Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Linear multistep methods for functional-differential equations
HTML articles powered by AMS MathViewer

by Maarten de Gee PDF
Math. Comp. 48 (1987), 633-649 Request permission

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
    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.
  • Herbert Arndt, Numerical solution of retarded initial value problems: local and global error and stepsize control, Numer. Math. 43 (1984), no. 3, 343–360. MR 738381, DOI 10.1007/BF01390178
  • 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.
  • Roland Bulirsch and Josef Stoer, Numerical treatment of ordinary differential equations by extrapolation methods, Numer. Math. 8 (1966), 1–13. MR 191095, DOI 10.1007/BF02165234
  • E. Fehlberg, Klassische Runge-Kutta-Formeln fünfter und siebenter Ordnung mit Schrittweiten-Kontrolle, Computing (Arch. Elektron. Rechnen) 4 (1969), 93–106 (German, with English summary). MR 260179, DOI 10.1007/bf02234758
  • Alan Feldstein and Kenneth W. Neves, High order methods for state-dependent delay differential equations with nonsmooth solutions, SIAM J. Numer. Anal. 21 (1984), no. 5, 844–863. MR 760621, DOI 10.1137/0721055
  • Maarten de Gee, Smoothness of solutions of functional-differential equations, J. Math. Anal. Appl. 107 (1985), no. 1, 103–121. MR 786015, DOI 10.1016/0022-247X(85)90356-7
  • Maarten de Gee, Asymptotic expansions for the midpoint rule applied to delay differential equations, SIAM J. Numer. Anal. 23 (1986), no. 6, 1254–1272. MR 865955, DOI 10.1137/0723085
  • M. de Gee, "The Gragg-Bulirsch-Stoer algorithm for delay differential equations." (To appear.)
  • B. A. Gottwald and G. Wanner, A reliable Rosenbrock integrator for stiff differential equations, Computing 26 (1981), no. 4, 355–360 (English, with German summary). MR 620404, DOI 10.1007/BF02237954
  • J. D. Lambert, Computational methods in ordinary differential equations, John Wiley & Sons, London-New York-Sydney, 1973. Introductory Mathematics for Scientists and Engineers. MR 0423815
  • H. J. Oberle and H. J. Pesch, Numerical treatment of delay differential equations by Hermite interpolation, Numer. Math. 37 (1981), no. 2, 235–255. MR 623043, DOI 10.1007/BF01398255
  • Jesper Oppelstrup, The RKFHB4 method for delay-differential equations, Numerical treatment of differential equations (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1976) Lecture Notes in Math., Vol. 631, Springer, Berlin, 1978, pp. 133–146. MR 0494955
  • Lucio Tavernini, Linear multistep methods for the numerical solution of Volterra functional differential equations, Applicable Anal. 3 (1973), 169–185. MR 398131, DOI 10.1080/00036817308839063
Similar Articles
  • Retrieve articles in Mathematics of Computation with MSC: 65Q05
  • Retrieve articles in all journals with MSC: 65Q05
Additional Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Math. Comp. 48 (1987), 633-649
  • MSC: Primary 65Q05
  • DOI: https://doi.org/10.1090/S0025-5718-1987-0878696-1
  • MathSciNet review: 878696