Ordinary Differential Equations

  • J. Stoer
  • R. Bulirsch

Abstract

Many problems in applied mathematics lead to ordinary differential equations. In the simplest case one seeks a differentiable function y = y(x) of one real variable x, whose derivative y’(x) is to satisfy an equation of the form y’(x) = f(x, y(x)), or more briefly,
$$y' = f\left( {x,y} \right); $$
(7.0.1)
one then speaks of an ordinary differential equation.

Keywords

Ordinary Differential Equation Discretization Error Multistep Method Linear Multistep Method Implicit Euler Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References for Chapter 7

  1. Babuška, I., Prager, M., Vitasék, E.: Numerical Processes in Differential Equations. New York: Interscience (1966).MATHGoogle Scholar
  2. Bulirsch, R.: Die Mehrzielmethode zur numerischen Lösung von nichtlinearen Randwertproblemen und Aufgaben der optimalen Steuerung. Report of the Carl-Cranz-Gesellschaft. 1971.Google Scholar
  3. Bulirsch, R., Stoer, J.: Numerical treatment of ordinary differential equations by extrapolation methods. Numer. Math. 8, 1–13 (1966).MathSciNetMATHCrossRefGoogle Scholar
  4. Bulirsch, R., Stoer, J., Deuflhard, P.: Numerical solution of nonlinear two-point boundary value problems 1. Handbook Series Approximation, Numer. Math. (to appear).Google Scholar
  5. Butcher, J. C.: On Runge-Kutta processes of high order. J. Austral Math. Soc. 4, 179–194 (1964).MathSciNetMATHCrossRefGoogle Scholar
  6. Ciarlet, P. G., Schultz, M. H., Varga, R. S.: Numerical methods of high order accuracy for nonlinear boundary value problems. Numer. Math. 9, 394–430 (1967).MathSciNetMATHCrossRefGoogle Scholar
  7. Ciarlet, P. G., Schultz, M. H., Varga, R. S., Wagschal, C.: Multipoint Taylor formulas and applications to the finite element method. Numer. Math. 17, 84–100 (1971).MathSciNetMATHCrossRefGoogle Scholar
  8. Clark, N. W.: A study of some numerical methods for the integration of systems of first order ordinary differential equations. Report ANL-7428, Argonne National Laboratory 1968.CrossRefGoogle Scholar
  9. Coddington, E. A., Levinson, N.: Theory of Ordinary Differential Equations. New York: McGraw-Hill (1955).MATHGoogle Scholar
  10. Collatz, L.: The Numerical Treatment of Differential Equations. Berlin: Springer-Verlag (1960).MATHCrossRefGoogle Scholar
  11. Collatz, L.: Functional Analysis and Numerical Mathematics. New York: Academic Press (1966).Google Scholar
  12. Collatz, L., Nicolovius, R.: Rand und Eigenwertprobleme bei gewöhnlichen und partiellen Differentialgleichungen und Integralgleichungen. In Sauer and Szabó (1969), 293–670.Google Scholar
  13. Crane, P. J., Fox, P. A.: A comparative study of computer programs for integrating differential equations. Numer. Math. Computer Program Library One—Basic Routines for General Use (Bell Telephone Laboratories Inc., New Jersey) 2, No. 2 (1969).Google Scholar
  14. Dahlquist, G.: Convergence and stability in the numerical integration of ordinary differential equations. Math. Scand. 4, 33–53 (1956).MathSciNetMATHGoogle Scholar
  15. Dahlquist, G.: Stability and error bounds in the numerical integration of ordinary differential equations. Trans. Roy. Inst. Tech. (Stockholm), No. 130 (1959).MATHGoogle Scholar
  16. Deuflhard, P.: A stepsize control for continuation methods with special application to multiple shooting techniques. Report TUM-Math.-7627, Technical University of Munich 1976.Google Scholar
  17. Diekhoff, H.-J., Lory, P., Oberle, H. J., Pesch, H.-J., Rentrop, P., Seydel, R.: Comparing routines for the numerical solution of initial value problems of ordinary differential equations in multiple shooting. Numer. Math. 27, 449–469 (1977).MathSciNetMATHCrossRefGoogle Scholar
  18. Enright, W. H., Hull, T. E., Lindberg, B.: Comparing numerical methods for stiff systems of ODEs. Tech. Rep. No. 69, Department of Computer Science, University of Toronto, Sept. 1974.Google Scholar
  19. Fehlberg, E.: New high-order Runge-Kutta formulas with stepsize control for systems of first- and second-order differential equations. Z. Angew. Math. Mech. 44, T17–T29 (1964).MathSciNetCrossRefGoogle Scholar
  20. Fehlberg, E.: New high-order Runge-Kutta formulas with an arbitrarily small truncation error. Z. Angew. Math. Mech. 46, 1–16 (1966).MathSciNetMATHCrossRefGoogle Scholar
  21. Fehlberg, E.: Klassische Runge-Kutta Formeln fünfter und siebenter Ordnung mit Schrittweiten-Kontrolle. Computing 4, 93–106 (1969).MathSciNetMATHCrossRefGoogle Scholar
  22. Fehlberg, E.: Klassische Runge-Kutta Formeln vierter und niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung auf Wärmeleitungsprobleme. Computing 6, 61–71 (1970).MathSciNetMATHCrossRefGoogle Scholar
  23. Fröberg, C. E.: Introduction into Numerical Analysis. London: Addison-Wesley (1966).Google Scholar
  24. Gantmacher, F. R.: Matrizenrechnung II. Berlin: VEB Deutscher Verlag der Wissenschaften (1969).Google Scholar
  25. Gear, C. W.: Numerical Initial Value Problems in Ordinary Differential Equations. Englewood Cliffs, N.J.: Prentice-Hall 1971.MATHGoogle Scholar
  26. George, J. A.: Computer implementation of the finite element method. Report CS 208, Computer Science Department, Stanford University 1971.Google Scholar
  27. George, J. A.: Nested dissection of a regular finite element mesh. SIAM J. Numer. Anal 10, 345–363 (1973).MathSciNetMATHCrossRefGoogle Scholar
  28. Gragg, W.: Repeated extrapolation to the limit in the numerical solution of ordinary differential equations. Thesis, UCLA 1963.Google Scholar
  29. Gragg, W.: On extrapolation algorithms for ordinary initial value problems. J. SIAM Numer. Anal. Ser. B 2, 384–403 (1965).MathSciNetGoogle Scholar
  30. Grigorieff, R. D.: Numerik gewöhnlicher Differentialgleichungen 1, 2. Stuttgart: Teubner 1972, 1977.Google Scholar
  31. Henrici, P.: Discrete Variable Methods in Ordinary Differential Equations. New York: John Wiley 1962.MATHGoogle Scholar
  32. Hestenes, M. R.: Calculus of Variations and Optimal Control Theory. New York: John Wiley 1966.MATHGoogle Scholar
  33. Heun, K.: Neue Methode zur approximativen Integration der Differentialgleichungen einer unabhängigen Variablen. Z. Math. Phys. 45, 23–38 (1900).MATHGoogle Scholar
  34. Hindmarsh, A. C.: GEAR-Ordinary differential equation solver. UCID-30001, Rev. 3, Lawrence Livermore Laboratory, University of California, Livermore, 1974.Google Scholar
  35. Hull, T. E., Enright, W. H., Fellen, B. M., Sedgwick, A. E.: Comparing numerical methods for ordinary differential equations. SIAM J. Numer. Anal. 9, 603–637 (1972). [Errata, ibid. 11, 681 (1974).]MathSciNetMATHCrossRefGoogle Scholar
  36. Isaacson, E., Keller, H. B.: Analysis of Numerical Methods. New York: John Wiley 1966.MATHGoogle Scholar
  37. Kaps, P., Rentrop, P.: Generalized Runge-Kutta methods of order four with step size control for stiff ordinary differential equations. Numer. Math. 33, 55–68, 1979.MathSciNetMATHCrossRefGoogle Scholar
  38. Keller, H. B.: Numerical Methods for Two-Point Boundary-Value Problems. London: Blaisdell 1968.MATHGoogle Scholar
  39. Krogh, F. T.: Changing step size in the integration of differential equations using modified divided differences. In: Proceedings of the Conference on the Numerical Solution of Ordinary Differential Equations, 22–71. Lecture Notes in Mathematics, No. 362. New York: Springer-Verlag 1974.CrossRefGoogle Scholar
  40. Kutta, W.: Beitrag zur näherungsweisen Integration totaler Differentialgleichungen. Z. Math. Phys. 46, 435–453 (1901).MATHGoogle Scholar
  41. Na, T. Y., Tang, S. C.: A method for the solution of conduction heat transfer with non-linear heat generation. Z. Angew. Math. Mech. 49, 45–52 (1969).MATHCrossRefGoogle Scholar
  42. Oden, J. T., Reddy, J. N.: An Introduction to the Mathematical Theory of Finite Elements. New York: Wiley 1976.MATHGoogle Scholar
  43. Osborne, M. R.: On shooting methods for boundary value problems. J. Math. Anal. Appl. 27, 417–433 (1969).MathSciNetMATHCrossRefGoogle Scholar
  44. Runge, C.: Über die numerische Auflösung von Differentialgleichungen. Math. Ann. 46, 167–178 (1895).MathSciNetMATHCrossRefGoogle Scholar
  45. Sauer, R., Szabó, I., Eds.: Mathematische Hilfsmittel des Ingenieurs, Part II. Berlin: Springer-Verlag 1969.MATHGoogle Scholar
  46. Sedgwick, A.: An effective variable order variable step Adams method. Technical Report No. 53, University of Toronto, Department of Computer Science 1973.Google Scholar
  47. Shampine, L. F., Gordon, M. K.: Computer Solution of Ordinary Differential Equations. The Initial Value Problem. San Francisco: Freeman 1975.MATHGoogle Scholar
  48. Shampine, L. F., Gordon, M. K., Watts, H. A., Davenport, S. M.: Solving nonstiff ordinary differential equations—the state of the art. SIAM Review 18, 376–411 (1976).MathSciNetMATHCrossRefGoogle Scholar
  49. Shanks, E. B.: Solution of differential equations by evaluation of functions. Math. Comp. 20, 21–38 (1966).MathSciNetMATHCrossRefGoogle Scholar
  50. Stetter, H. J.: Analysis of Discretization Methods for Ordinary Differential Equations. New York, Heidelberg, Berlin: Springer-Verlag 1973.MATHCrossRefGoogle Scholar
  51. Strang, G., Fix, G. J.: An Analysis of the Finite Element Method. Englewood Cliffs, N.J.: Prentice-Hall 1973.MATHGoogle Scholar
  52. Törnig, W.: Anfangswertprobleme bei gewöhnlichen und partiellen Differentialgleichungen. In: Sauer and Szabó (1969), 1–290 (1969).Google Scholar
  53. Troesch, B. A.: Intrinsic difficulties in the numerical solution of a boundary value problem. Internal Report NN-142, TRW, Inc. Redondo Beach, California, January 29, 1960.Google Scholar
  54. Troesch, B. A.: A simple approach to a sensitive two-point boundary value problem. J. Computational Phys. 21, 279–290 (1976).MathSciNetMATHCrossRefGoogle Scholar
  55. Willoughby, R. A., Ed.: Stiff Differential Systems. New York and London: Plenum Press 1974.Google Scholar
  56. Zlámal, M.: On the finite element method. Numer. Math. 12, 394–409 (1968).MathSciNetMATHCrossRefGoogle Scholar
  57. Zlámal, M.: On some finite element procedures for solving second order boundary value problems. Numer. Math. 14, 42–48 (1969).MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1980

Authors and Affiliations

  • J. Stoer
    • 1
  • R. Bulirsch
    • 2
  1. 1.Institut für Angewandte MathematikUniversität Würzburg am HublandWürzburgFederal Republic of Germany
  2. 2.Institut für MathematikTechnische UniversitätMünchenFederal Republic of Germany

Personalised recommendations