Jacobi-Picard iteration method for the numerical solution of nonlinear initial value problems

Mohammad Tafakkori-Bafghi, Ghasem Barid Loghmani, Mohammad Heydari, Xiaoli Bai

Research output: Contribution to journalArticle


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.

Original languageEnglish (US)
Pages (from-to)1084-1111
Number of pages28
JournalMathematical Methods in the Applied Sciences
Issue number3
StatePublished - Feb 1 2020

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Engineering(all)


  • Jacobi-Gauss quadrature formula
  • Picard iteration method
  • initial value problems
  • shifted Jacobi polynomials
  • vector-matrix form

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