Browse Journals

Functional differential equations arise in the modelling of hereditary systems such as ecological and biological systems, chemical and mechanical systems and many-many other. The long-term behaviour and stability of such systems are important areas for investigation. Analytical solutions of functional differential equations are generally unavailable and a lot of different numerical methods are adopted for obtaining approximate solutions. A quite natural question appears: “Do 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. Here the ability of difference analogues of the nonlinear integro-differential equation of convolution type to preserve the property of stability of solutions is studied. Several difference analogues are considered both with discrete and with continuous time. Besides, difference analogues of the considered differential equation under stochastic perturbations are studied too. For stability investigation, we employ the general method of constructing Lyapunov functionals. It is shown how the obtained research can be applied to the various mathematical models.

difference analogue, discrete and continuous time, Lyapunov functional, Ito’s stochastic differential equation, stability, controlled inverted pendulum, Nicholson’s blowflies equation.