Numerische Mathematik

, Volume 52, Issue 6, pp 605–619

Stability analysis of one-step methods for neutral delay-differential equations

  • A. Bellen
  • Z. Jackiewicz
  • M. Zennaro
Article

Summary

In this paper stability properties of one-step methods for neutral functional-differential equations are investigate. Stability regions are characterized for Runge-Kutta methods with respect to the linear test equation
$$\begin{gathered} y'\left( t \right) = ay\left( t \right) + by\left( {t - \tau } \right) + cy'\left( {t - \tau } \right),t \geqq 0, \hfill \\ y\left( t \right) = g\left( t \right), - \tau \leqq t \leqq 0, \hfill \\ \end{gathered} $$
τ>0, where,a, b, andc are complex parameters. In particular, it is shown that everyA-stable collocation method for ordinary differential equations can be extended to a method for neutrals delay-differential equations with analogous stability properties (the so called NP-stable method). We also investigate how the approximation to the derivative of the solution affects stability properties of numerical methods for neutral equations.

Subject Classifications

AMS(MOS): 65L20 CR: G1.7 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Barwell, V.K.: On the asymptotic behavior of the solution of a differential difference equation. Utilitas Math.6, 189–194 (1974)Google Scholar
  2. 2.
    Bellen, A.: Constrained mesh methods for functional differential equations. ISNM74, 52–70 (1985)Google Scholar
  3. 3.
    Grayton, R.K., Wiloughby, R.A.: On the numerical integration of a symmetric system of a difference-differential equations. J. Math. Anal. Appl.18, 182–189 (1967)Google Scholar
  4. 4.
    Castleton, R.N., Grimm, L.J.: A first order method for differential equations of neutral type. Math. Comput.27, 571–577 (1973)Google Scholar
  5. 5.
    Cryer, C.W.: Numerical methods for functional-differential equations. In: Delay and functional-differential equations and their applications (K. Schmitt, ed.), pp. 17–101. New York: Academic Press 1972Google Scholar
  6. 6.
    Dekker, K., Verwer, J.G.: Stability of Runge-Kutta methods for stiff nonlinear differential equations. Amsterdam: North-Holland 1984Google Scholar
  7. 7.
    Hornung, U.: Euler-Verfahren für neutrale Funktional-Differentialgleichungen. Numer. Math.24, 233–240 (1975)Google Scholar
  8. 8.
    Jackiewicz, Z.: One-step methods for the numerical solution of Volterra functional-differential equations of neutral type. Applicable Anal.12, 1–11 (1981)Google Scholar
  9. 9.
    Jackiewicz, Z.: The numerical solution of Volterra functional-differential equations of neutral type. SIAM J. Numer. Anal.18, 615–626, (1981)Google Scholar
  10. 10.
    Jackiewicz, Z.: Adams methods for neutral functional-differential equations. Numer. Math.39, 221–230 (1982)Google Scholar
  11. 11.
    Jackiewicz, Z.: One-step methods of any order for neutral functional-differential equations. SIAM J. Numer. Anal.21, 486–511 (1984)Google Scholar
  12. 12.
    Jackiewicz, Z.: Quasilinear multistep methods and variable-step predictor-corrector methods for neutral functional-differential equations. SIAM. J. Numer. Anal.23, 423–452 (1986)Google Scholar
  13. 13.
    Jackiewicz, Z.: One-step methods for neutral delay-differential equations with state dependent delays. Numerical Analysis Technical Report 65L05-2. University of Arkansas. Fayetteville 1985Google Scholar
  14. 14.
    Kamont, Z., Kwapisz, M.: On the Cauchy problem for differential-delay equations in a Banach space. Math. Nachr.74, 173–190 (1976)Google Scholar
  15. 15.
    Kappel, F., Kunisch, K.: Spline approximations for neutral functional-differential equations. SIAM J. Numer. Anal.18, 1058–1080, (1981)Google Scholar
  16. 16.
    Miranker, W.L.: Existence, uniqueness, and stability of solutions of systems of nonlinear difference-differential equations. J. Math. Mach.11, 101–108 (1962)Google Scholar
  17. 17.
    Pouzet, P.: Méthode d'intégration numérique des équations intégrales et intégro-différentielles du type Volterra de seconde expéce. Formules de Runge-Kutta. In: Symposium on the numerical treatment of ordinary differential equations, integral and integro-differential equations (Rome 1960), pp. 362–368. Basel: Birkhäuser 1960Google Scholar
  18. 18.
    Zennaro, M.: Natural continous extensions of Runge-Kutta methods. Math. Comput.46, 119–133 (1986)Google Scholar
  19. 19.
    Zennaro, M.:P-stability properties of Runge-Kutta methods for delay-differetial equations. Numer. Math.49, 305–318 (1986)Google Scholar
  20. 20.
    Zverkina, T.S.: A modification of finite difference methods for integrating ordinary differential equations with nonsmooth solutions (in Russian). Z. Vycisl. Mat.i Mat. Fiz. [Suppl.],4, 149–160 (1964)Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • A. Bellen
    • 1
  • Z. Jackiewicz
    • 2
  • M. Zennaro
    • 1
  1. 1.Dipartimento di Scienzes MatematicheUniversita degli Studi di TriesteTriesteItaly
  2. 2.Department of MathematicsArizona State UniversityTempeUSA

Personalised recommendations