Numerical Boundary Value ODEs pp 149-172 | Cite as

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

Chapter

## Abstract

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.

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### References

- [1]L. Abrahamsson and S. Osher (1982), “Monotone difference schemes for singular perturbation problems,” SIAM J. Num. Anal. 19, 979–992.Google Scholar
- [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]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]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]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]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]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]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]G. Dahlquist (1984), On Transformations of Graded Matrices with Applications to Stiff ODE’s preprint, Royal Institute of Technology, Stockholm.Google Scholar
- [10]F. R. deHoog and R. M. M. Mattheij (1983), On Dichotomy and Well-Conditioning in BVP Report, Catholic University, Nijmegen.Google Scholar
- [11]F. Diener (1981), Méthode du Plan d’Observabilité Thèse, Université Louis Pasteur, Strasbourg.Google Scholar
- [12]M. Diener (1981), Étude Générique des Canards These, Université Louis Pasteur, Strasbourg.Google Scholar
- [13]M. Diener (1984), “The canard unchained or how fast/ slow dynamical systems bifurcate,” Math. Intelligencer 6, no. 3, 38–49.Google Scholar
- [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]N. Fenichel (1979), “Geometric singular perturbation theory for ordinary differential equations,” J. Differential Equations 31, 53–98.Google Scholar
- [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]P. C. Fife (1979), Mathematical Aspects of Reacting and Diffusing Systems Lecture Notes in Biomathematics 28, Springer Verlag, Berlin.Google Scholar
- [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
- [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
- [21]P. Habets (1982), Singular Perturbations in Nonlinear Systems and Optimal Control report, Université Catholique de Louvain.Google Scholar
- [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
- [24]P. W. Hemker (1977), A Numerical Study of Stiff Two-Point Boundary Problems thesis, Mathematisch Centrum, Amsterdam.Google Scholar
- [25]P. W. Hemker (1982), Numerical Aspects of Singular Perturbation Problems report, Mathematisch Centrum, Amsterdam.Google Scholar
- [26]F. Hoppensteadt (1971), “Properties of solutions of ordinary differential equations with a small parameter,” Comm. Pure Appl. Math. 24, 807–840.Google Scholar
- [27]F. A. Howes (1978), “Boundary and interior layer behavior and their interaction,” Memoirs Amer. Math. Soc. 203.Google Scholar
- [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
- [31]F. A. Howes (1984), “Asymptotic structures in nonlinear dissipative and dispersive systems,” Physica D to appear.Google Scholar
- [32]F. A. Howes (1984), “Multi-dimensional reactionconvection-diffusion equations,” Proceedings, Dundee Conference on Differential Equations to appear.Google Scholar
- [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
- [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
- [36]N. Kopell (1983), “Invariant manifolds and the initialization problem for some atmospheric equations,” preprint, Northeastern University.Google Scholar
- [38]H. -O. Kreiss and N. Nichols (1975), Numerical Methods for Singular Perburbation Problems report, Uppsala University.Google Scholar
- [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
- [40]H. L. Kurland (1984), Singularly Perturbed Systems and the Morse-Conley Index preprint, Boston University.Google Scholar
- [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
- [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
- [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
- [44]J. Lorenz (1981), Nonlinear Singular Perturbation Problems and the Engquist-Osher Difference Scheme Report, Catholic University, Nijmegen.Google Scholar
- [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
- [46]J. Lorenz (1983), Stability and Monotonicity Properties of Stiff Quasilinear Boundary Problems preprint.Google Scholar
- [47]J. Lorenz (1983), Analysis of Difference Schemes for a Stationary Shock Problem preprint,Universitat Trier.Google Scholar
- [48]R. Lutz and M. Goze (1981), Non-Standard Analysis Lecture Notes in Math. 881, Springer-Verlag, Heidelberg.Google Scholar
- [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
- [50]A. Majda (1983), “The stability of multi-dimensional shock fronts,” Memoirs Amer. Math. Soc. 41, no. 275.Google Scholar
- [51]Mao Zu-fan (1982), Partitioning a Stiff Ordinary Differential Equation by a Scaling Technique report, Royal Institute of Technology, Stockholm.Google Scholar
- [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
- [53]R. M. M. Mattheij (1984), Decouplinq and Stability of Algorithms for Boundary Value Problems preprint, Catholic University, Nijmegen.Google Scholar
- [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
- [55]W. L. Miranker (1981), Numerical Methods for Stiff Equations Reidel, Dordrecht.Google Scholar
- [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
- [57]R. E. O’Malley, Jr. (1974), Introduction to Singular Perturbations Academic Press, New York.Google Scholar
- [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
- [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
- [60]S. Osher (1981), “Nonlinear singular perturbation problems and one-sided difference schemes,” SIAM J. Num. Anal. 18, 129–144.Google Scholar
- [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
- [64]A. Saberi and H. Khalil (1984), “Quadratic-type Liapunov functions for singularly perturbed systems,” IEEE Trans. Automatic Control 29, 542–550.Google Scholar
- [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
- [66]J. Smoller (1982), Shock Waves and Reaction-Diffusion Equations Springer-Verlag, New York.Google Scholar
- [671.V. A. Sobolev (1984), Integral Manifolds and Decomposition of Singularly Perturbed Systems preprint, Kuibyshev State University, USSR.Google Scholar
- [681.G. Söderlind and R. M. M. Mattheij (1984), “Stability and asymptotic estimates in nonautonomous linear differential equations,” SIAM J. Math. Anal.to appear.Google Scholar
- [69]J. J. Stoker (1950), Nonlinear Oscillations Wiley, New York.Google Scholar
- [70]E. Urlacher (1980), “Equations Differentielles du Type Ex”+f(x’)+x = 0 avec r Petit,“ report, Université Louis Pasteur, Strasbourg.Google Scholar
- [71]M. Van Dyke (19’64), Perturbation Methods in Fluid Dynamics Academic Press, New York.Google Scholar
- [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
- [73]A. B. Vasil’eva and V. F. Butuzov (1973), Asymptotic Expansions of Solutions of Singularly Perturbed Equations Nauka, Moscow.Google Scholar
- [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
- [75]W. Wasow (1965), Asymptotic Expansions for Ordinary Differential Equations Wiley, New York.Google Scholar
- [76]W. Wasow (1970), “The capriciousness of singular perturbations,” Nieuw Archief v. Wisk. 18, 190–210.MathSciNetMATHGoogle Scholar
- [77]W. Wasow (1984), Lectures on Linear Turning Point Theory manuscript, University of Wisconsin, Madison.Google Scholar
- [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