- Research
- Open Access
Solving nonlinear Volterra integro-differential equations of fractional order by using Euler wavelet method
- Yanxin Wang^{1}Email author and
- Li Zhu^{1, 2}
https://doi.org/10.1186/s13662-017-1085-6
© The Author(s) 2017
- Received: 7 October 2016
- Accepted: 14 January 2017
- Published: 25 January 2017
Abstract
In this paper, a wavelet numerical method for solving nonlinear Volterra integro-differential equations of fractional order is presented. The method is based upon Euler wavelet approximations. The Euler wavelet is first presented and an operational matrix of fractional-order integration is derived. By using the operational matrix, the nonlinear fractional integro-differential equations are reduced to a system of algebraic equations which is solved through known numerical algorithms. Also, various types of solutions, with smooth, non-smooth, and even singular behavior have been considered. Illustrative examples are included to demonstrate the validity and applicability of the technique.
Keywords
- Volterra integro-differential equations
- Euler wavelet
- operational matrix
- Caputo derivative
- numerical solution
1 Introduction
The fractional calculus is a mathematical discipline that is 300 years old, and it has developed progressively up to now. The concept of differentiation to fractional order was defined in the 19th century by Riemann and Liouville. In various problems of physics, mechanics, and engineering, fractional differential equations and fractional integral equations have been proved to be a valuable tool in modeling many phenomena [1, 2]. However, most fractional-order equations do not have analytic solutions. Therefore, there has been significant interest in developing numerical schemes for the solutions of fractional-order differential equations.
In the past 40 years, the theory and applications of the fractional-order partial differential equations (FPDEs) have become of increasing interest for the researchers to generalize the integer-order differential equations. Conventionally various technologies, e.g. modified homotopy analysis transform method (MHATM) [3], modified homotopy analysis Laplace transform method [4], homotopy analysis transform method (HATM) [5, 6], fractional homotopy analysis transform method (FHATM) [7], local fractional variational iteration algorithms [8] were used for the solutions of the FPDEs. Meanwhile, local fractional similarity solution for the diffusion equation was discussed in [9]. The inverse problems for the fractal steady heat transfer described by the local fractional Volterra integro-differential equations were considered in [10].
Recently, many effective methods for obtaining approximations or numerical solutions of fractional-order integro-differential equations have been presented. These methods include the variational iteration method [11–13], the adomian decomposition method [14], the fractional differential transform method [15], the reproducing kernel method [16], the collocation method [17, 18], and the wavelet method [19–24].
Wavelet theory is a relatively new and an emerging area in the field of applied science and engineering. Wavelets permit the accurate representation of a variety of functions and operators. Moreover, wavelets establish a connection with fast numerical algorithms [25]. So the wavelet method is a new numerical method for solving the fractional equations and it needs a small amount of calculation. However, the method will produce a singularity in the case of certain increased resolutions. Using wavelet numerical method has several advantages: (a) the main advantage is that after discretizing the coefficient matrix of the algebraic equation shows sparsity; (b) the wavelet method is computer oriented, thus solving a higher-order equation becomes a matter of dimension increasing; (c) the solution is a multi-resolution type; (d) the solution is convergent, even the size of the increment may be large [24]. Many researchers started using various wavelets for analyzing problems of high computational complexity. It is proved that wavelets are powerful tools to explore new directions in solving differential equations and integral equations.
In this paper, the main purpose is to introduce the Euler wavelet operational matrix method to solve the nonlinear Volterra integro-differential equations of fractional order. The Euler wavelet is first presented and it is constructed by Euler polynomials. The method is based on reducing the equation to a system of algebraic equations by expanding the solution as Euler wavelet with unknown coefficients. The characteristic of the operational method is to transform the integro-differential equations into the algebraic one. It not only simplifies the problem but also speeds up the computation. It is worth noting that the Euler polynomials are not based on orthogonal functions, nevertheless, they possess the operational matrix of integration. Also the Euler wavelet is superior to the Legendre wavelet and the Chebyshev wavelet for approximating an arbitrary function, which can be verified by numerical examples.
The structure of this paper is as follows: In Section 2, we recall some basic definitions and properties of the fractional calculus theory. In Section 3, the Euler wavelets are constructed and the operational matrix of the fractional integration is derived. In Section 4, we summarize the application of the Euler wavelet operational matrix method to the solution of the fractional integro-differential equations. Some numerical examples are provided to clarify the approach in Section 5. The conclusion is given in Section 6.
2 Fractional calculus
There are various definitions of fractional integration and derivatives. The widely used definition of a fractional integration is the Riemann-Liouville definition and the definition of a fractional derivative is the Caputo definition.
Definition 1
Definition 2
3 Euler wavelet operational matrix of the fractional integration
3.1 Wavelets and Euler wavelet
3.2 Function approximation
3.3 Convergence of Euler wavelets basis
Lemma 3
Proof
Theorem 4
Proof
3.4 Operational matrix of the fractional integration
4 Method of numerical solution
5 Numerical examples
In this section, six examples are given to demonstrate the applicability and accuracy of our method. Examples 1-5 have smooth solutions, while Example 6 has a non-smooth and singular solution. In all examples the package of Matlab 7.0 has been used to solve the test problems considered in this paper.
Example 1
The absolute errors of different k and \(\pmb{M=3}\) for Example 1
t | Euler | SCW | Euler | SCW | Euler | SCW |
---|---|---|---|---|---|---|
k = 4 | k = 4 | k = 5 | k = 5 | k = 6 | k = 6 | |
0.0 | 5.3744e-003 | 6.0169e-003 | 2.1912e-003 | 2.5058e-003 | 9.7161e-004 | 1.1272e-003 |
0.1 | 5.9284e-004 | 1.2504e-003 | 3.6354e-004 | 4.3350e-004 | 1.6976e-005 | 3.2831e-004 |
0.2 | 1.6221e-003 | 5.1499e-005 | 1.0167e-004 | 6.1143e-004 | 1.0476e-004 | 2.0079e-004 |
0.3 | 1.6909e-003 | 1.1123e-004 | 1.1436e-004 | 5.5204e-004 | 1.0746e-004 | 1.7628e-004 |
0.4 | 4.6755e-004 | 1.0434e-003 | 4.3188e-004 | 2.0180e-004 | 3.0374e-005 | 2.3828e-004 |
0.5 | 2.0212e-003 | 3.4768e-003 | 5.0356e-004 | 1.1110e-003 | 1.2551e-004 | 3.8232e-004 |
0.6 | 4.8121e-004 | 9.2652e-004 | 4.3422e-004 | 1.5066e-004 | 3.0730e-005 | 2.1590e-004 |
0.7 | 1.7295e-003 | 3.6739e-004 | 1.2103e-004 | 4.4277e-004 | 1.0861e-004 | 1.2863e-004 |
0.8 | 1.7247e-003 | 4.0781e-004 | 1.1954e-004 | 4.2369e-004 | 1.0818e-004 | 1.1993e-004 |
0.9 | 4.6700e-004 | 8.0375e-004 | 4.2982e-004 | 9.2562e-005 | 2.9460e-005 | 1.8947e-004 |
Approximate norm-2 of absolute errors for some k of the Euler and SCW
Example | Euler | SCW | ||||
---|---|---|---|---|---|---|
\(\boldsymbol {\|e_{8}\|_{2}}\) | \(\boldsymbol {\|e_{16}\|_{2}}\) | \(\boldsymbol {\|e_{32}\|_{2}}\) | \(\boldsymbol {\|e_{8}\|_{2}}\) | \(\boldsymbol {\|e_{16}\|_{2}}\) | \(\boldsymbol {\|e_{32}\|_{2}}\) | |
Example 1 | 8.3942e-007 | 7.2293e-008 | 7.1159e-009 | 7.1538e-007 | 2.3580e-007 | 5.7642e-008 |
Example 2 | 9.4203e-007 | 5.9209e-008 | 3.7129e-009 | 1.6350e-005 | 1.1839e-006 | 8.6352e-008 |
From Table 1, we find that the absolute errors become smaller and smaller with k increasing. Table 2 shows that the Euler wavelet method can reach a higher degree of accuracy than the SCW method.
Example 2
Example 3
Comparison of approximate norm-2 of absolute errors with reproducing kernel and CAS
Example 4
Example 5
Numerical results for Example 5 with comparison to CAS
x | α = 3.25 | α = 3.5 | α = 3.75 | |||
---|---|---|---|---|---|---|
CAS | Euler | CAS | Euler | CAS | Euler | |
0.0 | 1.0000 | 1.0006 | 1.0000 | 1.0006 | 1.0000 | 1.0006 |
0.1 | 1.1053 | 1.1060 | 1.1052 | 1.1059 | 1.1052 | 1.1059 |
0.2 | 1.2219 | 1.2228 | 1.2216 | 1.2224 | 1.2216 | 1.2223 |
0.3 | 1.3523 | 1.3527 | 1.3510 | 1.3516 | 1.3510 | 1.3510 |
0.4 | 1.4968 | 1.4972 | 1.4941 | 1.4948 | 1.4941 | 1.4934 |
0.5 | 1.6635 | 1.6581 | 1.6565 | 1.8295 | 1.8334 | 1.8248 |
0.7 | 2.0444 | 2.0346 | 2.0293 | 2.0240 | 2.0293 | 2.0167 |
0.8 | 2.2776 | 2.2534 | 2.2537 | 2.2386 | 2.2537 | 2.2281 |
0.9 | 2.5265 | 2.4943 | 2.4949 | 2.4747 | 2.4949 | 2.4603 |
Let us consider examples with non-smooth and singular solutions.
Example 6
In the examples above, we do not show the computational times of the different methods. In fact, the Euler wavelet method has the faster computing speed, compared with the CAS wavelet method and the second Chebyshev wavelet method. In Example 1, for instance, when \(k = 4,5,6\), the computational times of the second Chebyshev wavelet are 1.71 s, 1.94 s, and 4.21 s, while the computational times of the Euler wavelet are 0.79 s, 1.25 s, and 3.57 s. The same conclusion can be drawn from the other examples.
6 Conclusion
In this paper, we construct the Euler wavelet and derive the wavelet operational matrix of the fractional integration, and we use it to solve the fractional integro-differential equations. By solving the nonlinear system, approximate solutions are got. Graphical illustrations and tables of the numerical results with the aid of Euler wavelets indicate that the numerical results are well in agreement with exact solutions and superior to other results. Also the proposed method can be efficiently applied to a large number of similar fractional problems. Of course, the convergence of this algorithm has not been derived, which will be future research work.
Declarations
Acknowledgements
This work was supported by the KC Wong Education Foundation, Hong Kong, and the Project of Education of Zhejiang Province (No. Y201533324). The authors gratefully acknowledge the support of the KC Wong Education Foundation, Hong Kong. The authors are grateful to the editor and the anonymous referees for their constructive and helpful comments.
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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