Difference Equations as Difference Analogues of Differential Equations

Abstract

Functional differential equations arise in the modeling of hereditary systems such as ecological and biological systems, chemical and mechanical systems and many-many other. The long-term behavior and stability of such systems is an important area for investigation. For example, will a population decline to dangerously low levels? Could a small change in the environmental conditions have drastic consequences on the long-term survival of the population? There is a growing body of works devoted to such type of investigations. Analytical solutions of functional differential equations are generally unavailable and a lot of different numerical methods are adopted for obtaining approximate solutions. A natural question to ask is “do the numerical solutions preserve the stability properties of the exact solution?” Thus, to use numerical investigation of functional differential equations it is very important to know if the considered difference analogue of the original differential equation has the reliability to preserve some general properties of this equation, in particular, property of stability. In this chapter the capability of difference analogues of differential equations to save a property of stability of solutions of considered differential equations is studied. In particular, sufficient conditions for the step of discretization, at which stability of difference analogue solution is saved, are obtained for some known mathematical models, such that as the mathematical model of controlled inverted pendulum, Nicholson blowflies equation, predator-prey model, for difference analogue of an integro-differential equation of convolution type.

Keywords

Stability Region Trivial Solution Functional Differential Equation Inverted Pendulum Discrete Analogue

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