
  58. [66]
    J. Smoller (1982), Shock Waves and Reaction-Diffusion Equations Springer-Verlag, New York.Google Scholar
  59. [671.
    V. A. Sobolev (1984), Integral Manifolds and Decomposition of Singularly Perturbed Systems preprint, Kuibyshev State University, USSR.Google Scholar
  60. [681.
    G. Söderlind and R. M. M. Mattheij (1984), “Stability and asymptotic estimates in nonautonomous linear differential equations,” SIAM J. Math. appear.Google Scholar
  61. [69]
    J. J. Stoker (1950), Nonlinear Oscillations Wiley, New York.Google Scholar
  62. [70]
    E. Urlacher (1980), “Equations Differentielles du Type Ex”+f(x’)+x = 0 avec r Petit,“ report, Université Louis Pasteur, Strasbourg.Google Scholar
  63. [71]
    M. Van Dyke (19’64), Perturbation Methods in Fluid Dynamics Academic Press, New York.Google Scholar
  64. [72]
    A. van Harten and E. Vader-Burger (1984), Approximate Green Functions as a Tool to Prove Correctness of a Formal Approximation in a Model of Competing and Diffusing Species report, Utrecht University.Google Scholar
  65. [73]
    A. B. Vasil’eva and V. F. Butuzov (1973), Asymptotic Expansions of Solutions of Singularly Perturbed Equations Nauka, Moscow.Google Scholar
  66. [74]
    M. I. Vishik and L. A. Lyusternik (1960), “Initial jump for nonlinear differential equations containing a small parameter,” Soviet Math. Dokl. 1, 749–752.MATHGoogle Scholar
  67. [75]
    W. Wasow (1965), Asymptotic Expansions for Ordinary Differential Equations Wiley, New York.Google Scholar
  68. [76]
    W. Wasow (1970), “The capriciousness of singular perturbations,” Nieuw Archief v. Wisk. 18, 190–210.MathSciNetMATHGoogle Scholar
  69. [77]
    W. Wasow (1984), Lectures on Linear Turning Point Theory manuscript, University of Wisconsin, Madison.Google Scholar
  70. [78]
    R. Weiss (1984), “An analysis of the box and trapezoidal schemes for linear singularly perturbed boundary value problems,” Math. Comp. 42, 41–67.MathSciNetMATHCrossRefGoogle Scholar

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© Birkhäuser Boston, Inc. 1985

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  • Robert E. O’MalleyJr.

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