EIGENFUNCTION EXPANSIONS OF SECOND-ORDER ELLIPTIC OPERATORS

Jurayev Shaxzod Shuxratjonvich

Authors

Jurayev Shaxzod Shuxratjonvich Asia International University

Keywords:

second-order elliptic operators; spectral representation; eigenvalue problems; orthogonal expansions; boundary conditions; self-adjoint operators; Hilbert spaces; functional analysis.

Abstract

This study explores the theory of spectral representations for second-order elliptic operators. The existence and properties of eigenvalues and eigenfunctions in bounded regions are examined, and solutions are formulated through orthogonal basis functions in Hilbert spaces. The research further applies the obtained theoretical results to boundary value problems, investigates convergence behavior, and highlights their importance in numerical approximations. In addition, the effects of regularity conditions, parameter variations, and illustrative examples are discussed. The findings contribute to a deeper understanding of elliptic operator theory and open new directions for future investigations.

References

Буваев Қ. Т., Жўраев Ш. Ш. Уч каррали Фурье қаторининг октаэдрли қисмий йиғиндилар кетма-кетлиги учун Лебег ўзгармасини асимптотик ҳолати ҳақида INNOVATION IN THE MODERN EDUCATION SYSTEM jurn.2023,171-172-betlar.

Буваев Қ. Т., Жўраев Ш. Ш. “Уч каррали Фурье қаторининг октаэдрли қисмий йиғиндилар кетма-кетлигини текис яқинлашиши” ZAMONAVIY MATEMATIKANING DOLZARB MUAMMOLARI VA TATBIQLARI yosh olimlarning ilmiy konferensiyasi tezislari to‘plami.14-15 Mart 2024Toshkent54-56 betlar

Jo‘rayev S. S. (2026). ELLIPTIK DIFFERENSIAL OPERATORLAR CHIZIQLI SPEKTRAL YOYILMALARINING RAQAMLI IQTISODIYOTDA TUTGAN O‘RNI. Development Of Science, 1(1), pp. 229-236. .

Jo‘rayev S. S. (2025). MATEMATIKA FANINING RIVOJLANISHI ORQALI INSONLARNING KOINOT HAQIDAGI QARASHLARI. Development Of Science, 12(8), pp. 210-219.

Jurayev Shaxzod Shuxratjonvich, ENHANCING THE EFFECTIVENESS OF TEACHING METHODOLOGY FOR ELLIPTIC DIFFERENTIAL EQUATION SOLUTIONS BASED ON ARTIFICIAL INTELLIGENCE. (2025). Journal of Multidisciplinary Sciences and Innovations, 4(11), 1764-1768.

Jurayev Shaxzod Shuxratjonvich, THE ROLE OF THE LAPLACE OPERATOR IN ARTIFICIAL INTELLIGENCE , Journal of Multidisciplinary Sciences and Innovations: Vol. 4 No. 10 (2025): Journal of Multidisciplinary Sciences and Innovations.408-411

Jurayev Shaxzod Shuxratjonvich ,THE SIGNIFICANCE OF ELLIPTIC DIFFERENTIAL OPERATORS IN ARTIFICIAL INTELLIGENCE SYSTEMS,Journal of Applied SCIENCE AND SOCIAL SCIENCE.650-654

Elliptic Partial Differential Equations of Second Order — David Gilbarg and Neil S. Trudinger, Springer, 2001.

9 Behrndt, J., & Rohleder, J. (2015). Spectral analysis of selfadjoint elliptic differential operators via Dirichlet-to-Neumann maps and abstract Weyl functions. Advances in Mathematics, 274, 242283.

Methods of Modern Mathematical Physics, Vol. I: Functional Analysis — Michael Reed and Barry Simon, Academic Press, 1980.

Алимов Ш. А. Равномерная сходимость и суммируемость спектральных разложений функций из //Дифференц. уравнения.— 1973.— 9, № 4.— С. 669—681.