The modified Broyden-variational method for solving nonlinear elliptic differential equations Muhammed I. Syam UAE University, Department of Mathematical Sciences, College of Science, P.O. Box 17551, Al-Ain, United Arab Emirates Accepted 25 April 2006 Communicated by Professr Ji-Huan He Abstract In this paper, we present a numerical technique for solving nonlinear elliptic differential equations. It is based on the variational method which produces an underdetermined system of equations. We used the predictor–corrector tech- nique to solve this system. We modify the Broyden update method and we use it as a corrector. Numerical results and conclusions will be presented. Ó 2006 Elsevier Ltd. All rights reserved. 1. Introduction Many problems in science and engineering require the computation of a family of solutions of a nonlinear system of the form Gðu; kÞ¼ 0; u ¼ uðkÞ; ð1Þ where G : R nþ1 ! R n is continuously differentiable function, u represents the solution and k is a real parameter, i.e., Reynold’s number, load, etc. To solve these problems, we need to find the solution for some k-intervals, i.e., a path solutions, (u(k), k). Equations of the form (1) are called nonlinear elliptic eigenvalue problems if the operator G with k fixed is an elliptic differential operator. As a typical example of a nonlinear elliptic eigenvalue problems, we consider the following problem: Gðu; kÞ¼ Du þ kf ðuÞ¼ 0 in X; u ¼ 0 on oX ð2Þ Eq. (2) arises in many physical problems. For example, in chemical reactor theory, radiative heat transfer, combustion theory, and in modelling the expansion of the universe. The domain X is usually taken to be the unit interval [0, 1] in R, or the unite square [0, 1] · [0, 1] in R 2 , or the unit cube [0, 1] · [0, 1] · [0, 1] in R 3 . 0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.04.126 E-mail address: m.syam@uaeu.ac.ae. Chaos, Solitons and Fractals 32 (2007) 392–404 www.elsevier.com/locate/chaos