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Additional resources for Selected topics in the history of the two-dimensional biharmonic problem

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An interesting modification of the Bubnov–Galerkin procedure has been suggested by Biezeno and Koch ͓501͔; see also ͓5,25͔. They chose the representation of deflection in the clamped plate ͉ x ͉ рa, ͉ y ͉ рb as wϭ ͩ ͪͩ ͪ x2 Ϫ1 a2 2 y2 Ϫ1 b2 2 2 ͚ f kn k,nϭ1 ͩ ͪ ͩ ͪ x2 a2 nϪ1 y2 b2 kϪ1 (74) which satisfies the boundary conditions ͑4͒ and substituted it into governing Eq. ͑3͒. In this way, some fictitious load ¯p instead of p has appeared in the right-hand-side of Eq. ͑3͒. Then, the coefficients f kn are determined by the conditions that the integrals ͐͐ pdxdy and ͐͐ ¯p dxdy taken over wellchosen regions of the plate surface, are equal.

The first paper had been also published in German, according to the Soviet tradition of that time, but later, after Papkovich’s untimely death in 1946, was not even mentioned in the detailed list of his publications in the obituary. ͒ The second paper was based upon the dissertation ͓715͔, in which Fadle attributed this approach to To¨lke ͓479͔; note also the misprint in the year on the first page of ͓714͔. Papkovich ͓713͔ mentioned the possibility of applying the Gram-Schmidt orthonormalization process for constructing an orthonormal set of functions from a linearly independent set in order to define the complex unknown coefficients in series expansion ͑this paragraph for some reason was absent in the German version of the paper͒, but he did not provide any numerics.

This functional differential equation can be written as an integral equation and further analysis leads to both Poisson’s theory of mixed difference equations and the theory of certain linear integral equations and, finally, to an infinite system of linear algebraic equations. Dixon ͓7͔ noted that it is not clear whether this system could be truncated by the theory of infinite determinants, and said very little about the possible application of the whole theory to numerical calculation. Love28 suggested ͓626͔ an approach wherein the solution involves the conformal mapping of the rectangle onto the circular region of unit radius.

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Selected topics in the history of the two-dimensional biharmonic problem by VV Meleshko


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