Gauss-Legendre Iterative Methods and Their Applications on Nonlinear Systems and BVP-ODEs
Abstract: In this paper, a group of Gauss-Legendre iterative methods with cubic convergence for solving nonlinear systems are proposed. We construct the iterative schemes based on Gauss-Legendre quadrature formula. The cubic convergence and error equation are proved theoretically, and demonstrated numerically. Several numerical examples for solving the system of nonlinear equations and boundary-value problems of nonlinear ordinary differential equations (ODEs) are provided to illustrate the efficiency and performance of the suggested iterative methods.
Cite this paper: Liu, Z. and Sun, G. (2016) Gauss-Legendre Iterative Methods and Their Applications on Nonlinear Systems and BVP-ODEs. Journal of Applied Mathematics and Physics, 4, 2038-2046. doi: 10.4236/jamp.2016.411203.
References

   Ortega, J.M. and Rheinboldt, W.G. (1970) Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York.

   Traub, J.F. (1964) Iterative Methods for the Solution of Equations. Prentice-Hall, Englewood Cliffs.

   Noor, M.A. and Noor, K.I. (2006) Improved Iterative Methods for Solving Nonlinear Equations. Applied Mathematics and Computation, 183, 774-779.
https://doi.org/10.1016/j.amc.2006.05.084

   Chun, C. (2006) A New Ityerative Method for Solving Nonlinear Equations. Applied Mathematics and Computation, 178, 415-422.
https://doi.org/10.1016/j.amc.2005.11.055

   Weerakoon, S. and Fernando, T.G.I. (2000) A Variant of Newton’s Method with Accelerated Third-Order Convergence. Applied Mathematics Letters, 13, 87-93.
https://doi.org/10.1016/S0893-9659(00)00100-2

   Darvishi, M.T. and Barati, A. (2007) A Third-Order Newton-type Method to Solve Systems of Nonlinear Equations. Applied Mathematics and Computation, 187, 630-635.
https://doi.org/10.1016/j.amc.2006.08.080

   Frontini, M. and Sormani, E. (2004) Third-Order Methods from Quadrature Formulae for Solving Systems of Nonlinear Equations. Applied Mathematics and Computation, 149, 771-782.
https://doi.org/10.1016/S0096-3003(03)00178-4

   Cordero, A. and Torregrosa, J.R. (2007) Variants of Newtons Method Using Fifth-Order Quadrature Formulas. Applied Mathematics and Computation, 190, 686-698.
https://doi.org/10.1016/j.amc.2007.01.062

   Noor, M.A. and Wasteem, M. (2009) Some Iterative Methods for Solving a System of Nonlinear Equations. Computers and Mathematics with Applications, 57, 101-106.
https://doi.org/10.1016/j.camwa.2008.10.067

   Hafiz, M.A. and Bahgat, M.S.M. (2012) An Efficient Two-Step Iterative Method for Solving System of Nonlinear Equations. Journal of Mathematics Research, 4, 28-34.

   Khirallah, M.Q. and Hafiz, M.A. (2012) Novel Three Order Methods for Solving a System of Nonlinear Equations. Bulletin of Mathematical Sciences & Ap-plications, 1, 1-14.
https://doi.org/10.18052/www.scipress.com/BMSA.2.1

   Guan, Z. and Lu, J.F. (1998) Numerical Analysis. High Education Press, Beijing.

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