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Initial-Boundary Problem for a Three-Dimensional Inhomogeneous Equation of Parabolic-Hyperbolic Type

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Abstract

For an inhomogeneous three-dimensional equation of mixed parabolic-hyperbolic type in a rectangular parallelepiped, the initial-boundary problem is studied. A criterion for the uniqueness of a solution is established. The solution is constructed as the sum of an orthogonal series. In substantiating the convergence of the series, the problem of small denominators of two natural arguments arose. Estimates are established for the separation from zero of the small denominators with the corresponding asymptotics. These estimates made it possible to justify the convergence of the constructed series in the class of regular solutions of this equation.

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REFERENCES

  1. I. M. Gel’fand, ‘‘Some questions of analysis and differential equations,’’ Usp. Mat. Nauk 14 (3), 3–19 (1959).

    MathSciNet  Google Scholar 

  2. Ya. S. Uflyand, ‘‘On the question of the propagation of oscillations in compound electric lines,’’ Eng. Phys. J. 7, 89–92 (1964).

    Google Scholar 

  3. Ya. S. Uflyand and I. T. Lozanovskaya, ‘‘On a Class of Problems of Mathematical Physics with a Mixed Spectrum of Eigenvalues,’’ Dokl. Akad. Nauk SSSR 164, 1005–1007 (1965).

    MATH  Google Scholar 

  4. T. D. Dzhuraev, A. Sopuev, and M. Mamazhanov, Boundary Value Problems for Parabolic-Hyperbolic Type Equations (Fan, Tashkent, 1986) [in Russian].

  5. O. A. Ladyzhenskaya and L. Stupyalis, ‘‘On mixed-type equations,’’ Vestn. Leningr. Univ., Ser: Mat. Mekh. Astron. 19 (4), 38–46 (1965).

    Google Scholar 

  6. L. Stupyalis, ‘‘Initial-boundary value problems for equations of mixed type,’’ Proc. Steklov Inst. Math. 127, 135–170 (1975).

    MathSciNet  Google Scholar 

  7. K. B. Sabitov, ‘‘Tricomi problem for a mixed parabolic–hyperbolic equation in a rectangular domain,’’ Math. Notes 86, 249–254 (2009).

    Article  MathSciNet  Google Scholar 

  8. K. B. Sabitov, Direct and Inverse Problems for Equations of Mixed Parabolic-Hyperbolic Type (Nauka, Moscow, 2016) [in Russian].

    Google Scholar 

  9. K. B. Sabitov, ‘‘Initial boundary and inverse problems for the inhomogeneous equation of a mixed parabolic-hyperbolic equation,’’ Math. Notes 102, 378–395 (2017).

    Article  MathSciNet  Google Scholar 

  10. S. N. Sidorov, ‘‘Nonlocal problem for a degenerate parabolic-hyperbolic equation,’’ Rep. Adyghe Int. Acad. Sci. 14 (3), 34–44 (2012).

    Google Scholar 

  11. K. B. Sabitov and S. N. Sidorov, ‘‘On a nonlocal problem for a degenerating parabolic-hyperbolic equation,’’ Differ. Equat. 50, 352–361 (2014).

    Article  MathSciNet  Google Scholar 

  12. S. N. Sidorov, ‘‘Nonlocal problems for an equation of mixed parabolic-hyperbolic type with power degeneration,’’ Russ. Math. (Iz. VUZ) 59 (12), 46–55 (2015).

  13. K. B. Sabitov and S. N. Sidorov, ‘‘Initial boundary value problem for inhomogeneous degenerate equations of mixed parabolic-hyperbolic type,’’ Itogi Nauki Tekh., Ser.: Sovrem. Mat. Prilozh. Temat. Obz. 137, 26–60 (2017).

    Google Scholar 

  14. K. B. Sabitov and S. N. Sidorov, ‘‘Initial-boundary-value problem for inhomogeneous degenerate equations of mixed parabolic-hyperbolic type,’’ J. Math. Sci. 236, 603–640 (2019).

    Article  Google Scholar 

  15. K. B. Sabitov and E. M. Safin, ‘‘The inverse problem for an equation of mixed parabolic-hyperbolic type,’’ Math. Notes 87, 880–889 (2010).

    Article  MathSciNet  Google Scholar 

  16. K. B. Sabitov and S. N. Sidorov, ‘‘Inverse problem for degenerate parabolic-hyperbolic equation with nonlocal boundary condition,’’ Russ. Math. (Iz. VUZ) 59 (1), 39–50 (2015).

  17. S. N. Sidorov, ‘‘Inverse problems for a mixed parabolic-hyperbolic equation with a degenerate parabolic part,’’ Sib. Elektron. Mat. Izv. 16, 144–157 (2019).

    Article  MathSciNet  Google Scholar 

  18. S. N. Sidorov, ‘‘Inverse problems for a degenerate mixed parabolic-hyperbolic equation on finding time-depending factors in right hand sides,’’ Ufa Math. J. 11 (1), 75–89 (2019).

    Article  MathSciNet  Google Scholar 

  19. K. B. Sabitov, ‘‘Boundary-value problem for non-homogeneous mixed parabolic-hyperbolic type equation,’’ Lobachevskii J. Math. 38 (1), 137–147 (2017).

    Article  MathSciNet  Google Scholar 

  20. A. Ya. Khinchin, Continued Fractions (Dover, Mineola, New York, 1997).

    MATH  Google Scholar 

  21. A. A. Bukhshtab, Number Theory (Lan’, St. Petersburg, 2008) [in Russian].

  22. V. A. Il’in, V. A. Sadovnichiy, and B. L. Sendov, Mathematical Analysis (Mosk. Gos. Univ., Moscow, 1987), Vol. 2 [in Russian].

    Google Scholar 

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Funding

The reported study was funded by Russian Foundation for Basic Research, project no. 19-31-60016.

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Authors and Affiliations

  1. Sterlitamak Branch of Bashkir State University, 453103, Sterlitamak, Russia

    K. B. Sabitov & S. N. Sidorov

  2. Sterlitamak Branch of the Institute for Strategic Studies of the Republic of Bashkortostan, 453103, Sterlitamak, Russia

    K. B. Sabitov & S. N. Sidorov

Authors
  1. K. B. Sabitov
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  2. S. N. Sidorov
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Correspondence to K. B. Sabitov or S. N. Sidorov.

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(Submitted by F. G. Avkhadiev)

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Sabitov, K.B., Sidorov, S.N. Initial-Boundary Problem for a Three-Dimensional Inhomogeneous Equation of Parabolic-Hyperbolic Type. Lobachevskii J Math 41, 2257–2268 (2020). https://doi.org/10.1134/S1995080220110190

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  • DOI: https://doi.org/10.1134/S1995080220110190

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