We use the reproducing kernel method (RKM) with interpolation for finding approximate solutions of delay differential equations. Interpolation for delay differential equations has not been used by this method till now. The numerical approximation to the exact solution is computed. The comparison of the results with exact ones is made to confirm the validity and efficiency.

1. Introduction

In this paper we consider delay differential equations in the reproducing kernel space: where and   .

The theory of reproducing kernels [1] was used for the first time at the beginning of the 20th century by S. Zaremba in his work on boundary value problems for harmonic and biharmonic functions. In recent years, a lot of attention has been devoted to the study of RKM to investigate various scientific models. The RKM which accurately computes the series solution is of great interest to applied sciences. The method provides the solution in a rapidly convergent series with components that can be elegantly computed. The book [2] provides excellent overviews of the existing reproducing kernel methods for solving various model problems such as integral and integrodifferential equations.

The efficiency of the method was used by many authors to investigate several scientific applications. Geng and Cui [3] applied the RKM to handle the second-order boundary value problems. Wang et al. [4] investigated a class of singular boundary value problems by this method and the obtained results were good. Zhou et al. [5] used the RKM effectively to solve second-order boundary value problems. In [6], the method was used to solve nonlinear infinite-delay-differential equations. Wang and Chao [7] and Zhou and Cui [8] independently employed the RKM to variable-coefficient partial differential equations. Geng and Cui [9] and Du and Cui [10] researched the approximate solution of the forced Duffing equation with integral boundary conditions by combining the homotopy perturbation method and the RKM. Wu and Li [11] applied iterative reproducing kernel method to obtain the analytical approximate solution of a nonlinear oscillator with discontinuities. Yang et al. [12] used this method for solving the system of the linear Volterra integral equations with variable coefficients. A particular singular integral equation was solved by Du and Shen [13]. Barbieri and Meo [14] have studied evaluation of the integral terms in reproducing kernel methods. Third-order three-point boundary value problems were considered by Wu and Li [15]. Chen and Chen [16] investigated the exact solution of system of linear operator equations in reproducing kernel spaces. Akgül has investigated fractional order boundary value problems by RKM [17]. Inc et al. have solved ordinary and partial differential equations by RKM [1820].

The paper is organized as follows. Section 2 introduces several reproducing kernel spaces. The associated linear operator is presented in Section 3. Section 4 provides the main results. The exact and approximate solutions of problems and an iterative method are developed in the reproducing kernel space in this section. We have proved that the approximate solutions converge to the exact solutions uniformly. Some numerical experiments are illustrated in Section 5. Some conclusions are given in Section 6.

2. Preliminaries

2.1. Reproducing Kernel Spaces

In this section, we define some useful reproducing kernel spaces.

Definition 1 (reproducing kernel function). Let . A function is called areproducing kernel function of the Hilbert space if and only if (a) for all ;(b) for all and all .
The last condition is called “the reproducing property” as the value of the function at the point is reproduced by the inner product of with .

Definition 2. We define the space by The third derivative of exists almost everywhere since is absolutely continuous. The inner product and the norm in are defined by The space is called a reproducing kernel space, as, for each fixed and any , there exists a function such that

Definition 3. We define the space by The inner product and the norm in are defined by The space is a reproducing kernel space, and its reproducing kernel function is given by Cui and Lin [2]:

Lemma 4 (see [21]). The space is a reproducing kernel space, and its reproducing kernel function is given by where and coefficients can be found by Maple 16.

3. Solution Representation in

In this section, the solution of (1) is considered in the reproducing kernel space . On defining the linear operator as model problem (1) takes the form

In (9), since is sufficiently smooth, we see that is a bounded linear operator. For convenience, we write instead of in (10).

Theorem 5. The linear operator defined by (9) is a bounded linear operator.

Proof. We only need to prove , where is a positive constant. By (6), we have By reproducing property, we have so where is a positive constant; thus, Since we have where is a positive constant, so we have that is where is a positive constant. This completes the proof.

4. The Structure of the Solution and the Main Results

From (9), it is clear that is a bounded linear operator. Put and , where is conjugate operator of . The orthonormal system of can be derived from Gram-Schmidt orthogonalization process of :

Theorem 6. Let be dense in and . Then the sequence is a complete system in .

Proof. We have
The subscript by the operator indicates that the operator applies to the function of . Clearly, . For each fixed , let , which means that
Note that is dense in ; hence, . It follows that from the existence of . So the proof of Theorem 6 is completed.

Theorem 7. If is the exact solution of (10), then where is dense in .

Proof. From (19) and uniqueness of solution of (10), we have This completes the proof.

Now the approximate solution can be obtained from the -term intercept of the exact solution and

Lemma 8 (see [22]). If , , , and is continuous for , then

Lemma 9 (see [23]). For any fixed , suppose the following conditions are satisfied:  (i) (ii) is bounded; (iii) is dense in ; (iv) for any . Then in iterative formula (26) converges to the exact solution of (22) in and where is given by (27).

We assume that is dense in . Let be the exact solution of (1) and let be the -term approximation solution of (1). We set

Theorem 10. If , then Moreover, a sequence is monotonically decreasing in .

Proof. From (22) and (24), it follows that Thus, In addition, Clearly, is monotonically decreasing in .

Remark 11. Let us consider countable dense set and define Then coefficients can be found by

4.1. Interpolation for Reproducing Kernel Method

We used interpolation to find the numerical results by RKM with where . More details for interpolation can be found in [24].

5. Numerical Results

In this section, four numerical examples are provided to show the accuracy of the present method. We used interpolation for Examples 1214. The RKM does not require discretization of the variables, that is, time and space; it is not effected by computation round-off errors and one is not faced with necessity of large computer memory and time. The accuracy of the RKM for the delay differential equation is controllable and absolute errors are small with present choice of (see Tables 16). The numerical results we obtained justify the advantage of this methodology.

Example 12. Consider the equation where Thus, if the method described above is applied, then we find Table 1.

Example 13. We take notice of equation We use transformation to obtain Thus, if the method described above is applied, then we find Table 2.

Example 14. We regard the following equation: We use transformation to obtain Thus, if the method described above is applied, then we find Table 3.

Example 15. We consult equation We use transformation to obtain The exact solution of (45) is given as Thus, if the method described above is applied, then we find Tables 4, 5, and 6.

6. Conclusion

In this paper, we introduced an algorithm for finding approximate solutions of delay differential equations with RKM. For illustration purposes, four examples were selected to show the computational accuracy. It may be concluded that the RKM is very powerful and efficient in finding approximate solutions for wide classes of problems. Solutions obtained by the present method are uniformly convergent. As shown in Tables 16, results of numerical examples show that the present method is an accurate and reliable analytical method for these problems. The present study has confirmed that the RKM offers significant advantages in terms of its straightforward applicability, its computational effectiveness, and its accuracy to solve the strongly nonlinear equations.

Conflict of Interests

The authors declare that they do not have any competing interests or conflict of interests.