Abstracts

The perspectives of applications of mathematics in industry, economics, etc.

Full enough mathematical descriptions of laws of distribution of electric fields in electrolits, diffusion and electrode processes allow to create the mathematical models describing physical both electrochemical micro - and macrokinetic laws electrolyze metals from water solutions. The majority of practically significant problems of electrochemistry and electrochemical technology represent electrochemical systems in which the stream of the charged particles is defined mainly migratory and convective by components that results at their mathematical modelling to systems of the integral and differential equations of elliptic and parabolic type. The kind of system of the differential equations in partial derivatives is defined by type of environment in which electric and concentration fields are formed, namely its homogeneity or pseudo-homogeneity when to environment some average effective characteristics are appropriated. Complicated question at mathematical modelling processes of electrochemical sedimentation of metals on firm electrodes of electrochemical reactors is the question of formation and the mathematical description of boundary conditions, which, as a rule, represent boundary conditions of the mixed type including differential and integrated forms. The kind of a physical and chemical problem and its mathematical model in many respects defines a method of its decision. In case of passing of processes in homogeneous area, the researcher, as a rule, is interested of the processes occuring on border of the partition of phases, that is on electrodes of electrochemical system. In this case it is more favourable and in sense of accuracy of calculations, and in sense of minimization of time of the decision to use methods of the integral equations. In case of modelling electrochemical reactions in pseudo-homogeneous environments, it is necessary to use finite difference methods, or their combinations with methods of the integral equations. The situation is complicated with that the integral equations arising at modelling, frequently represent equations Fredgolm - I sorts, and physical effective parameters of model are from the decision of inverse problems of mathematical physics and not always have correct values. Besides in some cases of system of the differential equations (for example, for the description of pseudo-homogeneous reactions) are classically unstable, that complicates their decision. Nevertheless, application of existing methods of the decision of problems of mathematical physics and their specific updatings allow to describe well in many cases electrochemical processes on electrodes and in volume of electrolit, that, in turn, allows to put and solve problems of optimization and optimum control of electrochemical reactors. So, for example, the problem of optimum control of process of electrosedimentation of metal of the diluted galvanic solution on graphite fibrous electrode due to an optimum choice electric conductivity a material of an electrode, can be put as system from optimized functional and the restrictions consisting of the differential equations. To such statement the theory of optimum mathematical management, namely, a principle of a maximum of L.S.Pontrjagina is applicable. Calculations and experimental researches show efficiency of application of mathematical modelling and calculation for the decision of theoretical and practical problems of electrochemistry.

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