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The Dirichlet Problem for the Perturbed Elliptic Equation

Author

Listed:
  • Ulyana Yarka

    (Social Communication and Information Activities Department, Lviv Polytechnic National University, 79000 Lviv, Ukraine)

  • Solomiia Fedushko

    (Social Communication and Information Activities Department, Lviv Polytechnic National University, 79000 Lviv, Ukraine)

  • Peter Veselý

    (Faculty of Management, Comenius University in Bratislava, 81499 Bratislava, Slovakia)

Abstract

In this paper, the authors consider the construction of one class of perturbed problems to the Dirichlet problem for the elliptic equation. The operators of both problems are isospectral, which makes it possible to construct solutions to the perturbed problem using the Fourier method. This article focuses on the Dirichlet problem for the elliptic equation perturbed by the selected variable. We established the spectral properties of the perturbed operator. In this work, we found the eigenvalues and eigenfunctions of the perturbed task operator. Further, we proved the completeness, minimal spanning system, and Riesz basis system of eigenfunctions of the perturbed operator. Finally, we proved the theorem on the existence and uniqueness of the solution to the boundary value problem for a perturbed elliptic equation.

Suggested Citation

  • Ulyana Yarka & Solomiia Fedushko & Peter Veselý, 2020. "The Dirichlet Problem for the Perturbed Elliptic Equation," Mathematics, MDPI, vol. 8(12), pages 1-14, November.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:12:p:2108-:d:451081
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    References listed on IDEAS

    as
    1. Kryvinska, Natalia, 2008. "An analytical approach for the modeling of real-time services over IP network," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(4), pages 980-990.
    2. Oksana Hoshovska & Zhanna Poplavska & Natalia Kryvinska & Natalia Horbal, 2020. "Considering Random Factors in Modeling Complex Microeconomic Systems," Mathematics, MDPI, vol. 8(8), pages 1-18, July.
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    Cited by:

    1. Aleksandr I. Kozhanov & Oksana I. Bzheumikhova, 2022. "Elliptic and Parabolic Equations with Involution and Degeneration at Higher Derivatives," Mathematics, MDPI, vol. 10(18), pages 1-10, September.

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