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Numerische Mathematik

, Volume 37, Issue 2, pp 235–255 | Cite as

Numerical treatment of delay differential equations by Hermite Interpolation

  • H. J. Oberle
  • H. J. Pesch
Finite Difference Approximations of Solutions Periodic in Time of Hyperbolic Partial Differential Equations

Summary

A class of numerical methods for the treatment of delay differential equations is developed. These methods are based on the wellknown Runge-Kutta-Fehlberg methods. The retarded argument is approximated by an appropriate multipoint Hermite Interpolation. The inherent jump discontinuities in the various derivatives of the solution are considered automatically.

Problems with piecewise continuous right-hand side and initial function are treated too. Real-life problems are used for the numerical test and a comparison with other methods published in literature.

Subject Classifications

AMS(MOS): 65L05 65Q05 CR: 5.17 

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References

  1. 1.
    Bellman, R.: On the Computational Solution of Differential-Difference Equations. J. Math. Anal. and Appl.2, 108–110 (1961)Google Scholar
  2. 2.
    Bellman, R., Buell, J., Kalaba, R.: Numerical Integration of a Differential-Difference Equation with a Decreasing Time-Lag. Comm. ACM8, 227–228 (1965)Google Scholar
  3. 3.
    Bellman, R., Cooke, K.L.: On the Computational Solution of a Class of Functional Differential Equations. J. Math. Anal. Appl.12, 495–500 (1965)Google Scholar
  4. 4.
    Bellman, R., Cooke, K.L.: Differential-Difference Equations. New York-London: Academic Press 1963Google Scholar
  5. 5.
    Bulirsch, R.: Die Mehrzielmethode zur numerischen Lösung von nichtlinearen Randwertproblemen und Aufgaben der optimalen Steuerung. Report der Carl-Cranz-Gesellschaft, 1971Google Scholar
  6. 6.
    Bulirsch, R., Rutishauser, H.: Interpolation und genäherte Quadratur. In: Sauer, R., Szabó, I., (eds.): Mathematische Hilfsmittel des Ingenieurs. Berlin-Heidelberg-New York: Springer 1968Google Scholar
  7. 7.
    Driver, R.D.: Ordinary and Delay Differential Equations, Applied Mathematical Sciences. Vol. 20. Berlin-Heidelberg-New York: Springer 1977Google Scholar
  8. 8.
    Enright, W.H., Bedet, R., Farkas, I., Hull, T.E.: Test Results on Initial Value Methods for Nonstiff Ordinary Differential Equations. Tech. Rep. No. 68, Department of Computer Science, University of Toronto (1974)Google Scholar
  9. 9.
    Fehlberg, E.: Classical Fifth-, Sixth-, Seventh- and Eighth-Order Runge-Kutta Formulas with Stepsize Control. NASA Tech. Rep. No. 287, Huntsville (1968)Google Scholar
  10. 10.
    Fehlberg, E.: Klassische Runge-Kutta-Formeln fünfter und siebenter Ordnung mit Schrittweitenkontrolle. Computing4, 93–106 (1969)Google Scholar
  11. 11.
    Fehlberg, E.: Klassische Runge-Kutta-Formeln vierter und niedrigerer Ordnung mit Schritt-weitenkontrolle und ihre Anwendung auf Wärmeleitungsprobleme. Computing6, 61–71 (1970)Google Scholar
  12. 12.
    Feldstein, M.A.: Discretization Methods for Retarded Ordinary Differential Equations. University of California, Los Angeles. Dissertation (1964)Google Scholar
  13. 13.
    Hoppensteadt, F., Waltman, P.: A Problem in the Theory of Epidemics I. Math. Biosciences9, 71–91 (1970)Google Scholar
  14. 14.
    Jones, G.S.: On the Nonlinear Differential-Difference Equationf′(x)=−αf(x−1)(1+f(x)). J. Math. Anal. Appl.4, 440–469 (1962)Google Scholar
  15. 15.
    Kakutani, S., Markus, L.: On the Nonlinear Difference-Differential Equationy′(t)=(A−By(t−τ))y(t). In: Contributions to the Theory of Nonlinear Oscillations. Princeton, New Jersey: Princeton University Press 1–18 (1958)Google Scholar
  16. 16.
    MacDonald, N.: Time Lags in Biological Models. Lecture Notes in Biomathematics, Vol. 27. Berlin-Heidelberg-New York: Springer 1978Google Scholar
  17. 17.
    Minorsky, N.: Nonlinear Oscillations. Princeton, New Jersey: D. van Nostrand 1962Google Scholar
  18. 18.
    Neves, K.W.: Automatic Integration of Functional Differential Equations: An Approach. ACM Trans. Math. Software Vol.1, No. 4, 357–368 (1975)Google Scholar
  19. 19.
    Neves, K.W.: Automatic Integration of Functional Differential Equations. Collected Algorithms from ACM, Algorithm 497 (1975)Google Scholar
  20. 20.
    Oberle, H.J., Pesch, H.J.: A Seventh-Order-Integration Method for Delay Differential Equations, to be published; for FORTRAN-subroutines it is refered to ‘Numerical Treatment of Delay Differential Equations by Hermite Interpolation’, TUM-Report Nr. M8001, Technische Universität München (1980)Google Scholar
  21. 21.
    Oppelstrup, J.: The RKFHB4 Method for Delay-Differential Equations. In: Bulirsch, R., Grigorieff, R.D., Schröder, J. (eds.): Numerical Treatment of Differential Equations. Proceedings of a Conference held at Oberwolfach, 1976, Lecture Notes in Mathematics, Vol. 631, 133–146. Berlin-Heidelberg-New York: Springer 1978Google Scholar
  22. 22.
    Pinney, E.: Ordinary Differential-Difference Equations. Los Angeles, 1958Google Scholar
  23. 23.
    Shampine, L.F.: Quadrature and Runge-Kutta Formulas. Appl. Math. Comput.2, 161–171 (1976)Google Scholar
  24. 24.
    Stetter, H.J.: Numerische Lösung von Differentialgleichungen mit nacheilendem Argument. ZAMM45, T 79–80 (1965)Google Scholar
  25. 25.
    Wheldon, T.E., Kirk, J., Finlay, H.M.: Cyclical Granulopoiesis in Chronic Granulocytic Leukemia: A Simulation Study. Blood43, 379–387 (1974)Google Scholar
  26. 26.
    Wright, E.M.: A Functional Equation in the Heuristic Theory of Primes. Math. Gaz.45, 15–16 (1961)Google Scholar
  27. 27.
    Wright, E.M.: A Nonlinear Difference-Differential Equations. J. Reine Angew. Math.194, 66–87 (1955)Google Scholar

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • H. J. Oberle
    • 1
  • H. J. Pesch
    • 1
  1. 1.Institut für Mathematik der Technischen Universität MünchenMünchen 2Germany (Fed. Rep.)

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