Jacobi-Picard iteration method for the numerical solution of nonlinear initial value problems |
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Authors: | Mohammad Tafakkori-Bafghi,Ghasem Barid  Loghmani,Mohammad Heydari,Xiaoli Bai |
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Affiliation: | 1. Department of Mathematics, Yazd University, Yazd, Iran;2. Mechanical and Aerospace Engineering, Rutgers, The State University of New Jersey, Piscataway, 08854 NJ, USA |
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Abstract: | 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. |
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Keywords: | initial value problems Jacobi-Gauss quadrature formula Picard iteration method shifted Jacobi polynomials vector-matrix form |
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