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).
A mathematical model for the synthesis of yttrium aluminium garnet (Y 3 Al 5 O 12 , YAG) is presented in the article. The preparation of YAG by two synthesis methods, namely the sol-gel chemistry approach and the solid-state reaction method is considered. The model of YAG synthesis is based on the system of non-stationary diffusion equations containing the nonlinear terms related to the kinetics of the reaction and diffusion rates. The periodisation of the synthesis space is considered. In this study, the computer simulation tool was developed to solve the system of partial differential equations. The developed software is employed to investigate the influence of the concentration of YAG on diffusion rates and the synthesis duration as well as on the duration when the rate of change in reaction product mass is at a maximum.
A mathematical model of nitrate removal in woodchip denitrification bioreactor based on field experiment measurements was developed in this study. The approach of solving inverse problem for nonlinear system of differential convection-reaction equations was applied to optimize the efficiency of nitrate removal depending on bioreactor’s length and flow rate. The approach was realized through the developed algorithm containing a nonlocal condition with an incorporated PI controller. This allowed to adjust flow rate for varying inflow nitrate concentrations by using PI controller. The proposed model can serve as a useful tool for bioreactor design. The main outcome of the model is a mathematical relationship intended for bioreactor length selection when nitrate concentration at the inlet and the flow rate are known. Custom software was developed to solve the system of differential equations aiming to ensure the required nitrate removal efficiency.
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