Approximate solution of the system of nonlinear integral equation by Newton–Kantorovich method Z.K. Eshkuvatov a,c,⇑ , Anvarjon Ahmedov b,c , N.M.A. Nik Long a,c , O. Shafiq c a Department of Mathematics, Faculty of Science, Universiti Putra Malaysia (UPM), Malaysia b Department of Process and Food Engineering, Faculty of Engineering, UPM, Malaysia c Institute for Mathematical Research, UPM, Malaysia article info Keywords: Newton–Kantorovich method Rate of convergence Nonlinear operator Volterra integral equation Trapezoidal formula abstract The Newton–Kantorovich method is developed for solving the system of nonlinear integral equations. The existence and uniqueness of the solution are proved, and the rate of conver- gence of the approximate solution is established. Finally, numerical examples are provided to show the validity and the efficiency of the method presented. Crown Copyright Ó 2010 Published by Elsevier Inc. All rights reserved. 1. Introduction Solving the nonlinear integral equations (IEs) by linearization method is a popular approach by many researches [1,2,4– 6], and the necessity of Volterra IEs was emphasized in Baker [3]. For the system of nonlinear integral equations, Glushkov et al. [7] introduced the models of developing systems for describing a large class of problems in economics, ecology, med- icine, and other fields of applied mathematics. Boikov and Tynda [4] developed Newton–Kantorovich method for solving the system of nonlinear integral equations (named it two commodity models) as xðtÞ R t yðtÞ hðt; sÞgðsÞxðsÞds ¼ 0; R t yðtÞ kðt; sÞ½1 gðsÞxðsÞds ¼ f ðtÞ; 9 = ; ð1:1Þ where 0 < t 0 6 t 6 T, y(t)< t with given the functions hðt; sÞ; kðt; sÞ2 C ½t 0 ;1½t 0 ;1 f ðtÞ; gðtÞ2 C ½t 0 ;1 ð0 < gðtÞ < 1Þ and the unknown functions xðtÞ2 C t 0 ;1 ½ ; yðtÞ2 C 1 ½t 0 ;1 , and proved the existence, uniqueness and rate of convergence of the approx- imate solution for Eq. (1.1). In this work we further investigate the system of nonlinear integral equations of the form xðtÞ R t yðtÞ hðt; sÞx 2 ðsÞds ¼ 0; R t yðtÞ kðt; sÞx 2 ðsÞds ¼ f ðtÞ; 9 = ; ð1:2Þ where x(t) and y(t) are unknown functions defined on [t 0 , 1), t 0 > 0, and hðt; sÞ; kðt; sÞ2 C ½t 0 ;1½t 0 ;1 ; f ðtÞ2 C ½t 0 ;1 are given functions. Stable computational scheme is given in detail in Section 2. The existence and uniqueness of the solution and rate of con- vergence of the approximate solution are proved in Section 3. Discretization of the method is described in Section 4 and in Section 5, numerical examples are given. Finally we end up the theoretical findings and experimental works with conclu- sions in Section 6. 0096-3003/$ - see front matter Crown Copyright Ó 2010 Published by Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.09.068 ⇑ Corresponding author at: Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, UPM, Malaysia. E-mail addresses: ezaini@science.upm.edu.my (Z.K. Eshkuvatov), nmasri@math.upm.edu.my (N.M.A. Nik Long). Applied Mathematics and Computation 217 (2010) 3717–3725 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc