Fitted Reproducing Kernel Method for Solving a Class of Third-Order Periodic Boundary Value Problems

employed for solving a class of third-order periodic boundary value problem by using fitted reproducing kernel algorithm. The reproducing kernel function is built to get fast accurately and efficiently series solutions with easily computable coefficients throughout evolution the algorithm under constraint periodic conditions within required grid points. The analytic solution is formulated in a finite series form whilst the truncated series solution is given to converge uniformly to analytic solution. The reproducing kernel procedure is based upon generating orthonormal basis system over a compact dense interval in sobolev space to construct a suitable analytical-numerical solution. Furthermore, experiments results of some numerical examples are presented to illustrate the good performance of the presented algorithm. The results indicate that the reproducing kernel procedure is powerful tool for solving other problems of ordinary and partial differential equations arising in physics, computer and engineering fields.


Introduction
Periodic Boundary Value Problem (PBVP) is an active research of modern mathematics that can be found naturally in different branches of applied sciences, physics and engineering (Gregu, 1987;Minh, 1998;Ashyralyev et al., 2009). It has many applications due to the fact that a lot practical problems in mechanics, electromagnetic, astronomy and electrostatics may be converted directly to such PBVP. However, it is difficult generally to get a closed form solution for PBVP in terms of elementary functions, especially, for nonlinear and nonhomogeneous cases. So, PBVP has attracted much attention and has been studied by several authors (Kong and Wang, 2001;Chu and Zhou, 2006;Liu et al., 2007;Zehour et al., 2008;Yu and Pei, 2010;Abu Arqub and Al-Smadi, 2014). The purpose of this analysis is to develop analytical-numerical method for handling third-order, two-point PBVP with given periodic conditions by an application of the reproducing kernel theory. More specifically, consider the general form of third-order BVP: W [0, 1], 0 ≤ t ≤ 1, -∞ < v i < 1, i = 1, 2, 3, which is linear or nonlinear term depending on the problem discussed.
The numerical solvability of BVPs with periodic conditions of different order has been pursued in literature.
The structure of this article is organized as follows. In section 2, reproducing kernel spaces are described to compute its reproducing kernels functions in which every function satisfies the periodic conditions. In section 3 and 4, the analytical-numerical solutions of Equation 1 and 2 as well an iterative method for obtaining these solutions are presented with series formula in the space 4 2 W [0, 1]. The n-term numerical solution is obtained to converge uniformly to analytic solution. In section 5, some numerical examples are simulated to check the reasonableness of our theory and to demonstrate the high performance of the presented algorithm. Conclusions are summarized in the last section.

Construction of Reproducing Kernel Functions
In this section, we present some basic results and remarks in the reproducing kernel theory and its applications.

Lemma 1
The reproducing kernel function K s (t) of the Hilbert space 4 2 W [0, 1] can be given by: then y (i) (0) = y (i) (1), i = 0, 1, 2. Therefore: For each s, t ∈ [0, 1], assume K s (t) satisfy the following: Also, assume K s (t) satisfy that: Through the last computational results the unknown coefficients a i (s) and b i (s), i = 0, 1,..., 7 of K s (t) in Equation 4 can be obtained. However, the representation form of these coefficients using Maple 13 software package are provided by: However, Geng and Cui (2007) show that the reproducing kernel function Q s (t) of 1 2 W [0, 1] can be given by:

Representation of Analytical-Numerical Solutions
First, as in 2015;Al-Smadi and Altawallbeh, 2013;Abu Arqub, 2015), we transform the problem into a differential operator. To do so, we define an operator T from the space 4 2 W [0, 1] into 1 2 W [0, 1] such that Ty (t) = y′″ (t). Therefore, Equation 1 and 2 can be converted equivalently into following form: with periodic conditions:

Lemma 2
The operator T is linear bounded operator from

Proof
We want to show that

Proof
It is easy to note that ψ Whilst by Equation 9, we have that:  unknown. Thus, we can approximate B i using known A i . For computations, define the initial guess function y 0 (t 1 ) = 0, set y 0 (t 1 ) = y (t 1 ) and define the nth-order approximation y n (t) to y (t) as follows:

Ty t t t F t y t y t y t t t F t y t y t y t t T F t y t y t y t
where, the coefficients A i of ( ) i t ψ , i = 1, 2,..., n are obtained by:

Corollary 1
The numeric solution and its derivatives up to order three are converge uniformly to analytic solution and all its derivatives, respectively.

Proof
For each t ∈ [0, 1], we have:  = Ty (t j ). Next, we list two lemmas for convenience in order to prove the recent theorems.

Lemma 3
The numerical solution y n satisfies, Ty n (t j ) =

Computational Algorithm and Numerical Experiments
Using RKHS method, taking Example 2 Meditate in the following nonlinear diferential equation:

Example 3
Meditate in the following nonlinear differential equation: with periodic conditions: where, f(t) is given to obtain the exact solution as y(t) = cosh(t 2 -t).
The agreement between the analytical-numerical solutions is investigated for Examples 1, 2 and 3 at various t in [0, 1] by computing absolute errors and relative errors of numerically approximating their analytical solutions as shown in Tables 1 to 3, respectively.

Concluding Summary
In this article, we introduce the fitting reproducing kernel approach to enlarge its application range for treating a class of third-order periodic BVPs in a favorable reproducing kernel Hilbert space. The method does not require discretization of the variables as well as it provides best solutions in a less number of iterations and reduces the computational work. Further, we can conclude that the presented method is powerful and efficient technique in finding approximate solution for both linear and nonlinear problems. In the proposed algorithm, the solution and its approximation are represented in the form of series in 4 2 W [0, 1]. The approximate solution and its derivative converge uniformly to exact solution and its derivative, respectively.