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Convergence of Linear Multistep Methods and One-Leg Methods for Index-2 Differential-Algebraic Equations with a Variable Delay

Published online by Cambridge University Press:  03 June 2015

Hongliang Liu  and
Aiguo Xiao
Hongliang Liu*
Affiliation:
School of Mathematics and Computational Science, Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, Xiangtan University, Xiangtan, Hunan 411105, P. R. China
Aiguo Xiao*
Affiliation:
School of Mathematics and Computational Science, Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, Xiangtan University, Xiangtan, Hunan 411105, P. R. China
*
Email: [email protected]
Corresponding author. Email: [email protected]
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Abstract

Linear multistep methods and one-leg methods are applied to a class of index-2 nonlinear differential-algebraic equations with a variable delay. The corresponding convergence results are obtained and successfully confirmed by some numerical examples. The results obtained in this work extend the corresponding ones in literature.

Keywords

65L8065L20index-2 differential-algebraic equationsvariable delaylinear mutistep methodsone-leg methodsconvergence
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 4 , Issue 5 , October 2012 , pp. 636 - 646
Copyright
Copyright © Global-Science Press 2012

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References

[1]Arévalo, C., Führer, C. and Söderlind, G., Stabilized multistep methods for index 2 Euler-Lagrange differential-algebraic equations, BIT, 36 (1996), pp. 1–13.CrossRefGoogle Scholar
[2]Brenan, B. K. E. and Engquist, B. E., Backward differentiation approximations of nonlinear differential-algebraic equations and supplement, Math. Comput., 51 (1988), pp. 659–676.CrossRefGoogle Scholar
[3]Xiao, A. and Liu, J., Error analysis of parallel multivalue hybrid methods for index-2 differential-algebraic equations, Int. J. Comput. Sci. Math., 2 (2008), pp. 181–199.CrossRefGoogle Scholar
[4]Arévalo, C., Führer, C. and Söderlind, G., β-blocked multistep methods for Euler-Lagrange DAEs: linear analysis, Z. Angew. Math. Mech., 77 (1997), pp. 609–617.CrossRefGoogle Scholar
[5]Arévalo, C., Führer, C. and Söderlind, G., β-blocking of difference corrected multistep methods for nonstiff index-2 DAEs, Appl. Numer. Math., 35 (2000), pp. 293–305.CrossRefGoogle Scholar
[6]Cao, Y. and Li, Q., Highest order multistep formula for solving index 2 differential-algebraic equations, BIT, 38 (1998), pp. 644–662.CrossRefGoogle Scholar
[7]Hairer, E. and Wanner, G., Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Equations problems, 2nd Edition, Springer, Berlin, 1996.CrossRefGoogle Scholar
[8]Xu, Y. and Zhao, J., Stability of Runge-Kutta methods for neutral delay-integro-differential-algebraic system, Math. Comput. Simul., 79 (2008), pp. 571–583.CrossRefGoogle Scholar
[9]Li, H. and Li, J., Blocks implicit one-step methods for a class of retarded differential algebraic equation, J. Huazhong Univ. Sci. Tech., 31 (2003), pp. 111–113 (in Chinese).Google Scholar
[10]Xiao, F. and Zhang, C., Convergence analysis of one-leg methods for a class retarded differential algebraic equation, J. Numer. Meth. Comput. Appl., 29 (2008), pp. 217–225 (in Chinese).Google Scholar
[11]Zhu, W. and Pezold, L. R., Asymptotic stability of Hessenberg delay differential-algebraic equations of retarded or neutral type, Appl. Numer. Math., 27 (1998), pp. 309–325.CrossRefGoogle Scholar
[12]Ascher, U. M. and Pezold, L. R., The numerical solution of delay-differential-algebraic equations of retarded and neutral type, SIAM J. Mumer. Anal., 32 (1995), pp. 1635–1657.CrossRefGoogle Scholar
[13]Hauber, R., Numerical treatment of retarded differential-algebraic equations by collocation methods, Adv. Comput. Math., 7 (1997), pp. 573–592.CrossRefGoogle Scholar
[14]Liu, H. and Xiao, A., Convergence of backword differentiation formulas for index-2 differential-algebraic equations with variable delay, Chin. J. Eng. Math., 28 (2011), pp. 335–342 (in Chinese).Google Scholar
[15]Hairer, E.Nørsett, S. P., and Wanner, G., Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd Edition, Springer, Berlin, 1993.Google Scholar