Numerical investigations for systems of second-order periodic boundary value problems using reproducing kernel method

The reproducing kernel method is a numerical as well as analytical technique for solving a large variety of ordinary and partial differential equations associated to different kind of boundary conditions, and usually provides the solutions in term of rapidly convergent series in the appropriate Hilbert spaces with components that can be elegantly computed. The aim of the present analysis is to implement a relatively recent computational method, reproducing kernel Hilbert space, for obtaining the solutions for systems of second-order differential equations with periodic boundary conditions. A reproducing kernel space is constructed in which the periodic conditions of the systems are satisfied, whilst, the smooth kernel functions are used throughout the evolution of the method to obtain the required grid points. An efficient construction is given to obtain the approximate solutions for the systems together with an existence proof of the exact solutions is proposed based upon the reproducing kernel theory. Convergence analysis and error behavior of the presented method are also discussed. In this approach, computational results of some numerical examples are presented to illustrate the viability, simplicity, and applicability of the algorithm developed.