Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

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Loshkareva E. A., Gladyshev Y. A., Malyshev E. N. Application of the generalized degree method for constructing solutions of the quaternion variant of the Cauchy – Riemann system. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2023, vol. 23, iss. 1, pp. 11-23. DOI: 10.18500/1816-9791-2023-23-1-11-23, EDN: ZMDMGI

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
01.03.2023
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Language: 
Russian
Heading: 
Mathematics
Article type: 
Article
UDC: 
511.84
DOI: 
10.18500/1816-9791-2023-23-1-11-23
EDN: 
ZMDMGI

Application of the generalized degree method for constructing solutions of the quaternion variant of the Cauchy – Riemann system

Autors: 
Loshkareva Elena Anatolievna, Kaluga State University named after K. E. Tsiolkovski
Gladyshev Yuri A., Kaluga State University named after K. E. Tsiolkovski
Malyshev Evgeniy N., Bauman Moscow State Technical University (Kaluga Branch)
Abstract: 

This article indicates one of the ways to solve the generalized Cauchy – Riemann system for quaternionic functions in an eight-dimensional space. In previous works, some classes of solutions of this system were studied and it was stated that it is possible to use the method of generalized degrees to construct solutions of this system of differential equations. It is shown that the solution of the problem can be reduced to finding two arbitrary quaternionic harmonic functions in an eight-dimensional space. All 8 components of these functions $\varphi ,\psi$ must be harmonic functions, that is, be twice continuously differentiable over all 8 real variables $x_i$, $y_i$, where $i = \overline {1,4} $ solutions of the Laplace equation. In this article, the parametric method of generalized degrees is considered, which is applicable to individual equations of the second and higher orders.

Key words: 
generalized Bers degrees
quaternion
Cauchy – Riemann system
References: 
  1. Gladyshev Yu. A. On some classes of solutions to the generalized Cauchy – Riemann system problem. Problemy matematicheskogo analiza [Problems of Mathematical Analysis], 2021, vol. 109, pp. 59–64 (in Russian).
  2. Moisil G. Sur les quaternions monogenes. Bulletin of Mathematical Sciences, 1931, vol. 55, pp. 168–174 (in German).
  3. Fueter R. On the theory of regular functions of a quaternion variable. Monatshefte fur Mathematik und Physik, 1936, vol. 43, pp. 69–74 (in German). https://doi.org/10.1007/BF01707588
  4. Misyura N. Е. Kvaternionnye modeli v kinematike i dinamike tverdogo tela [Quaternion Models in Kinematics and Dynamics of a Solid]. Ekaterinburg, Ural University Publishing House, 1961. 260 p. (in Russian).
  5. Bers L. Matematicheskie osnovy dozvukovoy i transzvukovoy gazovoy dinamiki [Mathematical Foundations of Subsonic and Transonic Gas Dynamics]. Moscow, Publishing House of Foreign Literature, 1961. 206 p. (in Russian).
  6. Bers L., Gelbart А. On a class of differential equations in mechanics of continua. Quarterly of Applied Mathematics, 1943, vol. 1, pp. 168–188. https://doi.org/10.1090/qam/8556
  7. Bers L., Gelbart А. On a class of functions defined by partial differential equations. Transactions of the American Mathematical Society, 1944, vol. 56, iss. 1, pp. 67–93. https://doi.org/10.2307/1990278
  8. Landau L. D., Lifshitz E. M. Teoreticheskaya fizika. Т. 2. Teoriya polya [Theoretical Physics. Vol. 2. Field Theory]. Moscow, Nauka, 1973. 504 p. (in Russian).
  9. Gladyshev Yu. A., Loshkareva E. A. On methods of constructing complex generalized degrees of Bers. Vestnik of Kaluga University, 2020, iss. 2 (47), pp. 77–80 (in Russian). EDN: ZLPPKC
  10. Kalmanovich V. V., Stepovich M. A., Seregina E. V. On the numerical solution of heat and mass transfer problems using the matrix method and the method of generalized degrees of Bers. Teoreticheskie osnovy i konstruirovaniya chislennykh algoritmov resheniya zadach matematicheskoi fiziki [Theoretical foundations and design of numerical algorithms for solving problems of mathematical physics. Abstracts of the XII All-Russian Conference dedicated to the memory of K. I. Babenko]. Moscow, Keldysh Institute of Applied Mathematics Publ., 2018, pp. 51–52 (in Russian). EDN: YLOPJJ 
  11. Kalmanovich V. V., Stepovich M. A. On the possibility of joint application of the matrix method and the Bers apparatus to modeling the processes of heat and mass transfer caused by electromagnetic radiation in a multilayer planar medium. XXV Mezhdunarodnaia nauchno-tekhnicheskaia konferentsiia i shkola po fotoelektronike i priboram nochnogo videniia [XXV International Scientific and Technical Conference and School on Photoelectronics and Night Vision Devices. Proceedings of the Conference]. Moscow, NPP Orion, 2018, pp. 491–494 (in Russian).
  12. Kulikov A. N., Gorbunov A. K., Silaeva N. A., Korzhavy A. P. Modeling the behavior of hydrodynamic dispersion by solving boundary value problems. Journal Science Intensive Technologies, 2021, vol. 22, iss. 6, pp. 46–53 (in Russian). https://doi.org/10.18127/j19998465-202106-05
  13. Kalmanovich V. V., Stepovich M. A. On the joint application of the matrix method and the apparatus of generalized Bers degrees for mathematical modeling of heat and mass transfer processes in semiconductor materials of electronic technology. Problems of Development of Promising Micro- and Nanoelectronic Systems, 2018, iss. 3, pp. 194–201 (in Russian). https://doi.org/10.31114/2078-7707-2018-3-194-201
  14. Widder D. V. Some analogies from classical analysis in the theory of heat conduction. Archive for Rational Mechanics and Analysis, 1966, vol. 21, iss. 2, pp. 108–113 (in German). https://doi.org/10.1007/BF00266570
  15. Zhukov V. P., Barochkin A. E., Bobrova M. S., Belyakov A. N., Shuvalov S. I. Matrix method for solving the inverse problem of heat transfer in heat exchangers. Bulletin of the Ivanovo State Power Engineering University, 2021, iss. 2, pp. 62–69 (in Russian). https://doi.org/10.17588/2072-2672.2021.2.062-069
  16. Barochkin A. E. Matrix method for solving the inverse problem of heat transfer in contact devices taking into account the phase transition in heat carriers. Bulletin of the Ivanovo State Power Engineering University, 2021, iss. 5, pp. 68–75 (in Russian). https://doi.org/10.17588/2072-2672.2021.5.068-075
  17. Gladyshev Yu. A., Loshkareva E. A. On the use of the method of parametric generalized degrees for constructing solutions of one class of differential equations. In: Materialy Voronezhskoi zimnei matematicheskoi shkoly S. G. Crane [Materials of the International Conference “Voronezh Winter Mathematical School of S. G. Crane”], Voronezh, Publishing House of Voronezh State University, 2022, pp. 67–71 (in Russian). URL: https://vzms.kmm-vsu.ru/files/vzms2022.pdf (accessed 25 March 2022).
  18. Brinkman G. Primenenie spinornykh invariantov v atomnoi fizike [Application of Spinor Invariants in Atomic Physics]. Moscow, Publishing House of Foreign literature, 1959. 96 p. (in Russian).
Received: 
29.03.2022
Accepted: 
18.08.2022
Published: 
01.03.2023
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