On the numerical solutions of some Volterra equations on infinite intervals

Olavi Nevanlinna

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 views: 555

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.

Downloads

Download data is not yet available.

References

Brunner, H., Lambert, J. D., Stability of numerical methods for Volterra integro-differential equations. Computing (Arch. Elektron. Rechnen) 12 (1974), no. 1, 75-89, MR0418490, https://doi.org/10.1007/bf02239501

Dahlquist, Germund G., A special stability problem for linear multistep methods. Nordisk Tidskr. Informations-Behandling 3 1963 27-43, MR0170477, https://doi.org/10.1007/bf01963532

Dahlquist, Germund, Error analysis for a class of methods for stiff non-linear initial value problems. Numerical analysis (Proc. 6th Biennial Dundee Conf., Univ. Dundee, Dundee, 1975), pp. 60--72. Lecture Notes in Math., Vol. 506, Springer, Berlin, 1976, MR0448898.

Edwards, R. E., Fourier series: a modern introduction. Vol. II. Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London 1967 ix+318 pp., MR0222538.

Garey, L., The numerical solution of Volterra integral equations with singular kernels. Nordisk Tidskr. Informationsbehandling (BIT) 14 (1974), 33-39, MR0373345, https://doi.org/10.1007/bf01933115

de Hoog, F.; Weiss, R., Implicit Runge-Kutta methods for second kind Volterra integral equations. Numer. Math. 23 (1974/75), 199-213, MR0373349, https://doi.org/10.1007/bf01400303

Katznelson, Yitzhak, An introduction to harmonic analysis. John Wiley & Sons, Inc., New York-London-Sydney 1968 xiv+264 pp., MR0248482.

Malina, L'ubor, A-stable methods of high order for Volterra integral equations. Apl. Mat. 20 (1975), no. 5, 336-344, MR0386320, https://doi.org/10.21136/am.1975.103599

Matthys, J., A-stable linear multistep methods for Volterra integro-differential equations. Numer. Math. 27 (1976/77), no. 1, 85-94, MR0436638, https://doi.org/10.1007/bf01399087

Nevanlinna, Olavi, On error bounds for G-stable methods. Nordisk Tidskr. Informationsbehandling (BIT) 16 (1976), no. 1, 79-84, MR0488767, https://doi.org/10.1007/bf01940780

Nevanlinna, O., Positive quadratures for Volterra equations. (German summary) Computing 16 (1976), no. 4, 349-357, MR0408279, https://doi.org/10.1007/bf02252083

Staffans, Olof J., Nonlinear Volterra integral equations with positive definite kernels. Proc. Amer. Math. Soc. 51 (1975), 103-108, MR0370081, https://doi.org/10.1090/s0002-9939-1975-0370081-8

Staffans, Olof J., Positive definite measures with applications to a Volterra equation. Trans. Amer. Math. Soc. 218 (1976), 219-237, MR0458086, https://doi.org/10.1090/s0002-9947-1976-0458086-5

Stetter, Hans J., Analysis of discretization methods for ordinary differential equations. Springer Tracts in Natural Philosophy, Vol. 23. Springer-Verlag, New York-Heidelberg, 1973. xvi+388 pp., MR0426438 .

Stetter, Hans J., Discretizations of differential equations on infinite intervals and applications to function minimization. Topics in numerical analysis (Proc. Roy. Irish Acad. Conf., University Coll., Dublin, 1972), 277-284. Academic Press, London, 1973, MR0339501.

Widder, David Vernon, The Laplace Transform. Princeton Mathematical Series, v. 6. Princeton University Press, Princeton, N. J., 1941. x+406 pp., MR0005923 .