Numerical solution of two-point nonlinear boundary value problems via Legendre-Picard iteration method. (English) Zbl 07538453

Tafakkori-Bafghi, M.; Loghmani, G. B.; Heydari, M.

Math. Comput. Simul. 199, 133-159 (2022).

Summary: 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.

MSC:

65-XX	Numerical analysis
76-XX	Fluid mechanics

Keywords:

two-point boundary value problems; Picard iteration method; shifted Legendre polynomials; Legendre-Gauss quadrature formula; vector-matrix structure

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