# Frozen Jacobian Multistep Iterative Method for Solving Nonlinear IVPs and BVPs

Ahmad, F. and Rehman, S.U. and Ullah, M.Z. and Aljahdali, H.M. and Ahmad, S. and Alshomrani, A.S. and Carrasco, J.A. and Ahmad, S. and Sivasankaran, S. (2017) Frozen Jacobian Multistep Iterative Method for Solving Nonlinear IVPs and BVPs. Complexity, 2017. pp. 1-30. ISSN 1076-2787

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Official URL: http://dx.doi.org/10.1155/2017/9407656

## Abstract

In this paper, we present and illustrate a frozen Jacobian multistep iterative method to solve systems of nonlinear equations associated with initial value problems (IVPs) and boundary value problems (BVPs). We have used Jacobi-Gauss-Lobatto collocation (J-GL-C) methods to discretize the IVPs and BVPs. Frozen Jacobian multistep iterative methods are computationally very efficient. They require only one inversion of the Jacobian in the form of LU-factorization. The LU factors can then be used repeatedly in the multistep part to solve other linear systems. The convergence order of the proposed iterative method is , where is the number of steps. The validity, accuracy, and efficiency of our proposed frozen Jacobian multistep iterative method is illustrated by solving fifteen IVPs and BVPs. It has been observed that, in all the test problems, with one exception in this paper, a single application of the proposed method is enough to obtain highly accurate numerical solutions. In addition, we present a comprehensive comparison of J-GL-C methods on a collection of test problems.

Item Type: Article Frozen Jacobian Multistep Iterative Method; Solving; Nonlinear; IVPs; BVPs Q Science > QA Mathematics Faculty of Science > Institute of Mathematical Sciences Ms. Juhaida Abd Rahim 10 Jul 2017 07:35 10 Jul 2017 07:35 http://eprints.um.edu.my/id/eprint/17468