The numerical solution of boundary-value problems for differential equations with state dependent deviating arguments

Vernon L. Bakke; Zdzisław Jackiewicz

Aplikace matematiky (1989)

  • Volume: 34, Issue: 1, page 1-17
  • ISSN: 0862-7940

Abstract

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A numerical method for the solution of a second order boundary value problem for differential equation with state dependent deviating argument is studied. Second-order convergence is established and a theorem about the asymptotic expansion of global discretization error is given. This theorem makes it possible to improve the accuracy of the numerical solution by using Richardson extrapolation which results in a convergent method of order three. This is in contrast to boundary value problems for ordinary differential equations where the use of Richardson extrapolation results in a method of order four.

How to cite

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Bakke, Vernon L., and Jackiewicz, Zdzisław. "The numerical solution of boundary-value problems for differential equations with state dependent deviating arguments." Aplikace matematiky 34.1 (1989): 1-17. <http://eudml.org/doc/15560>.

@article{Bakke1989,
abstract = {A numerical method for the solution of a second order boundary value problem for differential equation with state dependent deviating argument is studied. Second-order convergence is established and a theorem about the asymptotic expansion of global discretization error is given. This theorem makes it possible to improve the accuracy of the numerical solution by using Richardson extrapolation which results in a convergent method of order three. This is in contrast to boundary value problems for ordinary differential equations where the use of Richardson extrapolation results in a method of order four.},
author = {Bakke, Vernon L., Jackiewicz, Zdzisław},
journal = {Aplikace matematiky},
keywords = {second order; difference operator; second order convergence; asymptotic expansion; global discretization error; numerical examples; boundary value problem; deviating argument; Richardson extrapolation; convergence of higher order; second order; difference operator; Second order convergence; asymptotic expansion; global discretization error; numerical examples},
language = {eng},
number = {1},
pages = {1-17},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The numerical solution of boundary-value problems for differential equations with state dependent deviating arguments},
url = {http://eudml.org/doc/15560},
volume = {34},
year = {1989},
}

TY - JOUR
AU - Bakke, Vernon L.
AU - Jackiewicz, Zdzisław
TI - The numerical solution of boundary-value problems for differential equations with state dependent deviating arguments
JO - Aplikace matematiky
PY - 1989
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 34
IS - 1
SP - 1
EP - 17
AB - A numerical method for the solution of a second order boundary value problem for differential equation with state dependent deviating argument is studied. Second-order convergence is established and a theorem about the asymptotic expansion of global discretization error is given. This theorem makes it possible to improve the accuracy of the numerical solution by using Richardson extrapolation which results in a convergent method of order three. This is in contrast to boundary value problems for ordinary differential equations where the use of Richardson extrapolation results in a method of order four.
LA - eng
KW - second order; difference operator; second order convergence; asymptotic expansion; global discretization error; numerical examples; boundary value problem; deviating argument; Richardson extrapolation; convergence of higher order; second order; difference operator; Second order convergence; asymptotic expansion; global discretization error; numerical examples
UR - http://eudml.org/doc/15560
ER -

References

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  1. V. L. Bakke Z. Jackiewicz, A note on the numerical computation of solutions to second order boundary value problems with state dependent deviating arguments, University of Arkansas Numerical Analysis Technical Report 65110-1, June, 1985. (1985) 
  2. B. Chartres R. Stepleman, Convergence of difference methods for initial and boundary value problems with discontinuous data, Math. Соmр., v. 25, 1971, pp. 724-732. (1971) MR0303739
  3. P. Chocholaty L. Slahor, 10.1007/BF01396496, Numer. Math., v. 33, 1979, pp. 69-75. (1979) MR0545743DOI10.1007/BF01396496
  4. K. De Nevers K. Schmitt, 10.1016/0022-247X(71)90041-2, J. Math. Anal. Appl., v. 36, 1971, pp. 588-597. (1971) MR0298166DOI10.1016/0022-247X(71)90041-2
  5. L. J. Grimm K. Schmitt, 10.1090/S0002-9904-1968-12114-7, Bull. Amer. Math. Soc., v. 74, 1968, pp. 997-1000. (1968) MR0228785DOI10.1090/S0002-9904-1968-12114-7
  6. L. J. Grimm K. Schmitt, 10.1007/BF01817758, Aequationes Math., v. 4, 1970, p. 176-190. (1970) MR0262632DOI10.1007/BF01817758
  7. G. A. Kamenskii S. B. Norkin L. E. Eľsgoľts, Some directions of investigation on the theory of differential equations with deviating arguments, (Russian). Trudy Sem. Tear. Diff. Urav. Otklon. Arg., v. 6, pp. 3-36. 
  8. H. B. Keller, Numerical methods for two-point boundary-value problems, Blaisdel Publishing Company, Waltham 1968. (1968) Zbl0172.19503MR0230476

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