An algorithm for computing unstable solutions of semilinear boundary value problems
type: Item
instance of: scholarly article
title: An algorithm for computing unstable solutions of semilinear boundary value problems
zbMATH Open document ID: 0793.65060
DOI: 10.1007/BF02238539
author: Philip Korman
published in: Computing
publication date: 1994-08-08T00:00:00Z
review text: This paper considers the problem of numerical resolution of a boundary value problem of the following form: \(-u''=f(x,u)\), \(u(a)=u(b)=0\) where \(u\) is so-called unstable solution of the problem; this means that \(u\) is an unstable steady state solution of the equation \(u_ t- u_{xx}=f(x,u)\), \(u(a,t)=u(b,t)=0\). Such unstable solutions are in general saddle points of the energy functional \(J(u)\). The author proposes the following interactive algorithm which is supposed to work under certain additional hypotheses. First, the differential equation is replaced by the standard finite difference one. First step of the algorithm consists in looking, in some class of points, for this minimizing the residual of the difference equation. If we are not happy enough with this approximation of the solution, then the second step of the algorithm tries to approach to the saddle point of the functional \(J(u)\) by minimizing the energy functional \(J\) with help of certain iterative process of gradient type. If we decide that the saddle point is near enough, then the third step of the algorithm minimizes the residual of the difference equation with help of the gradient method in the neighborhood of the supposed saddle point of \(J\). Numerical examples are given.
reviewed by: Krzysztof Moszyński
Mathematics Subject Classification ID: 65L10
Mathematics Subject Classification ID: 65L07
Mathematics Subject Classification ID: 65L12
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 490623
zbMATH Keywords: numerical examples
zbMATH Keywords: minimization of a functional
zbMATH Keywords: boundary value problem
zbMATH Keywords: unstable solutions
zbMATH Keywords: saddle points
zbMATH Keywords: energy functional
zbMATH Keywords: iterative process
zbMATH Keywords: difference equation
zbMATH Keywords: gradient method
MaRDI profile type: Publication
cites work: Large sparse continuation problems
cites work: On computation of solutions of fully nonlinear elliptic problems
cites work: Geometric theory of semilinear parabolic equations
cites work: Finite difference approximations to the Dirichlet problem for elliptic systems
cites work: On computation of solutions of elliptic systems
cites work: Positive Solutions of Negative Exponent Generalized Emden–Fowler Boundary Value Problems
cites work: Q3749777
cites work: Q5628581
cites work: A numerical study of spontaneous bifurcation
full work available at URL: https://doi.org/10.1007/bf02238539
OpenAlex ID: W56479968
Recommended article: Decoupling and Stability of Algorithms for Boundary Value Problems
Recommended article: A generalized midpoint-based boundary value method for unstable partial differential equations
Recommended article: Computational Solution of Blow-Up Problems for Semilinear Parabolic PDEs on Unbounded Domains
Recommended article: On algorithm of finding solutions of semiperiodical boundary value problem for systems of nonlinear hyperbolic equations
Recommended article: A numerical algorithm for solving a class of linear nonlocal boundary value problems
Recommended article: Q3824723
Recommended article: A new algorithm for solving nonlinear boundary value problems
Recommended article: A numerical algorithm for some singularly perturbed boundary value problems
Recommended article: Q3768316
Recommended article: An iterative method for semistable solutions
description: scientific article; zbMATH DE number 490623