Abstract
This 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.
1. 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 byin which and represent the nonlinear terms, for , and . are given functions defined in .
Concerning the existence and uniqueness of solution of problem (1), we refer to [1–3]. 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 [7–9], Peng et al. proposed an alternative iterative method called symplectic method to solve nonlinear BVPs. Furthermore, methods such as sinc-collocation 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 [14–21]. 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 well-known Lobatto orthogonal polynomials. NDM converges fast and LRKM avoids the Schmidt orthogonalization process and uses very few points. Therefore, QDM-LRKM is efficient and simple to implement.
The article is built up as follows. Several fundamental definitions about Fréchet derivative and Hilbert spaces are recalled in Section 2. The 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.
2. 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 . Actually, by a scaling from to , that is, , we can derive a system defined on as follows:where , , , , , , , , , and .
2.1. Several Reproducing Kernel Spaces and Hilbert Spaces
Denote by the space of polynomial functions with degrees no more than n. Let and be the well-known Legendre and Lobatto polynomials, respectively. To be more precise,satisfying the orthogonality condition on , which means
Furthermore, Lobatto polynomials are defined by
Clearly, Lobatto polynomials satisfy
Let and be the standard linear inner product space of polynomials with degrees not exceeding n, namely,and the norm
From Theorem 3.7 of [22], we obtain that the inner space is a Hilbert space as is a finite dimension closed subspace of .
Definition 1. Let be the standard orthonormal basis functions obtained by the Gram–Schmidt process of .
Lemma 1. is a reproducing kernel space with kernel function
Definition 2. We equip the following inner product and norm for :Under such definitions, we obtain that is a Hilbert space. Similarly, we can define a subspace of ,
2.2. Fréchet Derivative
Let X and Y be Banach spaces with norms and , respectively. denotes an operator from X to Y.
Definition 3. (see [23]). We say that the linear operator is the Fréchet derivative of at ifand is denoted by .
Let , , and , where and are also Banach spaces with norms and .
Definition 4. We say that the linear operator is the partial Fréchet derivative of on u at ifand is denoted by .
Similarly, we can define the partial Fréchet derivative of at with respect to by .
3. QNM-SRKM for Solving Equation (2)
In this section, we present the QNM-SRKM to solve the nonlinear system (2).
3.1. Analysis of QNM
By equation (1), we define as
Let
One can correspondingly rewrite equation (2) into the following operator form:where and .
Theorem 1. The Fréchet derivative of at is
To show Fréchet derivative of the nonlinear operator more clearly, we introduce two examples.
Example 1. Let , then , where is identity operator.
Example 2. Let , then .
Now, we will introduce the QNM to linearize the operator equation (18).
The Quasi-Newton’s Scheme: Finding such that
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. is linear and bounded.
Proof. Obviously, , and are linear operators. For , we have
The boundedness of results in the boundedness of the operator .
Letwherein are different points in , is the conjugate operator of , and R is the reproducing kernel of .
Lemma 3. .
Proof. From the reproducibility of R, we immediately deduce that
Similarly, we have .
Lemma 4. are linearly independent in .
Considering the numerical implement, we apply the SRKM proposed in [12, 13] to solve equation (21). We first denote a finite-dimensional subspace by
Let , where is the projection operator. Then, converges to in the sense of , which means that
As , there exits undetermined coefficients such that
The LRKM reads as follows: Finding such that
Once is obtained, one can get an approximate solution of by applying scaling from to :
Next, we will discuss the convergence of LRKM.
Theorem 2. and uniformly converge to and on as , respectively.
Proof
From the continuity of K, we have . Notice that as , and we conclude that
Analogously, . The proof is completed.
4. Numerical Experiments
Example 3. Let us consider a linear problem suggested in [5, 10]:Here, The exact solution is . Applying the LRKM, absolute errors are presented in Table 1 and Figure 1. Besides, we compare our method with RKM [5] and the sinc-collocation 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.
(a)
(b)
Example 4. Let us consider a nonlinear system proposed in [5, 10]:where and . The exact solution is . Selecting initial functions , as the polynomial that satisfies the boundary conditions and applying the technique of QNM-LRKM three times, we depict numerical results of and 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 in LRKM (28) and different iteration steps k, and absolute errors are shown in Table 5. Numerical results indicate that the convergence order of NDM is nearly 2.
(a)
(b)
5. Conclusions
This paper presents an efficient numerical method for solving second-order nonlinear system of BVPs. The 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
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by College Innovative Talent Foundation of Heilongjiang Province, China (Grant no. UNPYSCT-2017179).