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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
  • Two-Point Boundary-Value Problem
  • Finite Differences
  • Neumann Boundary Condition
  • Stationary Solution
  • Homotopy Continuation
  • Polynomial System Solving
  • Condition Number
  • Complexity
Disciplines
  • Computer Science
  • Mathematics

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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