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ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

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Two One-dimensional Boundary Value Problems of Nonlinear Diffusion Theory

Author(s):

Alexandra Viktorovna Grezina

Ph.D. Associate Professor of Lobachevsky State University of Nizhny Novgorod

aleksandra-grezina@yandex.ru

Vladimir Semionovich Metrikin

Ph.D. Associate Professor of Lobachevsky State University of Nizhny Novgorod

v.s.metrikin@mail.ru

Adolf Grigorievich Panasenko

Ph.D. Associate Professor of Lobachevsky State University of Nizhny Novgorod

a.g.panasenko@yandex.ru

Abstract:

This work is an extension of the authors earlier investigations into a nonlinear theory of propagation of impurities in solids. Two mathematical models governed by partial differential equations are studied in this paper. Both models describe the dynamics of the impurities propagation in a simi-infinite prismatic solid with an impenetrable side surface. The difference between the models is associated with the impurity source that is located at the boundary of the solid. In the first case the source is assumed to be present during a finite time interval, whereas in the second case the impurity source is assumed to be stationary and provide a constant impurity flow. An approach is presented in this paper that allows to transform the nonlinear partial differential equations into nonlinear ordinary differential equations and derive solutions to the latter. A methodology is presented for the formulation of the initial conditions for the nonlinear ordinary differential equations and criteria are formulated for evaluation of the correctness of those conditions. The obtained solutions provide information regarding the time-dependent concentration of the impurities and their finite propagation speed. It is also shown in the paper that the proposed solution approach is also efficient in application to the linear theory of diffusion.

Keywords

References:

  1. Gottstein G. Physical and chemical bases of material science. Per. with English. / Mr. Gottstein. — Moscow: BINOM, Laboratory of knowledge, 2011. - 400 p. [2] Mehrer, H. Diffusion in solids. Per. with English. / H. Merer. - Moscow: Intellect, 2011. - 536 p
  2. A. V. Zemskov, D. V. Tartakovskii Polar-symmetric problem of the elastic diffusion for multi-component environment. Problems of strength and plasticity. 2018. T. 80. № 1
  3. Theory and technology of nitriding / Yu. M. Lakhtin, Ya. d. Kogan, G. I. shpis, and Z. Bemer. — Moscow: Metallurgy, 1991. 320 p
  4. Lakhtin Yu. M. Chemical-heat treatment of metals /Yu. M. Lakhtin and B. N. Arzamasov. - Moscow: Metallurgy, 1985. - 256 p
  5. The process of mutual diffusion in alloys / I. B. Borovskii, K. P. Gurov, I. D. Marchukova, J. E. Ugaste. - Moscow: Nauka, 1973. - 360 p
  6. Gurov K. P. Mutual diffusion in multiphase metallic systems: a tutorial / K. P. Gurov, B. A. Kartashkin, and Yu. e. Ugaste. - Moscow: Nauka, 1981. - 350 p
  7. Popov A. A. Theoretical bases of chemical and thermal treatment of steel / A. A. Popov. - Moscow: Metallurgizdat, 1962. -120 p
  8. Cobble R. //J. Appl. Phys. 1963. V. 34. P. 1679
  9. Geguzin Y. E. Physics of sintering. - M. : Nauka. 1976
  10. Gorelik S. S. recrystallization of metals and alloys. - M.; Metallurgy. 1978
  11. Bokshtein S. Z., S. S. Ginzburg, S. T. Kishkin, Razumovskiy I. M., Stroganov, B. Autoradiography of interfaces and structural stability of the alloys. -M. : Metallurgy. 1987. 272 S
  12. J. Pout. Tu K., Meyer J. Thin film. Mutual diffusion and reactions.; Per. with English. - M.; Mir. 1982. 338 S
  13. Glickman, E. E., Nathan M. //J. Appl. Phys. 1999. V. 85. P. 3185
  14. Gleiter H. //Progress In Mater. Sci. 1989. V33 (4). P. 223
  15. Grain boundary diffusion and properties of nanostructured materials. / Ed. Yu. R. Kolobov, R. Z. Valieva. -Novosibirsk: Nauka. 2001. 232 S
  16. Baltex B. I. Diffusion in semiconductors. M. GOS. Izd-vo Fiz. -Mat. lit., 1961. 462 s
  17. Waxman, Y. F., Nizuc Y. A., V. V. Yatsun, A. S. Nasibov, P. V. Shapkin Optical absorption and diffusion of iron in ZnSe monocrystals // Physics and technics of semiconductors. 2010. Vol. 44. issue. 4. P. 463-466
  18. Umit Demirbas, Alphan Sennaroglu. Mehmet Somer. Synthesis and characterization of diffusion-doped Cr2+: ZnSe and Fe2+: ZnSe // Optical Materials. 2006. V28. P. 231-240
  19. Il'ichev N. N., Shapkin P. V., Gulyamova, E. S., Kulevi L. A., Nasibov A. S. Determination of the diffusion coefficient of Fe2+ in single-crystal ZnSe // Inorganic materials. 2010. Vol. 46. No. 2. P. 149-153
  20. Rusakov G. M., Lobanov M. L., Larionov, K. V. Calculation of impurity diffusion coefficients by decomposition of the concentration profile in the Fourier series//Physics of metals and metallography. 2001. volume 91. p. 14-16
  21. Tikhonov A. N., Samarskiy A. A. equations of mathematical physics. M. " Nauka ". Chapters'. ed. Fiz. -Mat. lit., 1966. 724 p
  22. Igumnov L. A., Grezina A. V., Metrikin V. S., Panasenko A. G. Numerical - analytical modeling of diffusion processes in limited multicomponent solids. Problems of strength and plasticity. Publishing house of the Nizhny Novgorod University. T. 80, No. 3, 2018. S. 336-348
  23. Kucheruk I. M., Gorbachuk I. T., Lutsik P. P. General course of physics. - Kiev: Tekhnika, 2006. - T. 2. [25 ] Grezina A. V., Metrikin V. S., Panasenko A. G Nelineynii effekti v teorii diffuzii primesi v tverdom tele// Differential equations and control processes- 2022, N. 4. С. 115-125. [26 ] Budak B. M., Samarsky A. A., Tikhonov A. P. Collection of problems in mathematical physics. “Science”, Main editorial office of physical and mathematical literature. Moscow 1972. 687 pp

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