An eighth order frozen Jacobian iterative method for solving nonlinear IVPs and BVPs

Volume 17, Issue 3, pp 378-399

Publication Date: 2017-07-19

http://dx.doi.org/10.22436/jmcs.017.03.04

Authors

Dina Abdullah Alrehaili - Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia.
Dalal Adnan Al-Maturi - Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia.
Salem Al-Aidarous - Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia.
Fayyaz Ahmad - Dipartimento di Scienza e Alta Tecnologia, Universita dell’Insubria, Via Valleggio 11, Como 22100, Italy.

Abstract

A frozen Jacobian iterative method is proposed for solving systems of nonlinear equations. In particular, we are interested in solving the systems of nonlinear equations associated with initial value problems (IVPs) and boundary value problems (BVPs). In a single instance of the proposed iterative method DEDF, we evaluate two Jacobians, one inversion of the Jacobian and four function evaluations. The direct inversion of the Jacobian is computationally expensive, so, for a moderate size, LU factorization is a good direct method to solve the linear system. We employed the LU factorization of the Jacobian to avoid the direct inversion. The convergence order of the proposed iterative method is at least eight, and it is nine for some particular classes of problems. The discretization of IVPs and BVPs is employed by using Jacobi-Gauss-Lobatto collocation (J-GL-C) method. A comparison of J-GL-C methods is presented in order to choose best collocation method. The validity, accuracy and the efficiency of our DEDF are shown by solving eleven IVPs and BVPs problems.

Keywords

Frozen Jacobian iterative methods, systems of nonlinear equations, nonlinear initial-boundary value problems, Jacobi-Gauss-Lobatto quadrature, collocation method.

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