Analytical solution of nonlinear second-order periodic boundary value problem using reproducing kernel method Nabil Shawagfeh a,1 , Omar Abu Arqub a , Shaher Momani b,* a Department of Mathematics, Al-Balqa Applied University, Salt 19117, Jordan b Department of Mathematics, The University of Jordan, Amman 11942, Jordan *Corresponding author: e-mail: s.momani@ju.edu.jo (Shaher Momani) ||||||||||||||||||||||||||||||||||||||||||||{ Abstract This paper investigates the numerical solution of nonlinear second-order periodic boundary value problems by using reproducing kernel Hilbert space method. The solution was calculated in the form of a convergent series in the space W 3 2 with easily computable components. In the proposed method, the n-term approximation is obtained and is proved to converge to the analytical solution. Meanwhile, the error of the approximate solution is monotone decreasing in the sense of the norm of W 3 2 . The proposed technique is applied to several examples to illustrate the accuracy, eciency, and applicability of the method. The results reveal that the method is very eective, straightforward, and simple. Keywords: periodic boundary value problems; Reproducing kernel Hilbert space method AMS Subject Classication: 34K28; 47B32; 34B15 ||||||||||||||||||||||||||||||||||||||||||||||||{ 1 Introduction Second-order boundary value problems (BVPs) for ordinary dierential equations arise very frequently in many branches of applied mathematics and physics such as atomic calculations, gas dynamics, nuclear physics, atomic structures, deformation of beams and plate deection theory, chemical reactions, and so on [1{4]. In recent years, the nonlinear second-order periodic BVPs which are a combination of second-order ordinary dierential equations and periodic boundary conditions have been widely studied by many authors [5{8], due to a wide range of applications in applied mathematics, physics, and engineering, particularly in the homogenization of composite materials with a periodic microstructure [9,10]. In most cases, nonlinear second-order periodic BVPs do not always have solutions which we can obtain using analytical methods. In fact, many of real physical phenomena encountered, are almost impossible to solve by this technique, these problems must be attacked by various approximate and numerical methods. This paper discusses and investigates the analytical approximate solution using reproducing kernel Hilbert space (RKHS) method for nonlinear second-order BVP with periodic boundary conditions which is as follows: u 00 (x)= F (x; u (x) ;u 0 (x)) , 0  x  1; (1) subject to the periodic boundary conditions u (0) = u (1) ; u 0 (0) = u 0 (1) ; (2) where u 2 W 3 2 [0; 1] is an unknown function to be determined, F (x; y; z) is continuous term in W 1 2 [0; 1] as y = y (x) ;z = z (x) 2 W 3 2 [0; 1], 0  x  1, 1 < y;z < 1, and is depending on the problem discussed, and W 1 2 [0; 1] ;W 3 2 [0; 1] are two reproducing kernel spaces. |||||||||||| 1 On sabbatical leave from Department of Mathematics, Faculty of Science, The University of Jordan, Amman- Jordan.