A Numerical Iterative Method for Solving Systems of First-Order Periodic Boundary Value Problems

Theobjective of this paper is to present a numerical iterativemethod for solving systems of first-order ordinary differential equations subject to periodic boundary conditions.This iterative technique is based on the use of the reproducing kernelHilbert spacemethod in which every function satisfies the periodic boundary conditions. The present method is accurate, needs less effort to achieve the results, and is especially developed for nonlinear case. Furthermore, the present method enables us to approximate the solutions and their derivatives at every point of the range of integration. Indeed, three numerical examples are provided to illustrate the effectiveness of the present method. Results obtained show that the numerical scheme is very effective and convenient for solving systems of first-order ordinary differential equations with periodic boundary conditions.


Introduction
Systems of ordinary differential equations with periodic boundary value conditions, the so-called periodic boundary value problems (BVPs), are well known for their applications in sciences and engineering [1][2][3][4][5].In this paper, we focus on finding approximate solutions to systems of first-order periodic BVPs, which are a combination of systems of firstorder ordinary differential equations and periodic boundary conditions.In fact, accurate and fast numerical solutions of systems of first-order periodic BVPs are of great importance due to their wide applications in scientific and engineering research.
Numerical methods are becoming more and more important in mathematical and engineering applications, simply not only because of the difficulties encountered in finding exact analytical solutions but also because of the ease with which numerical techniques can be used in conjunction with modern high-speed digital computers.A numerical procedure for solving systems of first-order periodic BVPs based on the use of reproducing kernel Hilbert space (RKHS) method is discussed in this work.
Among a substantial number of works dealing with systems of first-order periodic BVPs, we mention [6][7][8][9][10].The existence of solutions to systems of first-order periodic BVPs has been discussed as described in [6].In [7], the authors have discussed some existence and uniqueness results of periodic solutions for first-order periodic differential systems.Also, in [8] the authors have provided the existence, multiplicity, and nonexistence of positive periodic solutions for systems of first-order periodic BVPs.Furthermore, the existence of periodic solutions for the coupled first-order differential systems of Hamiltonian type is carried out in [9].Recently, the existence of positive solutions for systems of first-order periodic BVPs is proposed in [10].For more results on the solvability analysis of solutions for systems of firstorder periodic BVPs, we refer the reader to [11][12][13][14][15], and for numerical solvability of different categories of BVPs, one can consult [16][17][18][19].
Remark 2. Condition (2) in Definition 1 is called "the reproducing property, " which means that the value of the function  at the point  is reproducing by the inner product of (⋅) with (⋅, ).A Hilbert space which possesses a reproducing kernel is called a RKHS.
Definition 4 (see [23]).The Hilbert space An important subset of the RKHSs is the RKHSs associated with continuous kernel functions.These spaces have wide applications, including complex analysis, harmonic analysis, quantum mechanics, statistics, and machine learning.
Theorem 5.The Hilbert space  2 2 [0, 1] is a complete reproducing kernel and its reproducing kernel function   () can be written as where   () and   (),  = 1, 2, 3, 4, are unknown coefficients of   () and will be given in the following proof.

Formulation of Linear Operator
In this section, the formulation of a differential linear operator and the implementation method are presented in the reproducing kernel space  2 2 [0, 1].After that, we construct an orthogonal function system of the space  2 2 [0, 1] based on the use of the Gram-Schmidt orthogonalization process in order to obtain the exact and approximate solutions of ( 1) and ( 2) using RKHS method.
The internal structure of the following theorem is as follows: firstly, we will give the representation form of the exact solutions of ( 1) and (2) in the form of an infinite series in the space  2 2 [0, 1].After that, the convergence of approximate solutions  , () to the exact solutions   (),  = 1, 2, . . ., , will be proved.
, then the following hold: (i) the exact solutions of (9) could be represented by (ii) the approximate solutions of (9) and  ()  , (),  = 0, 1, are converging uniformly to the exact solutions   () and their derivatives as  → ∞, respectively.
We mention here that the approximate solutions  , () in ( 13) can be obtained directly by taking finitely many terms in the series representation for   () of (12).

Construction of Iterative Method
In this section, an iterative method of obtaining the solutions of (1) and ( 2) is represented in the reproducing kernel space  2 2 [0, 1] for linear and nonlinear cases.Initially, we will mention the following remark about the exact and approximate solutions of (1) and (2).
In order to apply the RKHS technique to solve (1) and (2), we have the following two cases based on the algebraic structure of the function   ,  = 1, 2, . . ., .
Case 1.If (1) is linear, then the exact and approximate solutions can be obtained directly from (12) and (13), respectively.Case 2. If (1) is nonlinear, then in this case the exact and approximate solutions can be obtained by using the following iterative algorithm.
Here, we note that, in the iterative process of (17), we can guarantee that the approximations  , () satisfy the periodic boundary conditions (2).Now, the approximate solutions   , () can be obtained by taking finitely many terms in the series representation of  , () and Now, we will proof that  , () in the iterative formula (17) are converged to the exact solutions   () of (1).In fact, this result is a fundamental in the RKHS theory and its applications.The next two lemmas are collected in order to prove the prerecent theorem.
According to the internal structure of the present method, it is obvious that if we let   () denote the exact solutions of ( 9),  , () denote the approximate solutions obtained by the RKHS method as given by ( 17), and  {}  () denote the difference between  , () and   (), where  ∈ [0, 1] and  = 1, 2, . . ., , then . Consequently, this shows the following theorem.

Numerical Examples
In this section, the theoretical results of the previous sections are illustrated by means of some numerical examples in order to illustrate the performance of the RKHS method for solving systems of first-order periodic BVPs and justify the accuracy and efficiency of the method.To do so, we consider the following three nonlinear examples.These examples have been solved by the presented method with different values of  and .Results obtained by the method are compared with the exact solution of each example by computing the absolute and relative errors and are found to be in good agreement with each other.In the process of computation, all experiments were performed in MAPLE 13 software package.
The present method enables us to approximate the solutions and their derivatives at every point of the range of integration.Hence, it is possible to pick any point in [0, 1] and as well the approximate solutions and their derivatives will be applicable.Next, new numerical results for Example 1 which include the absolute error at some selected gird points in [0, 1] for approximating   1 () and   2 (), where   = (−1)/(−1),  = 1, 2, . . ., ,  = 101, and  = 3, are given in Table 3.
From the previous tables, it can be seen that the RKHS method provides us with the accurate approximate solutions.On the other aspect as well, it is clear that the accuracy obtained using the mentioned method is advanced by using only a few tens of iterations.

Conclusions
Here, we use the RKHS method to solve systems of firstorder periodic BVPs.The solutions were calculated in the form of a convergent series in the space  2 2 [0, 1] with easily computable components.In the proposed method, the -term approximations are obtained and proved to converge to the exact solutions.Meanwhile, the error of the approximate solutions is monotone decreasing in the sense of the norm of  2 2 [0, 1].It is worthy to note that, in our work, the approximate solutions and their derivatives converge uniformly to the exact solutions and their derivatives, respectively.On the other aspect as well, the present method enables us to approximate the solutions and their derivatives at every point of the range of integration.The results show that the present method is an accurate and reliable analytical technique for solving systems of first-order periodic BVPs.