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Turtin D. V., Stepovich M. A., Kalmanovich V. V. On the application of the qualitative theory of differential equations to a problem of heat and mass transfer. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2023, vol. 23, iss. 1, pp. 48-57. DOI: 10.18500/1816-9791-2023-23-1-48-57, EDN: GENWZC
On the application of the qualitative theory of differential equations to a problem of heat and mass transfer
The possibilities of applying the qualitative theory of differential equations to one problem of heat and mass transfer in multilayer planar semiconducting structures are studied. The consideration is carried out on the example of a mathematical model of a stationary process of diffusion of nonequilibrium minority charge carriers generated by a wide excitation source. The use of a wide source of external influence makes it possible to reduce modeling problems to one-dimensional ones and describe these mathematical models by ordinary differential equations. These are the processes in various nanosystems exposed to wide beams of charged particles or electromagnetic radiation. The paper reviews the results of recent studies of such models. The main object of study was the questions of the correctness of the considered mathematical models, special attention is paid to the mathematical assessment of the influence of external factors on the state of the object under study. Previously, the methods of the qualitative theory of differential equations, in our case, the assessment of the influence of external influence on the distribution of nonequilibrium minority charge carriers as a result of their diffusion in a semiconductor, in combination with the consideration of the uniqueness of the solution of differential equations of heat and mass transfer and the correctness of the mathematical models used, were considered very rarely, and for wide electron beams, a quantitative analysis of such problems has not previously been carried out at all. In the present work, the main attention is paid to the influence of the right side of the differential equation, the excitation function of minority charge carriers, on the solution of the differential diffusion equation, which describes the distribution of nonequilibrium charge carriers that have diffused in each layer of such a structure. The uniqueness of the solution of the problem under consideration and the continuous dependence of the solution on the right side of the differential equation are proved. Estimates are obtained for the influence of external factors on the diffusion of generated carriers in each layer of a multilayer planar semiconductor structure.
- Nakhushev A. M. On some new boundary value problems for hyperbolic equations and mixed type equations. Differential Equations, 1969, vol. 5, iss. 1, pp. 44–59 (in Russian).
- Nakhushev A. M. New boundary value problem for one degenerate hyperbolic equation. Doklady Mathematics, 1969, vol. 187, iss. 4, pp. 736–739 (in Russian).
- Nakhushev A. M. On Nonlocal Boundary Value Problems with Displacement and Their Relationship with Loaded Equations. Differential Equations, 1985, vol. 21, iss. 1, pp. 92–101 (in Russian).
- Nakhushev A. M. Zadachi so smeshcheniem dlya uravnenii v chastnykh proizvodnykh [Problems with Shifts for Partial Differential Equations]. Moscow, Nauka, 2006. 287 p. (in Russian).
- Nemitskii V. V., Stepanov V. V. Kachestvennaya teoriya differentsial’nykh uravneniy [Qualitative Theory of Differential Equations]. Moscow, State Publishing House of Technical and Theoretical Literature, 1947. 448 p. (in Russian).
- Bonch-Bruevich V. L., Kalashnikov S. G. Fizika poluprovodnikov [The Physics of Semiconductors]. Moscow, Nauka, 1990. 686 p. (in Russian).
- Smith R. A. Semiconductors. 2nd ed. Cambridge, Cambridge University Press, 1978. 523 p. (Russ. ed.: Moscow, Mir, 1982. 560 p.).
- Wittry D. B., Kyser D. F. Measurements of diffusion lengths in direct-gap semiconductors by electron beam excitation. Journal of Applied Physics, 1967, vol. 38, iss. 1, pp. 375–382. https://doi.org/10.1063/1.1708984
- Rao-Sahib T. S., Wittry D. B. Measurements of diffusion lengths in p-type gallium arsenide by electron beam excitation. Journal of Applied Physics, 1969, vol. 40, iss. 9, pp. 3745–3750. https://doi.org/10.1063/1.1658265
- Petrovskii I. G. Lektsii po teorii obyknovennykh differentsial’nykh uravneniy [Lectures on the Theory of Ordinary Differential Equations]. Moscow, Nauka, 1964. 272 p. (in Russian).
- Pontryagin L. S. Obyknovennye differentsial’nye uravneniya [Ordinary Differential Equations]. Moscow, Nauka, 1965. 332 p. (in Russian).
- Polyakov A. N., Smirnova A. N., Stepovich M. A., Turtin D. V. Mathematical model of qualitative properties of exciton diffusion generated by electron probe in a homogeneous semiconductor material. Lobachevskii Journal of Mathematics, 2018, vol. 39, iss. 2, pp. 259–262. https://doi.org/10.1134/S199508021802021X
- Turtin D. V., Seregina E. V., Stepovich M. A. Qualitative Analysis of a Class of Differential Equations of Heat and Mass Transfer in a Condensed Material. Journal of Mathematical Sciences (United States), 2020, vol. 259, iss. 1, pp. 166–174. https://doi.org/10.1007/s10958-020-05008-4
- Turtin D. V., Stepovich M. A., Kalmanovich V. V., Kartanov A. A. On the correctness of mathematical models of diffusion and cathodoluminescence. Taurida Journal of Computer Science Theory and Mathematics, 2021, iss. 1 (50), pp. 81–100 (in Russian). https://doi.org/10.37279/1729-3901-2021-20-1-81-100, EDN: CJKSHG
- Kalmanovich V. V., Seregina E. V., Stepovich M. A. Mathematical modeling of heat and mass transfer phenomena caused by interaction between electron beams and planar semiconductor multilayers. Bulletin of the Russian Academy of Sciences: Physics, 2020, vol. 84, iss. 7, pp. 844–850. https://doi.org/10.3103/S1062873820070138
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