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

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

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Fractional Polynomial Differential Equations: Discrete Groups and Solutions Throw the Transcendent of the 1st Painleve Equation

Author(s):

Zilya Nailevna Khakimova

Mozhaisky Military Space Academy, Saint-Petersburg
Candidate of Science in Physics and Mathematics, docent
Associate professor of the department of mathematics
in Mozhaisky Military Space Academy, Saint-Petersburg
Russia, 197198, St. Petersburg, Zhdanovskaya st., 13

vka@mil.ru

Oleg Valentinovich Zaytsev

BSTU «VOENMEH» named after D.F. Ustinov, Saint-Petersburg
Russia, 190005, St. Petersburg, 1st Krasnoarmeyskaya st., 1

zaytsev_oleg1997@mail.ru

Abstract:

For a class of the second-order ordinary differential equations with fractional polynomial right-hand side, discrete group analysis is carried out. Transformations are found that are closed on this class of equations or on its subclasses. The dihedral group of transformations of the 12th order and the graph of this group for the class of equations with fractional polynomial right-hand side are constructed.Using this transformation dihedral group a transformation pseudogroup of the 120th order for the 1st Painlevé equation, as well as the graph of this pseudogroup, are constructed. Exact solutions through the 1st Painlevé transcendent are found for 119 equations of polynomial and fractional polynomial types related with the 1st Painlevé equation by transformations of this pseudogroup.

Keywords

References:

  1. Zaitsev, V. F., Polianin, A. D. Spravochnik po nelinejnym differencial'nym uravneniyam. Prilozheniya v mekhanike, tochnye resheniya [Handbook of Nonlinear Differential Equations. Applications in mechanics, exact solutions]. Moscow, Nauka Publ., 1993. 464 p
  2. Zaitsev, V. F., Polianin, A. D. Spravochnik po obyknovennym differencial'nym uravneniyam: tochnye resheniya [Ordinary Differential Equations Handbook: Exact Solutions]. Moscow, Fizmatlit Publ., 1995. 560 p
  3. Zaitsev, V. F. [Discrete-group analysis of ordinary differential equations]. Differencial'nye uravneniya, 1989, vol. 25, no. 3, pp. 379-389. (In Russian)
  4. Zaitsev, V. F. [On discrete-group analysis of ordinary differential equations]. Doklady akademii nauk SSSR, vol. 299, no. 3, pp. 542-545. (In Russian)
  5. Zaitsev, V. F. Diskretno-gruppovye metody teorii differencial'nyh uravnenij, ch. 1 [Discrete-group methods of the theory of differential equations, part 1]. Leningrad, Leningrad St. Univ. Publ., 1982. 130 p
  6. Zaitsev, V. F., Kormilitsina, T. V. Diskretno-gruppovye metody teorii differencial'nyh uravnenij, ch. 2 [Discrete-group methods of the theory of differential equations, part 2]. Leningrad, Leningrad St. Univ. Publ., 1985. 150 p
  7. Zaitsev, V. F., Polianin, A. D. Diskretno-gruppovoj metod integrirovaniya uravnenij nelinejnoj mekhaniki [Discrete-group method of integration of equations of nonlinear mechanics]. Moscow, Institute for Problems in Mechanics AS USSR Publ., preprint no. 339, 1988. 44 p
  8. Zaitsev, V. F., Flegontov, A. V. Diskretno-gruppovye metody integrirovaniya obyknovennyh differencial'nyh uravnenij [Discrete-group methods of integration of ordinary differential equations]. Leningrad, Leningrad St. Institute of Informatics and Automation AS USSR Publ., 1991. 240 p
  9. Zaitsev, V. F., Flegontov, A. V., Khakimova, Z. N. Diskretno-gruppovoj analiz differencial'nyh uravnenij. Tochnye resheniya uravneniya [Discrete-group analysis of differential equations. Exact Equation Solutions]. Leningrad, Leningrad St. Institute of Informatics and Automation AS USSR Publ., preprint no. 105, 1989. 61 p
  10. Zaitsev, V. F., Khakimova, Z. N. O diskretno-gruppovom analize uravneniya !!!! ERROR!!! IMAGE IS NOT ALLOWED! [On discrete-group analysis of an equation !!!! ERROR!!! IMAGE IS NOT ALLOWED! ]. Leningrad, Leningrad St. Univ. Publ., 1986. 31 p
  11. Zaitsev, V. F., Khakimova, Z. N. [Bä cklund transformations closed on one class of equations]. Differencial'nye uravneniya i prikladnye zadachi, mezhvuzovskij sbornik nauchnyh trudov [Proc. of the Tula Polytechnic Institute], 1990, pp. 5-12. (In Russian)
  12. Khakimova, Z. N. [On transformations of the Painlevé equation]. Differencial'nye uravneniya s chastnoj proizvodnoj, mezhvuzovskij sbornik nauchnyh trudov [Proc. of the Leningrad State Pedagogical Institute], 1989, pp. 150-158. (In Russian)
  13. Khakimova, Z. N. [On point transformations of a nonlinear second-order equation]. Differencial'nye uravneniya s chastnoj proizvodnoj, mezhvuzovskij sbornik nauchnyh trudov [Proc. of the Leningrad State Pedagogical Institute], 1988, pp. 85-90
  14. Khakimova, Z. N. Diskretno-gruppovoj analiz nelinejnyh obyknovennyh differencial'nyh uravnenij 2-go poryadka s algebraicheskoj pravoj chast'yu. Cand Diss. [Discrete-group analysis of nonlinear ordinary differential equations of the second order with an algebraic right-hand side. Cand Diss. ]. St. Petersburg, The Herzen State Pedagogical University of Russia Publ., 1997. 119 p
  15. Zaitsev, V. F., Polianin, A. D. Spravochnik po obyknovennym differencial'nym uravneniyam [Ordinary Differential Equations Handbook]. Moscow, Fizmatlit Publ., 2001. 576 p
  16. Zaitsev, V. F., Polianin, A. D. Spravochnik po nelinejnym obyknovennym differencial'nym uravneniyam [A handbook of nonlinear ordinary differential equations]. Moscow, Faktorial Publ., 1997. 512 p
  17. Polyanin, A. D., Zaitsev, V. F. Handbook of Exact Solutions for Ordinary Differential Equations. CRC Press, Boca Raton-New York, 2003
  18. Zaitsev, V. F., Polianin, A. D. Obyknovennye differencial'nye uravneniya v 2 chastyah. Spravochnik dlya akademicheskogo bakalavriata. - 2-e izdanie, ispravlennoe i dopolnennoe [Ordinary differential equations in 2 parts. Handbook for Academic Baccalaureate. - 2nd edition, revised and enlarged]. Moscow, Yurajt Publ., 2018
  19. Polyanin, A. D., Zaytsev, V. F. Handbook of Ordinary Differential Equations: Exact Solutions, Methods, and Problems. - CRC Press. Boca Raton - London, 2018
  20. Khakimova, Z. N., Zaitsev, O. V. [Classification of new solvable cases in the class of polynomial differential equations]. Aktual'nye voprosy sovremennoj nauki, 2014, no. 3, pp. 3-11. (In Russian)
  21. Khakimova, Z. N. [Integration of discrete invariants in the class of polynomial differential equations of the 2nd class]. Trudy Voenno-kosmicheskoj akademii im. A. F. Mozhaiskogo [Proc. of the Mozhaisky Military Space Academy], St. Petersburg, 2014, pp. 8-16. (In Russian)
  22. Zaitsev, O. V. [On discrete symmetries and new solvable cases in the class of polynomial differential equations]. Nauka XXI veka: novyj podhod: materialy IX molodezhnoj mezhdunarodnoj nauchno-prakticheskoj konferencii studentov, aspirantov i molodyh uchenyh [Proc. Int. Conf. «Science of the XXI century], St. Petersburg, Otkrytie Publ., 2014, pp. 8-16
  23. Zaitsev, V. F., Zaitsev, O. V. [One application of the nesting method]. Nekotorye aktual'nye problemy sovremennoj matematiki i matematicheskogo obrazovaniya: materialy nauchnoj konferencii «Gercenovskie chteniya - 2015» [Proc. Conf. «Herzen Readings - 2015»], St. Petersburg, The Herzen St. Pedagogical University of Russia, 2014, pp. 30-33. (In Russian)
  24. Khakimova, Z. N. [Selection of a class of differential equations for finding new solvable cases]. Nekotorye aktual'nye problemy sovremennoj matematiki i matematicheskogo obrazovaniya: materialy nauchnoj konferencii «Gercenovskie chteniya - 2017» [Proc. Conf. «Herzen Readings - 2017»], St. Petersburg, The Herzen St. Pedagogical University of Russia, 2017, pp. 112-117. (In Russian)
  25. Khakimova, Z. N. [Legendre type transformation and new solvable differential equations of the 2nd and 3rd orders]. Nekotorye aktual'nye problemy sovremennoj matematiki i matematicheskogo obrazovaniya: materialy nauchnoj konferencii «Gercenovskie chteniya - 2018» [Proc. Conf. «Herzen Readings - 2018»], St. Petersburg, The Herzen St. Pedagogical University of Russia, 2018, pp. 119-122. (In Russian)
  26. Khakimova, Z. N. [Power-law differential equations integrable in polynomials]. Nekotorye aktual'nye problemy sovremennoj matematiki i matematicheskogo obrazovaniya: materialy nauchnoj konferencii «Gercenovskie chteniya - 2019» [Proc. Conf. «Herzen Readings - 2019»], St. Petersburg, The Herzen St. Pedagogical University of Russia, 2019, pp. 97-101. (In Russian)
  27. Khakimova, Z. N. [Integration of fractional-polynomial differential equations]. Nekotorye aktual'nye problemy sovremennoj matematiki i matematicheskogo obrazovaniya: materialy nauchnoj konferencii «Gercenovskie chteniya - 2020» [Proc. Conf. «Herzen Readings - 2020»], St. Petersburg, The Herzen St. Pedagogical University of Russia, 2020, pp. 115-118. (In Russian)
  28. Linchuk, L. V. [Parametric polynomial solutions of one class of ordinary differential equations of the second order]. Nekotorye aktual'nye problemy sovremennoj matematiki i matematicheskogo obrazovaniya: materialy nauchnoj konferencii «Gercenovskie chteniya - 2018» [Proc. Conf. «Herzen Readings - 2018»], St. Petersburg, The Herzen St. Pedagogical University of Russia, 2018, pp. 85-90. (In Russian)
  29. Coxeter, H. S. M, Moser, W. O. J. Generators and relations for discrete groups. Fourth edition. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 14. Springer-Verlag, Berlin-New York, 1980. ix+169 pp
  30. Ince, Edward L. Ordinary differential equations. - Dover Publications, New York, 1926

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