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On a Boundary-value Problem for a Parabolic-Hyperbolic Equation with Fractional Order Caputo Operator in Rectangular Domain

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Abstract

In this paper we study a new problem for a parabolic-hyperbolic equation with fractional order Caputo operator in rectangular domain. There are many works devoted to study problems for the second order mixed parabolic-hyperbolic and elliptic-hyperbolic type equations in rectangular domains with two gluing conditions with respect to second argument and with boundary value conditions on all borders of the domain. In studying the unique solvability of this problem, it becomes necessary to specify an additional condition on the hyperbolic boundary of the domain. For this reason, the considering problem became unresolved in an arbitrary rectangular domain. In this paper, we were able to remove this restriction by setting three gluing conditions for the second argument.

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REFERENCES

  1. M. M. Dzharbashyan and A. B. Nersesyan, ‘‘Fractional derivatives and Cauchy problems for fractional differential equations,’’ Izv. Akad. Nauk Arm SSR, Mat. 3, 3–28 (1968).

    Google Scholar 

  2. M. M. Dzharbashyan, Integral Transformations and Representations of Functions in a Complex Domain (Nauka, Moscow, 1966) [in Russian].

    Google Scholar 

  3. R. Gorenflo, Y. F. Luchko, and S. R. Umarov, ‘‘On the Cauchy and multipoint problems for partial pseudo-differential equations of fractional order,’’ Fract. Calc. Appl. Anal. 3, 249–275 (2000).

    MathSciNet  MATH  Google Scholar 

  4. R. Gorenflo, Y. F. Luchko, and F. Mainardi, ‘‘Wright fractions as scale-invariant solutions of the diffusion-wave equation,’’ J. Comput. Appl. Math. 118, 175–191 (2000).

    Article  MathSciNet  Google Scholar 

  5. A. A. Kilbas and S. A. Marzan, ‘‘Cauchy problem for differential equation with Caputo derivative,’’ Fract. Calc. Appl. Anal. 7, 297–321 (2004).

    MathSciNet  MATH  Google Scholar 

  6. A. V. Pskhu, ‘‘The solution of the first boundary value problem for the diffusion equation of fractional and continuum order,’’ Differ. Equat. 39, 1359–1363 (2003).

    Article  MathSciNet  Google Scholar 

  7. A. V. Pskhu, Boundary Value Problems for Fractional and Continuum Partial Differential Equations (Inst. Prikl. Mat. Avtomat. Kab.-Balk. Nauch. Tsentr RAN, Nalchik, 2005) [in Russian].

    Google Scholar 

  8. B. Zh. Kadirkulov and B. Kh. Turmetov, ‘‘On a generalization of the heat equation,’’ Uzb. Mat. Zh., No. 3, 40–45 (2006).

  9. V. A. Il’in, ‘‘The uniqueness and belonging of the classical solution of the mixed problem for a self-adjoint hyperbolic equation,’’ Mat. Zam. 17, 93–103 (1975).

    Google Scholar 

  10. M. M. Khachev, ‘‘The Dirichlet problem for the generalized Lavrentiev-Bitsadze equation in a rectangular domain,’’ Differ. Uravn. 14, 136–139 (1978).

    MathSciNet  Google Scholar 

  11. K. B. Sabitov, ‘‘Dirichlet problem for mixed-type equations in a rectangular domain,’’ Dokl. Math. 75, 193–196 (2007).

    Article  MathSciNet  Google Scholar 

  12. K. B. Sabitov, ‘‘A boundary value problem for the mixed type equations of of the third order in a rectangular region,’’ Differ. Equat. 47, 706–714 (2011).

    Article  Google Scholar 

  13. N. Yu. Kapustin and E. I. Moiseev, ‘‘On the evaluation of the solution of one problem for parabolic-hyperbolic equation using Fourier series,’’ Differ. Equat. 39, 694–700 (2003).

    Article  Google Scholar 

  14. K. B. Sabitov, ‘‘On the theory of the mixed parabolic-hyperbolic type equations of with spectral parameter,’’ Differ. Equat. 25, 93–100 (1989).

    MathSciNet  MATH  Google Scholar 

  15. M. A. Sadybekov and G. D. Toyzhanova, ‘‘Spectral properties of a class of boundary value problems for a parabolic-hyperbolic equation,’’ Differ. Uravn. 28, 179–179 (1992).

    MathSciNet  Google Scholar 

  16. N. Yu. Kapustin and E. I. Moiseev, ‘‘On spectral problems with a spectral parameter in the boundary condition,’’ Differ. Equat. 33, 116–120 (1997).

    MathSciNet  MATH  Google Scholar 

  17. E. I. Egorov, E. S. Efimova, and I. M. Tikhonova, ‘‘Fredholm solvability of the first boundary-value problem for a second-order equation of mixed type with a spectral parameter,’’ Mat. Zam. Sev.-Zap. Fed. Univ. 25, 15–24 (2018).

    MATH  Google Scholar 

  18. T. K. Yuldashev, ‘‘On a mixed type fourth-order differential equation ,’’ Izv. Inst. Mat. Inform. UdGU 47 (1), 119–128 (2016).

    Google Scholar 

  19. T. K. Yuldashev and A. V. Bagrova, ‘‘Nonlocal problem for a mixed type fourth order differential equation in three dimensional domain,’’ J. Srednevolzh. Mat. Ob-va 18 (3), 70–79 (2016).

    Google Scholar 

  20. T. K. Yuldashev, ‘‘Nonlocal problem for a mixed type differential equation in rectangular domain,’’ Proc. Yerevan State Univ., Phys. Math. Sci., No. 3, 70–78 (2016).

  21. T. K. Yuldashev, ‘‘On an integro-differential equation of pseudoparabolic-pseudohyperbolic type with degenerate kernels,’’ Proc. Yerevan State Univ., Phys. Math. Sci. 52 (1), 19–26 (2018).

  22. T. K. Yuldashev, ‘‘Solvability of a boundary value problem for a differential equation of the Boussinesq type,’’ Differ. Equat. 54, 1384–1393 (2018).

    Article  MathSciNet  Google Scholar 

  23. T. K. Yuldashev, ‘‘On a Boundary value problem for Boussinesq type nonlinear integro-differential equation with reflecting argument,’’ Lobachevskii J. Math. 41, 111–123 (2020).

    Article  MathSciNet  Google Scholar 

  24. M. S. Salakhitdinov and E. T. Karimov, ‘‘On a nonlocal problem with conjugation conditions of an integral form for a parabolic-hyperbolic with the Caputo operator,’’ Dokl. Akad. Nauk Uzbekist., No. 4, 6–9 (2014).

  25. E. T. Karimov and J. S. Ahatov, ‘‘A boundary problem with integral gluing condition for a parabolic-hyperbolic equation involving the Caputo fractional derivative,’’ Elect. J. Diff. Equat. 2014 (14), 1–6 (2014).

    Article  MathSciNet  Google Scholar 

  26. B. I. Islomov and U. Sh. Ubaidullaev, ‘‘A boundary value problem for the parabolic-hyperbolic type equation with a fractional order operator in the sense of Caputo in a rectangular domain,’’ Nauch. Vestn., Mat., No. 5, 25–30 (2017).

  27. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Nauka Tekhnol., Minsk, 1987; CRC, Boca Raton, FL, 1993).

  28. Sh. A. Alimov, ‘‘About solutions of one boundary value problem,’’ Uzb. Mat. Zh., No. 1, 3–9 (1999).

  29. Sh. A. Alimov, ‘‘On the solvability of one ill-posed problem,’’ Uzb. Mat. Zh., No. 3, 19–28 (1999).

  30. Sh. A. Alimov, ‘‘On solutions of one ill-posed boundary value problem,’’ Dokl. Akad. Nauk Uzbekist., No. 9, 7–11 (1999).

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Authors and Affiliations

  1. Department of Differential Equations and Mathematical Physics, National University of Uzbekistan, 100174, Tashkent, Uzbekistan

    B. I. Islomov & U. Sh. Ubaydullayev

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  1. B. I. Islomov
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  2. U. Sh. Ubaydullayev
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Correspondence to B. I. Islomov or U. Sh. Ubaydullayev.

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(Submitted by A. M. Elizarov)

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Islomov, B.I., Ubaydullayev, U.S. On a Boundary-value Problem for a Parabolic-Hyperbolic Equation with Fractional Order Caputo Operator in Rectangular Domain. Lobachevskii J Math 41, 1801–1810 (2020). https://doi.org/10.1134/S1995080220090115

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