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- Open Access
Sine-cosine wavelet operational matrix of fractional order integration and its applications in solving the fractional order Riccati differential equations
- Yanxin Wang^{1},
- Tianhe Yin^{1}Email author and
- Li Zhu^{1, 2}
https://doi.org/10.1186/s13662-017-1270-7
© The Author(s) 2017
Received: 19 February 2017
Accepted: 6 July 2017
Published: 3 August 2017
Abstract
In this paper, the sine-cosine wavelet method is presented for solving Riccati differential equations. The sine-cosine wavelet operational matrix of fractional integration is derived and utilized to transform the equations to system of algebraic equations. Also, the error analysis of the sine-cosine wavelet bases is given. The proposed method can be used to solve not only the classical Riccati differential equations but also the fractional ones. Some examples are included to demonstrate the validity and applicability of the technique.
Keywords
- fractional calculus
- sine-cosine wavelet
- operational matrix
- Riccati differential equations
- numerical method
1 Introduction
Fractional calculus is an extension of derivatives and integrals to non-integer orders and it has been widely used to model engineering and scientific problems. Many physical problems are governed by fractional differential and integral equations, and finding the solutions of these equations has been the subject of many investigators in recent years. However, it is difficult to derive the analytical solutions to most of the fractional equations. Therefore, there has been significant interest in developing numerical schemes for their solutions. Some numerical methods include the homotopy perturbation method (HPM) [1], the homotopy analysis method (HAM) [2], the variational iteration method (VIM) [3], the Adomian decomposition method (ADM) [4], and different wavelet methods, such as the Legendre wavelet [5, 6], Haar wavelet [7], Chebyshev wavelet [8–14], Bernoulli wavelet [15], and ultraspherical wavelet methods [16, 17].
The Riccati equations play a significant role in many fields of engineering and applied science such as the theory of random processes, diffusion problems, transmission-line phenomena and optimal control theory. Thus, the solving methods for the Riccati differential equations are important. There have been several methods for solving the Riccati differential equations, such as He’s variational iteration method (HVIM) [18], ADM [19], HPM [20] and piecewise VIM [21]. Moreover, Saha et al. [22] used the modified ADM method to solve the fractional Riccati differential equations, Odibat and Momani [23], Hosseinnia et al. [24] and Khan et al. [25] used the modified homotopy perturbation method (MHPM) to solve the fractional Riccati differential equations. Khan [26] used the Laplace-Adomian-Padé method, Abd-Elhameed et al. [27] used the spectral wavelets algorithms, and Mehmet et al. [28] applied an iterative reproducing kernel Hilbert space method to get the solutions of fractional Riccati differential equations.
We notice that the sine-cosine wavelet was constructed by sine and cosine functions, and it is more suitable to solve periodic solution problem. Moreover, since the basis functions used to construct the sine-cosine wavelet are orthogonal and have compact support, it makes the more useful and simple in actual computations. Also, the numbers of mother wavelet’s components are restricted to one, so they do not lead to the growth of complexity of calculations comparing with other wavelets. It is worthy to mention here that the CAS wavelet [31] has similar properties to the sine-cosine wavelet, but they have completely different constructs and expressions. In this paper, the sine-cosine wavelet operational matrix of fractional integration is derived firstly and used to solve the Riccati differential equations, the wavelet operational matrix method is computer oriented. The efficiency and accuracy of the presented method are shown by several examples.
The rest of the paper is organized as follows. In the next section, some necessary definitions and mathematical preliminaries of the fractional calculus are introduced. In Section 3 the error analysis of the sine-cosine wavelet bases and the sine-cosine wavelet operational matrix of fractional integral are obtained. The sine-cosine wavelet method for solving (1) with initial conditions (2) is presented in Section 4. Numerical examples are presented in Section 5. A conclusion is given in Section 6.
2 Preliminaries and notations
In this section, we present some definitions, notations and preliminaries of the fractional calculus theory which will be used in this paper.
3 Sine-cosine wavelet and operational matrix of the fractional integration
3.1 Wavelet and sine-cosine wavelet
3.2 Function approximation
3.3 Error analysis of the sine-cosine wavelet bases
In this section, the error analysis of the sine-cosine wavelet is derived. We can conclude that the sine-cosine wavelet expansion of a function \(u(t)\), with bounded second derivative, converges uniformly to \(u(t)\).
Lemma 3.1
If the sine-cosine wavelet expansion of a continuous function \(u(t)\) converges uniformly, then the sine-cosine wavelet expansion converges to the function \(u(t)\).
Proof
Theorem 3.2
Proof
3.4 Sine-cosine wavelet operational matrix of the fractional integration
4 Applications of the operational matrix of fractional integration
5 Numerical examples
In this section, we demonstrate the effectiveness and simplicity of the proposed method with three examples.
Example 1
Approximate norm-2 of absolute error for some k and L
Example 2
Comparison between sine-cosine, HPM and UWCM for Example 2 for \(\pmb{\alpha= 1/2, 3/4}\)
t | \(\boldsymbol{\alpha= \frac{1}{2}}\) | \(\boldsymbol{\alpha= \frac{3}{4}}\) | ||||
---|---|---|---|---|---|---|
Sine-cosine | HPM | UWCM | Sine-cosine | HPM | UWCM | |
0.1 | 0.62587 | 0.321730 | 0.58092 | 0.26346 | 0.216866 | 0.24456 |
0.3 | 1.18291 | 0.940941 | 1.12057 | 0.71969 | 0.654614 | 0.71031 |
0.5 | 1.50602 | 1.549439 | 1.45668 | 1.20886 | 1.132763 | 1.15155 |
0.7 | 1.64338 | 2.066523 | 1.63391 | 1.48589 | 1.594278 | 1.49335 |
0.9 | 1.75292 | 2.396839 | 1.75008 | 1.72284 | 1.962239 | 1.73018 |
Example 3
6 Conclusion
In this paper, we proposed the sine-cosine wavelet operational matrix method to solve nonlinear fractional Riccati differential equations. The sine-cosine wavelet operational matrix of fractional order integration is obtained. Compared to ADM, HPM, VIM, MHPM and NHPM, the sine-cosine wavelet method is simple and easy to implement; moreover, it enables us to approximate the solution more accurate in a bigger interval. However, we have also noticed that the sine-cosine wavelet is constructed from the trigonometric polynomials and has periodicity. It is more suitable for solving the periodic problem.
Declarations
Acknowledgements
This work was supported by the K.C. Wong Education Foundation, Hong Kong, the Natural Science Foundation of Ningbo City, China (Grant No. 2017A610143), and the Project of Education of Zhejiang Province (No. Y201533324), the Natural Science Foundation of Zhejiang Province, China (Grant No. LQ15D010002), the Natural Science Foundation of Ningbo City, China (Grant No. 2014A610068), the Science and Technology Benefit People of Ningbo City, China (Grant No. 2015C50052) and the Zhejiang Provincial Research Plan for Public Science and Technology, China (Grant No. 2014C31076).
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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