A High-Order Numerical Method for a Nonlinear System of Second-Order Boundary Value Problems

*is paper is concerned with a high-order numerical scheme for nonlinear systems of second-order boundary value problems (BVPs). First, by utilizing quasi-Newton’s method (QNM), the nonlinear system can be transformed into linear ones. Based on the standard Lobatto orthogonal polynomials, we introduce a high-order Lobatto reproducing kernel method (LRKM) to solve these linear equations. Numerical experiments are performed to investigate the reliability and efficiency of the presented method.


Introduction
Nonlinear systems of second-order BVPs are widely used in applied physics, mechanical engineering, biology, etc. In this article, we mainly focus on numerical solutions for such problems. In general, these systems can be described by in which F and G represent the nonlinear terms, a i (x), b i (x) ∈ C[a, b] for i � 1, 2, 3, and f, g. are given functions defined in [a, b].
Concerning the existence and uniqueness of solution of problem (1), we refer to [1][2][3]. Nevertheless, to the best of our knowledge, research studies on numerical methods for nonlinear systems of second-order BVPs were seldom reported. Recently, several techniques for approximate solutions of such systems have been presented. Arqub and Abo-Hammour [4] suggested a continuous genetic algorithm to deal with second-order linear systems. In [5,6], the authors combined homotopy perturbation methods and iterative RKM to solve problem (1). In [7][8][9], Peng et al. proposed an alternative iterative method called symplectic method to solve nonlinear BVPs. Furthermore, methods such as sinccollocation methods and Jacobi matrix methods were also considered in [10,11].
In this work, QNM [12] and simplified reproducing kernel method (SRKM) [13] are combined to design a numerical method for solving nonlinear systems (1). It is worth noting that QNM acts directly on operator equations to linearize nonlinear problems by Fréchet derivative. Here we extend QNM to solve nonlinear systems (1). Many improved versions of RKM have been developed to solve linear second-order BVPs [14][15][16][17][18][19][20][21]. However, the high-order RKM has been seldom discussed. Motivated by designing RKM of high-order accuracy, we establish a reproducing kernel space with polynomial form by applying the wellknown Lobatto orthogonal polynomials. NDM converges fast and LRKM avoids the Schmidt orthogonalization process and uses very few points. erefore, QDM-LRKM is efficient and simple to implement. e article is built up as follows. Several fundamental definitions about Fréchet derivative and Hilbert spaces are recalled in Section 2.
e QNM-LRKM is introduced in Section 3. Some numerical examples validate the robustness and effectiveness in Section 4. Finally, a summary is provided in the last section.

Preliminaries
To simplify our discussion and demonstrate the idea clearly, we rewrite system (1) into a system with homogenous boundary conditions and the domain defined on [− 1, 1]. Actually, by a scaling from [a, b] to [− 1, 1], that is, x � ((b − a)/2)t + ((b + a)/2), we can derive a system defined on [− 1, 1] as follows:

Several Reproducing Kernel Spaces and Hilbert Spaces.
Denote by P n the space of polynomial functions with degrees no more than n. Let P n and L n be the well-known Legendre and Lobatto polynomials, respectively. To be more precise, satisfying the orthogonality condition on [− 1, 1], which means Furthermore, Lobatto polynomials are defined by Clearly, Lobatto polynomials satisfy be the standard linear inner product space of polynomials with degrees not exceeding n, namely, and the norm From eorem 3.7 of [22], we obtain that the inner space Let e i n i�2 be the standard orthonormal basis functions obtained by the Gram-Schmidt process of L i n i�2 .
Definition 2 We equip the following inner product and norm for Mathematical Problems in Engineering Under such definitions, we obtain that W n [− 1, 1] is a Hilbert space. Similarly, we can define a subspace W 0 2.2. Fréchet Derivative. Let X and Y be Banach spaces with norms ‖·‖ X and ‖·‖ Y , respectively. F denotes an operator from X to Y.
Definition 3 (see [23]). We say that the linear operator A is and A is denoted by F ′ (u 0 ).
where X 1 and X 2 are also Banach spaces with norms ‖·‖ X 1 and ‖·‖ X 2 .

Definition 4. We say that the linear operator
and
To show Fréchet derivative of the nonlinear operator more clearly, we introduce two examples.
Now, we will introduce the QNM to linearize the operator equation (18).
e Quasi-Newton's Scheme: 3.2. Analysis of LRKM. By improving the RKM, an effective and simple algorithm is derived to solve the linear system (20). For convenience, we rewrite (20) into the following form: Lemma 2. L is linear and bounded.
Proof. Obviously, L 11 , L 12 , L 21 , and L 22 are linear operators. For ∀ U ∈ W 0 n , we have e boundedness of L ij 2 i,j�1 results in the boundedness of the operator L.

Let
wherein x i n i�1 are different points in [− 1, 1], L * is the conjugate operator of L, and R is the reproducing kernel of Proof. From the reproducibility of R, we immediately deduce that Similarly, Considering the numerical implement, we apply the SRKM proposed in [12,13] to solve equation (21). We first denote a finite-dimensional subspace S 2n by Let U n ≜ P 2n+4 U, where P 2n : W ∞ ⟶ S 2n is the projection operator. en, U n converges to U in the sense of ‖·‖ W n [− 1,1] , which means that As U n ∈ W n [− 1, 1], there exits undetermined coefficients a i1 and a i2 (i � 1, 2 . . . n) such that e LRKM reads as follows: Once U n is obtained, one can get an approximate solution U n of U by applying scaling from [− 1, 1] to [a, b]: Next, we will discuss the convergence of LRKM.
From the continuity of K, we have ‖K‖ W n ≤ M. Notice that ‖U n − U‖ W n ⟶ 0 as n ⟶ ∞, and we conclude that

Numerical Experiments
Example 3. Let us consider a linear problem suggested in [5,10]: e exact solution is u(x) � 2(1 − x)sin(x), v(x) � sin(πx). Applying the LRKM, absolute errors are presented in Table 1 and Figure 1. Besides, we compare our method with RKM [5] and the sinccollocation method [10] in Table 2. Numerical results show that LRKM has high accuracy in solving the linear equation, but uses the least number of points.
Example 4. Let us consider a nonlinear system proposed in [5,10]: e exact solution is u � x − x 2 ; v � sin(πx). Selecting initial functions u 0 , v 0 as the polynomial that satisfies the boundary conditions and applying the technique of QNM-LRKM three times, we depict numerical results of u(x i ) and v(x i ) in Table 3 and Figure 2. Numerical comparison with homotopy analysis and collocation method for problem 2 is shown in Table 4. Numerical results indicate that our method uses fewer nodes but obtains the more accurate numerical solution. We take n � 11 in LRKM (28) and different iteration steps    [5] and the sinc-collocation method [10] for Example 3.   Mathematical Problems in Engineering k, and absolute errors are shown in Table 5. Numerical results indicate that the convergence order of NDM is nearly 2.

Conclusions
is paper presents an efficient numerical method for solving second-order nonlinear system of BVPs. e algorithm is easy to implement. Numerical results verify that the QNM-LRKM is a reliable numerical technique of high-order accuracy. Precisely, NDM ensures the speed of iteration, while LRKM guarantees the accuracy of the algorithm.

Data Availability
No datasets were generated or analyzed during the current study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.