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

Agus Yodi Gunawan, Kasbawati Kasbawati, Kuntjoro Adji Sidarto


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.


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

Full Text:



Kasbawati, Gunawan, A.Y., Hertadi, R. & Sidarto, K.A., Effects of Time Delay on the Dynamics of a Kinetic Model of a Microbial Fermentation Process, The ANZIAM J., 55, pp. 336-356, 2014.

Wright, E.M., Solution of the Equation . Bull. Am. Math. Soc., 65, 1959.

Asl, F.M. & Ulsoy, A.G., Analysis of a System of Linear Delay Differential Equations, Journal of Dynamic Systems, Measurement, and Control, 125, pp. 215-233, 2003.

Ulsoy, G., Solution of a System of Linear Delay Differential Equations Using the Matrix Lambert Function, American Control Conference, 4, pp. 2433-2438, 2006.

Nelson, P.W., Yi, S. & Ulsoy, A.G., Survey on Analysis of Time Delayed Systems via the Lambert w Function, Dynamics of Continuous, Discrete and Impulsive Systems, Series A, 14, pp. 296-301, 2007.

Ivanoviene, I. & Rimas, J., Analysis of Delay Differential Equations Using the Lambert Function, Mathematical Modelling and Analysis, x, pp. 1-15, 2010.

Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J. & Knuth, D.E., On the Lambert W Function, Advances in Computational Mathematics, 5, pp. 329-359, 1996.

Yi, S., Nelson, P.W. & Ulsoy, A.G., Analysis and Control Using the Lambert W Function, Time-delay Systems, World Scientific, 2010.

Shampine, L.F. & Reichelt, M.W., The MATLAB ODE Suite, SIAM Journal on Scientific Computing, 18, pp. 1-22, 1997.

Shampine, L.F., Solving ODEs and DDEs with Residual Control, Applied Numerical Mathematics, 52, pp. 113-127, 2005.

Yang, X-S. & Deb, S., Cuckoo Search via Levy Flights, In: Proc. of World Congress on Nature & Biologically Inspired Computing (NaBIC 2009), IEEE Publications, pp. 210-214, 2009.

Yang, X-S. & Deb, S., Engineering Optimization by Cuckoo Search, Int. J. Mathematical Modelling and Numerical Optimization, 1, pp. 330-343, 2010.



  • There are currently no refbacks.

View my Stats

Creative Commons License
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.