Chebyshev spectral collocation method for stochastic delay differential equations
- Zhengwei Yin^{1, 2} and
- Siqing Gan^{1}Email author
https://doi.org/10.1186/s13662-015-0447-1
© Yin and Gan; licensee Springer. 2015
Received: 10 December 2014
Accepted: 19 March 2015
Published: 8 April 2015
Abstract
The purpose of the paper is to propose the Chebyshev spectral collocation method to solve a certain type of stochastic delay differential equations. Based on a spectral collocation method, the scheme is constructed by applying the differentiation matrix \(D_{N}\) to approximate the differential operator \(\frac{d}{dt}\). \(D_{N}\) is obtained by taking the derivative of the interpolation polynomial \(P_{N}(t)\), which is interpolated by choosing the first kind of Chebyshev-Gauss-Lobatto points. Finally, numerical experiments are reported to show the accuracy and effectiveness of the method.
Keywords
spectral collocation method stochastic delay differential equations Lamperti-type transformation Chebyshev-Gauss-Lobatto nodes1 Introduction
Deterministic differential models require that the parameters involved be completely known, though in the original problem, one often has insufficient information on parameter values. These may fluctuate due to some external or internal ‘noise’, which is random. Thus, it is necessary to move from deterministic problems to stochastic problems. Stochastic differential equations (SDEs) play a prominent role in a range of application areas, such as biology, chemistry, epidemiology, mechanics, microelectronics, economics and so on [1, 2]. For SDEs, roughly speaking, there are two major types of numerical methods, explicit numerical methods [3, 4] and implicit numerical methods [5, 6]. One can refer to [7] for an overview of the numerical solution of SDEs.
Our approach is derived by constructing the interpolating polynomial of degree N based on a spectral collocation method and applying the differentiation matrix to approximate the differential operator arising in SDDEs. The interpolating polynomial of degree N is constructed by applying the Chebyshev-Gauss-Lobatto (C-G-L) points as interpolation points and the Lagrange polynomial as a trial function. To the best of our knowledge, they have not been utilized in solving SDDEs. Finally, we would like to mention that the idea of the spectral collocation was previously employed in [13, 14] to construct methods for SODEs and DDEs. The authors in [13] propose a spectral collocation method for SODEs. Inspired by the idea, we construct the Chebyshev spectral collocation method for SDDEs.
This paper is organized as follows. In the next section, some fundamental knowledge is reviewed and the derivation of the Chebyshev spectral collocation for solving SDDEs is introduced. Section 3 is devoted to reporting some numerical experiments to confirm the accuracy and effectiveness of the method. At the end of the article, conclusions are made briefly.
2 Construction of the Chebyshev spectral collocation method
2.1 The Lamperti-type transformation
2.2 Review on Chebyshev interpolation polynomials
Chebyshev polynomials are a well-known family of orthogonal polynomials on the interval \([-1,1]\). These polynomials present, among others, very good properties in the approximation of functions. Therefore, Chebyshev polynomials appear frequently in several fields of mathematics, physics and engineering. In this subsection, we will recall the Chebyshev interpolation polynomial for a given function \(x(t)\in C^{k}(-1,1)\), where \(C^{k}\) is the space of all functions whose k times derivatives are continuous on the interval \((-1,1)\). More details can be found in [16].
Remark 2.1
The differentiation matrix \(D_{N}\) is not dependent on the problem itself but dependent on the C-G-L nodes. Therefore, the differentiation matrices can be obtained before a problem setting.
2.3 Chebyshev spectral collocation method for DDEs
Spectral method is one of the three technologies for numerical solutions of partial differential equations. The other two are finite difference methods (FDMs) and finite element methods (FEMs). The spectral methods based on Chebyshev polynomials as basis functions for solving numerical differential equations [16–18] with smooth coefficients and simple domain have been well applied by many authors. Furthermore, they can often achieve ten digits of accuracy while FDMs and FEMs would get two or three. An interested reader can refer to references [19, 20]. Later, the spectral methods are developed to solve neutral differential equations [14] or special DDEs [18]. In this subsection, we will introduce the spectral collocation method for DDEs.
Remark 2.4
Remark 2.5
2.4 The Chebyshev spectral collocation method for SDDEs
In this subsection, we first give the theorem to guarantee the existence and uniqueness of the exact solution of SDDE (1.2).
Theorem 2.6
Remark 2.7
To find the derivative of function \(x(t)\) on C-G-L nodes, \(D_{N}x\) is of high accuracy only if \(x(t)\) is smooth enough. But the standard Wiener process \(w(t)\) is a nowhere differentiable process, \(D_{N}w\) behaves very badly. However, if the coefficient of diffusion term of SDDEs is a constant, we can avoid \(D_{N}w\) as above (see [13]).
3 Numerical experiments
The theoretical discussion of numerical processes is intended to provide an insight into the performance of numerical methods in practice. Therefore, in this section, some numerical experiments are reported to test the accuracy and the effectiveness of the spectral collocation method.
Example 1
Mean-square errors of the numerical solutions for ( 3.1 )
N | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|
I ϵ | 4.38e−002 | 3.24e−002 | 5.7e−003 | 5.5e−003 | 9.1e−003 |
II ϵ | 3.1e−003 | 1.7349e−05 | 5.5209e−06 | 5.6e−003 | 2.2626e−05 |
We apply the spectral collocation method to solve (3.1) under the set of coefficients I: \(a=-0.9\), \(b=0.1\), \(\beta_{1}=0.1\) and II: \(a=-2\), \(b=0.1\), \(\beta_{1}=1\). The numerical results are shown in Table 1. In Table 1, we denote N by the number of the C-G-L nodes. The approximation errors reported in Table 1 show that the Chebyshev spectral collocation method works very well for SDDEs and has high accuracy and effectiveness.
Example 2
Errors for equation ( 3.3 ) in the case of \(\pmb{a=-0.9}\) , \(\pmb{b=1}\)
N | \(\boldsymbol {\|x-X\|_{L^{\infty}(0,1)}}\) |
---|---|
3 | 2.3e−003 |
4 | 1.082e−004 |
5 | 4.169e−006 |
6 | 1.356e−007 |
7 | 3.831e−009 |
8 | 9.597e−011 |
9 | 2.160e−012 |
10 | 4.430e−014 |
11 | 6.661e−016 |
12 | 4.441e−016 |
Errors for equation ( 3.3 ) in the case of \(\pmb{a=-2}\) , \(\pmb{b=0.1}\)
N | \(\boldsymbol {\|x-X\|_{L^{\infty}(0,1)}}\) |
---|---|
3 | 1.10e−002 |
4 | 1.3e−003 |
5 | 1.160e−004 |
6 | 8.647e−006 |
7 | 5.543e−007 |
8 | 3.129e−8 |
9 | 1.581e−009 |
10 | 7.242e−011 |
11 | 3.034e−012 |
12 | 1.170e−013 |
4 Conclusions
In the paper, the Chebyshev spectral collocation is proposed to solve a certain type of SDDEs. The most important step to construct the scheme is the Lamperti-type transformation. This transformation allows to shift non-linearities from the diffusion coefficient into the drift coefficient. Then, based on the spectral collocation method, we construct the Nth degree interpolating polynomials to approximate the SDDEs. The numerical results confirm that the scheme is effective and easy to implement. However, the convergence of the method is not obtained for its complexity. It will be our future work.
Declarations
Acknowledgements
The authors would like to thank the anonymous referees for their valuable and insightful comments which have improved the paper. This work is supported by the National Natural Science Foundation of China (No. 11171352, No. 11301550), the New Teachers’ Specialized Research Fund for the Doctoral Program from the Ministry of Education of China (No. 20120162120096) and Mathematics and Interdisciplinary Sciences Project, Central South University.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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