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

Nachman Aronszajn, R. D. Brown, R. S. Butcher (1973)

Annales de l'institut Fourier

A technique is developed for constructing the solution of Δ 2 u = F in R = { ( x , y ) : | x | < a , | y | < b } , subject to boundary conditions u = φ , ∂ u ∂ n = ψ on ∂ R . The problem is reduced to that of finding the orthogonal projection P w of w in L 2 ( R ) onto the subspace H of square integrable functions harmonic in R . This problem is solved by decomposition H into the closed direct (not orthogonal) sum of two subspaces H ( 1 ) , 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 H ( 1 ) , H ( 2 ) respectively. The resulting construction...

Existence of solutions to the Poisson equation in L p -weighted spaces

Joanna Rencławowicz, Wojciech M. Zajączkowski (2010)

Applicationes Mathematicae

We examine the Poisson equation with boundary conditions on a cylinder in a weighted space of L p , p≥ 3, type. The weight is a positive power of the distance from a distinguished plane. To prove the existence of solutions we use our result on existence in a weighted L₂ space.

Existence of solutions to the Poisson equation in L₂-weighted spaces

Joanna Rencławowicz, Wojciech M. Zajączkowski (2010)

Applicationes Mathematicae

We consider the Poisson equation with the Dirichlet and the Neumann boundary conditions in weighted Sobolev spaces. The weight is a positive power of the distance to a distinguished plane. We prove the existence of solutions in a suitably defined weighted space.

Internal finite element approximation in the dual variational method for the biharmonic problem

Ivan Hlaváček, Michal Křížek (1985)

Aplikace matematiky

A conformal finite element method is investigated for a dual variational formulation of the biharmonic problem with mixed boundary conditions on domains with piecewise smooth curved boundary. Thus in the problem of elastic plate the bending moments are calculated directly. For the construction of finite elements a vector potential is used together with C 0 -elements. The convergence of the method is proved and an algorithm described.

Method of fundamental solutions for biharmonic equation based on Almansi-type decomposition

Koya Sakakibara (2017)

Applications of Mathematics

The aim of this paper is to analyze mathematically the method of fundamental solutions applied to the biharmonic problem. The key idea is to use Almansi-type decomposition of biharmonic functions, which enables us to represent the biharmonic function in terms of two harmonic functions. Based on this decomposition, we prove that an approximate solution exists uniquely and that the approximation error decays exponentially with respect to the number of the singular points. We finally present results...

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