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

Authors: Mohammed AL-Smadi; Omar Abu Arqub; Ahmad El-Ajou
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

The objective of this paper is to present a numerical iterative method 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 kernel Hilbert space method 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.


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## Body

## 1. 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 [1–5]. In this paper, we focus on finding approximate solutions to systems of first-order periodic BVPs, which are a combination of systems of first-order 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 [6–10]. 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 first-order periodic BVPs, we refer the reader to [11–15], and for numerical solvability of different categories of BVPs, one can consult [16–19].Investigation about systems of first-order periodic BVPs numerically is scarce. In this paper, we utilize a methodical way to solve these types of differential systems. In fact, we provide criteria for finding the approximate and exact solutions to the following system:(1)u1′(x)=F1(x,u1(x),u2(x),…,un(x)),u2′(x)=F2(x,u1(x),u2(x),…,un(x)),⋮un′(x)=Fn(x,u1(x),u2(x),…,un(x)),
subject to the periodic boundary conditions
(2)u1(0)=u1(1),u2(0)=u2(1),⋮un(0)=un(1),
where x∈[0,1], us∈W22[0,1] are unknown functions to be determined, Fs(x,v1,v1,…,vn) are continuous terms in W21[0,1] as vs=vs(x)∈W22[0,1], 0≤x≤1, -∞<vs<∞ in which s=1,2,…,n, and W21[0,1], W22[0,1] are two reproducing kernel spaces. Here, we assume that (1) subject to the periodic boundary conditions (2) has a unique solution on [0,1].Reproducing kernel theory has important applications in numerical analysis, differential equations, integral equations, probability and statistics, and so forth [20–22]. In the last years, extensive work has been done using RKHS method which provides numerical approximations for linear and nonlinear equations. This method has been implemented in several operator, differential, integral, and integrodifferential equations side by side with their theories. The reader is kindly requested to go through [23–35] in order to know more details about RKHS method, including its history, its modification for use, its applications, and its characteristics.The rest of the paper is organized as follows. In the next section, two reproducing kernel spaces are described in order to formulate the reproducing kernel functions. In Section3, some essential results are introduced and a method for the existence of solutions for (1) and (2) is described. In Section 4, we give an iterative method to solve (1) and (2) numerically. Numerical examples are presented in Section 5. Section 6 ends this paper with brief conclusions.


## 2. Construct of Reproducing Kernel Functions

In this section, two reproducing kernels needed are constructed in order to solve (1) and (2) using RKHS method. Before the construction, we utilize the reproducing kernel concept. Throughout this paper, C is the set of complex numbers, L2[a,b]={u∣∫ab‍u2(x)dx<∞}, and l2={A∣∑i=1∞‍(Ai)2<∞}.Definition 1 (see [23]).
LetE be a nonempty abstract set. A function R:E×E→C is a reproducing kernel of the Hilbert space H if(1)
for eachx∈E, R(·,x)∈H,(2)
for eachx∈E and φ∈H, 〈φ(·),R(·,x)〉=φ(x).Remark 2.
Condition(2) in Definition 1 is called “the reproducing property,” which means that the value of the function φ at the point x is reproducing by the inner product of φ(·) with R(·,x). A Hilbert space which possesses a reproducing kernel is called a RKHS.To solve (1) and (2) using RKHS method, we first define and construct a reproducing kernel space W22[0,1] in which every function satisfies the periodic boundary condition u(0)=u(1). After that, we utilize the reproducing kernel space W21[0,1].Definition 3.
The inner product spaceW22[0,1] is defined as W22[0,1]={u(x)∣u,u′ are absolutely continuous real-valued functions on [0,1], u,u′,u′′∈L2[0,1], and u(0)=u(1)}. On the other hand, the inner product and the norm in W22[0,1] are defined, respectively, by
(3)〈u,v〉W22=∑i=01‍u(i)(0)v(i)(0)+∫01‍u′′(t)v′′(t)dt,
and ∥u∥W22=〈u,u〉W22, where u,v∈W22[0,1].It is easy to see that〈u,v〉W22 satisfies all the requirements for the inner product. First, 〈u,u〉W22≥0. Second, 〈u,v〉W22=〈v,u〉W22. Third, 〈γu,v〉W22=γ〈u,v〉W22. Fourth, 〈u+w,v〉W22=〈u,v〉W22+〈w,v〉W22, where u,v,w∈W22[0,1]. It therefore remains only to prove that 〈u,u〉W22=0 if and only if u=0. In fact, it is obvious that when u=0, then 〈u,u〉W22=0. On the other hand, if 〈u,u〉W22=0, then by (3), we have 〈u,u〉W22=∑i=01‍(u(i)(0))2+∫01‍(u′′(t))2dt=0; therefore, u(0)=u′(0)=0 and u′′(t)=0. Then, we can obtain u=0.Definition 4 (see [23]).
The Hilbert spaceW22[0,1] is called a reproducing kernel if, for each fixed x∈[0,1], there exist R(x,y)∈W22[0,1] (simply Rx(y)) such that 〈u(y),Rx(y)〉W22=u(x) for any u(y)∈W22[0,1] and y∈[0,1].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 spaceW22[0,1] is a complete reproducing kernel and its reproducing kernel function Rx(y) can be written as
(4)Rx(y)={p1(x)+p2(x)y+p3(x)y2+p4(x)y3,y≤x,q1(x)+q2(x)y+q3(x)y2+q4(x)y3,y>x,
where pi(x) and qi(x), i=1,2,3,4, are unknown coefficients of Rx(y) and will be given in the following proof.Proof.
The proof of the completeness and reproducing property ofW22[0,1] is similar to the proof in [24]. Now, let us find out the expression form of the reproducing kernel function Rx(y) in the space W22[0,1]. Through several integration by parts, we have ∫01‍u′′(y)∂y3Rx(y)dy=∑i=01‍(-1)1-iu(i)(y)∂y3-iRx(y)|y=0y=1+∫01‍u(y)∂y4Rx(y)dy. Thus, from (3), we can write 〈u(y),Rx(y)〉W22=∑i=01‍u(i)(0)[∂yiRx(0)+(-1)i∂y3-iRx(0)] + ∑i=01‍(-1)1-iu(i)(1)∂y3-iRx(1) + ∫01‍u(y)∂y4Rx(y)dy. Since Rx(y)∈W22[0,1], it follows that Rx(0)=Rx(1); also since u(x)∈W22[0,1], it follows that u(0)=u(1). Then
(5)〈u(y),Rx(y)〉W22=∑i=01‍u(i)(0)[∂yiRx(0)+(-1)i∂y3-iRx(0)]+∑i=01‍(-1)i+1u(i)(1)∂y3-iRx(1)+∫01u(y)∂y4Rx(y)dy+c1(u(0)-u(1)).
But on the other aspect as well, if ∂y2Rx(1)=0, Rx(0)+∂y3Rx(0)+c1=0, ∂y1Rx(0)-∂y2Rx(0)=0, and ∂y3Rx(1)+c1=0, then (5) implies that 〈u(y),Rx(y)〉W22=∫01‍u(y)∂y4Rx(y)dy. Now, for any x∈[0,1], if Rx(y) satisfies(6)∂y4Rx(y)=-δ(x-y),δdirac-deltafunction,
then 〈u(y),Rx(y)〉W22=u(x). Obviously, Rx(y) is the reproducing kernel function of the space W22[0,1]. Next, we give the expression form of the reproducing kernel function Rx(y). The characteristic formula of (6) is given by λ4=0. Then the characteristic values are λ=0 with multiplicity 4. So, let the expression form of the reproducing kernel function Rx(y) be as defined in (4). On the other hand, for (6), let Rx(y) satisfy the equation ∂ymRx(x+0)=∂ymRx(x-0), m=0,1,2. Integrating ∂y6Rx(y)=-δ(x-y) from x-ε to x+ε with respect to y and letting ε→0, we have the jump degree of ∂y5Rx(y) at y=x given by ∂y3Rx(x+0)-∂y3Rx(x-0)=-1. Through the last descriptions, the unknown coefficients of (4) can be obtained. However, by using MAPLE 13 software package, the representation form of the reproducing kernel function Rx(y) is provided by(7)Rx(y)={148(x3y(6+3y-y2)+3x2y(-6-3yy2)+6xy(2+y+y2)-8(-6+y3)),y≤x,148(48+6xy(2-3y+y2)+3x2y(2-3y+y2)-x3(8-6y-3y2+y3)),y>x,This completes the proof.Definition 6 (see [25]).
The inner product spaceW21[0,1] is defined as W21[0,1]={u(x)∣u is absolutely continuous real-valued function on [0,1] and u′∈L2[0,1]}. On the other hand, the inner product and the norm in W21[0,1] are defined, respectively, by 〈u(x),v(x)〉W21=u(0)v(0)+∫01‍u′(x)v′(x)dx, and ∥u∥W21=〈u,u〉W21, where u,v∈W21[0,1].Theorem 7 (see [25]).
The Hilbert spaceW21[0,1] is a complete reproducing kernel and its reproducing kernel function Gx(y) can be written as
(8)Gx(y)={1+y,y≤x,1+x,y>x.Reproducing kernel functions possess some important properties such as being symmetric, unique, and nonnegative. The reader is asked to refer to [23–35] in order to know more details about reproducing kernel functions, including their mathematical and geometrical properties, their types and kinds, and their applications and method of calculations.


## 3. Formulation of Linear Operator

In this section, the formulation of a differential linear operator and the implementation method are presented in the reproducing kernel spaceW22[0,1]. After that, we construct an orthogonal function system of the space W22[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.First, as in [23–35], we transform the problem into a differential operator. To do this, we define a differential operator L as L:W22[0,1]→W21[0,1] such that Lu(x)=u′(x). As a result, (1) and (2) can be converted into the equivalent form as follows:
(9)Lus(x)=Fs(x,u1(x),u2(x),…,un(x)),us(0)-us(1)=0,
where 0≤x≤1 and s=1,2,…,n in which us(x)∈W22[0,1] and Fs(x,v1,v1,…,vn)∈W21[0,1] for vs=vs(x)∈W22[0,1], -∞<vs<∞, and 0≤x≤1. It is easy to show that L is a bounded linear operator from the space W22[0,1] into the space W21[0,1].Initially, we construct an orthogonal function system ofW22[0,1]. To do so, put φi(x)=Gxi(x) and ψi(x)=L*φi(x), where {xi}i=1∞ is dense on [0,1] and L* is the adjoint operator of L. In terms of the properties of reproducing kernel function Gx(y), one obtains 〈us(x),ψi(x)〉W22=〈us(x),L*φi(x)〉W22=〈Lus(x),φi(x)〉W21=Lus(xi), i=1,2,…, s=1,2,…,n.For the orthonormal function system{ψ¯i(x)}i=1∞ of the space W22[0,1], it can be derived from the Gram-Schmidt orthogonalization process of {ψi(x)}i=1∞ as follows:
(10)ψ¯i(x)=∑k=1i‍βikψk(x),
where βik are orthogonalization coefficients and are given as
(11)βij=1∥ψ1∥,fori=j=1,βij=1∥ψi∥2-∑k=1i-1‍(〈ψi,ψ¯k〉W22)2,fori=j≠1,βij=-1∥ψi∥2-∑k=1i-1‍(cik)2∑k=ji-1〈ψi,ψ¯k〉W22βkj,mmmmmmmmmmmmmmmmmlllfori>j.Clearly,ψi(x)=L*φi(x)=〈L*φi(x),Rx(y)〉W22=〈φi(x),LyRx(y)〉W21=LyRx(y)|y=xi∈W22[0,1]. Thus, ψi(x) can be written in the form ψi(x)=LyRx(y)|y=xi, where Ly indicates that the operator L applies to the function of y.Theorem 8.
If{xi}i=1∞ is dense on [0,1], then {ψi(x)}i=1∞ is a complete function system of the space W22[0,1].Proof.
For each fixedus(x)∈W22[0,1], let 〈us(x),ψi(x)〉W22=0. In other words, one can write 〈us(x),ψi(x)〉W22=〈us(x),L*φi(x)〉W22=〈Lus(x),φi(x)〉W21=Lus(xi)=0. Note that {xi}i=1∞ is dense on [0,1]; therefore Lus(x)=0. It follows that us(x)=0, s=1,2,…,n, from the existence of L-1. So, the proof of the theorem is complete.Lemma 9.
Ifus(x)∈W22[0,1], then there exist positive constants M{s} such that ∥us(i)(x)∥C≤M{s}∥us(x)∥W22, i=0,1, s=1,2,…,n, where ∥us(x)∥C=max⁡0≤x≤1⁡|us(x)|.Proof.
For anyx,y∈[0,1], we have us(i)(x)=〈us(y),∂xiRx(y)〉W22. By the expression form of the kernel function Rx(y), it follows that ∥∂xiRx(y)∥W22≤Mi{s}. Thus, |us(i)(x)|=|〈us(x),∂xiRx(x)〉W22|≤∥∂xiRx(x)∥W22∥us(x)∥W22≤Mi{s}∥us(x)∥W22. Hence, ∥us(i)(x)∥C≤max⁡i=0,1⁡{Mi{s}}∥us(x)∥W22, i=0,1, s=1,2,…,n.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 W22[0,1]. After that, the convergence of approximate solutions us,m(x) to the exact solutions us(x), s=1,2,…,n, will be proved.Theorem 10.
For eachus, s=1,2,…,n in the space W22[0,1], the series ∑i=1∞‍〈us(x),ψ¯i(x)〉ψ¯i(x) is convergent in the sense of the norm of W22[0,1]. On the other hand, if {xi}i=1∞ is dense on [0,1], then the following hold:(i)
the exact solutions of (9) could be represented by
(12)us(x)=∑i=1∞∑k=1iβikFs(xk,u1(xk),u2(xk),…,un(xk))ψ¯i(x),(ii)
the approximate solutions of (9)
(13)us,m(x)=∑i=1m∑k=1iβikFs(xk,u1(xk),u2(xk),…,un(xk))ψ¯i(x),andus,m(i)(x), i=0,1, are converging uniformly to the exact solutions us(x) and their derivatives as m→∞, respectively.Proof.
For the first part, letus(x) be solutions of (9) in the space W22[0,1]. Since us(x)∈W22[0,1], ∑i=1∞〈us(x),ψ¯i(x)〉ψ¯i(x) is the Fourier series expansion about normal orthogonal system {ψ¯i(x)}i=1∞, and W22[0,1] is the Hilbert space, then the series ∑i=1∞〈us(x),ψ¯i(x)〉ψ¯i(x) is convergent in the sense of ∥·∥W22. On the other hand, using (10), it easy to see that
(14)us(x)=∑i=1∞‍〈us(x),ψ¯i(x)〉W22ψ¯i(x)=∑i=1∞∑k=1i‍βik〈us(x),ψk(x)〉W22ψ¯i(x)=∑i=1∞∑k=1i‍βik〈us(x),L*φk(x)〉W22ψ¯i(x)=∑i=1∞∑k=1i‍βik〈Lus(x),φk(x)〉W21ψ¯i(x)=∑i=1∞∑k=1i‍βik〈Fs(x,u1(x),u2(x),…,un(x)),φk(x)〉W21ψ¯i(x)=∑i=1∞∑k=1i‍βikFs(xk,u1(xk),u2(xk),…,mmmmmmmmmmmllun(xk))ψ¯i(x).
Therefore, the form of (12) is the exact solutions of (9). For the second part, it is easy to see that by Lemma 9, for any x∈[0,1],
(15)|us,m(i)(x)-us(i)(x)|=|〈us,m(x)-us(x),Rx(i)(x)〉W22|≤∥∂xiRx(x)∥W22∥us,m(x)-us(x)∥W22≤Mi{s}∥us,m(x)-us(x)∥W22,
where i=0,1 and Mi{s} are positive constants. Hence, if ∥us,m(x)-us(x)∥W22→0 as m→∞, the approximate solutions us,m(x) and us,m(i)(x), i=0,1, s=1,2,…,n, are converged uniformly to the exact solutions us(x) and their derivatives, respectively. So, the proof of the theorem is complete.We mention here that the approximate solutionsus,m(x) in (13) can be obtained directly by taking finitely many terms in the series representation for us(x) of (12).


## 4. 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 W22[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 Fs, s=1,2,…,n.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.Algorithm 11.
According to (12), the representation form of the solutions of (1) can be denoted by
(16)us(x)=∑i=1∞‍Bi{s}ψ¯i(x),s=1,2,…,n,
where Bi{s}=∑k=1i‍βikFs(xk,u1,k-1(xk),u2,k-1(xk),…, un,k-1(xk)). In fact, Bi{s} in (16) are unknown; one will approximate Bi{s} using known Ai{s}. For numerical computations, one defines the initial functions us,0(x1)=0, put us,0(x1)=us(x1), and define the m-term approximations to us(x) by
(17)us,m(x)=∑i=1m‍Ai{s}ψ¯i(x),s=1,2,…,n,
where the coefficients Ai{s} of ψ¯i(x), i=1,2,…,n, s=1,2,…,n, are given as
(18)A1{s}=β11Fs(x1,u1,0(x1),u2,0(x1),…,un,0(x1)),us,1(x)=A1{s}ψ¯1(x),A2{s}=∑k=12‍β2kFs(xk,u1,k-1(xk),u2,k-1(xk),…,un,k-1(xk)),us,2(x)=∑i=12‍Ai{s}ψ¯i(x),⋮An{s}=∑k=1m‍βmkFs(xk,u1,k-1(xk),u2,k-1(xk),…,un,k-1(xk)),us,m(x)=∑i=1m-1‍Ai{s}ψ¯i(x).
Here, we note that, in the iterative process of (17), we can guarantee that the approximations us,m(x) satisfy the periodic boundary conditions (2). Now, the approximate solutions us,mM(x) can be obtained by taking finitely many terms in the series representation of us,m(x) and
(19)us,mM(x)=∑i=1M‍∑k=1i‍βikFs(xk,u1,m-1(xk),u2,m-1(xk),…,un,m-1(xk))ψ¯i(x),mmmmmmmmmmmmmmmmmmls=1,2,…,n.Now, we will proof thatus,m(x) in the iterative formula (17) are converged to the exact solutions us(x) 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.Lemma 12.
If∥us,m(x)-us(x)∥W22→0, xm→y as m→∞, and Fs(x,v1,v2,…,vn) is continuous in [0,1] with respect to x,vi, for x∈[0,1] and vi∈(-∞,∞), then Fs(xm,u1,m-1(xm),u2,m-1(xm),…,un,m-1(xm))→Fs(y,u1(y),u2(y),…,un(y)), s=1,2,…,n as m→∞.Proof.
Firstly, we will prove thatus,m-1(xm)→us(y) in the sense of ∥·∥W22. Since
(20)|us,m-1(xm)-us(y)|=|us,m-1(xm)-us,m-1(y)+us,m-1(y)-us(y)|≤|us,m-1(xm)-us,m-1(y)|+|us,m-1(y)-us(y)|.
By reproducing property of Rx(y), we have us,m-1(xm)=〈us,m-1(x),Rxm(x)〉 and us,m-1(y)=〈us,m-1(x),Ry(x)〉. Thus, |us,m-1(xm)-us,m-1(y)|=|〈us,m-1(x),Rxm(x)-Ry(x)〉W22|≤∥us,m-1(x)∥W22∥Rxm(x)-Ry(x)∥W22. From the symmetry of Rx(y), it follows that ∥Rxm(x)-Ry(x)∥W22→0 as m→∞. Hence, |us,m-1(xm)-us,m-1(y)|→0 as soon as xm→y. On the other hand, by Theorem 10 part (ii), for any y∈[0,1], it holds that |us,m-1(y)-us(y)|→0 as m→∞. Therefore, us,m-1(xm)→us(y) in the sense of ∥·∥W22 as xm→y and m→∞. Thus, by means of the continuation of Fs, it is obtained that Fs(xm,u1,m-1(xm),u2,m-1(xm),…,un,m-1(xm))→Fs(y,u1(y),u2(y),…,un(y)), s=1,2,…,n as m→∞.Lemma 13.
Forj≤m, one has Lus,m(xj)=Lus(xj)=Fs(xj,u1,j-1(xj),u2,j-1(xj),…,un,j-1(xj)), s=1,2,…,n.Proof.
The proof ofLus,m(xj)=Fs(xj,u1,j-1(xj),u2,j-1(xj),…,un,j-1(xj)) will be obtained by induction as follows: if j≤m, then Lus,m(xj)=∑i=1m‍Ai{s}Lψ¯i(xj)=∑i=1m‍Ai{s}〈Lψ¯i(x),φj(x)〉W21=∑i=1m‍Ai{s}〈ψ¯i(x),Lj*φ(x)〉W22=∑i=1m‍Ai{s}〈ψ¯i(x),ψj(x)〉W22. Using the orthogonality of {ψ¯i(x)}i=1∞, it yields that
(21)∑l=1j‍βjlLus,m(xl)=∑i=1m‍Ai{s}〈ψ¯i(x),∑l=1j‍βjlψl(x)〉W22=∑i=1m‍Ai{s}〈ψ¯i(x),ψ¯j(x)〉W22=Aj{s}=∑l=1j‍βjlFs(xl,u1,l-1(xl),u2,l-1(xl),…,un,l-1(xl)).
Now, if j=1, then Lus,m(x1)=Fs(x1,u1,0(x1),u2,0(x1),…,un,0(x1)). Again, if j=2, then β21Lus,m(x1)+β22Lus,m(x2)=β21Fs(x1,u1,0(x1),u2,0(x1),…,un,0(x1))+β22Fs(x2,u1,1(x2),u2,1(x2),…,un,1(x2)). Thus, Lus,m(x2)=Fs(x2,u1,1(x2),u2,1(x2),…,un,1(x2)). Indeed, it is easy to see by using mathematical induction that Lus,m(xj)=Fs(xj,u1,j-1(xj),u2,j-1(xj),…,un,j-1(xj)), s=1,2,…,n. But on the other hand, from Theorem 10, us,m(x) converge uniformly to us(x). It follows that, on taking limits in (17), us(x)=∑i=1∞‍Ai{s}ψ¯i(x). Therefore, us,m(x)=Pmus(x), where Pm is an orthogonal projector from the space W22[0,1] to Span {ψ1,ψ2,…,ψm}. Thus,
(22)Lus,m(xj)=〈Lus,m(x),φj(x)〉W21=〈us,m(x),Lj*φ(x)〉W22=〈Pmus(x),ψj(x)〉W22=〈us(x),Pmψj(x)〉W22=〈us(x),ψj(x)〉W22=〈Lus(x),φj(x)〉W21=Lus(xj),
as j≤m and s=1,2,…,n.Theorem 14.
If∥us,m∥W22 is bounded and {xi}i=1∞ is dense on [0,1], then the m-term approximate solutions us,m(x) in the iterative formula (17) converge to the exact solutions us(x) of (9) in the space W22[0,1] and us(x)=∑i=1∞‍Ai{s}ψ¯i(x), s=1,2,…,n, where Ai{s} is given by (18).Proof.
The proof consists of the following three steps. Firstly, we will prove that the sequence{us,m}m=1∞ in (17) is monotone increasing in the sense of ∥·∥W22. By Theorem 8, {ψ¯i}i=1∞ is the complete orthonormal system in the space W22[0,1]. Hence, we have ∥us,m∥W222=〈us,m(x),us,m(x)〉W22=〈∑i=1m‍Ai{s}ψ¯i(x),∑i=1m‍Ai{s}ψ¯i(x)〉W22=∑i=1m‍(Ai{s})2. Therefore, ∥us,m∥W22, s=1,2,…,n, is monotone increasing. Secondly, we will prove the convergence of us,m(x). From (17), we have us,m+1(x)=us,m(x)+Am+1{s}ψ¯m+1(x). From the orthogonality of {ψ¯i(x)}i=1∞, it follows that ∥us,m+1∥W222=∥us,m∥W222+(Am+1{s})2=∥us,m-1∥W222+(Am{s})2+(Am+1{s})2=⋯=∥us,0∥W222+∑i=1m+1‍(Ai{s})2. Since, the sequence {us,m}m=1∞ is monotone increasing in the sense of ∥·∥W22. Due to the condition that ∥us,m∥W22 is bounded, ∥us,m∥W22 is convergent as m→∞. Then, there exist constants c{s} such that ∑i=1∞‍(Ai{s})2=c{s}. It implies that Ai{s}=∑k=1i‍βikFs(xk,u1,k-1(xk),u2,k-1(xk),…,un,k-1(xk))∈l2, i=1,2,…. On the other hand, since (us,m-us,m-1)⊥(us,m-1-us,m-2)⊥⋯⊥(us,m+1-us,m) it follows for l>m that
(23)∥us,l(x)-us,m(x)∥W222=∥us,l(x)-us,l-1(x)+us,l-1(x)-⋯+us,l+1(x)-us,m(x)∥W222=∥us,l(x)-us,l-1(x)∥W222+⋯+∥us,l+1(x)-us,m(x)∥W222.
Furthermore, ∥us,l(x)-us,l-1(x)∥W222=(Al{s})2. Consequently, as l,m→∞, we have ∥us,l(x)-us,m(x)∥W222=∑i=m+1l‍(Ai{s})2→0. Considering the completeness of W22[0,1], there exists us(x)∈W22[0,1] such that us,l(x)→us(x), s=1,2,…,n as l→∞ in the sense of ∥·∥W22. Thirdly, we will prove that us(x) are the solutions of (9). Since {xi}i=1∞ is dense on [0,1], for any x∈[0,1], there exists subsequence {xmj}j=1∞, such that xmj→x as j→∞. From Lemma 13, it is clear that Lus(xmj)=Fs(xmj,u1,mj-1(xk),u2,mj-1(xk),…,un,mj-1(xk)). Hence, let j→∞; by Lemma 12 and the continuity of Fs, we have Lus(x)=Fs(x,u1(x),u2(x),…,un(x)). That is, us(x) satisfies (1). Also, since ψ¯i(x)∈W22[0,1], clearly, us(x) satisfies the periodic boundary conditions (2). In other words, us(x) are the solutions of (1) and (2), where us(x)=∑i=1∞‍Ai{s}ψ¯i(x) and Ai{s} are given by (18). The proof is complete.According to the internal structure of the present method, it is obvious that if we letus(x) denote the exact solutions of (9), us,m(x) denote the approximate solutions obtained by the RKHS method as given by (17), and rm{s}(x) denote the difference between us,m(x) and us(x), where x∈[0,1] and s=1,2,…,n, then ∥rm{s}(x)∥W222=∥us(x)-us,m(x)∥W222=∥∑i=m+1∞‍Ai{s}ψ¯i(x)∥W222=∑i=m+1∞‍(Ai{s})2 and ∥rm-1{s}(x)∥W222=∑i=m∞‍(Ai{s})2 or ∥rm{s}(x)∥W22≤∥rm-1{s}(x)∥W22. Consequently, this shows the following theorem.Theorem 15.
The differencerm{s}(x), s=1,2,…,n, is monotone decreasing in the sense of the norm of W22[0,1].


## 5. 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 ofm and M. 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.Example 1.
Consider the following first-order nonlinear differential system:(24)u1′(x)-u1(x)+(u2(x))3=f1(x),u2′(x)-sinh⁡(u1(x))u2(x)=f2(x),f1(x)=(x-1)(cos⁡x-sinx)+sinx+e3x(x-1),f2(x)=(sinh⁡(sin(x)(1-x))+2x-1)ex(x-1),
subject to the periodic boundary conditions
(25)u1(0)=u1(1),u2(0)=u2(1).
The exact solutions are u1(x)=(x-1)sin(x) and u2(x)=ex(x-1).Using RKHS method, takexi=(i-1)/(M-1), i=1,2,…,M, on [0,1]. The numerical results at some selected grid points for M=101 and m=3 are given in Tables 1 and 2 for the dependent variables u1(x) and u2(x), respectively.Table 1
Numerical results ofu1(x) for Example 1.
x
Exact solution
Approximate solution
Absolute error
Relative error
0.16
-
0.133827
-
0.13382630119666272
9.92359
×
1
0
-
7
7.41522
×
1
0
-
6
0.32
-
0.213905
-
0.21390423277976867
1.02844
×
1
0
-
6
4.80792
×
1
0
-
6
0.48
-
0.240125
-
0.24012413380235342
1.03748
×
1
0
-
6
4.32058
×
1
0
-
6
0.64
-
0.214990
-
0.21498933621279104
1.02268
×
1
0
-
6
4.75685
×
1
0
-
6
0.80
-
0.143471
-
0.14347022966680445
9.88513
×
1
0
-
7
6.88997
×
1
0
-
6
0.96
-
0.032768
-
0.03276672205464815
9.40677
×
1
0
-
7
2.87075
×
1
0
-
5Table 2
Numerical results ofu2(x) for Example 1.
x
Exact solution
Approximate solution
Absolute error
Relative error
0.16
0.874240
0.8742398572490666
4.41286
×
1
0
-
7
5.04765
×
1
0
-
7
0.32
0.804447
0.8044464859485744
6.70233
×
1
0
-
7
8.33160
×
1
0
-
7
0.48
0.779112
0.7791116154224935
7.50275
×
1
0
-
7
9.62986
×
1
0
-
7
0.64
0.794216
0.7942151498560056
7.02761
×
1
0
-
7
8.84848
×
1
0
-
7
0.80
0.852144
0.8521432738935479
5.15073
×
1
0
-
7
6.04443
×
1
0
-
7
0.96
0.962328
0.9623277968729329
1.35849
×
1
0
-
7
1.41167
×
1
0
-
7The 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 u1′(x) and u2′(x), where xi=(i-1)/(M-1), i=1,2,…,M, M=101, and m=3, are given in Table 3.Table 3
Absolute error of approximatingu1′(x) and u2′(x) for Example 1.
Derivative
x
=
0.16
x
=
0.48
x
=
0.64
x
=
0.96
u
1
′
(
x
)
3.96943
×
1
0
-
6
4.14991
×
1
0
-
6
4.09071
×
1
0
-
6
3.7627
×
1
0
-
6
u
2
′
(
x
)
8.88178
×
1
0
-
7
3.15362
×
1
0
-
6
1.11022
×
1
0
-
6
2.10942
×
1
0
-
7Example 2.
Consider the following first-order nonlinear differential system:(26)u1′(x)+u1(x)+1u2(x)=f1(x),v2′(x)-u1(x)(u2(x))2+(u2(x))2=f2(x),f1(x)=(x4-2x3+x2+1)-1/2+4x3-6x2+2x,f2(x)=-x4+2x3-5x2+2x-1(x4-2x3+x2+1)2,
subject to the periodic boundary conditions
(27)u1(0)=u1(1),u2(0)=u2(1).
The exact solutions are u1(x)=(x(x-1))2 and u2(x)=1/((x(x-1))2+1).Using RKHS method, takexi=(i-1)/(M-1), i=1,2,…,M, on [0,1]. The numerical results at some selected grid points for M=101 and m=3 are given in Tables 4 and 5 for the dependent variables u1(x) and u2(x), respectively.Table 4
Numerical results ofu1(x) for Example 2.
x
Exact solution
Approximate solution
Absolute error
Relative error
0.16
0.0180634
0.01806252000000006
8.39999
×
1
0
-
7
4.65029
×
1
0
-
5
0.32
0.0473498
0.04734840000000003
1.35999
×
1
0
-
6
2.87224
×
1
0
-
5
0.48
0.0623002
0.06229859999999997
1.56000
×
1
0
-
6
2.50401
×
1
0
-
5
0.64
0.0530842
0.05308272000000007
1.43999
×
1
0
-
6
2.71267
×
1
0
-
5
0.80
0.0256000
0.02559900000000005
9.99999
×
1
0
-
7
3.90625
×
1
0
-
5
0.96
0.0014746
0.00147432000000012
2.39999
×
1
0
-
7
1.62760
×
1
0
-
4Table 5
Numerical results ofu2(x) for Example 2.
x
Exact solution
Approximate solution
Absolute error
Relative error
0.16
0.982257
0.982258015821409
8.80077
×
1
0
-
7
8.95974
×
1
0
-
7
0.32
0.954791
0.954792235061675
1.35412
×
1
0
-
6
1.41824
×
1
0
-
6
0.48
0.941354
0.941355026122341
1.50133
×
1
0
-
6
1.59487
×
1
0
-
6
0.64
0.949592
0.949593136843461
1.41570
×
1
0
-
6
1.49085
×
1
0
-
6
0.80
0.975039
0.975040039211652
1.03765
×
1
0
-
6
1.06422
×
1
0
-
6
0.96
0.998528
0.998527858662958
2.47537
×
1
0
-
7
2.47902
×
1
0
-
7Example 3.
Consider the following first-order nonlinear differential system:(28)u1′(x)+u3(x)eu1(x)+(u2(x))2=f1(x),u2′(x)-u2(x)e-u1(x)+(u3(x))2=f2(x),u3′(x)-u1(x)u2(x)u3(x)=f3(x),f1(x)=(cosh⁡(x(x-1))+x(x-1))(x(x-1)+1)+e-2x(x-1)+2x-1x(x-1)+1,f2(x)=(cosh⁡(x(x-1))+x(x-1))2-e-x(x-1)(2x-1)-e-x(x-1)x(x-1)+1,f3(x)=sinh⁡(x(x-1))(2x-1)-(cosh⁡(x(x-1))+x(x-1))×ln⁡(x(x-1)+1)e-x(x-1)+2x-1,
subject to the periodic boundary conditions
(29)u1(0)=u1(1),u2(0)=u2(1),u3(0)=u3(1).
The exact solutions are u1(x)=ln⁡(x(x-1)+1), u2(x)=ex(1-x), and u3(x)=x(x-1)+cosh⁡(x(x-1)).Using RKHS method, takexi=(i-1)/(M-1), i=1,2,…,M, on [0,1]. The numerical results at some selected grid points for M=101 and m=3 are given in Tables 6, 7, and 8 for the dependent variables u1(x), u2(x), and u3(x), respectively.Table 6
Numerical results ofu1(x) for Example 3.
x
Exact solution
Approximate solution
Absolute error
Relative error
0.16
-
0.144332
-
0.1443317288548306
6.42035
×
1
0
-
7
4.44831
×
1
0
-
6
0.32
-
0.245389
-
0.2453879153282803
1.24493
×
1
0
-
6
5.07330
×
1
0
-
6
0.48
-
0.287149
-
0.2871473420226861
1.53927
×
1
0
-
6
5.36052
×
1
0
-
6
0.64
-
0.261884
-
0.2618830218641385
1.35777
×
1
0
-
6
5.18460
×
1
0
-
6
0.80
-
0.174353
-
0.1743525808946390
8.06250
×
1
0
-
7
4.62423
×
1
0
-
6
0.96
-
0.0391567
-
0.0391565628696831
1.52332
×
1
0
-
7
3.89030
×
1
0
-
6Table 7
Numerical results ofu2(x) for Example 3.
x
Exact solution
Approximate solution
Absolute error
Relative error
0.16
1.14385
1.143849565677054
7.02579
×
1
0
-
7
6.14223
×
1
0
-
7
0.32
1.24309
1.243088505342329
1.22692
×
1
0
-
6
9.86995
×
1
0
-
7
0.48
1.28351
1.283510460180459
1.44905
×
1
0
-
6
1.12897
×
1
0
-
6
0.64
1.25910
1.259102236286934
1.31435
×
1
0
-
6
1.04388
×
1
0
-
6
0.80
1.17351
1.173510014918347
8.56073
×
1
0
-
7
7.29498
×
1
0
-
7
0.96
1.03915
1.039146624444131
1.84037
×
1
0
-
7
1.77104
×
1
0
-
7Table 8
Numerical results ofu3(x) for Example 3.
x
Exact solution
Approximate solution
Absolute error
Relative error
0.16
0.874645
0.8746445398520759
7.43544
×
1
0
-
7
8.50109
×
1
0
-
7
0.32
0.806168
0.8061672349997103
1.20922
×
1
0
-
6
1.49996
×
1
0
-
6
0.48
0.781712
0.7817107460286419
1.39143
×
1
0
-
6
1.77998
×
1
0
-
6
0.64
0.796259
0.7962584197762563
1.28185
×
1
0
-
6
1.60984
×
1
0
-
6
0.80
0.852827
0.8528264441003238
8.85879
×
1
0
-
7
1.03876
×
1
0
-
6
0.96
0.962337
0.9623371578286029
2.12773
×
1
0
-
7
2.21100
×
1
0
-
7From 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.


## 6. Conclusions

Here, we use the RKHS method to solve systems of first-order periodic BVPs. The solutions were calculated in the form of a convergent series in the spaceW22[0,1] with easily computable components. In the proposed method, the m-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 W22[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.



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*Source: 135465-2014-03-25.xml*