On the numerical solutions of some Volterra equations on infinite intervals

Authors

  • Olavi Nevanlinna Helsinki University of Technology, Institute of Mathematics, Netherlands

Keywords:

Volterrra equations, positive quadratures, \(A\)-stable multistep methods, monotone mappings. MSC, 65R99, 65L04, 65L20.

Abstract

The paper discusses long time behavior and error bounds for discretized Volterra equations. A key property is positivity of the quadrature which is combined with monotonicity properties of the nonlinearities in the equations. It is shown how the positivity of discretization quadrature is linked with \(A\)-stability property of linear multistep methods for ordinary differential equations. Some of the results are new when applied to differential equations with monotone nonlinearities and \(A\)-stable discretizations.

References

[1] Brunner, H. and Lambert, J.D., Stability of numerical methods for Voterra integro-differential equations. Computing, 12, 75 (1974).

[2] Dahlquist, G., A special stability problem for linear mutlistep methods, BIT, 3, 27, (1963).

[3] Dahlquist, G., Error analysis for a class of methods for stiff non-linear initial value problems. To be published in The Proceedings of the conference on numerical analysis, Dundee, 1975.

[4] Edwards, R.E., Fourier series: A modern introduction. Vol. II. New York: Holt Rinehart and Winston, Inc., 1967.

[5] Garey, L., The numerical solution of Volterra integral equations with singular kernels. BIT, 14, 33 (1974).

[6] De Hoog, F. and Weiss, R., Implicit Runge-Kutta methods for second kind Volterra integral equations. Numer. Math., 23, 199 (1975).

[7] Katznelson, Y., An introduction to harmonic analysis. New-York: Wiley Inc., 1968.

[8] Malina, L., \(A\)-stable methods of high order for Volterra integral equations. Aplikace matematiky, 20, 336 (1975).

[9] Matthys, S., \(A\)-stable linear multistep methods for Volterra integro-differential equations. To appear in Numer. Math.

[10] Nevanlinna, O., On error bounds for \(G\)-stable methods. To appear in BIT.

[11] Nevanlinna, O., Positive quadratures for Volterra equations. To appear in Computing.

[12] Staffans, O. J., Nonlinear Volterra integral equations with positive definite kernels. Proc. Amer. Math. Soc., 51, 103 (1975).
http://dx.doi.org/10.1090/S0002-9939-1975-0370081-8

[13] Staffans, O.J., Positive definite measures with applications to a Volterra equation. To appear in Trans. Amer. Math. Soc.

[14] Stetter, H.J., Analysis of discretization methods for ordinary differential equations. Berlin: Springer-Verlag, 1973.

[15] Stetter, H.J., Discretizations of differential equations on infinite intervals and applications to function minimization. In Topics in numerical analysis, Proceedings of the conference on numerical analysis, 1972, ed. J. J. H. Miller, New-York: Academic Press, 1973.

[16] Widder, D.V., The Laplace transform. Princeton: Princeton University Press, 1946.

Downloads

Published

1976-02-01

How to Cite

Nevanlinna, O. (1976). On the numerical solutions of some Volterra equations on infinite intervals. Anal. Numér. ThéOr. Approx., 5(1), 31-57. Retrieved from https://ictp.acad.ro/jnaat/journal/article/view/1976-vol5-no1-art4

Issue

Section

Articles