Qualitative behaviour of numerical approximations to Volterra integro-differential equations

Authors:
Yihong Song;Christopher T. H. Baker
Affiliations:
Department of Mathematics, Suzhou University, Suzhou, Jiangsu 215006, PR China;Department of Mathematics, The Victoria University of Manchester, Oxford Road, Manchester M13 9PL, UK and University College Chester, CH1 4BJ, UK
Venue:
Journal of Computational and Applied Mathematics
Year:
2004

Citing 3
Cited 3

The numerical treatment of Volterra integro-differential equations with unbounded delay

Journal of Computational and Applied Mathematics
An introduction to difference equations

An introduction to difference equations
Stability analysis of continuous implicit Runge-Kutta methods for Volterra integro-differential systems with unbounded delays

Selected papers of the second international conference on Numerical solution of Volterra and delay equations : Volterra centennial: Volterra centennial

Stability for a class of difference equations

Journal of Computational and Applied Mathematics
Stochastic Lotka-Volterra system with infinite delay

Journal of Computational and Applied Mathematics
Suppression of functional system with Markovian switching

ICONIP'12 Proceedings of the 19th international conference on Neural Information Processing - Volume Part II

Quantified Score

Hi-index	7.29

Visualization

Abstract

In this paper, we investigate the qualitative behaviour of numerical approximations to a nonlinear Volterra integro-differential equation with unbounded delay. We consider the simple single-species growth model d/dt N(t) = λN(t)(1 - c-1 ∫t-∞t k(t - s)N(s)ds). We apply the (composite) θ-rule as a quadrature to discretize the equation. We are particularly concerned with the way in which the long-term qualitative properties of the analytical solution can be preserved in the numerical approximation. Using results in (S.N. Elaydi and S. Murakami, J. Differ. Equations Appl. 2 (1996) 401; Y. Song and C.T.H. Baker, J. Differ. Equations Appl. 10 (2004) 379) for Volterra difference equations, we show that, for a small bounded initial function and a small step size, the corresponding numerical solutions display the same qualitative properties as found in the original problem. In the final section of this paper, we discuss how the analysis can be extended to a wider class of Volterra integral equations and Volterra integro-differential equations with fading memory.