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Solving boundary value problems for delay differential equations by a fixed-point method

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2011年第6期236卷 1576-1590页

作者： Bartoszewski, Z. Gdansk Univ Technol Dept Appl Phys & Math PL-80233 Gdansk Poland

A general linear boundary value problem for a nonlinear system of delay differential equations (DDE in short) is reduced to a fixed-point problem v = Av with a properly chosen (generally nonlinear) operator A. The unknown fixed-point v is approximated by piecewise linear function v(h) defined by its values v(i) = v(h)(t(i)) at grid points t(i), i = 0, 1, ... , N, where N is a given positive integer and h = max(1 <= i <= N)(t(i) - t(i-1)). Under suitable assumptions, the existence of a fixed-point of A is equivalent to existence of so called epsilon(h)-approximate fixed-points of v(h) = Av(h), which can be found by minimization of L-2((n))) norm of residuum v(h) - Av(h), with respect to the variables v(i) These epsilon(h)-approximate fixed-points are used for obtaining approximate solutions of the original boundary value problem for a system of DDE. numerical experiments with the boundary value problem for a system of delay differential equations of population dynamics as well as with two periodic boundary value problems: one for the periodic distributed delay Lotka-Volterra competition system and the second one modeling two coupled identical neurons with time-delayed connections show an efficiency of this kind of approach. (C) 2011 Elsevier B.V. All rights reserved.

关键词： boundary value problems for differential equations with deviated arguments Fixed-point problems numerical methods for boundary value problems for differential equations with deviated arguments

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