Approximation of the solution of certain nonlinear ODEs with linear complexity

Authors:
Ezequiel Dratman
Affiliations:
Instituto de Ciencias, Universidad Nacional de General Sarmiento, Juan M. Gutiérrez 1150 (B1613GSX) Los Polvorines, Buenos Aires, Argentina
Venue:
Journal of Computational and Applied Mathematics
Year:
2010

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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 @e-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.