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Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 518406, 13 pages
Research Article

Reproducing Kernel Algorithm for the Analytical-Numerical Solutions of Nonlinear Systems of Singular Periodic Boundary Value Problems

Department of Mathematics, Al-Balqa Applied University, Salt 19117, Jordan

Received 8 December 2014; Accepted 2 May 2015

Academic Editor: Francesco Tornabene

Copyright © 2015 Omar Abu Arqub. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The reproducing kernel algorithm is described in order to obtain the efficient analytical-numerical solutions to nonlinear systems of two point, second-order periodic boundary value problems with finitely many singularities. The analytical-numerical solutions are obtained in the form of an infinite convergent series for appropriate periodic boundary conditions in the space , whilst two smooth reproducing kernel functions are used throughout the evolution of the algorithm to obtain the required nodal values. An efficient computational algorithm is provided to guarantee the procedure and to confirm the performance of the proposed approach. The main characteristic feature of the utilized algorithm is that the global approximation can be established on the whole solution domain, in contrast with other numerical methods like onestep and multistep methods, and the convergence is uniform. Two numerical experiments are carried out to verify the mathematical results, whereas the theoretical statements for the solutions are supported by the results of numerical experiments. Our results reveal that the present algorithm is a very effective and straightforward way of formulating the analytical-numerical solutions for such nonlinear periodic singular systems.