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Volume 61 >>> № 3 MAY — JUNE 2025

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UDC 519.8

A.M. Shutovskyi¹

¹ Lesya Ukrainka Volyn National University, Lutsk, Ukraine

INTEGRAL REPRESENTATIONS OF POLYHARMONIC OPERATORS

Abstract. The study is devoted to establishing optimal mathematical models in the context of system analysis problems. Namely, nontrivial boundary conditions are applied to the problem of integrating polyharmonic equations in polar coordinates. The function, which is triharmonic in a unit disk, is presented in the form of an integral with a delta-like kernel. The question of the existence of a structural connection between solutions to the triharmonic equations in polar coordinates and positive operators, which are solutions to other partial differential equations, is considered. It is shown that the triharmonic Poisson integral for a unit disk can be represented as an average value of the solution to the Laplace equation in polar coordinates.

Keywords: Fourier series, triharmonic equation, unit disk, triharmonic Poisson integral, Abel–Poisson operator.

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