Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



Stability of the discretized pantograph differential equation

Authors: Martin Buhmann and Arieh Iserles
Journal: Math. Comp. 60 (1993), 575-589
MSC: Primary 65L20; Secondary 34K20
MathSciNet review: 1176707
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study discretizations of the general pantograph equation

$\displaystyle y'(t) = ay(t) + by(\theta (t)) + cy'(\phi (t)),\quad t \geq 0,\quad y(0) = {y_0},$

, where a, b, c, and $ {y_0}$ are complex numbers and where $ \theta $ and $ \phi $ are strictly increasing functions on the nonnegative reals with $ \theta (0) = \phi (0) = 0$ and $ \theta (t) < t, \phi (t) < t$ for positive t. Our purpose is an analysis of the stability of the numerical solution with trapezoidal rule discretizations, and we will identify conditions on a, b, c and the stepsize which imply that the solution sequence $ \{ {y_n}\} _{n=0}^\infty $ is bounded or that it tends to zero algebraically, as a negative power of n.

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

  • [1] R. Bellman and K. L. Cooke, Asymptotic behavior of solutions of differential-difference equations, Mem. Amer. Math. Soc. No. 35 (1959). MR 0111917 (22:2775)
  • [2] -, Differential-difference equations, Academic Press, New York, 1963. MR 0146481 (26:4003)
  • [3] M. D. Buhmann and A. Iserles, On the dynamics of a discretized neutral equation, IMA J. Numer. Anal. 12 (1992), 339-363. MR 1181255 (93h:34142)
  • [4] -, Numerical analysis of functional equations with a variable delay, Numerical Analysis 1991 (D. F. Griffiths and G. A. Watson, eds.), Longman, Harlow, 1992, pp. 17-33. MR 1177226 (93g:65162)
  • [5] J. Carr and J. Dyson, The functional differential equation $ y'(x) = ay(\lambda x) + by(x)$, Proc. Roy. Soc. Edinburgh Sect. A 74 (1974-75), 165-174. MR 0442421 (56:803)
  • [6] G. A. Derfel, Kato problem for functional-differential equations and difference Schrödinger operators, Operator Theory 46 (1990), 319-321. MR 1124676 (92g:34093)
  • [7] A. Feldstein, A. Iserles, and D. Levin, Embedding of delay equations into an infinite-dimensional ODE system, Technical Report DAMTP NA21, University of Cambridge, 1991.
  • [8] A. Feldstein and Z. Jackiewicz, Unstable neutral functional differential equations, Arizona State University Technical Report (1989). MR 1091347 (91k:34099)
  • [9] L. Fox, D. F. Mayers, J. R. Ockendon, and A. B. Tayler, On a function differential equation, IMA J. Appl. Math. 8 (1971), 271-307. MR 0301330 (46:488)
  • [10] J. Hale, Theory of functional differential equations, Springer-Verlag, New York, 1977. MR 0508721 (58:22904)
  • [11] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Clarendon Press, Oxford, 1954. MR 0067125 (16:673c)
  • [12] A. Iserles, On the generalized pantograph functional-differential equation, European J. Appl. Math. (to appear). MR 1208418 (94f:34127)
  • [13] A. Iserles and J. Terjéki, Stability and asymptotic stability of functional-differential equations, Technical Report DAMTP NA1, University of Cambridge, 1992.
  • [14] T. Kato and J. B. McLeod, The functional-differential equation $ y'(x) = ay(\lambda x) + by(x)$, Bull. Amer. Math. Soc. 77 (1971), 891-937. MR 0283338 (44:570)
  • [15] K. Mahler, On a special functional equation, J. London Math. Soc. 15 (1940), 115-123. MR 0002921 (2:133e)
  • [16] G. Meinardus and G. Nürnberger, Approximation theory and numerical methods for delay equations, Delay Equations, Approximation and Application (G. Meinardus and G. Nürnberger, eds.), Birkhäuser-Verlag, Basel, 1986.
  • [17] G. R. Morris, A. Feldstein, and E. W. Bowen, The Phragmén-Lindelöf principle and a class of functional-differential equations, Ordinary Differential Equations (L. Weiss, ed.), Academic Press, New York, 1972. MR 0427771 (55:801)
  • [18] J. R. Ockendon and A. B. Tayler, The dynamics of a current collection system for an electric locomotive, Proc. Roy. Soc. London Ser. A 322 (1971), 447-468.
  • [19] G. Pólya and G. Szegő, Problems and theorems in analysis. Vol. 1, Springer-Verlag, Berlin and New York, 1972.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65L20, 34K20

Retrieve articles in all journals with MSC: 65L20, 34K20

Additional Information

Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society