CORRECTNESS OF A MIXED PROBLEM FOR DEGENERATE THREE-DIMENSIONAL HYPERBOLIC-PARABOLIC EQUATIONS


Cite item

Abstract

In mathematical modeling of electromagnetic fields in space, the nature of electromagnetic process is determined by the properties of the medium. If the medium is non-conducting, we obtain degenerate three-dimensional hyperbolic equations. If the medium has a high conductivity, then we come to degenerate
three-dimensional parabolic equations. Consequently, the analysis of electromagnetic fields in complex media (for example, if the medium’s conductivity changes) is reduced to degenerate three-dimensional hyperbolic-parabolic equations. The mixed problem for multidimensional hyperbolic equations is well studied and has been previously considered in the works of various authors. In the articles of Professor S.A. Aldashev, the unique solvability of the mixed problem for degenerate multidimensional hyperbolic equations is proved. It is known that mixed problems for multidimensional hyperbolic-parabolic equations have not been studied much. The paper finds a new class of degenerate three-dimensional hyperbolic-parabolic equations for which the mixed problem has a unique solution and gives an explicit representation of its classical solution.

About the authors

S. A. Aldashev

Institute of Mathematics, Physics and Informatics, Abai Kazakh National Pedagogical University, 13, Dostyk ave., Almaty, 050100, Republic of Kazakhstan.

Author for correspondence.
Email: morenov.sv@ssau.ru
ORCID iD: 0000-0002-8223-6900

Doctor of Physical and Mathematical Sciences, full professor

Kazakhstan

Z. N. Kanapyanova

Institute of Mathematics, Physics and Computer Science, Abai Kazakh National Pedagogical University, 13, Dostyk ave., Almaty, 050100, Republic of Kazakhstan.

Email: morenov.sv@ssau.ru
ORCID iD: 0000-0003-2544-8197

PhD student of the 2nd year of study

Kazakhstan

References

  1. Ladyzhenskaya O.A. Smeshannaya zadacha dlya giperbolicheskogo uravneniya . Moscow: Gostekhizdat, 1953, 279 p. Available at: http://bookre.org/reader?file=579384 .
  2. Ladyzhenskaya O.A. Kraevye zadachi matematicheskoi fiziki . Moscow: Nauka, 1973, 407 p. Available at: http://bookre.org/reader?file=442669 .
  3. Krasnov M.L. Smeshannye kraevye zadachi dlya vyrozhdayushchikhsya lineinykh giperbolicheskikh differentsial’nykh uravnenii vtorogo poryadka . Matem. sb. , 1959, vol. 49 (91), pp. 29–84. Available at: http://www.mathnet.ru/links/aee615ef85b73c2fbc4c0f4cd201c7f7/sm4910.pdf .
  4. Baranovskii F.T. Smeshannaya zadacha dlya lineinogo giperbolicheskogo uravneniya vtorogo poryadka, vyrozhdayushchegosya na nachal’noi ploskosti . Uchenye zapiski Leningr. ped. instituta, 1958, vol. 183, pp. 23–58 .
  5. Aldashev S.A. Well-posedness of the mixed problem for degenerate multi-dimensional hyperbolic equations. Materials of the international conferences, Modern Problems of Mathematical Modeling, Computational Methods and Information Technologies. Kyiv: Kyiv National University named after T. Shevchenko, 2018, pp. 14–15.
  6. Aldashev S.A. Korrektnost’ smeshannoi zadachi v tsilindricheskoi oblasti dlya odnogo klassa mnogomernykh giperbolo-parabolicheskikh uravnenii . Ukr. mat. zhurn. , 2020, vol. 72, no. 2, pp. 280–288. Available at: http://umj.imath.kiev.ua/index.php/umj/article/view/870. .
  7. Kolmogorov A.N., Fomin S.V. Elementy teorii funktsii i funktsional’nogo analiza . Moscow: Nauka, 1976, 543 p. Available at: http://mat.net.ua/mat/Kolmogorov-Funkanaliz.htm. .
  8. Aldashev S.A. Korrektnost’ zadachi Dirikhle dlya vyrozhdayushchikhsya mnogomernykh giperbolo-parabolicheskikh uravnenii . Nauchnye vedomosti BelGU. Ser.: Matematika, fizika , 2016, no. 27 (248), issue 45, pр. 16–25. Available at: https://elibrary.ru/item.asp?id=29201717 .
  9. Aldashev S.A., Kanapyanova Z.N. Korrektnost’ smeshannykh zadach dlya odnogo klassa vyrozhdayushchikhsya trekhmernykh giperbolicheskikh uravnenii . In: Tezisy dokladov Uzbeksko-Rossiiskoi nauchnoi konferentsii. Neklassicheskie uravneniya matematicheskoi fiziki i ikh prilozheniya . Tashkent, 2019, 96 p. Available at: http://numf2019.nuu.uz/numf2019.pdf .
  10. Smirnov V.I. Kurs vysshei matematiki. T. 4. Ch. 2 . Moscow: Nauka, 1981, 550 p. Available at: https://obuchalka.org/2012030763879/kurs-visshei-matematiki-tom-4-chast-2-smirnov-v-i-1974.html. .
  11. Friedman A. Uravneniya s chastnymi proizvodnymi parabolicheskogo tipa . Moscow: Mir, 1968, 527 p. Available at: http://bookre.org/reader?file=579715. .

Copyright (c) 2020 С. А. Алдашев, З. Н. Канапьянова

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies