RESEARCH ARTICLE

Convergence of Runge-Kutta methods for neutral Volterra delay-integro-differential equations

  • Wansheng WANG , 1 ,
  • Shoufu LI 2
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  • 1. School of Mathematics and Computational Science, Changsha University of Science & Technology, Changsha 410004, China
  • 2. School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China

Received date: 21 May 2008

Accepted date: 28 Nov 2008

Published date: 05 Mar 2009

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg

Abstract

In this paper, we focus on the error behavior of Runge-Kutta methods for nonlinear neutral Volterra delay-integro-differential equations (NVDIDEs) with constant delay. The convergence properties of the Runge- Kutta methods with two classes of quadrature technique, compound quadrature rule and Pouzet type quadrature technique, are investigated.

Cite this article

Wansheng WANG , Shoufu LI . Convergence of Runge-Kutta methods for neutral Volterra delay-integro-differential equations[J]. Frontiers of Mathematics in China, 2009 , 4(1) : 195 -216 . DOI: 10.1007/s11464-009-0021-9

1
Baker C T H. A perspective on the numerical treatment of Volterra equations. J Comput Appl Math, 2000, 125: 217-249

DOI

2
Baker C T H, Ford N J. Stability properties of a scheme for the approximate solution of a delay integro-differential equation. Appl Numer Math, 1992, 9: 357-370

DOI

3
Brunner H. The numerical solutions of neutral Volterra integro-differential equations with delay arguments. Ann Numer Math, 1994, 1: 309-322

4
Brunner H. Collocation Methods for Volterra Integral and Related Functional Differential Equations. Cambridge: Cambridge University Press, 2004

5
Brunner H. High-order collocation methods for singular Volterra functional equations of neutral type. Appl Numer Math, 2007, 57: 533-548

DOI

6
Brunner H, Houwen P J van der. The Numerical Solution of Volterra Equations. CWI Monographs. Amsterdam: Elsevier Science Publishers B V, 1986

7
Burrage K, Butcher J C. Non-linear stability of a general class of differential equation methods. BIT, 1980, 20: 185-203

DOI

8
Butcher J C. The Numerical Analysis of Ordinary Differential Equations. New York: John Wiley, 1987

9
Enright WH, Hu M. Continuous Runge-Kutta methods for neutral Volterra integrodifferential equations with delay. Appl Numer Math, 1997, 24: 175-190

DOI

10
Hairer E, Wanner G. Solving Ordinary Differential Equations II: Stiff and Differential Algebraic Problems. Berlin: Springer-Verlag, 1991

11
Huang C M, Fu H Y, Li S F, Chen G N, D-convergence of Runge-Kutta methods for stiff delay differential equations. J Comput Math, 2001, 19: 259-268

12
Huang C M, Li S F, Fu H Y, Chen G N. Stability and error analysis of one-leg methods for nonlinear delay differential equations. J Comput Appl Math, 1999, 103: 263-279

DOI

13
Jackiewicz Z. One-step methods of any order for neutral functional differential equations. SIAM J Numer Anal, 1984, 21: 486-511

DOI

14
Jackiewicz Z. Quasilinear multistep methods and variable step predictor-corrector methods for neutral functional differential equations. SIAM J Numer Anal, 1986, 23: 423-452

DOI

15
Jackiewicz Z, Kwapisz M, Lo E. Waveform relaxation methods for functional differential systems of neutral type. J Math Anal Appl, 1997, 207: 255-285

DOI

16
Kolmanovskii V B, Myshkis A. Introduction to the Theory and Applications of Functional Differential Equations. Dordrecht: Kluwer Academy, 1999

17
Koto T. Stability of Runge-Kutta methods for delay integro-differential equations. J Comput Appl Math, 2002, 145: 483-492

DOI

18
Li S F. Theory of Computational Methods for Stiff Differential Equations. Changsha: Hunan Science and Technology Publisher, 1997

19
Li S F. B-theory of Runge-Kutta methods for stiff Volterra functional differential equations. Science in China, Ser A, 2003, 46: 662-674

DOI

20
Li S F. B-theory of general linear methods for stiff Volterra functional differential equations. Appl Numer Math, 2005, 53: 57-72

DOI

21
Vermiglio R. Natural continuous extensions of Runge-Kutta methods for Volterra integro-differential equations. Numer Math, 1988, 53: 439-458

DOI

22
Wang W S. Numerical Analysis of Nonlinear Neutral Functional Differential Equations. Ph D Thesis. Xiangtan: Xiangtan University, 2008

23
Wang W S, Li S F. Convergence of one-leg methods for nonlinear neutral delay integro-differential equations. Science in China, Ser A (in press)

24
Zhang C J, Vandewalle S. Stability analysis of Runge-Kutta methods for nonlinear Volterra delay-integro-differential equations. IMA J Numer Anal, 2004, 24: 193-214

DOI

25
Zhang C J, Zhou S Z. Nonlinear stability and D-convergence of Runge-Kutta methods for DDEs. J Comput Appl Math, 1997, 85: 225-237

DOI

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