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Jacobi‐Picard iteration method for the numerical solution of nonlinear initial value problems
Author(s) -
TafakkoriBafghi Mohammad,
Barid Loghmani Ghasem,
Heydari Mohammad,
Bai Xiaoli
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5900
Subject(s) - mathematics , jacobi method , nonlinear system , iterative method , initial value problem , fixed point iteration , jacobi polynomials , numerical analysis , quadrature (astronomy) , mathematical analysis , algorithm , orthogonal polynomials , fixed point , physics , quantum mechanics , electrical engineering , engineering
In this paper, an effective numerical iterative method for solving nonlinear initial value problems (IVPs) is presented. The proposed iterative scheme, called the Jacobi‐Picard iteration (JPI) method, is based on the Picard iteration technique, orthogonal shifted Jacobi polynomials, and shifted Jacobi‐Gauss quadrature formula. In comparison with traditional methods, the JPI method uses an iterative formula for updating next step approximations and calculating integrals of the shifted Jacobi polynomials are performed via an exact relation. Also, a vector‐matrix form of the JPI method is provided in details which reduce the CPU time. The performance of the presented method has been investigated by solving several nonlinear IVPs. Numerical results show the efficiency and the accuracy of the proposed iterative method.