Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: RELIABILITY OF DIFFERENCE ANALOGUES TO PRESERVE STABILITY PROPERTIES OF STOCHASTIC VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS | RELIABILITY OF DIFFERENCE ANALOGUES TO PRESERVE STABILITY PROPERTIES OF STOCHASTIC VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS LEONID E. SHAIKHET AND JASON A. ROBERTS Received 2 August 2004 Revised 16 January 2005 Accepted 10 April 2005 We consider the reliability of some numerical methods in preserving the stability properties of the linear stochastic functional differential equation dx t ax t p J0tx s ds dt ơx t - T dW t where a p ơ T 0 are real constants and W t is a standard Wiener process. The areas of the regions of asymptotic stability for the class of methods considered indicated by the sufficient conditions for the discrete system are shown to be equal in size to each other and we show that an upper bound can be put on the time-step parameter for the numerical method for which the system is asymptotically mean-square stable. We illustrate our results by means of numerical experiments and various stability diagrams. We examine the extent to which the continuous system can tolerate stochastic perturbations before losing its stability properties and we illustrate how one may accurately choose a numerical method to preserve the stability properties of the original problem in the numerical solution. Our numerical experiments also indicate that the quality of the sufficient conditions is very high. Copyright 2006 L. E. Shaikhet and J. A. Roberts. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction Volterra integro-differential equations arise in the modelling of hereditary systems . systems where the past influences the present such as population growth pollution financial markets and mechanical systems see . 1 4 . The long-term behaviour and stability of such systems is an important area for investigation. For example will a population decline to dangerously low levels Could a small .
