Grosser Systeme Echtzeitoptimierung Schwerpunktprogramm Der Deutschen Forschungsgemeinschaft

In this paper the development of a simulation envir nment for time delay di#er ential equations will bepr sented. After the analysis of the mathematical foundations, the pr ctical implementation using the C++ pr gr amming language will be discussed. Simulations of a biomathematical time delay model of the human gr nulopoiesis ar pr sented and the paper concludes with an er2 r analysis and a discussion of simulation execution times. 1 Introducti6 Delay di#erential equations (DDE) are used to model a variety of phenomena in the physical andnaturalsciences[1,2,3,4,5,6,7,8,9,10,11]. Alsointhemodelingofbiological systems time delays occur which can be modeled using DDEs.BC7 use of this large number of applications it is of special interest to develop a simulation tool for DDEs. In previous work of Feldstein and Neves [1] high-order methods for the integration of DDEs with time and state dependent time delays have been derived. In this paper we concentrate on the numerical implementation of these methods using the C++ programming language where the class concept will be presented. The analysis of dicretization errors will show high accuracy while simulation results show excellent e#ciency of our implementation. We consider time delay di#erential equations with state vector x(t)=(x 1 (t), ..., x n (t)) T and state space representation x i (t)=f i (t, x(t),x(# 1 (t, x(t))),...,x(# r (t, x(t)))) for t [t 0 ,t e ] ,i=1, ..., n (1) with r retarding functions # j (t, x(t)) [t 0 ,t e ],j =1, ..., r (2) # Cor79 onding author Phone ++49 731 50-26323, Fax ++49 731 50-26301, Email frank.lehn@ chnik.uni-ulm.de which are allowed to depend on time and state. The initial value problem is given by an initial function x(t)=#(t)fort t 0 (3) which describes the history of th..