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.
Published in | American Journal of Applied Mathematics (Volume 11, Issue 4) |
DOI | 10.11648/j.ajam.20231104.11 |
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), 2023. Published by Science Publishing Group |
Integral Representation of Functions, S. L. Sobolev Space, Periodic Function, Homogeneous Function, Odd Function, Even Function, Function with Many Variables, Boundary Value Problem
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APA Style
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
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
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
@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} }
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 -