# Approximation of the solution of certain nonlinear ODEs with linear complexity

- Authors
- Journal
- Journal of Computational and Applied Mathematics 0377-0427
- Publisher
- Elsevier
- Publication Date
- Volume
- 233
- Issue
- 9
- Identifiers
- DOI: 10.1016/j.cam.2009.10.019
- Keywords
- Disciplines

## Abstract

Abstract We study the positive stationary solutions of a standard finite-difference discretization of the semilinear heat equation with nonlinear Neumann boundary conditions. We prove that there exists a unique solution of such a discretization, which approximates the unique positive stationary solution of the “continuous” equation. Furthermore, we exhibit an algorithm computing an ε -approximation of such a solution by means of a homotopy continuation method. The cost of our algorithm is linear in the number of nodes involved in the discretization and the logarithm of the number of digits of approximation required.

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