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Cybernetics and programming
Reference:

The Regression Method of Reducing the Boundary Equation to the Cauchy Problem When Modeling Chemical Fiber Treatment

Kalabin Aleksandr Leonidovich

Doctor of Physics and Mathematics

professor of the Department of Computer Software at TvSTU (Tver State University)

170026, Russia, Tver Region, Tver, nab. Afanasiya Nikitina, 22

kalabin@tstu.tver.ru
Udalov Evgeniy Vadimovich

head of the Division of Enterprise Development at GUTA BANK

170100, Russia, Tver Region, Tver, Tverskoi prospekt, 6, of. 220

evgeny.udalov@mail.ru

DOI:

10.7256/2306-4196.2016.4.19735

Review date:

13-07-2016


Publish date:

26-08-2016


Abstract: The subject of the research is the numerical method implemented in the exploratory research program system applied during technical procedures of chemical fiber treatment. The aforesaid method allows to execute the selection of indeterminate initial conditions based on determinate boundary conditions, i.e. reduce the boundary equation to the Cauchy problem. The method also includes verification of numerical solutions through calculating the initial equation of force balance. This is a task that appears when solving a set of non-linear ordinary differential equations of second order (mathematical model of chemical fiber treatment) by using the 4th-order Runge-Kutta equation. The authors note that it is possible to apply a determinate shooting method together with dichotomy methods and Newton's (tangent) methods. The authors pay attention to the analysis of the shooting method, in particular, describe dependencies of the method iteration number on some initial conditions, of the initial rate - on the finishing fiber speed. The authors also give an insight into the regression method and compare efficiency of the regression method to the shooting method. The authors give a mathematical description of the process of chemical fiber treatment and select a mathematical model. Based on this model, the author carried out a computing experiment using the shooting method. The analysis of the results of the numerical modeling using the shooting method allowed to offer the regression model. The authors also describe drawbacks of using a determinate shooting method reducing the boundary equation to the Cauchy problem when modeling the process of chemical fiber treatment. The authors define regression equations relating a determinate boundary condition and indeterminate initial condition. The authors offer the regression method that converges by approximately 30 % faster than determinate dichotomy and Newton's methods. 


Keywords: treatment modeling, chemical fibers, computing experiment, numerical method, boundary equation, Cauchy Problem, regression method, exploratory research program, shooting method, mathematical modeling
This article written in Russian. You can find full text of article in Russian here .

References
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Kalabin A.L. Elongatsionnoe techenie strui rastvorov i rasplavov polimerov. Tver': TvGTU, 2011. 144 s.
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Zyabitskii A. Teoreticheskie osnovy formovaniya volokna. M.: Khimiya, 1979. 504 c.
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Kalabin, A.L. Programmnaya sistema issledovanii dinamiki tekhnologicheskikh protsessov formovaniya khimicheskikh volokon / A.L. Kalabin, E.V. Udalov, A.R. Khabarov // Programmnye produkty i sistemy. 2015. Vyp. 1. S. 139-144.
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Kalitkin N.N. Chislennye metody: uch. posobie. 2-e izd., ispr. SPb.: BKhV-Peterburg, 2011. 592 s.: il.
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Bakhvalov N.S., Zhidkov N.P., Kobel'kov G.M. Chislennye metody. M.: Laboratoriya Bazovykh Znanii, 2000. 624 s.
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Kalabin A.L., Udalov E.V. Dynamic characteristics of filament melt-spinning // Fibre Chemistry. 2013. 44. № 6. r. 356-360.
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Kalabin A.L., Udalov E.V. Modeling the Dynamics of the Aerodynamic Forming of Fibers // Fibre Chemistry. 2014. 46. № 1. p. 10-15.
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