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Ukrainian Mathematical Journal

, Volume 70, Issue 9, pp 1467–1476 | Cite as

Third Boundary-Value Problem for a Third-Order Differential Equation with Multiple Characteristics

  • Yu. P. Apakov
  • A. Kh. Zhuraev
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We prove the unique solvability of the third boundary-value problem for a third-order differential equation with multiple characteristics containing the second time derivative in a rectangular domain.

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References

  1. 1.
    O. S. Ryzhov, “Asymptotic picture of the flow around bodies of revolution by a sonic flow of a viscous and thermally conducting gas,” Prikl. Mat. Mekh., 29, Issue 6, 1004–1014 (1965).zbMATHGoogle Scholar
  2. 2.
    V. N. Diesperov, “On the Green function of a linearized viscous transonic equation,” Zh. Vychisl. Mat. Mat. Fiz., 12, No. 5, 1265–1279 (1972).MathSciNetGoogle Scholar
  3. 3.
    H. Block, “Sur les equations lineaires aux derivees partielles a carateristiques multiples,” Ark. Mat., Astron. Fis. Note 1, 7, No. 13, 1–34 (1912); Ark. Mat., Astron. Fis. Note 2, 7, No. 21, 1–30 (1912); Ark. Mat., Astron. Fis. Note 3, 8, No. 23, 1–51 (1912–1913).Google Scholar
  4. 4.
    L. Cattabriga, “Potenziali di linea e di dominio per equazioni non paraboliche in due variabili a caratteristiche multiple,” Rend. Semin. Mat. Univ. Padova, 31, 1–45 (1961).MathSciNetzbMATHGoogle Scholar
  5. 5.
    T. D. Dzhuraev and Yu. P. Apakov, “On the self-similar solution of a third-order equation with multiple characteristics,” Vestn. Samar. Gos. Tekh. Univ., Ser. Fiz.-Mat. Nauk., No. 2(15), 18–26 (2007).CrossRefGoogle Scholar
  6. 6.
    T. D. Dzhuraev and Yu. P. Apakov, “On the theory of the third-order equation with multiple characteristics containing the second time derivative,” Ukr. Mat. Zh., 62, No. 1, 40–51 (2010); English translation : Ukr. Math. J., 62, No. 1, 43–55 (2010).Google Scholar
  7. 7.
    Yu. P. Apakov and S. Rutkauskas, “On a boundary problem to third order PDE with multiple characteristics,” Nonlin. Anal. Model. Control, 16, No. 3, 255–269 (2011).MathSciNetzbMATHGoogle Scholar
  8. 8.
    Yu. P. Apakov, “On the solution of a boundary-value problem for a third-order equation with multiple characteristics,” Ukr. Mat. Zh., 64, No. 1, 3–13 (2012); English translation : Ukr. Math. J., 64, No. 1, 1–12 (2012).Google Scholar
  9. 9.
    Yu. P. Apakov and B. Yu. Irgashev, “Boundary-value problem for a degenerate high-odd-order equation,” Ukr. Mat. Zh., 66, No. 10, 1318–1331 (2014); English translation : Ukr. Math. J., 66, No. 10, 1475–1490 (2015).Google Scholar
  10. 10.
    Zh. A. Balkizov and A. Kh. Kadzakov, “On the representation of the solution of a boundary-value problem for the third-order inhomogeneous equation with multiple characteristics,” Izv. Kabardino-Balkar. Nauch. Centr. Ros. Akad Nauk, No. 4, 64–69 (2010).Google Scholar
  11. 11.
    T. K. Yuldashev, “Inverse problem for one integrodifferential Fredholm equation with third-order partial derivatives,” Vestn. Samar. Gos. Tekh. Univ., Ser. Fiz.-Mat. Nauk, No. 1(34), 56–65 (2014).Google Scholar
  12. 12.
    V. V. Shubin, “Boundary-value problems for third-order equations with discontinuous coefficients,” Vestn. NGU, Ser. Mat. Mekh. Inform., 12, No. 1, 126–138 (2012).zbMATHGoogle Scholar
  13. 13.
    A. Ashyralyev, N. Aggez, and F. Hezenci, “Boundary-value problem for a third order partial differential equation,” in: First Internat. Conf. on Analysis and Applied Mathematics (ICAAM 2012): AIP Conference Proceedings, 1470 (2012), pp. 130–133.Google Scholar
  14. 14.
    A. Ashyralyev and S. N. Simsek, “An operator method for a third order partial differential equation,” Numer. Funct. Anal. Optim., 38, No. 10, 1341–1360 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    A. Ashyralyev, Kh. Belakroum, and A. Guezane-Lakoud, “Stability of boundary-value problems for third-order partial differential equations,” Electron. J. Different. Equat., 2017, No. 53, 1–11 (2017).MathSciNetzbMATHGoogle Scholar
  16. 16.
    A. Ashyralyev, Kh. Belakroum, and A. Guezane-Lakoud, “Numerical algorithm for the third-order partial differential equation with local boundary conditions,” in: AIP Conf. Proc., 1880, 040008 (2017).Google Scholar
  17. 17.
    A. Ashyralyev, Kh. Belakroum, and A. Guezane-Lakoud, “Numerical algorithm for the third-order partial differential equation with nonlocal boundary conditions,” in: AIP Conf. Proc., 1880, 040012 (2017).Google Scholar
  18. 18.
    A. I. Kozhanov and G. A. Lukina, “Space-nonlocal problems with integral conditions for third-order differential equations,” Differents. Uravn., 53, No. 7, 906–917 (2017).zbMATHGoogle Scholar
  19. 19.
    Yu. P. Apakov, “Solution of boundary-value problems for a third-order equation with multiple characteristics by the method of separation of variables,” Uzb. Mat. Zh., No. 1, 14–23 (2007).MathSciNetzbMATHGoogle Scholar
  20. 20.
    V. A. Il’in and É. G. Poznyak, Foundations of Mathematical Analysis [in Russian], Vol. 2, Nauka, Moscow (1980).Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Yu. P. Apakov
    • 1
  • A. Kh. Zhuraev
    • 1
  1. 1.Namangan Construction Engineering InstituteNamanganUzbekistan

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