Superconvergence in collocation methods on quasi-graded meshes for functional differential equations with vanishing delays

BIT Numerical Mathematics

Volumn 46, Issue 2, 2006, Pages 229-247

  Bellen, A ^a   Brunner, H ^b   Maset, S ^a   Torelli, L ^a  

^a UNIVERSITY OF TRIESTE   (Italy)

^b MEMORIAL UNIVERSITY OF NEWFOUNDLAND   (Canada)

Author keywords

Collocation methods; Functional differential equations; Optimal order of superconvergence; Order reduction; Quasi graded meshes; Vanishing delays; Volterra integro differential equations

Indexed keywords

EID: 33745322100     PISSN: 00063835     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10543-006-0055-2     Document Type: Article

References (26)

1. G. Andreoli, Sulle equazioni integrali, Rend. Circ. Mat. Palermo, II. Ser., 37 (1914), 76-112.
2. A. Bellen, One-step collocation for delay differential equations, J. Comput. Appl. Math., 10 (1984), 275-283.
3. A. Bellen, Constrained mesh methods for functional-differential equations, in Delay Equations, Approximation and Application, G. Meinardus and G. Nürnberger, eds., Int. Ser. Numer. Math., vol. 74, pp. 52-70, Birkhäuser, Basel-Boston, 1985.
4. A. Bellen, Preservation of superconvergence in the numerical integration of delay differential equations with proportional delay, IMA J. Numer. Anal., 22 (2002), 529-536.
5. A. Bellen, N. Guglielmi, and L. Torelli, Asymptotic stability properties of θ-methods for the pantograph equation, Appl. Numer. Math., 24 (1997), 275-293.
6. A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations, Oxford University Press, Oxford, 2003.
7. J. M. Bownds, J. M. Gushing, and R. Schutte, Existence, uniqueness, and extendibility of solutions to Volterra integral systems with multiple variable delays, Funke. Ekvacioj, Ser. Int., 19 (1976), 101-111.
8. H. Brunner, The numerical solution of neutral Volterra integro-differential equations with delay arguments, Ann. Numer. Math., 1 (1994), 309-322.
9. H. Brunner, On the discretization of differential and Volterra integral equations with variable delay, BIT, 37 (1997), 1-12.
10. H. Brunner, Collocation Methods for Volterra Integral and Related Functional Equations, Cambridge University Press, Cambridge, 2004.
11. H. Brunner and Q.-Y. Hu, Superconvergence of iterated collocation solutions for Volterra integral equations with variable delays, SIAM J. Numer. Anal., 43 (2005), 1943-1949.
12. H. Brunner, Q.-Y. Hu, and Q. Lin, Geometric meshes in collocation methods for Volterra integral equations with proportional delays, IMA J. Numer. Anal., 21 (2001), 783-798.
13. λxg(x, y, f(y))dy, Int. J. Math. Math. Sci., 14 (1990), 27-44.
14. A. M. Denisov and A. Lorenzi, On a special Volterra integral equation of the first kind, Boll. Unione Mat. Ital., VII. Ser., B (7), 9 (1995), 443-157.
15. A. Iserles, Numerical analysis of delay differential equations with variable delays, Ann. Numer. Math., 1 (1994), 133-152.
16. A. Iserles and Y. Liu, On pantograph integro-differential equations, J. Integral Equations Appl., 6 (1994), 213-237.
17. E. Ishiwata, On the attainable order of collocation methods for the neutral functional-differential equations with proportional delays, Computing, 64 (2000), 207-222.
18. Y. Liu, On θ-methods for delay differential equations with infinite lag, J. Comput. Appl. Math., 71 (1996), 177-190.
19. S. Maset, L. Torelli, and R. Vermiglio, Runge-Kutta methods for retarded functional differential equations, Math. Models Methods Appl. Sci., 15 (2005), 1203-1251.
20. Y. Muroya, E. Ishiwata, and H. Brunner, On the attainable order of collocation methods for pantograph integro-differential equations, J. Comput. Appl. Math., 152 (2003), 347-366.
21. N. Takama, Y. Muroya, and E. Ishiwata, On the attainable order of collocation methods for the delay differential equations with proportional delay, BIT, 40 (2000), 374-394.
22. V. Volterra, Sopra alcune questioni di inversione di integrale definiti, Ann. Mat. Pure Appl. (2), 25 (1897), 139-178.
23. V. Volterra, Leçons sure les équations intégrales et les équations intégro-différentielles, Gauthier-Villars, Paris, 1913.
24. M. Zennaro, Natural continuous extensions of Runge-Kutta methods, Math. Comput., 46 (1986), 119-133.
25. W. Zhang, Numerical Analysis of Delay Differential and Integro-Differental Equations, Ph.D. Thesis, Dept. of Mathematics & Statistics, Memorial University of Newfoundland, St. John's, NL, 1998.
26. W. Zhang and H. Brunner, Collocation approximations for second-order differential equations and Volterra integro-differential equations with variable delays, Canad. Appl. Math. Quart., 6 (1998), 269-285.