Approximation of the solution of certain nonlinear ODEs with linear complexity

Newton’s method and a mesh-independence principle for certain semilinear boundary-value problems

“…In [7,3,4,8] we exhibited an algorithm which, for a given j ∈ N and ϵ ′ > 0, computes an ϵ ′ -approximation of the discrete system F j = 0, i.e., a point…”

“…Combining an algorithm of [4] or [8] for the approximation of the discrete solutions of (1) with mesh size h * and estimates provided by our mesh-independence principle we obtain an algorithm which computes a starting point (Theorem 36). Using this starting point and a discrete Newton iteration we obtain an ϵ-approximation of the positive solution of (1) with O((1/ϵ) 1/2 log 2 log 2 (1/ϵ)) flops and function evaluations (Theorem 37).…”

“…For a given mesh size, the solutions of the discretization of (1) were studied in [7,3,4,8,9], for different conditions concerning g and f . In this paper we consider the behavior of the discretization of the solutions (1) as the mesh size tends to zero, that is, we aim to approximate the solutions of (1).…”

“…Such solutions arise as the stationary solutions of the semilinear heat equation with Neumann boundary conditions. In particular, the dynamics of the latter are usually described in terms of the solutions of the former (see, e.g., [1][2][3][4]). …”

Efficient approximation of the solution of certain nonlinear reaction–diffusion equations with large absorption

“…Very little is known concerning the study of the stationary solutions of (3) and the comparison between the stationary solutions of (3) and (1). In [FR01], [DM09] and [Dra10] there is a complete study of the positive solutions of (4) for the particular case g 1 (x) := x p and g 2 (x) := x q , i.e., a complete study of the positive solutions of (5)…”