The Convergence and Stability of Splitting Finite-Difference Schemes for Nonlinear Evolutionary Equations

We consider a splitting finite-difference scheme for an initial-boundary value problem for a two-dimensional nonlinear evolutionary equation. The problem is split into nonlinear and linear parts. The linear part is also split into locally one-dimensional equations. We prove the convergence and stability of the scheme in L ₂ and C norms. Printed in Lietuvos Matematikos Rinkinys, Vol. 45, No. 3, pp. 413–434, July–September, 2005.     

Computational Modelling of the Behaviour of Potentiometric Membrane Biosensors

This paper presents a mathematical model of a potentiometric biosensor based on a potentiometric electrode covered with an enzyme membrane. The model is based on the reaction–diffusion equations containing a non-linear term related to theMichaelis–Menten kinetics of the enzymatic reaction. Using computer simulation the influence of the thickness of the enzyme membrane on the biosensor response was investigated. The digital simulation was performed using the finite difference technique. Results of the numerical simulation were compared with known analytical solutions.      

Modelling Dynamics of Amperometric Biosensors in Batch and Flow Injection Analysis

A mathematical model of amperometric biosensors has been developed. The model is based on non-stationary diffusion equations containing a non-linear term related to Michaelis–Menten kinetic of the enzymatic reaction. Using digital simulation, the influence of the substrate concentration as well as maximal enzymatic rate on the biosensor response was investigated. The digital simulation was carried out using the finite difference technique. The model describes the biosensor action in batch and flow injection regimes.     

On the modelling of solid state reactions.Synthesis of YAG

There is a model of yttrium aluminium garnet (YAG) synthesis presented in this article. The developed model is based on nonlinear reaction–diffusion partial differential equations. The solution was carried out numerically using finite difference techniques. We got dependability curves for diffusion and reaction rates and offered possible method to localize values of diffusion and reaction rate constants precisely enough.AMS subject classification: 35K57, 65M06     

Changes of the solution pH due to exposure by high-voltage electric pulses

The change of the pH of a NaCl solution (139-149 mM NaCl) buffered with 5-15 mM sodium phosphates (pH 7.4) during electromanipulation was studied. It has been determined that an increase in the pH value of electroporation solution of a whole chamber volume, caused by the application of electric field pulses, commonly used in cell electromanipulation procedures, can exceed 1-2 pH units. Several materials for the cathode were tested. In all cases a stainless steel anode was utilized. The aluminum cathode gave a two-fold greater DeltapH in comparison with platinum, copper or stainless steel cathodes. In addition, a substantial release of aluminum (up to 1 mg/l) from the cathode was observed. It has also been found that the shift in pH depended on the medium conductivity: DeltapH of the solution, in which sucrose was substituted for NaCl, was about 5 times less. On the basis of the results obtained here, to avoid the plausible undesirable consequences of the cathodic electrolysis processes, in particular under the conditions of strong electric treatment, it could be recommended that chambers with aluminum electrodes not be utilized and one should use strongly buffered solutions of low conductivity and alternating current (sine or square wave) bipolar electric pulses.     

Computer simulation of the steady state currents at enzyme doped carbon paste electrodes

The plate-gap model of porous enzyme doped electrode has been proposed and analyzed. It was suggested that reaction diffusion conditions in pores of bulk electrode resemble particular conditions in thin gap between parallel conducting plates. The model is based on the diffusion equations containing a nonlinear term related to the Michaelis–Menten kinetic of the enzymatic reaction inside gap. Steady state current was calculated for the wide range of given parameters and substrate concentrations. All dependences of current on substrate concentration were approximated by hyperbolas in order to obtain “apparent” parameters (maximal currents and apparent Michaelis constants) of modelled biosensors. Simple approximate relationships between given and apparent parameters were derived. The applicability of theoretical plate-gap model was tested for the case of carbon paste electrodes which were doped with PQQ – dependent glucose dehydrogenase. It was found, that soluble glucose dehydrogenase based biosensors exhibit characteristic features of the theoretical plate-gap biosensors.     

Comparison characterization of copper selenide thin layers prepared on polyamide 6 films by sorption-diffusion method

Some earlier synthesized copper selenide (Cu _x Se) layers formed on the surface of polyamide 6 by sorption-diffusion method using potassium selenotrithionate (K₂SeS₂O₆) as precursor of selenium were characterized by the XRD, XPS and SEM methods. According to the results of the SEM studies, the most uniform Cu _x Se layers form at the 2.5 h polyamide seleniumized duration at the temperature of 60°C. The thickness of layers, which dependeds on the duration of seleniumization, changed in the range of 0.8–3.2 µm. The XRD patterns of not previously studied Cu _x Se layers showed their phase composition of six copper selenides: Cu₂Se, two phases of CuSe₂, Cu₃Se₂, berzellianite, Cu_2-x Se, and bellidoite Cu₂Se. Analysis of the XRD and XPS data shows that the macrostructure and composition of the Cu_xSe layers depend on the conditions of formation of these layers.      

Weak Euler Approximation for Itô Diffusion and Jump Processes

This article studies the rate of convergence of the weak Euler approximation for Itô diffusion and jump processes with Hölder-continuous generators. It covers a number of stochastic processes including the nondegenerate diffusion processes and a class of stochastic differential equations driven by stable processes. To estimate the rate of convergence, the existence of a unique solution to the corresponding backward Kolmogorov equation in Hölder space is first proved. It then shows that the Euler scheme yields positive weak order of convergence.     

Modeling of electrocatalysis at polyaniline-modified electrodes: autoacceleration of the process by the reaction product

We extend our previous model taking into account a possible autoacceleration mechanism of electrochemical processes. It is supposed that protons generated in an electrochemical oxidation of ascorbate are responsible for an increase of electric conductivity of polyaniline. Accordingly, the current–time profiles have been calculated taking into account a “threshold” value for proton concentration to increase the electric conductivity of polyaniline by one or more orders of magnitude. Different kinds of corresponding profiles, including ones with a sigmoid character and possessing sharp current flashes as well, were obtained and analyzed.