An algorithm for computing unstable solutions of semilinear boundary value problems
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.
scientific article; zbMATH DE number 490623
scientific article