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Mixed Problem with Integral Condition for a Loaded Third Order Pseudoparabolic Equation

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We consider a nonlocal problem with an integral condition for a loaded third order pseudoparabolic equation in a rectangular domain. We prove the unique solvability of the problem with the help of an auxiliary Goursat problem for a pseudoparabolic equation. Based on an integral representation of the solution to the Goursat problem, we reduce the original problem to a Volterra integral equation of the second kind and, using the contraction mapping principle, establish the solvability of the original problem.

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References

  1. D. G. Gordeziani and G. A. Avalishvili, ‘Investigation of the nonlocal initial-boundary value problems for some hyperbolic equations,” Hiroshima Math. J. 31, No. 3, 345–366 (2001).

    Article  MathSciNet  Google Scholar 

  2. A. Bouziani, “A mixed problem with only integral boundary conditions for a hyperbolic equation,” Int. J. Math. Math. Sci. 2004, No. 21–24, 1279–1291 (2004).

    Article  MathSciNet  Google Scholar 

  3. O. S. Zikirov, “On boundary value problem for hyperbolic type equation of the third order,” Lith. Math. J. 47, No. 4, 484–495 (2007).

    Article  MathSciNet  Google Scholar 

  4. Dao-Qing Dai and Yu. Huang, “Nonlocal boundary problems for a third-order onedimensional nonlinear pseudoparabolic equation,” Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 66, No. 1, 179–191 (2007).

  5. J. Jachimavičienè et al., “On the stability of explicit finite difference schemes for a pseudoparabolic equation with nonlocal conditions,” Nonlinear Anal., Model. Control 19, No. 2, 225–240 (2014).

  6. S. Roman and A. Stikonas, “Green’s functions for stationary problems with nonlocal boundary conditions,” Lith. Math. J. 49, No. 2, 190–202 (2009).

    Article  MathSciNet  Google Scholar 

  7. S. Roman and A. Stikonas, “Third-order linear differential equation with three additional conditions and formula for Green’s function,” Lith. Math. J. 50 (4), 426–466 (2010).

    Article  MathSciNet  Google Scholar 

  8. M. H. Beshtokov, “Riemann method for solution of nonlocal boundary value problems for pseudoparabolic equations of the third order” [in Russian], Vestn. Samar. Gos. Tekh. Univ., Ser. Fiz-Mat. Nauk 33, No. 4, 15–24 (2013).

  9. M. Kh. Shkhanukov. “Boundary value problems for a third-order equation occurring in the modeling of water filtration in porous media,” Differ. Equations 18, 509–517 (1982).

    MathSciNet  Google Scholar 

  10. A. P. Soldatov and M. Kh. Shkhanukov, “Boundary value problems with general nonlocal Samarskij condition for pseudoparabolic equations of higher order,” Sov. Math., Dokl. 36, No. 3, 507–511 (1988).

  11. A. I. Kozhanov, “On a nonlocal boundary value problem with variable coefficients for the heat equation and the Aller equation,” Differ. Equ. 40, No. 6, 815–826 (2004).

    Article  MathSciNet  Google Scholar 

  12. A. I. Kozhanov and N. S. Popov, “Solvability of nonlocal boundary value problems for pseudoparabolic equations,” J. Math. Sci. 186, No. 3, 438–452 (2012).

    Article  MathSciNet  Google Scholar 

  13. A. I. Kozhanov and A. V. Dyuzheva, “Solvability of a nonlocal problem with integral conditions for third-order Sobolev-type equations,” Mat. Zamet. SVFU 27, No. 4, 30–42 (2020).

    Google Scholar 

  14. O. M. Dzhokhadze, “The Influence of lower terms on the well-posedness of the formulation of characteristic problems for third-order hyperbolic equations,” Mat. Notes 74, No. 4, 491–501 (2003).

    Article  MathSciNet  Google Scholar 

  15. O. S. Zikirov. “On solvability of the Dirichlet problem for the third-order hyperbolic equation,” Lith. Math. J. 50, No. 2, 239–247 (2010).

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Sobolev Institute of Mathematics SB RAS, 4, Universitetskii pr., Novosibirsk, 630090, Russia

    Alexandr Kozhanov

  2. Novosibirsk State University, 1, Pirogova St., Novosibirsk, 630090, Russia

    Alexandr Kozhanov

  3. Mirzo Ulugbek National University of Uzbekistan, 4, Universitetskaya St., 100174, Tashkent, Uzbekistan

    Obidjan Zikirov

  4. Tashkent University of Architecture and Construction, 9, Yangi Shaxar St., 100194, Tashkent, Uzbekistan

    Dilshod Kholikov

Authors
  1. Alexandr Kozhanov
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  2. Obidjan Zikirov
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  3. Dilshod Kholikov
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Correspondence to Alexandr Kozhanov.

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International Mathematical Schools. Vol. 6. Mathematical Schools in Uzbekistan

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Kozhanov, A., Zikirov, O. & Kholikov, D. Mixed Problem with Integral Condition for a Loaded Third Order Pseudoparabolic Equation. J Math Sci 277, 420–430 (2023). https://doi.org/10.1007/s10958-023-06845-9

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  • DOI: https://doi.org/10.1007/s10958-023-06845-9

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