# Approximate Solutions of Linearized Delay Differential Equations Arising from a Microbial Fermentation Process Using the Matrix Lambert Function

## Authors

• Agus Yodi Gunawan Industrial & Financial Mathematics Research Group, Institut Teknologi Bandung, Jl. Ganesha 10, Bandung 40132, Indonesia
• Kasbawati Kasbawati Industrial & Financial Mathematics Research Group, Institut Teknologi Bandung, Jl. Ganesha 10, Bandung 40132, Indonesia Department of Mathematics, Hasanuddin University, Jl. P. Kemerdekaan Km. 10 Makassar 90245, Indonesia
• Kuntjoro Adji Sidarto Industrial & Financial Mathematics Research Group, Institut Teknologi Bandung, Jl. Ganesha 10, Bandung 40132, Indonesia

## Keywords:

Cuckoo Search algorithm, linearized delay differential system, microbial fermentation process, the Lambert function, the zeroth mode

## Abstract

In this paper we present approximate solutions of linearized delay differential equations using the matrix Lambert function. The equations arise from a microbial fermentation process in a metabolic system. The delay term appears due to the existence of a rate-limiting step in the fermentation pathway. We find that approximate solutions can be written as a linear combination of the Lambert function solutions in all branches. Simulations are presented for three cases of the ratio of the rate of glucose supply to the maximum reaction rate of the enzyme that experienced delay. The simulations are worked out by taking the principal branch of the matrix Lambert function as the most dominant mode. Our present numerical results show that the zeroth mode approach is quite reliable compared to the results given by classical numerical simulations using the Runge-Kutta method.

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2016-04-01

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