Numerical solution of two-point nonlinear boundary value problems via Legendre–Picard iteration method

Abstract

The aim of the present work is to introduce an effective numerical method for solving two-point nonlinear boundary value problems. The proposed iterative scheme, called the Legendre–Picard iteration method is based on the Picard iteration technique, shifted Legendre polynomials and Legendre–Gauss quadrature formula. In the Legendre–Picard iteration method, the boundary value problem is reduced to an iterative formula for updating the coefficients of the approximate solution in each step and with a straightforward manner, the integrals of the shifted Legendre polynomials are calculated. In addition, to reduce the CPU time, a vector–matrix scheme of the Legendre–Picard iteration method is constructed. The convergence analysis of the method is studied. Five nonlinear boundary value problems are given to illustrate the validity of the Legendre–Picard iteration method. Numerical results indicate the good performance and the precision of the proposed procedure.