Chebyshev reproducing kernel method: application to two-point boundary value problems
- M Khaleghi^{1}View ORCID ID profile,
- E Babolian^{1} and
- S Abbasbandy^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s13662-017-1089-2
© The Author(s) 2017
Received: 29 September 2016
Accepted: 17 January 2017
Published: 25 January 2017
Abstract
In this paper, a new implementation of the reproducing kernel method is proposed in order to obtain the accurate numerical solution of two-point boundary value problems with Dirichlet boundary conditions. Based on reproducing kernel theory, reproducing kernel functions with polynomial form will be constructed in the reproducing kernel spaces spanned by the Chebyshev basis polynomials. Convergence analysis and an error estimation for the present method in \(L_{w}^{2}\) space is also discussed. The numerical solutions obtained by this method are compared with the exact solutions. The results reveal that the proposed method is quite efficient and accurate.
Keywords
1 Introduction
Boundary value problems (BVPs) associated with different kinds of differential equations play important roles in modeling a wide variety of natural phenomena. Therefore, these problems have attracted much attention and have been fascinating to a number of researchers. Two-point boundary value problems associated with second order differential equations have been investigated in a wide variety of problems in science and engineering. Many approaches for solving ordinary boundary value problems numerically are available [1–12]. Recently, reproducing kernel methods (RKMs) were used to solving a variety of BVPs [13–24].
According to these references, we see that the implementation of the reproducing kernel method for solving a problem consists of four stages.
First, we carefully identify a solution space. An inappropriate choice is an obstacle to achieve the desired solution.
Second, we construct the reproducing kernel function. In all of the above mentioned papers, this function is constructed by solving a boundary value problem and a subsequent linear system of equations. Explicit formulas for two kinds of reproducing kernel functions are introduced in [25].
Third, we produce a set of orthonormal basis functions for the space solution by using the kernel function, a boundary operator, a dense sequence of nodal points in the domain of solution space and Gram-Schmidt orthogonalization process.
Finally, we represent the exact solution of the problem by an infinite sum of orthonormal basis functions achieved from the last stage above. We use a truncated series of the exact solution series by N terms as an approximate solution.
2 Basis functions and polynomial reproducing kernel function
2.1 Basis functions
Lemma 2.1
[26]
Proposition 2.1
Proof
Use Lemma 2.1 and induction on i, which completes the proof. □
2.2 Polynomial reproducing kernel function
Definition 2.1
- 1.
\(R(x,\cdot)\in\mathcal{H}\), \(\forall x\in\mathcal{X}\),
- 2.
\(\langle\varphi(\cdot),R(x,\cdot) \rangle_{\mathcal{H}}=\varphi(x)\), \(\forall\varphi\in\mathcal{H}\), \(\forall x \in\mathcal{X}\) (reproducing property).
Theorem 2.1
[28], Theorem 3.7
- 1.
each bounded sequence in M has a subsequence that converges to a point in M;
- 2.
M is closed;
- 3.
M is complete;
- 4.
suppose \(\{x_{1}, x_{2}, \ldots, x_{n}\}\) is a basis for M, \(y_{k}=\sum_{i=1}^{n} \alpha_{ki}x_{i}\), and \(y=\sum_{1}^{n} \alpha_{i}x_{i}\). Then \(y_{k} \rightarrow y\) if and only if \(\alpha_{ki} \rightarrow\alpha_{i}\) for \(i=1, 2, \ldots, n\).
Theorem 2.2
[13], Theorem 1.1.2
Theorem 2.3
The function space \(\Pi_{w}^{m}[a,b]\) by its inner product and norm (mentioned above) is a reproducing kernel Hilbert space.
Proof
It is clear that \(\Pi_{w}^{m}[a,b]\) is a finite-dimensional inner product space, so by Theorems 2.1 and 2.2, \(\Pi_{w}^{m}[a,b]\) is a RKHS, which completes the proof. □
For practical use of the RKM method, it is necessary to define a closed subspace of \(\Pi_{w}^{m}[a,b]\) by imposing required homogeneous boundary conditions on it.
Definition 2.2
So similar to the proof of Theorem 2.3, by using equation (5), we can prove that the function space \({}^{o}\Pi_{w}^{m}[a,b]\) is a reproducing kernel Hilbert space.
Theorem 2.4
[13], Theorem 1.3.5
The reproducing kernel space \({}^{o}\Pi_{w}^{m}[a,b]\) is a closed subspace of \(\Pi_{w}^{m}[a,b]\).
3 The Chebyshev reproducing kernel method (C-RKM)
3.1 Representation of exact solution in \({}^{o}\Pi_{w}^{m}[a,b]\)
Theorem 3.1
For \(m \geq2\), let \(\{x_{i}\}_{i=0}^{m-2}\) be any \((m-1)\)-distinct points in \((a,b)\), then \(\{\psi_{i}^{m}\}_{i=0}^{m-2}\) is a basis for \({}^{o}\Pi_{w}^{m}[a,b]\).
Proof
Theorem 3.1 shows that, in our method (C-RKM), use of a finite sequence of nodal points is sufficient. So, implementation of C-RKM for solving problems does not need a dense sequence of nodal points.
Theorem 3.2
Proof
Theorem 3.3
[24] If \(u\in{}^{o}\Pi_{w}^{m}[a,b]\), then \(|u(x)| \leq C \|u\|_{{}^{o}\Pi _{w}^{m}}\) and \(|u^{(k)}(x)| \leq C \|u\|_{{}^{o}\Pi_{w}^{m}}\) for \(1\leq k \leq m-1\), where C is a constant.
3.2 Convergence analysis and error estimation in \({}^{o}L_{w}^{2}[a,b]\)
3.2.1 Convergence analysis
Theorem 3.4
Proof
Theorem 3.5
[13], Theorem 1.3.4
If \(u_{m}(x)\) converges to \(u(x)\) in the sense of \(\|\cdot\|_{{}^{o}L_{w}^{2}}\), then \(u_{m}^{(k)}(x)\) converges to \(u^{(k)}(x)\) uniformly for \(0 \leq k \leq m-1\).
3.2.2 Error analysis
Theorem 3.6
Proof
Corollary 3.1
Corollary 3.2
4 Numerical examples
In this section, some numerical examples are considered to illustrate the performance and accuracy of the C-RKM. Results obtained by C-RKM are compared with the exact solution of each example and are found to be in good agreement with each other. In the process of computation, all the symbolic and numerical computations are performed by using Mathematica 10.
Example 4.1
Numerical results for Example 4.1
x | Absolute errors: \(\boldsymbol{|u-u_{m}|}\) | ||||
---|---|---|---|---|---|
m = 2 | m = 4 | m = 6 | m = 8 | m = 9 | |
0.1 | 3.7948E − 03 | 1.9500E − 05 | 4.2625E − 08 | 7.7160E − 11 | 8.3641E − 13 |
0.2 | 3.8926E − 03 | 3.1593E − 06 | 2.3047E − 09 | 4.0493E − 11 | 6.8001E − 13 |
0.3 | 1.3057E − 03 | 8.9693E − 06 | 1.0733E − 08 | 5.8828E − 11 | 7.2092E − 13 |
0.4 | 2.9333E − 03 | 5.7392E − 06 | 1.8736E − 08 | 4.3152E − 11 | 6.5603E − 13 |
0.5 | 7.7603E − 03 | 5.4270E − 06 | 7.8240E − 09 | 5.6790E − 11 | 6.6636E − 13 |
0.6 | 1.2072E − 02 | 9.1774E − 06 | 7.4446E − 09 | 8.4509E − 11 | 5.8942E − 13 |
0.7 | 1.4711E − 02 | 6.7151E − 06 | 2.4443E − 08 | 8.5180E − 11 | 6.4981E − 13 |
0.8 | 1.4458E − 02 | 3.9780E − 05 | 4.6549E − 09 | 8.4175E − 11 | 3.6804E − 13 |
0.9 | 1.0016E − 02 | 6.0631E − 05 | 1.2269E − 07 | 2.7533E − 13 | 2.0268E − 12 |
\(\|\varepsilon_{m}\|_{{}^{o}L_{w}^{2}}\) | 9.6803E − 03 | 3.1956E − 05 | 7.0121E − 08 | 9.1748E − 11 | 1.9859E − 12 |
It is worth noting here that the obtained numerical solution in this example is very accurate, although the number of basis functions in the expansion of the obtained result is very low.
Example 4.2
[7]
Summarizing, RKM is proposed in order to obtain the accurate numerical solution of two-point boundary value problems with the Dirichlet boundary conditions. Chebyshev basis polynomials are used. A convergence analysis is discussed. The numerical solutions obtained by this method are compared with the exact solutions. The results reveal that the proposed method is quite efficient and accurate.
5 Conclusions
In the method of this paper, retaining important property of the reproducing kernel we solved two-point boundary value problems. In fact, with increasing m (the number of nodal points), the solution space and associated reproducing kernel function are improved. Also, the absolute error of the approximate solution is rapidly decreasing with m. This leads to using less nodal points and applying unstable Gram-Schmidt process a moderate number of times.
Declarations
Acknowledgements
The authors would like to thank the referee for valuable comments and suggestions, which improved the paper in its present form.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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