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Section
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Mathematics
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Title
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A new approach to the reaction-diffusion systems modelling
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Author(-s)
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Nazarov M.N.a
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Affiliations
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National Research University of Electronic Technologya
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Abstract
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We consider a new technique for modelling the reaction-diffusion systems based on systems of ordinary differential equations. In contrary to the specialized numerical methods such as straight line method, this new technique is positioned at model level as a full alternative for partial differential equations. The description of this new method is quite similar to the description of finite volume method, except that it uses statistical simplifications and principles of geometric probability to describe diffusion processes. The main goal of this approach is to simplify the qualitative analysis of reaction-diffusion systems and to increase the efficiency of the numerical implementation. The first task is successfully resolved because of the fact that for the qualitative analysis of model dynamics based on ordinary differential equations it is possible to use the apparatus of the classical theory of dynamical systems. The second task is solved only partially, because the gain in efficiency while maintaining acceptable accuracy for numerical implementation will be considerable only for certain simple initial distribution of molecules, as well as for certain diffusion coefficients. To determine the criteria for practical application of this technique we also estimate the model error in general.
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Keywords
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reaction-diffusion, alternative to partial derivatives, dynamic systems theory
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UDC
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517.958
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MSC
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35K57, 76R50
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DOI
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10.20537/vm140407
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Received
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23 September 2014
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Language
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Russian
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Citation
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Nazarov M.N. A new approach to the reaction-diffusion systems modelling, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 4, pp. 84-94.
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References
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- Shishkin G.I., Shishkina L.P. Improved difference scheme of the solution decomposition method for a singularly perturbed reaction-diffusion equation, Proceedings of the Steklov Institute of Mathematics, 2011, vol. 272, no. 1, pp. 197-214.
- Makarov V.L., Samarskii A.A. Use of exact difference schemes for estimating the rate of convergence of the method of straight lines, USSR Computational Mathematics and Mathematical Physics, 1980, vol. 20, no. 2, pp. 102-119.
- Guzhev D.S., Seifert P., Kalitkin N.N., Shirkov P.D. Numerical methods for problems of chemical kinetics with diffusion, Matem. Mod., 1992, vol. 4, no. 1, pp. 98-110 (in Russian).
- Gur'eva Ya.L., Il'in V.P. Computational aspects of finite volume methods, Computational Mathematics and Mathematical Physics, 1998, vol. 38, no. 11, pp. 1783-1800.
- Eymard R., Gallouet T., Herbin R. The finite volume method, Handbook of Numerical Analysis, 2000, vol. 7, pp. 713-1020.
- Bussing T., Murman E. Finite-volume method for the calculation of compressible chemically reacting flows, AIAA Journal, 1998, vol. 26, no. 9, pp. 1070-1078.
- Nazarov M.N. Generalization of coarse-grained models with introduction of three-dimensional space, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2011, no. 4, pp. 110-117 (in Russian).
- Nazarov M.N. On alternative to partial differential equations for the modelling of reaction-diffusion systems, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2013, no. 2, pp. 35-47 (in Russian).
- Nazarov M.N. New approaches to the analysis of the elementary reactions kinetics, J. Sib. Fed. Univ. Math. Phys., 2014, vol. 7, no. 3, pp. 373-382.
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Full text
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