Abstract

It is attempted to provide the stability and convergence analysis of the reproducing kernel space method for solving the Duffing equation with with boundary integral conditions. We will prove that the reproducing space method is stable. Moreover, after introducing the method, it is shown that it has convergence order two.

1. Introduction

Reproducing kernel space method is a very powerful method for solving linear and nonlinear equation such as initial or boundary differential equation and integral equations [1–3]. This technique has been used not only for well-posed problems [4–6], but also for ill-posed problems [7]. In other words, the flexibility of choosing some tools in dealing with the given equation can be considered as the main reason for designing the solution method. Based on these features, one can use reproducing kernel space method efficiently to approximate the solution in any accuracy. In addition, it should be noted that, in fact, the applications of reproducing kernel Hilbert space method in the numerical analysis field are not new and on the other side possessing some of the well-known advantages; for example [8–10],(i)it is accurate, with needless effort to achieve the results,(ii)it is possible to pick any point in the interval of integration and as well the approximate solutions and their derivatives will be applicable,(iii)the method does not require discretization of the variables, and it is not affected by computation round off errors and one is not faced with necessity of large computer memory and time,(iv)it is of global nature in terms of the solutions obtained as well as its ability to solve other mathematical, physical, and engineering problems.

Duffing equation springs from modeling some different branches of sciences and engineerings such as chemical engineering, thermoelasticity, periodic orbit extraction, nonlinear mechanical oscillators, and prediction of diseases [11–13]. To solve this equation, some variants of it have been investigated in recent years. One of them is due to Du and Cui, who applied an efficient method based on reproducing kernel space method (RKSM) [14]. Indeed, this technique is of great importance in solving linear and nonlinear equations [1]. Du and Cui used RKSM for solving the forced Duffing equation with boundary conditions [14] given bywhere , are continuous functions and , are nonnegative constants.

To approximate the solution of the forced Duffing equation (1), however, based on our best knowledge, accuracy and stability have not been studied yet. In this work, it is attempted to study these issues. The rest of this paper is organised as follows.

Section 2 concerns reviewing some preliminaries. In Section 3, accuracy, convergence order, and stability are established. We confine ourselves to reporting the numerical implementation since they have been carried out in [14].

2. Preliminaries

In this section, we recall some basics which have been taken from [1]. We start with recalling the definition of where is a positive integer. This space is the core of RKSM.

Definition 1. One hasThe inner product and norm in are defined, respectively, bywhere .

Also we need the following.

Definition 2. is an absolutely continuous real function; .

The inner product and norm in are defined as mentioned above for any .

Definition 3 (reproducing kernel space, reproducing kernel). The function space is called a reproducing kernel space ifMoreover, , or , is called the reproducing kernel.

Theorem 4 (see [15]). The reproducing kernel in is conjugate symmetric; that is, . It is also unique. Moreover, , for each , and if and only if .

It has been proven that the reproducing kernel space is a complete space. Furthermore, for instance, the reproducing kernel of and is given [1], respectively, by

3. Accuracy and Convergence Analysis

Here, we study the convergence order of the RKSM for solving (1). We will prove that this technique has convergence order two. Let , where ; then, (1) can be written as follows:where and . Therefore, is a linear and bounded operator on interval .

To apply the RKSM, first of all, an orthogonal system of functions is constructed. Let , and then , where is the conjugate operator of . Consequently, because of the properties of the reproducing kernel, we have the following.

Lemma 5. One has .

Proof. Consider

If is dense on , then is a complete system of and [1]. Applying the well-known Gram-Schmidt process, an orthonormal system, for example, in , is generated from bywhere are the orthogonalization coefficients, .

According to [14], we have the following solution method.

Theorem 6 (see [14]). If is dense on , and is the solution of (1), thenwhere .

It is worth nothing that when is nonlinear, this method can not be used directly in action. Therefore, an iterative modified version of it has been introduced as follows.

Theorem 7 (see [14]). If is dense on , is given, and is the solution of (1), thenwhere .

To obtain the approximate solution , a proper truncated series of is used bywhere or are given as before.

The main contribution of [14] says that, under the given conditions (see Theorems 3.1 and  3.2 in [14]), the approximate solution converges to the exact solution . Nevertheless, applying the numerical results by Du and Cui in [14], they converge quadratically. Surprisingly, this fact has been neither stated nor proved already. So, we state and prove it formally here. First, we need the following lemma.

Lemma 8. Let the conditions of the Theorem 6 be held. Moreover, suppose that is independent of . Then,

Proof. Because of the properties of reproducing kernel definition and assumptions, we haveUsing this relation with orthonormality and definition of , we haveOn the other hand, based on the definition of in Theorem 6 and the assumption that is independent of , we have . It is now sufficient to equate right-hand sides of these two relations for definition of , when varies. Then, the proof is complete.

In what follows, we provide a priori and a posteriori error estimations.

Theorem 9. Suppose that and are the approximate and the exact solution of (1), generated by RKSM in Theorem 6, , and . If , , , and , thenwhere .

Proof. By Lemma 10, we have . If interpolates at nodes and , then . Therefore,where is between and . Thus, we have , for some constant and . This completes the first assertion. Furthermore, since is a bounded linear operator, it is invertible, and, therefore, and the second estimation follows.

Very similar to the above argument, we have the following.

Lemma 10. Let the conditions of Theorem 7 be held. Then,

Similar to Theorem 9, we can conclude the following.

Theorem 11. Suppose that and are the approximate and the exact solution of (1), generated by RKSM in Theorem 7, , and . If , , and , thenwhere .

Now, we deal with the stability of RKHS method for the solution of , where the operator is given in (6). For this purpose, suppose that the right-hand side has perturbation. We indicate variation of the approximate solution is bounded by a constant multiple of . In other words, approximate solution depends continuously on the right-hand side. We need the following.

Lemma 12 (see [15]). If , then there is a constant such that

Theorem 13. Consider the problem , which has a unique solution, and is bounded linear. Then, the approximate solution obtained from RKHS method (9) is stable.

Proof. Suppose that is the approximate solution of the abovementioned equation obtained from RKHS method; that is,where , , and are orthonormal bases and coefficient obtained from Gram-Schmidt orthogonalization process. Moreover, suppose that is the approximate solution of , where and is bounded. We prove that there exists constant such that . According to the definition of and , we haveOn the other hand, exists and . Therefore,Since the right-hand sides of relations (21) and (22) are equal, then Since is continuous on , it is bounded and we have Hence, with , we conclude that . Based on Lemma 12, and therefore .

Similarly, we have the following theorem.

Theorem 14. Consider the problem , which has a unique solution, and is bounded and linear. Then, the approximate solution obtained from RKHS method (10) is stable.

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

Also, the authors acknowledge Hamedan Branch of Isalmic Azad University for their support during conducting this research.