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Journal of Applied Mathematics
Volume 2015, Article ID 864238, 8 pages
Research Article

Perturbation and Truncation of Probability Generating Function Methods for Stiff Chemical Reactions

Department of Mathematical Sciences, School of Natural Science, Ulsan National Institute of Science and Technology (UNIST), Ulsan Metropolitan City 689-798, Republic of Korea

Received 15 September 2014; Revised 3 February 2015; Accepted 6 February 2015

Academic Editor: Lotfollah Najjar

Copyright © 2015 Soyeong Jeong et al. 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.


One can reformulate chemical master equations of the stochastic reaction network into a partial differential equation (PDE) of a probability generating function (PGF). In this paper, we present two improvements in such PGF-PDE approach, based on perturbation and double-truncation, respectively. The stiff system that involves fast and slow reactions together often requires high computational cost. By applying the perturbation method to PGF-PDEs, we expand the equation in terms of a small reaction rate which is often responsible for such stiffness of the system. Also by doubly truncating, we dump relatively small terms and reduce the computational load significantly at each time step. The terms corresponding to rare events are sieved out through truncations of Taylor expansion. It is shown through numerical examples of enzyme kinetics, transition model, and Brusselator model that the suggested method is accurate and efficient for approximation of the state probabilities.