Numerical continuation and the Gelfand problem
type: Item
instance of: scholarly article
title: Numerical continuation and the Gelfand problem
zbMATH Open document ID: 0908.65094
DOI: 10.1016/S0096-3003(97)81660-8
author: Jeff S. McGough
published in: Applied Mathematics and Computation
publication date: 1998-08-02T00:00:00Z
review text: This paper is concerned with the Gelfand problem \[ \Delta u+\lambda e^u= 0\quad\text{in }\Omega,\quad u= 0\quad\text{on }\partial\Omega, \] where \(\lambda>0\), and where \(\Omega\in\mathbb{R}^3\) denotes a bounded domain. The author is interested in finding the number of positive solutions in terms of the `bifurcation diagram' where the sup norm of the solutions \(u\) is plotted against \(\lambda\). To calculate the diagram a numerical approach is proposed which is based on finite differences together with an arclength method. Results are presented for a number of special domains as cube and punctured cubes where smaller cubes are deleted.
Mathematics Subject Classification ID: 65N25
Mathematics Subject Classification ID: 65N06
Mathematics Subject Classification ID: 35P30
zbMATH DE Number: 1183199
zbMATH Keywords: numerical continuation
zbMATH Keywords: nonlinear eigenvalue problem
zbMATH Keywords: bifurcation diagram
zbMATH Keywords: Gelfand problem
zbMATH Keywords: positive solutions
zbMATH Keywords: finite differences
zbMATH Keywords: arclength method
reviewed by: Waldemar Velte
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1183199