Integral Representations of a Function in the S. L. Sobolev Space and Their Application to Boundary Problems
Ilgar Gurbat Mamedov,
Aynura Jabbar Abdullayeva
Issue:
Volume 11, Issue 4, August 2023
Pages:
58-70
Received:
17 June 2023
Accepted:
25 August 2023
Published:
8 September 2023
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.
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. Sobo...
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Student’s CGPA Versus Skill Comparison in Bipolar Fuzzy Soft Domain
Issue:
Volume 11, Issue 4, August 2023
Pages:
71-76
Received:
3 September 2023
Accepted:
19 September 2023
Published:
12 October 2023
Abstract: In the realm of education, traditional methods of evaluating students often fall short when it comes to assessing their true abilities and potential. Merely acquiring knowledge is insufficient in fulfilling the objectives of learning; it is imperative that students apply their skills and abilities effectively. The Bloom's Taxonomy, a renowned classification system, places a greater emphasis on the development of skills over the mere absorption of content. This research delves into the assessment of students, taking into account both their skills and the conventional CGPA (Cumulative Grade Point Average) system. This study introduces a novel approach by incorporating bipolar fuzzy soft numbers to establish a comprehensive ranking system. Bipolar fuzzy soft numbers provide a versatile and nuanced framework for evaluating students, considering not only their achievements but also their strengths and weaknesses. The research employs the bipolar fuzzy soft weighted arithmetic averaging operator to aggregate these multifaceted evaluations, resulting in a holistic ranking of students. The final phase of the study involves a comparative analysis of the rank list based on the conventional CGPA system and the one derived from the assessment of skills parameters. This comparison will shed light on the effectiveness of the traditional grading system versus a more skill-oriented approach, providing valuable insights for educators and institutions seeking to enhance their evaluation methods and better nurture their students' talents.
Abstract: In the realm of education, traditional methods of evaluating students often fall short when it comes to assessing their true abilities and potential. Merely acquiring knowledge is insufficient in fulfilling the objectives of learning; it is imperative that students apply their skills and abilities effectively. The Bloom's Taxonomy, a renowned class...
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