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Summary:
Some problems of plane elasticity lead to the solution of biharmonic problem. Many methods have been developped to the solution of this problem (the method of finite differences, the finite element method, classical variational methods, methods based on the theory of functions of a complex variable, etc.). In this paper, the method of least squares on the boundary is presented, having its specific preferences. In the first part, the algorithm of this method and a numerical example are given. This part is mainly intended for "consumers" of mathematics and is written in more detail. In the second part, the proof of convergence of the method is given. This part is mainly intended for mathematicians.
Applied to the solution of the biharmonic problem, the method takes an essential use of the form of equation. As to its idea itself, it can be applied - in proper modifications - also to the solution of other problems.
References:
[1] Nečas J.: Les méthodes dlrectes en théorie des équations elliptiques. Praha, Academia 1967. MR 0227584
[2] Babuška I., Rektorys K., Vyčichlo F.: Matematická teorie rovinné pružnosti. Praha NČSAV 1955. (Mathematische Elastizitätstheorie der ebenen Probleme. Berlin, Akademie-verlag 1960.) MR 0115343
[3] Rektorys K.: Variational methods in engineering problems and in those of mathematical physics. (Variační metody v inženýrských problémech a v problémech matematické fyziky.) In Czech: Praha, SNTL 1974. In English: Dordrecht (Holland)-Boston, Reidel Co, to appear in 1976. MR 0487652
[4] Бондаренко В. А.: Полигармонические полиномы. Ташкент, ФАН 1968. Zbl 1171.62301
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