Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Iterated collocation methods for Volterra integral equations with delay arguments


Author: Hermann Brunner
Journal: Math. Comp. 62 (1994), 581-599
MSC: Primary 65R20; Secondary 45L10
DOI: https://doi.org/10.1090/S0025-5718-1994-1213835-8
MathSciNet review: 1213835
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we give a complete analysis of the global convergence and local superconvergence properties of piecewise polynomial collocation for Volterra integral equations with constant delay. This analysis includes continuous collocation-based Volterra-Runge-Kutta methods as well as iterated collocation methods and their discretizations.


References [Enhancements On Off] (What's this?)

  • [1] H. Arndt and C. T. H. Baker, Runge-Kutta formulae applied to Volterra functional equations with fixed delay, Numerical Treatment of Differential Equations: 4th Internat. Seminar NUMDIFF (Halle-Wittenberg, 1987), Teubner-Texte Math., vol. 104, Leipzig, 1988, pp. 19-30. MR 1065161 (91g:65173)
  • [2] C. T. H. Baker and M. S. Derakhshan, R-K formulae applied to Volterra equations with delay, J. Comput. Appl. Math. 29 (1990), 293-310. MR 1051791 (91i:65207)
  • [3] J. Banaś, An existence theorem for nonlinear Volterra integral equation with deviating argument, Rend. Circ. Mat. Palermo (2) 35 (1986), 82-89. MR 880665 (88i:45005)
  • [4] A. Bellen, One-step collocation for delay differential equations, J. Comput. Appl. Math. 10 (1984), 275-283. MR 755804 (85m:65061)
  • [5] -, Constrained mesh methods for functional differential equations, Delay Equations, Approximation and Applications (G. Meinardus and G. Nürnberger, eds.), Internat. Ser. Numer. Math., vol. 74, Birkhäuser Verlag, Basel, 1985, pp. 52-70. MR 899088 (88f:65226)
  • [6] J. G. Blom and H. Brunner, The numerical solution of nonlinear Volterra integral equations of the second kind by collocation and iterated collocation methods, SIAM J. Sci. Statist. Comput. 8 (1987), 806-830. MR 902744 (89f:65138)
  • [7] J. M. Bownds, J. M. Cushing, and R. Schutte, Existence, uniqueness, and extendibility of solutions of Volterra integral systems with multiple, variable lags, Funkcial. Ekvac. 19 (1976), 101-111. MR 0427975 (55:1005)
  • [8] H. Brunner, Iterated collocation methods and their discretizations for Volterra integral equations, SIAM J. Numer. Anal. 21 (1984), 1132-1145. MR 765511 (86d:65160)
  • [9] -, On discrete superconvergence properties of spline collocation methods for nonlinear Volterra integral equations, J. Comput. Math. 10 (1992), 348-357. MR 1187686 (93k:65107)
  • [10] H. Brunner and P. J. van der Houwen, The numerical solution of Volterra equations, CWI Monograph 3, North-Holland, Amsterdam, 1986. MR 871871 (88g:65136)
  • [11] B. Cahlon, On the numerical stability of Volterra integral equations with delay argument, J. Comput. Appl. Math. 33 (1990), 97-104. MR 1081245 (92a:65343)
  • [12] B. Cahlon and L. J. Nachman, Numerical solutions of Volterra integral equations with a solution dependent delay, J. Math. Anal. Appl. 112 (1985), 541-562. MR 813619 (87h:65219)
  • [13] K. L. Cooke, An epidemic equation with immigration, Math. Biosci. 29 (1976), 135-158. MR 0682252 (58:33130)
  • [14] R. Esser, Numerische Behandlung einer Volterraschen Integralgleichung, Computing 19 (1978), 269-284. MR 0520367 (58:25036)
  • [15] S. Sugiyama, On functional integral equations, Mem. School Sci. Engrg. Waseda Univ. 41 (1977), 135-153. MR 536577 (80g:45004a)
  • [16] L. Torelli and R. Vermiglio, On the stability of continuous quadrature rules for differential equations with r delays, IMA J. Numer. Anal. 13 (1993), 291-302. MR 1210827 (93m:65099)
  • [17] P. Vâţa, Convergence theorems of some numerical approximation schemes for the class of non-linear integral equations, Bul. Univ. Galaţi Fasc. II Mat. Fiz. Mec. Teoret. 1 (1978), 25-33. MR 571913 (81h:65135)
  • [18] R. Vermiglio, A one-step subregion method for delay differential equations, Calcolo 22 (1985), 429-455. MR 859086 (88a:65087)
  • [19] -, On the stability of Runge-Kutta methods for delay integral equations, Numer. Math. 61 (1992), 561-577. MR 1155339 (93b:65214)
  • [20] M. Zennaro, Natural extensions of Runge-Kutta methods, Math. Comp. 46 (1986), 119-133. MR 815835 (86m:65083)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65R20, 45L10

Retrieve articles in all journals with MSC: 65R20, 45L10


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1994-1213835-8
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society