Reaction–diffusion equation with nonlocal boundary condition subject to PID-controlled bioreactor
Feliksas Ivanauskas
Vilnius University
Valdas Laurinavičius
Vilnius University
Mifodijus Sapagovas
Vilnius University
Anatolij Neciporenko
Vilnius University
Published 2017-03-15


mathematical modeling
nonlocal boundary condition,
immobilized enzyme
PID controller

How to Cite

Ivanauskas F., Laurinavičius V., Sapagovas M. and Neciporenko A. (2017) “Reaction–diffusion equation with nonlocal boundary condition subject to PID-controlled bioreactor”, Nonlinear Analysis: Modelling and Control, 22(2), pp. 261-272. doi: 10.15388/NA.2017.2.8.


We study a system of two parabolic nonlinear reaction–diffusion equations subject to a nonlocal boundary condition. This system of nonlinear equations is used for mathematical modeling of biosensors and bioreactors. The integral-type nonlocal boundary condition links the solution on the system boundary to the integral of the solution within the system inner range. This integral plays an important role in the nonlocal boundary condition and in the general formulation of the boundary value problem. The solution at boundary points is calculated using the integral combined with the proportional-integral-derivative controller algorithm. The mathematical model was applied for the modeling and control of drug delivery systems when prodrug is converted into active form in the enzyme-containing bioreactor before the delivering into body. The linear, exponential, and stepwise protocols of drug delivery were investigated, and the corresponding mathematical models for the prodrug delivery were created.

*The research was partially supported by the Research Council of Lithuania (grant No. MIP-047/2014).

Creative Commons License

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

Please read the Copyright Notice in Journal Policy

Most read articles by the same author(s)