Abstract
This paper deals with the asymptotic stability of theoretical solutions and numerical methods for systems of neutral differential equationsx′=Ax′(t−τ)+Bx(t)+Cx(t−τ), whereA, B, andC are constant complexN ×N matrices, and τ>0. A necessary and sufficient condition such that the differential equations are asymptotically stable is derived. We also focus on the numerical stability properties of adaptations of one-parameter methods. Further, we investigate carefully the characterization of the stability region.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. K. Barwell,Special stability problems for functional differential equations BIT, 15 (1975), pp. 130–135.
R. Bellman and K. L. Cooke,Differential-Difference Equations Academic Press, New York, 1963.
R. K. Brayton and R. A. Willoughby,On the numerical integration of a symmetric system of difference-differential equations of neutral type J. Math. An. Appl., 18 (1967), pp. 182–189.
J. K. Hale,Theory of Functional Differential Equations Springer Verlag, New York, 1977.
P. Henrici,Applied and Computational Complex Analysis. Volume 1, John Wiley & Sons, New York, 1974.
K. J. In't Hout and M. N. Spijker,Stability analysis of numerical methods for delay differential equations Numer. Math., 59 (1991), pp. 807–814.
K. J. In't Hout,The stability of θ-methods for systems of delay differential equations, Report Series No. 282, March 1993, ISSN 0112-4021, Univ. Auckland, 1993.
K. Knopp,Funktionen Theorie II Gruyter, Berlin, 1955.
P. Lancaster and M. Tismenetsky,The Theory of Matrices second edition, Academic Press, New York, 1985.
M. Z. Liu and M. N. Spijker,The stability of the θ-methods in the numerical solution of delay differential equations IMA J. Numer. Anal., 10 (1991), pp. 31–48.
L. H. Lu,The stability of the block θ-methods IMA J. Numer. Anal., 13 (1993), pp. 101–114.
W. L. Miranker,The wave equation with a nonlinear interface condition IBM J. Res. Developm., 25 (1961), pp. 2–24.
W. L. Miranker,Existence, uniqueness, and stability of solutions of systems of nonlinear difference-differential equations J. Math. Mech., 11 (1962), pp. 101–108.
W. Snow,Existence, uniqueness, and stability for nonlinear differential-difference equations in the neutral case, N.Y.U., Courant Inst. Math. Sci. Rep. IMM-NYU 328, 1965.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kuang, J.X., Xiang, J.X. & Tian, H.J. The asymptotic stability of one-parameter methods for neutral differential equations. BIT 34, 400–408 (1994). https://doi.org/10.1007/BF01935649
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01935649