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Construction of the Solution of a Boundary-Value Problem for the Third-Order Equation with Lower Terms with the Help of the Green Function

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We consider the first boundary-value problem in a rectangular domain for an inhomogeneous third-order equation with lower terms. The uniqueness of the solution to the stated problem is proved by the method of energy integrals. The solution is represented in terms of the constructed Green function.

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References

  1. O. S. Ryzhov, “Asymptotic picture of a sonic flow of viscous and heat-conducting gas around the bodies of revolution,” Prikl. Mat. Mekh., 29, Issue 6, 1004–1014 (1965).

    MATH  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  3. H. Block, “Sur les equations lineaires aux derivees partielles a carateristiques multiples,” Ark. Mat., Astron. Fus. Note 1, 7(13), 1–34 (1912); Ark. Mat., Astron. Fus. Note 2, 7(21), 1–30 (1912); Ark. Mat., Astron. Fus. Note 3, 8(23), 1–51 (1912–1913).

  4. E. Del Vicchio, “Sulleequazioni,” Mem. R. Accad. Sci., Ser. 2, 66, 1–41 (1915).

  5. L. Cattabriga, “Potenziali di linea e di dominio per equazioni non paraboliche in due variabilia caratteristiche multiple,” Rend. Sem. Mat. Univ. Padova, 31, 1–45 (1961).

    MathSciNet  MATH  Google Scholar 

  6. T. D. Dzhuraev and Yu. P. Apakov, “On the selfsimilar solution of a third-order equation with multiple characteristics,” Vestn. Samarsk. Gos. Tekh. Univ., Ser. Fiz.-Mat. Nauk., No. 2(15), 18–26 (2007).

  7. 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).

  8. Y. 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).

    Article  MathSciNet  MATH  Google Scholar 

  9. 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).

  10. Yu. P. Apakov and A. Kh. Zhuraev, “On the solution of a boundary-value problem for the third-order equation by using the Green function,” Uzb. Mat. Zh., No. 3, 36–42 (2011).

  11. T. K. Yuldashev, Y. P. Apakov, and A. Kh. Zhuraev, “Boundary value problem for third order partial integro-differential equation with a degenerate kernel,” Lobachevskii J. Math., 42, No. 6, 1316–1326 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  12. B. Yu. Irgashev, “Boundary-value problem for one degenerate high-order equation with lower terms,” Bull. Inst. Math., No. 6 23–29 (2019).

  13. K. B. Sabitov, “Dirichlet problem for a third-order equation of mixed type,” Dokl. Akad. Nauk Ros., 427, No. 5, 593–596 (2009).

    Google Scholar 

  14. Zh. A. Balkizov and A. Kh. Kadzakov, “On representation of the solution to the boundary-value problem for the third-order inhomogeneous equation with multiple characteristics,” Izv. Kabard.-Balkar. Nauch. Tsentr. Ros. Akad Nauk, No. 4, 64–69 (2010).

  15. G. A. Lukina, “Boundary-value problems with integral boundary conditions for the linearized Korteweg–de Vries equation,” Vestn. Yuzhn.-Ural. Gos. Univ., Ser. Mat. Model. Program., 234, No. 17, 52–61 (2011).

  16. V. V. Shubin, “Boundary-value problems for third-order equations with discontinuous coefficients,” Vest. Novosibirsk. Gos. Univ., Ser. Mat. Mekh. Inform., 12, No. 1, 126–138 (2012).

  17. A. Ashyralyev, K. 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).

    MathSciNet  MATH  Google Scholar 

  18. A. I. Kozhanov and A. V. Dyuzheva, “Nonlocal problems with integral condition for third-order differential equations,” Vestn. Samarsk. Gos. Tekh. Univ., Ser. Fiz.-Mat. Nauk., 24, No. 4, 607–620 (2020).

  19. Yu. P. Apakov and A. Kh. Zhuraev, “Third boundary-value problem for a third-order differential equation with multiple characteristics,” Ukr. Mat. Zh., 70, No. 9, 1274–1281 (2018); English translation: Ukr. Math. J., 70, No. 9, 70, 1467–1476 (2019).

  20. A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1972).

    Google Scholar 

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Authors and Affiliations

  1. Romanovskii Institute of Mathematics, Academy of Sciences of Uzbekistan, Mirzo Ulugbek Str., 81, 100170, Tashkent, Uzbekistan

    Yu. P. Apakov

  2. Namangan Engineering-Construction Institute, Islam Karimov Str., 12, 160100, Namangan, Uzbekistan

    Yu. P. Apakov & R. A. Umarov

Authors
  1. Yu. P. Apakov
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  2. R. A. Umarov
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Correspondence to R. A. Umarov.

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Translated from Neliniini Kolyvannya, Vol. 25, No. 2-3, pp. 161–173, April–September, 2022.

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Apakov, Y.P., Umarov, R.A. Construction of the Solution of a Boundary-Value Problem for the Third-Order Equation with Lower Terms with the Help of the Green Function. J Math Sci 274, 807–821 (2023). https://doi.org/10.1007/s10958-023-06644-2

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  • DOI: https://doi.org/10.1007/s10958-023-06644-2

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