In this paper we consider the problem of approximating the dynamical system that models reliability of a system consisting of two machines separated by a finite storage buffer. The system is described as a distributed parameter system defined by a coupled partial and ordinary differential equations and formulated as an abstract Cauchy problem. To derive the dynamical solution and some instantaneous indexes of the model, we present a simple finite difference scheme and establish the convergence of this scheme by employing Trotter-Kato Theorem. Numerical results are given to illustrate the effectiveness of the scheme.
Bibliographical noteFunding Information:
The authors thank Professor John Burns for his valuable help and advice on this paper. This work is supported by NSFC ( 11001013 ) and supported by the Interdisciplinary Center for Applied Mathematics during the first author’s visit at Virginia Tech. This work is also supported by the Air Force Office of Scientific Research Grants FA9550-07-1-0273 and FA9550-10-1-0201 and by the Environmental Security Technology Certification Program (ESTCP) under a subcontract from United Technologies. The authors would also like to thank the referees for their helpful comments and suggestions.
- Dynamical solution
- Trotter-Kato theorem