On the solution of the polynomial systems arising in the discretization of certain ODEs
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.
scientific article; zbMATH DE number 5602555
scientific article