# On a stable calculation of the normal to a surface given approximately

**Authors:**Laneev E.B.^{1}, Baaj O.^{1}-
**Affiliations:**- Peoples’ Friendship University of Russia (RUDN University)

**Issue:**Vol 31, No 3 (2023)**Pages:**228-241**Section:**Articles**URL:**https://journals.rudn.ru/miph/article/view/35919**DOI:**https://doi.org/10.22363/2658-4670-2023-31-3-228-241**EDN:**https://elibrary.ru/KNQAEY

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## Abstract

The paper proposes a stable method for constructing a normal to a surface given approximately. The normal is calculated as the gradient of the function in the surface equation. As is known, the problem of calculating the derivative is ill-posed. In the paper, an approach is adopted to solving this problem as to the problem of calculating the values of an unbounded operator. To construct its stable solution, the principle of minimum of the smoothing functional in Morozov’s formulation is used. The normal is obtained in the form of a Fourier series in the expansion in terms of eigenfunctions of the Laplace operator in a rectangle with boundary conditions of the second kind. The functional stabilizer uses the Laplacian, which makes it possible to obtain a normal in the form of a Fourier series that converges uniformly to the exact normal vector as the error in the surface definition tends to zero. The resulting approximate normal vector can be used to solve various problems of mathematical physics using surface integrals, normal derivatives, simple and double layer potentials.

## Full Text

1. Introduction When solving many problems of mathematical physics, which are boundary value problems for partial differential equations, there is a need to calculate the normal to the surface, in particular, when calculating the normal derivative. For example, when calculating the potentials of a simple and double layer, as well as other surface integrals. In the case when the surface is known “exactly”, that is, for example, it is given by an equation with an exactly known function## About the authors

### Evgeniy B. Laneev

Peoples’ Friendship University of Russia (RUDN University)
Email: elaneev@yandex.ru

ORCID iD: 0000-0002-4255-9393

Doctor of Physical and Mathematical Sciences, professor of Mathematical Department

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation### Obaida Baaj

Peoples’ Friendship University of Russia (RUDN University)
**Author for correspondence.**

Email: 1042175025@rudn.ru

ORCID iD: 0000-0003-4813-7981

Post-Graduate Student of Mathematical Department

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation## References

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