Abstract
In recent works, exact and nonstandard finite difference schemes for scalar first order linear delay differential equations have been proposed. The aim of the present work is to extend these previous results to systems of coupled delay differential equations X ′ ( t ) = A X ( t ) + B X ( t - τ ) , where X is a vector, and A and B are commuting real matrices, in general not simultaneously diagonalizable. Based on a constructive expression for the exact solution of the vector equation, an exact scheme is obtained, and different nonstandard numerical schemes of increasing order are proposed. Dynamic consistency properties of the new nonstandard schemes are illustrated with numerical examples, and proved for a class of methods.
Highlights
Due to the presence of time lags in the dynamics of most real systems, delay differential equations (DDE) have become basic instruments in the mathematical modelling of a wide range of problems in science and engineering, such as in population biology, physiology, epidemiology, economics, and control problems, and special methods have been developed to compute numerical solutions for DDE [6]
Despite the growing interest in nonstandard finite difference (NSFD) methods, including their application to some problems with delay, the scheme presented in Theorem 2 is possibly the first example of an exact scheme for a system of delay differential equations, generalising to systems of linear DDE with commuting matrix coefficients the results presented in [23] for scalar linear DDE problems
The families of F M and T M schemes defined in Theorem 3 allow the computation of numerical solutions for problem (3)–(4) with high accuracy and low computational costs for extended time intervals, showing good dynamic consistency properties
Summary
Due to the presence of time lags in the dynamics of most real systems, delay differential equations (DDE) have become basic instruments in the mathematical modelling of a wide range of problems in science and engineering, such as in population biology, physiology, epidemiology, economics, and control problems (see, e.g., [1,2,3,4,5], and references therein), and special methods have been developed to compute numerical solutions for DDE [6]. The aim, and main contribution, of the present work is to make available, for a wide class of coupled linear delay differential systems, NSFD methods that possess analogous advantages to those in the scalar setting, exhibiting similar properties in terms of accuracy and dynamic consistency. Once the new NSFD schemes are defined, the process of proving dynamical properties, which is much more complex than in the scalar case To overcome these difficulties, the key point is to assume commutativity of the coupled coefficient matrices, a property considered in other problems involving delay systems [24]. The key point is to assume commutativity of the coupled coefficient matrices, a property considered in other problems involving delay systems [24] With this assumption, a compact expression for the exact solution of problem (3)–(4), analogous to the scalar case, can be obtained.
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