Construction of the solutions of boundary value problems for the biharmonic operator in a rectangle

Aronszajn, Nachman, Brown, R. D., and Butcher, R. S.. "Construction of the solutions of boundary value problems for the biharmonic operator in a rectangle." Annales de l'institut Fourier 23.3 (1973): 49-89. <http://eudml.org/doc/74142>.

@article{Aronszajn1973,
abstract = {A technique is developed for constructing the solution of $\Delta ^2u=F$ in $R=\lbrace (x,y):\vert x\vert &lt; a,\;\vert y\vert &lt; b\rbrace $, subject to boundary conditions $u=\varphi $, $\{\partial u\over \partial n\}=\psi $ on $\partial R$. The problem is reduced to that of finding the orthogonal projection $Pw$ of $w$ in $L^2(R)$ onto the subspace $\{\bf H\}$ of square integrable functions harmonic in $\{\bf R\}$. This problem is solved by decomposition $\{\bf H\}$ into the closed direct (not orthogonal) sum of two subspaces $\{\bf H\}^\{(1)\},\{\bf H\}^\{(2)\}$ for which complete orthogonal bases are known. $P$ is expressed in terms of the projections $P^\{(1)\}$, $P^\{(2)\}$ of $L^2(R)$ onto $\{\bf H\}^\{(1)\}$, $\{\bf H\}^\{(2)\}$ respectively. The resulting construction yields an approximation technique with both a priori and a posteriori error bounds (the latter very precise). In a short appendix the numerical results are given of the application of the technique in some specific examples and the a posteriori error evaluated.},
author = {Aronszajn, Nachman, Brown, R. D., Butcher, R. S.},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {3},
pages = {49-89},
publisher = {Association des Annales de l'Institut Fourier},
title = {Construction of the solutions of boundary value problems for the biharmonic operator in a rectangle},
url = {http://eudml.org/doc/74142},
volume = {23},
year = {1973},
}

TY - JOUR
AU - Aronszajn, Nachman
AU - Brown, R. D.
AU - Butcher, R. S.
TI - Construction of the solutions of boundary value problems for the biharmonic operator in a rectangle
JO - Annales de l'institut Fourier
PY - 1973
PB - Association des Annales de l'Institut Fourier
VL - 23
IS - 3
SP - 49
EP - 89
AB - A technique is developed for constructing the solution of $\Delta ^2u=F$ in $R=\lbrace (x,y):\vert x\vert &lt; a,\;\vert y\vert &lt; b\rbrace $, subject to boundary conditions $u=\varphi $, ${\partial u\over \partial n}=\psi $ on $\partial R$. The problem is reduced to that of finding the orthogonal projection $Pw$ of $w$ in $L^2(R)$ onto the subspace ${\bf H}$ of square integrable functions harmonic in ${\bf R}$. This problem is solved by decomposition ${\bf H}$ into the closed direct (not orthogonal) sum of two subspaces ${\bf H}^{(1)},{\bf H}^{(2)}$ for which complete orthogonal bases are known. $P$ is expressed in terms of the projections $P^{(1)}$, $P^{(2)}$ of $L^2(R)$ onto ${\bf H}^{(1)}$, ${\bf H}^{(2)}$ respectively. The resulting construction yields an approximation technique with both a priori and a posteriori error bounds (the latter very precise). In a short appendix the numerical results are given of the application of the technique in some specific examples and the a posteriori error evaluated.
LA - eng
UR - http://eudml.org/doc/74142
ER -