RELIABLE APPROXIMATE SOLUTION OF SYSTEMS OF VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS WITH TIME-DEPENDENT DELAYS

Volterra integro-differential equations with time-dependent delay arguments can provide us with realistic models of many real-world phenomena. Delayed Lokta--Volterra predator-prey systems arise in Ecology and are well-known examples of delay Volterra integro-differential equations (DVIDEs) first in...

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Bibliographic Details
Published in:SIAM journal on scientific computing 2011-01, Vol.33 (3-4), p.1134-1158
Main Authors: SHAKOURIFAR, M, ENRIGHT, W. H
Format: Article
Language:English
Subjects:
Approximation
Convergence
Defects
Delay
Differential equations
Exact sciences and technology
Mathematical analysis
Mathematical models
Mathematics
Measure and integration
Methods of scientific computing (including symbolic computation, algebraic computation)
Monitoring
Nonlinear algebraic and transcendental equations
Numerical analysis
Numerical analysis. Scientific computation
Ordinary differential equations
Sciences and techniques of general use
Strategy
Studies
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Summary:Volterra integro-differential equations with time-dependent delay arguments can provide us with realistic models of many real-world phenomena. Delayed Lokta--Volterra predator-prey systems arise in Ecology and are well-known examples of delay Volterra integro-differential equations (DVIDEs) first introduced by Volterra in 1928. We investigate the numerical solution of systems of DVIDEs using an adaptive stepsize selection strategy. We will present a generic variable stepsize approach for solving systems of neutral DVIDEs based on an explicit continuous Runge--Kutta method using defect error control and study the convergence of the resulting numerical method for various kinds of delay arguments. We will show that the global error of the numerical solution can be effectively and reliably controlled by monitoring the size of the defect of the approximate solution and adjusting the stepsize on each step of the integration. Numerical results will be presented to demonstrate the effectiveness of this approach. [PUBLICATION ABSTRACT]
ISSN:1064-8275
1095-7197
DOI:10.1137/100793098