An efficient multi-step iterative method for computing the numerical solution of systems of nonlinear equations associated with ODEs Malik Zaka Ullah a,c,⇑ , Stefano Serra-Capizzano a , Fayyaz Ahmad b a Dipartimento di Scienza e Alta Tecnologia, Universita dell’Insubria, Via Valleggio 11, 22100 Como, Italy b Dept. de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Comte d’Urgell 187, 08036 Barcelona, Spain c Department of Mathematics, Faculty of Sciences, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia article info Keywords: Nonlinear systems Nonlinear ordinary differential equations Higher order Frechet derivative Multi-step abstract We developed multi-step iterative method for computing the numerical solution of nonlin- ear systems, associated with ordinary differential equations (ODEs) of the form LðxðtÞÞ þ f ðxðtÞÞ ¼ gðtÞ: here LðÞ is a linear differential operator and f ðÞ is a nonlinear smooth function. The proposed iterative scheme only requires one inversion of Jacobian which is computationally very efficient if either LU-decomposition or GMRES-type methods are employed. The higher-order Frechet derivatives of the nonlinear system stemming from the considered ODEs are diagonal matrices. We used the higher-order Frechet derivatives to enhance the convergence-order of the iterative schemes proposed in this note and indeed the use of a multi-step method dramatically increases the convergence-order. The second- order Frechet derivative is used in the first step of an iterative technique which produced third-order convergence. In a second step we constructed matrix polynomial to enhance the convergence-order by three. Finally, we freeze the product of a matrix polynomial by the Jacobian inverse to generate the multi-step method. Each additional step will increase the convergence-order by three, with minimal computational effort. The convergence-order (CO) obeys the formula CO ¼ 3m, where m is the number of steps per full-cycle of the consid- ered iterative scheme. Few numerical experiments and conclusive remarks end the paper. Ó 2014 Elsevier Inc. All rights reserved. 1. Preliminaries In this study, we consider the scalar ordinary differential equations of the form: LðxðtÞÞ þ f ðxðtÞÞ ¼ gðtÞ where t 2 D; ð1Þ where LðÞ is a linear differential operator, f ðÞ is a differentiable nonlinear function and D is an interval subset of R. The linear operator and the nonlinear function are well defined over the domain of problem. Furthermore, we suppose that A is the discrete approximation of the linear differential operator L over a partition ft 1 ; t 2 ; t 3 ; ... ; t n g of domain D and x ¼½xðt 1 Þ; xðt 2 Þ; ... ; xðt n Þ T : ð2Þ http://dx.doi.org/10.1016/j.amc.2014.10.103 0096-3003/Ó 2014 Elsevier Inc. All rights reserved. ⇑ Corresponding author. E-mail addresses: malik.zakaullah@uninsubria.it (M.Z. Ullah), stefano.serrac@uninsubria.it (S. Serra-Capizzano), fayyaz.ahmad@upc.edu (F. Ahmad). Applied Mathematics and Computation 250 (2015) 249–259 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc