Klyachin A.A., Klyachin V.A. The Approximation of the Fourth-Order Partial Differential Equations in the Class of the Piecewise Polynomial Functions on the Triangular Grid

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https://doi.org/10.15688/mpcm.jvolsu.2019.2.5

Aleksey Aleksandrovich Klyachin
Doctor of Physical and Mathematical Sciences,
Head of the Department of Mathematical Analysis and Function Theory,
Volgograd State University
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https://orcid.org/0000-0003-3293-9066
Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation

Vladimir Aleksandrovich Klyachin
Doctor of Physical and Mathematical Sciences,
Head of the Department of Computer Sciences and Experimental Mathematics,
Volgograd State University
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https://orcid.org/0000-0003-1922-7849
Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation

Abstract. The present work determines the deviation of the piecewise cubic almost-solution of the biharmonic equation and derives the general formula (3) for its calculation. Based on this concept, we obtained an approximation of the equation. A number of numerical calculations were carried out in order to confirm the obtained formula (see pictures 1 and 2) experimentally. In general, for all selected biharmonic functions, the deviation value εΔΔ(u) turned out to be, as expected, rather small even with a relatively small number of triangulation nodes. On average, with 25 ≤ N ≤ 35, the absolute error was about 0,0001, which gives an approximately asymptotic estimate of O(h2) when the partitioning step is h→0.

Key words: piecewise cubic function, almost solution, biharmonic equation, approximation of the equation, deviation of the piecewise cubic almost solution.

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The Approximation of the Fourth-Order Partial Differential  Equations in the Class of the Piecewise Polynomial Functions on the Triangular Grid by Klyachin A.A., Klyachin V.A. is licensed under a Creative Commons Attribution 4.0 International License.

Citation in English: Mathematical Physics and Computer Simulation. Vol. 22 No. 2 2019, pp. 65-72

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