Durkin A.A., Yermolenko A.V., Shadrin L.S. Legendre Polynomial Basis for Constructing the Green’s Function of the Biharmonic Operator on a Square

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https://doi.org/10.15688/mpcm.jvolsu.2026.2.1

Anatoliy A. Durkin
Postgraduate Student, Department of Applied Mathematics and Computer Science, Pitirim Sorokin Syktyvkar State University
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https://orcid.org/0009-0001-3332-2124

Prosp. Oktyabrsky, 55, 167001 Syktyvkar, Russian Federation

Andrei V. Yermolenko
Candidate of Sciences (Physics and Mathematics), Head, Department of Applied Mathematics and Computer Science, Pitirim Sorokin Syktyvkar State University
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https://orcid.org/0000-0002-2904-1717

Prosp. Oktyabrsky, 55, 167001 Syktyvkar, Russian Federation

Lev S. Shadrin
Postgraduate Student, Assistant Lecturer, Department of Applied Mathematics and Computer Science, Pitirim Sorokin Syktyvkar State University
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https://orcid.org/0009-0000-7316-2463

Prosp. Oktyabrsky, 55, 167001 Syktyvkar, Russian Federation

Abstract. This paper is devoted to the numerical construction of the Green’s function for the biharmonic operator on the unit square under clamped boundary conditions. The problem models the bending of a thin elastic Kirchhoff – Love plate subjected to an arbitrary transverse load. Within the framework of the Galerkin – Ritz projection method, a comparison is made between the trigonometric basis of sine squares and a polynomial system based on shifted Legendre polynomials with a weight “bubble” factor that exactly satisfies the homogeneous boundary conditions. The completeness of the polynomial system in the energy space H₀²(Ω), the quasi-optimality of the approximation by Cea’s lemma, the exactness of the method on polynomial data, and the exponential (spectral) convergence rate for analytic solutions are demonstrated. It is shown that the trigonometric system of sine squares is not complete in H₀²(Ω); the closure of its linear span in the energy norm coincides with the subspace of functions whose second derivative is symmetric with respect to the midpoint of the interval. Consequently, the Galerkin approximations with this basis converge only for solutions with a symmetric Laplacian and stagnates otherwise. Numerical experiments confirm the theoretical conclusions and verify the algorithm against classical reference coefficients for the deflection of a clamped square plate (uniform load and concentrated force). The Legendre polynomial basis provides high-accuracy approximation of higher derivatives, which is unattainable for the trigonometric analogue.

Key words: Ritz – Galerkin method, clamped boundary, plate bending, energy space, spectral convergence, condition number, completeness of a coordinate system.

Durkin A.A., Yermolenko A.V., Shadrin L.S. Legendre Polynomial Basis for Constructing the Green’s Function of the Biharmonic Operator on a Square is licensed under a Creative Commons Attribution 4.0 International License.

Citation in English: Mathematical Physics and Computer Simulation. Vol. 29 No. 2 2026, pp. 5-19

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