American Journal of Applied Mathematics

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Integral Representations of a Function in the S. L. Sobolev Space and Their Application to Boundary Problems

Received: 17 June 2023    Accepted: 25 August 2023    Published: 8 September 2023
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Abstract

First, we prove a theorem on the integral representation of functions of three variables at the middle of a domain in S. L. Sobolev space with a dominant mixed derivative on a three-dimensional parallelepiped. Further, an integral representation of periodic functions of three variables is given at the middle of the domain in the space of S. L. Sobolev with a dominant mixed derivative. A theorem is also given on the integral representation of homogeneous functions of three variables at the middle of a domain in S. L. Sobolev with a dominant mixed derivative. In addition, a theorem is given on the integral representation of odd functions of three variables at the middle of a domain in S. L. Sobolev with a dominant mixed derivative. Next, we present a theorem on the integral representation of even functions of three variables at the middle of a domain in S. L. Sobolev with a dominant mixed derivative. The above theorems are directly applicable to the qualitative theory of differential equations. In this article, in the most general form, an integral representation of functions of several variables at the middle of a domain in S. L. Sobolev with a dominant mixed derivative on a multidimensional parallelepiped. In this article, such an integral representation of functions in Sobolev space is used to study a boundary value problem in the middle of a domain for the Bianchi integro-differential equation, which is a class of dominating mixed differential equations. For the Bianchi integro-differential equation, the boundary value problem in the middle of the domain in the classical form is reduced to a nonclassical boundary value problem. In this setting, no additional conditions such as matching are required. Then the non-classical boundary value problem posed in the middle of the region is reduced to an operator equation. With the method of integral representations of functions for the boundary value problem, an equivalent integral equation is constructed. Using this integral equation, we prove the homeomorphism theorem. By definition, this theorem is demonstrated by the correct solvability of the considered boundary value problem in the middle of the domain.

DOI 10.11648/j.ajam.20231104.11
Published in American Journal of Applied Mathematics (Volume 11, Issue 4, August 2023)
Page(s) 58-70
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Integral Representation of Functions, S. L. Sobolev Space, Periodic Function, Homogeneous Function, Odd Function, Even Function, Function with Many Variables, Boundary Value Problem

References
[1] Amanov Tuleybay Idrisovich (1976). Spaces of differentiable functions with a dominating mixed derivative. Monograph, Nauka publishing (in Russian).
[2] Akhiev Seyidali Seydi (1976). On the general form of linear bounded functionals in a certain functional space of S. L. Sobolev type. Dokl. AN. Azerbaijan SSR (35) (in Russian).
[3] Besov Oleq Vladimirovich, Il’in Valentin Petrovich, and Nikol’skii Sergey Mikhaylovich (1975). Integral Representations of Functions and Embedding Theorems. Monograph, Nauka publishing (in Russian).
[4] Jabrailov Allahveren Jabrail (1964). On some function spaces. Direct and inverse embedding theorems. Dokl. AN SSSR (159) (in Russian).
[5] Lizorkin Petr Ivanovich, Nikol’skii Sergey Mikhaylovich (1965). Classification of differentiable functions based on spaces with a dominant derivative. Proceedings of the USSR Academy of Sciences (77) (in Russian).
[6] Mamedov Ilgar Gurbat, Abdullayeva Aynura Jabbar (2018). On the correct solvability of a boundary value problem in a non-classical interpretation of a domain given in the middle for a 3D Bianchi integro-differential equation. Journal of Contemporary Applied Mathematics (8) (in Russian).
[7] Mamedov Ilgar Gurbat, Abdullayeva Aynura Jabbar (2019). Optimal control problem for one 3D Bianchi integro-differential equation with non-smooth coefficients under conditions on the arithmetic middle of the domain. Sumgayit State University: Nauchnye Izvestia, Series: Natural and technical sciences (19) (in Russian).
[8] Mamedov Ilgar Gurbat (1999). On a decomposition for a continuous function of several variables. Bulletin of the Baku State University. Ser. Phys.-Math. Sciences (3) (in Russian).
[9] Najafov Alik Malik (2005). On integral representations of functions from spaces with a dominating mixed derivative. Bulletin of the Baku State University. Ser. Phys.-Math. Nauk (3) (in Russian).
[10] Nikol’skii Sergey Mikhaylovich (1969). Approximation of functions of several variables and embedding theorems. Monograph. Nauka Nauka publishing (in Russian).
[11] Zhegalov Valentin Ivanovich, Mironova Lubov Borisovna (2020). The Goursat and Cauchy problems for three-dimensional Bianchi equation. Russian Math.
[12] Mironova Lubov Borisovna (2020). A problem for a factorized equation with a pseudoparabolic differential operator. Russian Math.
[13] Mironov Alexei Nikolaevich, Mironova Lubov Borisovna, Yakovleva Juliya Oleqovna (2021). The Riemann method for equations with a dominant partial derivative. Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki (in Russian).
[14] Mironov Alexei Nikolaevich (2021). Construction of the Riemann–Hadamard function for the three-dimensional Bianchi equation. Russian Math.
[15] Mironov Alexei Nikolaevich, Volkov Aleksandr Petrovich (2022). On the Darboux problem for a hyperbolic system of equations with multiple characteristics. Russian Math.
[16] Mironov Alexei Nikolaevich, Mironova Lubov Borisovna (2023) On the Darboux Problem for Hyperbolic Systems. Differential Equations.
[17] Mironov Alexei Nikolaevich, Mironova Lubov Borisovna (2021). Riemann–Hadamard Method for One System in Three-Dimensional Space. Differential Equations.
[18] Mironov Alexei Nikolaevich., Yakovleva Juliya Oleqovna (2021) Constructing the Riemann–Hadamard Function for a Fourth-Order Bianchi Equation. Differential Equations.
[19] Mironov Alexei Nikolaevich (2021). Darboux Problem for the Fourth-Order Bianchi Equation. Differential Equations.
[20] Mamedov Ilgar Gurbat (2014). On a nonclassical interpretation of the Dirichlet problem for a fourth-order pseudoparabolic equation. Differential Equations.
[21] Mamedov Ilgar Gurbat (2015). On the well-posed solvability of the Dirichlet problem for a generalized Mangeron equation with nonsmooth coefficients. Differential Equations.
[22] Mamedov Ilgar Gurbat, Mardanov Misir Dzhumail, Melikov Telman Kuli and Bandaliev Rovshan Alifaga (2019) Well-posed solvability of the Neumann problem for a generalized Mangeron equation with nonsmooth coefficients. Differential Equations.
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    Ilgar Gurbat Mamedov, Aynura Jabbar Abdullayeva. (2023). Integral Representations of a Function in the S. L. Sobolev Space and Their Application to Boundary Problems. American Journal of Applied Mathematics, 11(4), 58-70. https://doi.org/10.11648/j.ajam.20231104.11

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    ACS Style

    Ilgar Gurbat Mamedov; Aynura Jabbar Abdullayeva. Integral Representations of a Function in the S. L. Sobolev Space and Their Application to Boundary Problems. Am. J. Appl. Math. 2023, 11(4), 58-70. doi: 10.11648/j.ajam.20231104.11

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    AMA Style

    Ilgar Gurbat Mamedov, Aynura Jabbar Abdullayeva. Integral Representations of a Function in the S. L. Sobolev Space and Their Application to Boundary Problems. Am J Appl Math. 2023;11(4):58-70. doi: 10.11648/j.ajam.20231104.11

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  • @article{10.11648/j.ajam.20231104.11,
      author = {Ilgar Gurbat Mamedov and Aynura Jabbar Abdullayeva},
      title = {Integral Representations of a Function in the S. L. Sobolev Space and Their Application to Boundary Problems},
      journal = {American Journal of Applied Mathematics},
      volume = {11},
      number = {4},
      pages = {58-70},
      doi = {10.11648/j.ajam.20231104.11},
      url = {https://doi.org/10.11648/j.ajam.20231104.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20231104.11},
      abstract = {First, we prove a theorem on the integral representation of functions of three variables at the middle of a domain in S. L. Sobolev space with a dominant mixed derivative on a three-dimensional parallelepiped. Further, an integral representation of periodic functions of three variables is given at the middle of the domain in the space of S. L. Sobolev with a dominant mixed derivative. A theorem is also given on the integral representation of homogeneous functions of three variables at the middle of a domain in S. L. Sobolev with a dominant mixed derivative. In addition, a theorem is given on the integral representation of odd functions of three variables at the middle of a domain in S. L. Sobolev with a dominant mixed derivative. Next, we present a theorem on the integral representation of even functions of three variables at the middle of a domain in S. L. Sobolev with a dominant mixed derivative. The above theorems are directly applicable to the qualitative theory of differential equations. In this article, in the most general form, an integral representation of functions of several variables at the middle of a domain in S. L. Sobolev with a dominant mixed derivative on a multidimensional parallelepiped. In this article, such an integral representation of functions in Sobolev space is used to study a boundary value problem in the middle of a domain for the Bianchi integro-differential equation, which is a class of dominating mixed differential equations. For the Bianchi integro-differential equation, the boundary value problem in the middle of the domain in the classical form is reduced to a nonclassical boundary value problem. In this setting, no additional conditions such as matching are required. Then the non-classical boundary value problem posed in the middle of the region is reduced to an operator equation. With the method of integral representations of functions for the boundary value problem, an equivalent integral equation is constructed. Using this integral equation, we prove the homeomorphism theorem. By definition, this theorem is demonstrated by the correct solvability of the considered boundary value problem in the middle of the domain.},
     year = {2023}
    }
    

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  • TY  - JOUR
    T1  - Integral Representations of a Function in the S. L. Sobolev Space and Their Application to Boundary Problems
    AU  - Ilgar Gurbat Mamedov
    AU  - Aynura Jabbar Abdullayeva
    Y1  - 2023/09/08
    PY  - 2023
    N1  - https://doi.org/10.11648/j.ajam.20231104.11
    DO  - 10.11648/j.ajam.20231104.11
    T2  - American Journal of Applied Mathematics
    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
    SP  - 58
    EP  - 70
    PB  - Science Publishing Group
    SN  - 2330-006X
    UR  - https://doi.org/10.11648/j.ajam.20231104.11
    AB  - First, we prove a theorem on the integral representation of functions of three variables at the middle of a domain in S. L. Sobolev space with a dominant mixed derivative on a three-dimensional parallelepiped. Further, an integral representation of periodic functions of three variables is given at the middle of the domain in the space of S. L. Sobolev with a dominant mixed derivative. A theorem is also given on the integral representation of homogeneous functions of three variables at the middle of a domain in S. L. Sobolev with a dominant mixed derivative. In addition, a theorem is given on the integral representation of odd functions of three variables at the middle of a domain in S. L. Sobolev with a dominant mixed derivative. Next, we present a theorem on the integral representation of even functions of three variables at the middle of a domain in S. L. Sobolev with a dominant mixed derivative. The above theorems are directly applicable to the qualitative theory of differential equations. In this article, in the most general form, an integral representation of functions of several variables at the middle of a domain in S. L. Sobolev with a dominant mixed derivative on a multidimensional parallelepiped. In this article, such an integral representation of functions in Sobolev space is used to study a boundary value problem in the middle of a domain for the Bianchi integro-differential equation, which is a class of dominating mixed differential equations. For the Bianchi integro-differential equation, the boundary value problem in the middle of the domain in the classical form is reduced to a nonclassical boundary value problem. In this setting, no additional conditions such as matching are required. Then the non-classical boundary value problem posed in the middle of the region is reduced to an operator equation. With the method of integral representations of functions for the boundary value problem, an equivalent integral equation is constructed. Using this integral equation, we prove the homeomorphism theorem. By definition, this theorem is demonstrated by the correct solvability of the considered boundary value problem in the middle of the domain.
    VL  - 11
    IS  - 4
    ER  - 

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Author Information
  • Laboratory of Mathematical Analysis of Control in Physical, Biological and Economic Processes, Institute of Control Systems of the Ministry of Science and Education of Republic of Azerbaijan, Baku, Azerbaijan

  • Military-Scientific Research Institute, Azerbaijan National Defense University, Baku, Azerbaijan

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