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«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2026. Vol 55

On the Collocation Method in Constructing a Solution to the Bending Equation for a Long Rectangular Nanoplate

Author(s)

Oksana V. Germider1, Vasilii N. Popov1

1 Northern (Arctic) Federal University named after M.V. Lomonosov, Arkhangelsk, Russian Federation

Abstract
Within the framework of the theory of microstructural deformation, a new approach is proposed for constructing a solution to the bending equation of a long rectangular nanoplate that is under the influence of a transverse load. The proposed approach is based on the collocation method using a system of orthogonal Chebyshev polynomials of the first kind. The bending function is represented as a partial sum of a series of these polynomials. The roots of Chebyshev polynomials of the first kind are chosen as the collocation points. By sequentially multiplying the left and right sides of the resulting matrix equation by the inverse matrix to the matrix with the values of Chebyshev polynomials at the collocation points and by the generalized inverse matrix to the degenerate matrix of differentiation of these polynomials, the equation of the bending surface, taking into account boundary conditions, is reduced to a system of linear algebraic equations with respect to unknown coefficients in the representation of the solution. In this case, the elements of each of these matrices are presented explicitly. An estimate of the error of the constructed solution based on an infinite norm is obtained. The results of the conducted computational experiments are presented, which demonstrate the effectiveness of the proposed approach.
About the Authors

Vasilii N. Popov, Dr. Sci. (Phys.-Math.), Prof., Northern (Arctic) Federal University named after M. V. Lomonosov, Arkhangelsk, 163002, Russian Federation, v.popov@narfu.ru

Oksana V. Germider, Cand. Sci. (Phys.-Math.), Assoc. Prof., Northern (Arctic) Federal University named after M. V. Lomonosov, Arkhangelsk, 163002, Russian Federation, o.germider@narfu.ru

For citation
Germider O. V., Popov V. N. On the Collocation Method in Constructing a Solution to the Bending Equation for a Long Rectangular Nanoplate. The Bulletin of Irkutsk State University. Series Mathematics, 2026, vol. 55, pp. 15–30. (in Russian) https://doi.org/10.26516/1997-7670.2026.55.15
Keywords
microstructural deformation of thin plates, bending equation for a long nanoplate, collocation method, Chebyshev polynomials
UDC
517.958
MSC
34B60
DOI
https://doi.org/10.26516/1997-7670.2026.55.15
References
  1. Bakhvalov N.S., Zhidkov N.P., Kobelkov G.M. Numerical methods. Moscow, Knowledge Laboratory Publ., 2020, 636 p. (in Russian) 
  2. Varin V.P. Approximation of Differential Operators with Boundary Conditions. Comput. Math. and Math. Phys., 2023, vol. 63, pp. 1381–1400. https://doi.org/10.1134/S0965542523080158
  3. Germider O.V., Popov V.N. Mathematical modeling of bending of a thin orthotropic plate clamped along the contour. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 2024, vol. 20, no. 3, pp. 310–323. https://doi.org/10.21638/spbu10.2024.301 (in Russian) 
  4. Germider O.V., Popov V.N. On the Collocation Method in Constructing a Solution to the Volterra Integral Equation of the Second Kind Using Chebyshev and Legendre Polynomials. The Bulletin of Irkutsk State University. Series Mathematics, 2024, vol. 50, pp. 19–35. https://doi.org/10.26516/1997-7670.2024.50.1 (in Russian) 
  5. Zveryaev E.M., Kovalenko M.D., Abrukov D.A., Menshova I.V., Kerzhaev A.P. Examples of exact solutions for problems of bending plate with free face planes. Keldysh Institute preprints, 2019, vol. 46, pp. 1–17. https://doi.org/10.20948/prepr-2019-46 (in Russian) 
  6. Mikhasev G.I. Long-wave flexural vibrations and deformation of a small size beam considering curface effects. Prikl. Mekh. Tekh. Fiz., 2024, vol. 65, no. 2, pp. 214– 225. https://doi.org/10.15372/PMTF202315379 (in Russian) 
  7. Sevastianov L.A., Lovetskiy K.P., Kulyabov D.S. A new approach to the formation of systems of linear algebraic equations for solving ordinary differential equations by the collocation method. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2023, vol. 23, no. 1, pp. 36–47 https://doi.org/10.18500/1816-9791-2023-23-1-36-47 (in Russian)
  8. Amiraslani A., Corless R.M., Gunasingam M. Differentiation matrices for univariate polynomials. Numerical Algorithms, 2020, vol. 83, pp. 1–31. https://doi.org/10.1007/s11075-019-00668-z 
  9. Babu B., Patel B.P. Analytical solution for strain gradient elastic Kirchhoff rectangular plates under transverse static loading. European Journal of Mechanics / A Solids, 2019, vol. 73, pp. 101–111. https://doi.org/10.1016/j.euromechsol.2018.07.007
  10. Bejjani N., Sab K., Bodgi J., Lebee A. The bending-gradient theory for thick plates: existence and uniqueness results. Journal of Elasticity, 2018, vol. 133, pp. 37–72. https://doi.org/10.1007/s10659-017-9669-7
  11. Corless R.M., Jeffrey D.J. The Turing factorization of a rectangular matrix. ACM SIGSAM Bulletin, 1997, vol. 31, no. 3, pp. 20–30. https://doi.org/10.1145/271130.271135
  12. Hu X., Wang Z., Hu B. A collocation method based on roots of Chebyshev polynomial for solving Volterra integral equations of the second kind. Applied Mathematics Letters, 2023, vol. 29, no. 108804, pp. 1–8. https://doi.org/10.1016/j.aml.2023.108804
  13. Golub G.H., Van Loan C.F. Matrix Computations. 3rd Edition. Baltimore, Johns Hopkins University Press, 1996, 728 p. 
  14. Mason J., Handscomb D. Chebyshev polynomials. Florida, CRC Press, 2003, 335 p. 
  15. Niiranen J., Niemi A.H. Variational formulations and general boundary conditions for sixth-order boundary value problems of gradient-elastic Kirchhoff plates. European Journal of Mechanics / A Solids, 2017, vol. 61, pp. 164–179. http://dx.doi.org/10.1016/j.euromechsol.2016.09.001
  16. Papargyri-Beskou S., Giannakopoulos A.E., Beskos D.E. Variational analysis of gradient elastic flexural plates under static loading. Int. J. Solids Struct., 2010, vol. 47, no. 20, pp. 2755–2766. https://doi.org/10.1016/j.ijsolstr.2010.06.003
  17. Siddique M.U.M., Nazmul I.M. Static bending analysis of BDFG nanobeams by nonlocal couple stress theory and nonlocal strain gradient theory. Forces in Mechanics, 2024, vol. 17, no. 100289, pp. 1–16. https://doi.org/10.1016/j.finmec.2024.100289
  18. Timoshenko S., Woinowsky-Krieger S. Theory of Plates and Shells. New York, McGraw-Hill Press, 1959, 580 p. 
  19. Zhou Y., Huang K. On simplified deformation gradient theory of modified gradient elastic Kirchhoff–Love plate. European Journal of Mechanics / A Solids, 2023, vol. 100, no. 105014. pp. 1–14. https://doi.org/10.1016/j.euromechsol.2023.105014

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