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

MATHEMATICS AND COMPUTERS IN SIMULATION 2022年 199卷 133-159页

作者： Tafakkori-Bafghi, M. Loghmani, G. B. Heydari, M. Yazd Univ Dept Math Yazd Iran

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.(c) 2022 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.

关键词： Two-point boundary value problems Picard iteration method Shifted Legendre polynomials Legendre-Gauss quadrature formula vector-matrix structure

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Piecewise Jacobi-Picard Iteration Method for Solving Nonlinear Initial Value Problems on Large Domains

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IRANIAN JOURNAL OF SCIENCE AND TECHNOLOGY TRANSACTION A-SCIENCE 2022年第2期46卷 595-613页

作者： Tafakkori-Bafghi, M. Loghmani, G. B. Heydari, M. Bai, X. Yazd Univ Dept Math Yazd Iran Rutgers State Univ Mech & Aerosp Engn 98 Brett RdEN B-222 Piscataway NJ 08854 USA

In this investigation, an effective numerical technique, called the piecewise Jacobi-Picard iteration (PJPI) method, to find the approximate solution of nonlinear initial value problems (IVPs) with high accuracy on large domains is proposed. This method is based on the Jacobi-Picard iteration method combined with the domain decomposition technique. To reduce the CPU time, a vector-matrix structure of the proposed method is designed. Based on this vector-matrix representation, the convergence analysis of the PJPI method is studied. Several applied problems in different fields of science and engineering such as physics and biology are presented to demonstrate the capability and efficiency of the PJPI method by comparing it with the fourth- and fifth-order Runge-Kutta-Fehlberg (RKF45) method and some other existing methods in the literature.

关键词： Picard iteration method Jacobi polynomials Jacobi-Gauss quadrature Piecewise Jacobi-Picard iteration vector-matrix structure Initial value problems

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