Low-dimensional qualitative and numerical approximation of bifurcation in a semilinear elliptic problem
type: Item
instance of: scholarly article
title: Low-dimensional qualitative and numerical approximation of bifurcation in a semilinear elliptic problem
zbMATH Open document ID: 0648.65077
DOI: 10.1016/0377-0427(87)90200-7
published in: Journal of Computational and Applied Mathematics
publication date: 1987-01-01T00:00:00Z
review text: In this interesting paper, bifurcation phenomena are investigated for the following nonlinear boundary value problem, treated as a model for more complicated situations: \(u_{xx}+u_{yy}+g(u)=0,\) (x,y)\(\in \Omega\); \(u(x,y)=0\), (x,y)\(\in \partial \Omega\). Here the function g is of some special form and may depend on parameters. (One parameter is the dimension of the rectangle \(\Omega)\). The considered problem admits always the trivial solution but also some nonlocal bifurcations (non- trivial solutions, appearing spontaneously for certain values of the parameters) are theoretically expected. Developing the solution as double Fourier series with respect to the eigenfunctions of the Laplace operator on the rectangle \(\Omega\), a reduction process is performed. This process reduces the original problem to some finite-dimensional one. First, the author investigates a Galerkin approximation of the reduced problem, taking into account a subspace generated by only one eigenfunction. Then the same problem is treated numerically, and the results are compared. The paper contains a rather detailed description of the algorithm applied to the numerical solution of a nonlinear parameter- dependent system of equations, resulting after the reduction process. Some results of computations are presented.
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 35J65
zbMATH DE Number: 4058788
zbMATH Keywords: eigenfunction expansion
zbMATH Keywords: Fourier series expansion
zbMATH Keywords: numerical examples
zbMATH Keywords: nonlocal bifurcations
zbMATH Keywords: eigenfunctions of the Laplace operator
zbMATH Keywords: reduction process
zbMATH Keywords: Galerkin approximation
author: Michael W. Smiley
describes a project that uses: PITCON
MaRDI profile type: Publication
full work available at URL: 0377-0427(87)90200-7
OpenAlex ID: W2017551237
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description: scientific article; zbMATH DE number 4058788