Frontiers of Mathematics in China >
Convergence of Runge-Kutta methods for neutral Volterra delay-integro-differential equations∗
Received date: 21 May 2008
Accepted date: 28 Nov 2008
Published date: 05 Mar 2009
Copyright
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.
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
|
| 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
|
| 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
|
| 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
|
| 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
|
| 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
|
| 13 |
Jackiewicz Z. One-step methods of any order for neutral functional differential equations. SIAM J Numer Anal, 1984, 21: 486-511
|
| 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
|
| 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
|
| 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
|
| 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
|
| 20 |
Li S F. B-theory of general linear methods for stiff Volterra functional differential equations. Appl Numer Math, 2005, 53: 57-72
|
| 21 |
Vermiglio R. Natural continuous extensions of Runge-Kutta methods for Volterra integro-differential equations. Numer Math, 1988, 53: 439-458
|
| 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
|
| 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
|
/
| 〈 |
|
〉 |