# Vestnik КRAUNC. Fiz.-Mat. nauki. 2024. vol. 46. no. 1. P. 118-133. ISSN 2079-6641

**INFORMATION AND COMPUTATION TECHNOLOGIES**

https://doi.org/10.26117/2079-6641-2024-46-1-118-133

Research Article

Full text in English

MSC 65D05, 65L60

**Construction of Basis Functions for Finite Element Methods in a Hilbert Space**

**A. R. Hayotov^{\ast \bf{123}}, N. N. Doniyorov^{\ast \bf{145}}**

^1V. I. Romanovskiy Institute of Mathematics, 9, University str., Tashkent, 100174, Uzbekistan

^2Tashkent State Transport University, Temiryo‘lchilar str., Tashkent, 100167, Uzbekistan

^3Central Asian University, 264, Milliy bog str., Tashkent, 111221, Uzbekistan

^4National University of Uzbekistan named after Mirzo Ulugbek, 4, University str., Tashkent, 100174, Uzbekistan

^5Bukhara State University, 11, Muhammad Ikbol str., Bukhara, 200114, Uzbekistan

Abstract. The present work is devoted to construction of the optimal interpolation formula exact for

trigonometric functions sin(ωx) and cos(ωx). Here the analytical representations of the coefficients of the optimal interpolation formula in a certain Hilbert space are obtained using the discrete analogue of the differential operator. Taking the coefficients of the optimal interpolation formula as basis functions, in the finite element methods the boundary value problems for ordinary differential equations of the second order are approximately solved. In particular, it is shown that the coefficients of the optimal interpolation formula can serve as a set of effective basis functions. Approximate solutions of the differential equations are compared using the constructed basis functions and known basis functions. In particular, we have obtained numerical results for the cases when the numbers of basis functions are 6 and 11. In both cases, we have got that the accuracy of the approximate solution to the boundary value problems for second-order ordinary differential equations found using our basis functions is higher than the accuracy of the approximate solution found using known basis functions. It is proven that the accuracy of the approximate solution increases with increasing the number of basis functions.

*Key words: basis functions, ordinary differential equation, boundary value problem, finite element **method, interpolation.*

Received: 03.01.2024; Revised: 25.02.2024; Accepted: 29.02.2024; First online: 07.03.2024

**For citation.** Hayotov A. R., Doniyorov N. N. Construction of basis functions for finite element methods in a Hilbert space. Vestnik KRAUNC. Fiz.-mat. nauki. 2024, 46: 1, 118-133. EDN: EUIRSM. https://doi.org/10.26117/2079-6641-2024-46-1-118-133.

**Funding.** The work was not carried out within the framework of funds

**Competing interests.** There are no conflicts of interest regarding authorship and publication.

**Contribution and Responsibility.** All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.

^\astCorrespondence: E-mail: hayotov@mail.ru, doniyorovnn@mail.ru

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Hayotov A. R., Doniyorov N. N., 2024

© Institute of Cosmophysical Research and Radio Wave Propagation, 2024 (original layout, design, compilation)

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**Information about authors**

**Hayotov Abdullo Rahmonovich** – D. Sci. (Phys. & Math.), Professor, Head of the Computational Mathematics Laboratory, V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, Tashkent, Uzbekistan, ORCID 0000-0002-2756-9542.

**Doniyorov Negmurod Normurodovich** – (PhD) student, the Computational Mathematics Laboratory, V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, Tashkent, Uzbekistan, ORCID 0009-0001-3889-1641.