On the solution of the polynomial systems arising in the discretization of certain ODEs
type: Item
instance of: scholarly article
title: On the solution of the polynomial systems arising in the discretization of certain ODEs
zbMATH Open document ID: 1173.65035
author: Ezequiel Dratman
author: Guillermo Matera
published in: Computing
publication date: 2009-09-09T00:00:00Z
review text: The paper is concerned with the study of positive stationary solutions of a standard finite-difference discretization of the semilinear heat equation in one dimension, with nonlinear Neumann boundary conditions. In order to describe the dynamic behaviour of the solutions of this boundary-value problem, it is usually necessary to analyze the behaviour of the corresponding stationary solutions. By a second-order finite difference semi-discretization, an initial-value problem is obtained and the stationary solutions of this problem are considered. The convergence of the positive solutions of the obtained initial-value problem to the corresponding solutions of the boundary-value problem is proved. An algorithm which computes an \(\epsilon\)-approximation of the stationary solutions, in a special case, is described. This algorithm is a homotopy continuation method [see \textit{J. Ortega} and \textit{W. Rheinboldt}, Iterative solutions of nonlinear equations in several variables, Classics in Applied Mathematics. 30. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics (2000; Zbl 0949.65053)]. Also, the existence and the uniqueness of the stationary solutions are discussed. The cost of the presented algorithm is polynomial in the number of nodes involved in the discretization and the logarithm of the number of digits of approximation required. Many other questions concerning the stationary solutions of the boundary-value problem are discussed: the bounds for the solutions, the bound for the inverse of the Jacobian matrix, the estimation of the condition number for positive solutions, etc.
reviewed by: Iulian Coroian
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 65L10
Mathematics Subject Classification ID: 65L12
Mathematics Subject Classification ID: 65H20
Mathematics Subject Classification ID: 65Y20
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 65M20
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 5602555
zbMATH Keywords: two-point boundary-value problem
zbMATH Keywords: finite-differences
zbMATH Keywords: nonlinear Neumann boundary condition
zbMATH Keywords: stationary solutions
zbMATH Keywords: homotopy continuation method
zbMATH Keywords: condition number
zbMATH Keywords: complexity
zbMATH Keywords: semilinear heat equation
zbMATH Keywords: semi-discretization
zbMATH Keywords: initial-value problem
zbMATH Keywords: convergence
zbMATH Keywords: positive solutions
zbMATH Keywords: algorithm
MaRDI profile type: Publication
full work available at URL: s00607-009-0046-7
OpenAlex ID: W2112734325
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DOI: 10.1007/S00607-009-0046-7
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