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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 959346, 7 pages
A Novel Approach to Calculation of Reproducing Kernel on Infinite Interval and Applications to Boundary Value Problems
1School of Mathematics and Sciences, Harbin Normal University, Harbin 150025, China
2Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
Received 23 January 2013; Revised 4 July 2013; Accepted 18 August 2013
Academic Editor: Yong Hong Wu
Copyright © 2013 Jing Niu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A new analytical method for the computation of reproducing kernel is proposed and tested on some examples. The expression of reproducing kernel on infinite interval is obtained concisely in polynomial form for the first time. Furthermore, as a particular effective application of this method, we give an explicit representation formula for calculation of reproducing kernel in reproducing kernel space with boundary value conditions.
It is well known that reproducing kernel theory has been used in many research fields such as complex analysis, dilation of linear operators, stochastic processes [1–5], and solution of various differential and integral equations [6–10]. Recently, there are a rather small number of methods for calculating the reproducing kernel expression. One is using Green’s function for a differential operator to construct a reproducing kernel [11, 12]. Another very standard method involves boundary value conditions depending on the property of function [13, 14]. The disadvantages of the above two approaches are clear. In the first method, the expression of reproducing kernel including integral form is complicated as a result of Green’s function. Because function is highly abstractive in the second approach, it is quite difficult to calculate the kernel in the procedure.
The purpose of this paper is to avoid the complex operation of Green’s function and function and simply give representation of reproducing kernel in polynomial form. The principal step of the procedure consists of classifying and discussing the infinite interval to satisfy the reproducing property. Therefore, our approach has the advantages that no additional condition is required in order to solve the kernel and a much simpler formalism which is in contrast to the previous two methods. In effect, the universal formula can be obtained in the case of finite interval.
Numerically solving an initial and boundary value problem for a differential equation by the reproducing kernel method can be described as follows: construct reproducing kernel spaces which can absorb initial or boundary value conditions, and then, transfer the initial and boundary value problem into an operator equation in the reproducing kernel space where the exact solution to the initial and boundary value problem is expressed by the reproducing kernel, and at last solve the operator equation by approximation. It is obvious that constructing reproducing kernel space which satisfies the initial or boundary conditions and effectively solving for the reproducing kernel become the key to apply reproducing kernel method for initial and boundary value problems.
In this work, in order to apply the new approach to solving differential equations with multiform boundary value problems, the explicit formula for calculation of reproducing kernel in the appropriate reproducing kernel space is provided successfully by using the orthogonal decomposition property.
The rest of the paper is organized as follows. In Section 2, a new reproducing kernel space on infinite interval is presented. Section 3 shows for representation of class one how reproducing kernel can be expressed in polynomial form and gives some examples. Then, these basic ideas are shown to extend to cases involving reproducing kernel space with boundary value condition in Section 4. Finally, in Section 5, we give a brief conclusion and discuss extensions and generalizations of the present work.
2. A New Reproducing Kernel Space on Infinite Interval
Definition 1 (see ). Let be a Hilbert function space on a set . is called a reproducing kernel space if and only if for any , there exists a unique function , such that for any . Meanwhile, is called a reproducing kernel function.
Definition 2. . The inner product and norm are given, respectively, by
Theorem 3. is a Hilbert reproducing kernel space; namely, it is complete, and for any fixed and , there exists a , such that .
Proof. Suppose that is a Cauchy sequence in . Then it holds that
Therefore, we have which indicate that the sequences and are Cauchy sequences, respectively, in and . So, there exist unique real number and function , such that
One easily sees that
Because , we get . Moreover, it holds which means that is a Hilbert space.
Meanwhile, we can introduce that where and are positive numbers.
Then, we also introduce that . This completes the proof of Theorem 3.
3. Calculation of Reproducing Kernel on Infinite Interval
In this section, we will give a novel method to calculate the reproducing kernel of . Suppose is the reproducing kernel function. According to the definition of inner, for any fixed , we have
According to the formula it follows that if then which indicates that is the reproducing kernel function.
It is now clear how to compute by solving (11). Moreover, the resulting function is represented locally by polynomials.
Example 4. endowed inner product
By applying the procedure of the previous statement, we can construct the equations:
Then, the reproducing kernel of is given by the following.(1)When (2)When , .(3)When
It can be obviously obtained that for any , we have
Example 5. endowed inner product
Similarly, the of satisfying the following: can be obtained; namely,
4. A Concrete Application to Boundary Value Problems
4.1. Application to Initial Value Problems
Actually, when we apply reproducing kernel theory to solve problems with boundary value conditions [16, 17], it is important to find the representation of reproducing kernel in the appropriate reproducing kernel space. In this section, we show how to express reproducing kernel function in terms of reproducing kernel space with boundary value condition on infinite interval.
Set . Let be a reproducing kernel Hilbert space and its reproducing kernel function. As is well known, is the closed subspace of .
Lemma 6. If is the proper subspace of , then for any ; namely, .
Proof. The proof is by contradiction. Suppose that . Then for a fixed it holds . Choosing , we have which infers . It is conflict about proper subspace.
Lemma 7. The orthogonal complement space of about is
Proof. Due to the only restriction of , it means that is one dimensional space. Moreover, for any we get Then, it indicates . Therefore, we have , and satisfies (22).
By virtue of the orthogonal decomposition property of reproducing kernel, the reproducing kernel function of can be represented as
So under the assumption that , we can introduce
Here, according to Lemma 6, we know that .
Now, the formula of reproducing kernel function of can be obtained by
4.2. Application to Multipoint Boundary Value Problem
To see that the previous results can be generalized, let be a reproducing kernel space with kernel . For any finite point set , we get which is the closed subspace of . Because, for any , we know that , where is dimensional orthogonal complement space of .
Lemma 9. If, for any and differential points , the valuation functional of reproducing kernel space are linearly independent, then the reproducing kernel function is positive definite.
Proof. Suppose that is not positive definite. Then there exist differential points and nonzero vector such that
It follows . Therefore, for any , we have
It indicates that are linearly dependent, which can be a contradiction.
Theorem 10. The reproducing kernel function of has the form
Proof. It needs to be proven that is the basis of ; namely, are linearly independent.
Let . Then . Applying Lemma 9, we get that matrix is positive definite. So, , and formula (32) holds. Meanwhile, according to , we can obtain the uniquely defined .
Example 11. Here we apply the proposed method to solve the following simplified model of propagation of nonadiabatic flames inside long tubes [18–22]:
Firstly, due to the complex boundary conditions of (33), the following reproducing kernel space is constructed.
. The inner product is defined, respectively, by
Suppose that is the reproducing kernel of . According to the previous method for computation of reproducing kernel, we can get
For , the concrete expression of is given as follows:
Secondly, we need to establish the subspace of :
Now, we use the formula (32) to get the reproducing kernel of : where and is the reproducing kernel function of .
Thirdly, we will prove that in (38) is the reproducing kernel of . For any , there holds that
For , and , the concrete expression of is given as follows
Compared with the procedure for computation of the reproducing kernel in , we can see that our method is easier to implement, and it avoids the complexity of function.
5. Conclusions and Future Work
To summarize, in this paper, a new method for the calculation of reproducing kernel on infinite interval was introduced, and the representation in polynomial form was obtained for the first time. The scheme was then used to generate the formula for the reproducing kernel in reproducing kernel space with boundary value conditions. We end this paper by mentioning the following applications. On one hand, the approach detailed here can be readily adapted to the case of reproducing kernel on the finite interval. According to the former method in , which cannot represent reproducing kernel on the infinite interval in polynomial form, the advantages of the present approach are that we use theory of elementary to avoid the complex operation and numerical algorithm will be much more timesaving. On the other hand, the formula in Section 4 can be used to solve multipoint boundary value problems on the positive half-line, such as in [23–25].
The authors appreciate the constructive comments and suggestions provided from the kind referees and editor. This work was supported by Academic Foundation for Youth of Harbin Normal University (KGB201226).
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