Abstract
This paper is concerned with the numerical stability of backward differential formulae for the second order differential equations with a lagging argument. A sufficient condition such that this kind of system is asymptotically stable is derived. Furthermore, it is proved that every A-stable BDF method preserves the stable properties of exact solutions for the second order delay differential equations. Finally, some numerical experiments are given to demonstrate the main conclusions.
This paper is supported by the Natural Science Foundation of Heilongjiang Province (A200602), the project (HITC2007) supported by Science Research Foundation and the project (HITQNJS.2006.053) supported by Development Program for Outstanding Young Teachers in Harbin Institute of Technology.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Jerri, A.: Introduction to Integral Equations with Applications. Wiley, New York (1999)
Zhao, J.J., Xu, Y., Liu, M.Z.: Stability analysis of numerical methods for linear neutral Volterra delay integro-differential system. Appl. Math. Comput. 167, 1062–1079 (2005)
Zhang, C.J., Liao, X.X.: Stability of BDF methods for nonlinear Volterra integral equations with delay. Comput. Math. Appl. 43, 95–102 (2002)
Cahlon, B.: On the stability of Volterra integral equations with a lagging argument. BIT 35, 19–29 (1995)
Xu, Y., Zhao, J.J., Liu, M.Z.: TH-stability of θ-method for second order delay differential equation. Math. Numer. Sin. 26, 189–192 (2004)
Bownds, J.M., Cushing, J.M., Schutte, R.: Existence, uniqueness and extendibility of solutions of Volterra integral systems with multiple variable lags. Funkcial. Ekvac. 19, 101–111 (1976)
Hairer, E., Wanner, G.: Solving Ordinary Differential Equations: Stiff and Differential-Algebraic Problems. Springer, Berlin (1991)
Bellen, A., Zennaro, M.: Numerical Methods for Delay Differential Equations. Oxford University Press, New York (2003)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Xu, Y., Zhao, J.J. (2007). Stability of Backward Differential Formulae for Second Order Delay Differential Equations. In: Park, JW., Kim, T.G., Kim, YB. (eds) AsiaSim 2007. AsiaSim 2007. Communications in Computer and Information Science, vol 5. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-77600-0_18
Download citation
DOI: https://doi.org/10.1007/978-3-540-77600-0_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-77599-7
Online ISBN: 978-3-540-77600-0
eBook Packages: Computer ScienceComputer Science (R0)