Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



Linear multistep methods for functional-differential equations

Author: Maarten de Gee
Journal: Math. Comp. 48 (1987), 633-649
MSC: Primary 65Q05
MathSciNet review: 878696
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A new way to define linear multistep methods for functional differential equations is presented, and some of their properties are analyzed. The asymptotic behavior of the global discretization error is investigated. Finally, Milne's device is generalized to functional differential equations. The effect of the nonsmoothness of the exact solution is taken into account.

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

  • [1] H. Arndt, "The influence of interpolation on the global discretization error in retarded differential equations," in Differential Difference Equations (L. Collatz, G. Meinardus and W. Wetterling, eds.), ISNM 62, Birkhäuser Verlag, Basel, 1983.
  • [2] H. Arndt, "Numerical solution of retarded initial value problems: Local and global error and stepsize control," Numer. Math., v. 43, 1984, pp. 343-360. MR 738381 (85h:65142)
  • [3] H. G. Bock & J. Schlöder, "Numerical solution of retarded differential equations with state-dependent time-lags," Z. Angew. Math. Mech., v. 61, 1981, pp. 269-271.
  • [4] R. Bulirsch & J. Stoer, "Numerical treatment of ordinary differential equations by extrapolation methods," Numer. Math., v. 8, 1966, pp. 1-13. MR 0191095 (32:8504)
  • [5] E. Fehlberg, "Klassische Runge-Kutta-Formeln fünfter und siebenter Ordnung mit Schrittweiten-Kontrolle," Computing, v. 4, 1969, pp. 93-106. MR 0260179 (41:4807)
  • [6] A. Feldstein & K. W. Neves, "High order methods for state-dependent delay differential equations with nonsmooth solutions," SIAM J. Numer. Anal., v. 21, 1984, pp. 844-863. MR 760621 (86c:34127)
  • [7] M. de Gee, "Smoothness of solutions of functional differential equations," J. Math. Anal. Appl., v. 107, 1985, pp. 103-121. MR 786015 (86f:34144)
  • [8] M. de Gee, "Asymptotic expansions for the midpoint rule applied to delay differential equations," SIAM J. Numer. Anal., v. 23, 1986, pp. 1254-1272. MR 865955 (88c:65106)
  • [9] M. de Gee, "The Gragg-Bulirsch-Stoer algorithm for delay differential equations." (To appear.)
  • [10] B. A. Gottwald & G. Wanner, "A reliable Rosenbrock integrator for stiff differential equations," Computing, v. 26, 1981, pp. 355-360. MR 620404 (83d:65206)
  • [11] J. D. Lambert, Computational Methods in Ordinary Differential Equations, Wiley, New York, 1973. MR 0423815 (54:11789)
  • [12] H. J. Oberle & H. J. Pesch, "Numerical treatment of delay differential equations by Hermite interpolation," Numer. Math., v. 37, 1981, pp. 235-255. MR 623043 (83a:65077)
  • [13] J. Oppelstrup, The RKFHB4 Method for Delay Differential Equations, Lecture Notes in Math., vol. 631, Springer-Verlag, Berlin and New York, 1978. MR 0494955 (58:13730)
  • [14] L. Tavernini, "Linear multistep methods for the numerical solution of Volterra functional differential equations," Applicable Anal., v. 1, 1973, pp. 169-185. MR 0398131 (53:1986)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65Q05

Retrieve articles in all journals with MSC: 65Q05

Additional Information

Keywords: Functional differential equations, delay differential equations, linear multistep methods, predictor-corrector methods, Milne's device
Article copyright: © Copyright 1987 American Mathematical Society

American Mathematical Society