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Mathematical Problems in Engineering
Volume 2011, Article ID 521342, 20 pages
http://dx.doi.org/10.1155/2011/521342
Research Article

Combining Legendre's Polynomials and Genetic Algorithm in the Solution of Nonlinear Initial-Value Problems

Escola Politécnica da Universidade de São Paulo, Avenida Prof. Luciano Gualberto, travessa 3, No. 158, 05508-900 São Paulo, SP, Brazil

Received 16 January 2011; Revised 9 May 2011; Accepted 24 May 2011

Academic Editor: Wei-Chiang Hong

Copyright © 2011 Osvaldo Guimarães et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This work develops a method for solving ordinary differential equations, that is, initial-value problems, with solutions approximated by using Legendre's polynomials. An iterative procedure for the adjustment of the polynomial coefficients is developed, based on the genetic algorithm. This procedure is applied to several examples providing comparisons between its results and the best polynomial fitting when numerical solutions by the traditional Runge-Kutta or Adams methods are available. The resulting algorithm provides reliable solutions even if the numerical solutions are not available, that is, when the mass matrix is singular or the equation produces unstable running processes.