- Research
- Open Access
A fractional-order Legendre collocation method for solving the Bagley-Torvik equations
- Fakhrodin Mohammadi^{1}Email author and
- Syed Tauseef Mohyud-Din^{2}
https://doi.org/10.1186/s13662-016-0989-x
© Mohammadi and Mohyud-Din 2016
- Received: 5 September 2016
- Accepted: 7 October 2016
- Published: 22 October 2016
Abstract
In this article, a numerical method based on the fractional-order shifted Legendre polynomials (FSLPs) and their operational matrix of fractional integration is introduced for solving the fractional Bagley-Torvik equations. The main advantage of the presented method is that it can reduce a solution of the initial and boundary value problems for the fractional Bagley-Torvik differential equations to a system of algebraic equations. In order to confirm the efficiency and superiority of the presented method, some numerical examples are provided and a comparison is presented between the obtained results and those results achieved from other existing methods in the literature.
Keywords
- Bagley-Torvik equations
- Riemann-Liouville fractional integration
- fractional-order Legendre polynomials
- operational matrix
- collocation method
MSC
- 26A33
- 34A08
- 65N35
1 Introduction
Fractional calculus, the theory of differentiation and integration to non-integer order, is very useful for the description of various physical phenomena, such as damping laws, diffusion process, etc. Fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes [1–10]. Especially, fractional differential equations provide outstanding tools for illustration of many engineering and physical problems. Since most fractional differential equations do not have exact and analytic solutions, the accurate numerical techniques for solving these fractional equations are a challenging and motivational research area in mathematics and engineering.
The fractional Bagley-Torvik equation was originally formulated in a description of a real material by the use of fractional calculus. Moreover, the Bagley-Torvik equation has appeared in simulating the motion of a rigid plate immersed in a Newtonian fluid [11–13]. This equation has been studied both analytically and numerically in [3]. Diethelm [14] transformed this equation into a system of fractional differential equation and solved the problem with the Adams predictor and the corrector method. Recently, considerable attention has been devoted to numerical solutions of the fractional Bagley-Torvik equation. For example the spectral tau method [15, 16], the operational formulation of collocation methods [17, 18], collocation methods [19–21], wavelet methods [22, 23], pseudospectral methods [24], differential transform methods [25], hybrid functions methods [26], and fractional Taylor methods [27] have been used to solve this fractional differential equation. In this study, a fractional-order Legendre collocation method is proposed for solving the Bagley-Torvik equations.
Applications of orthogonal functions and polynomials for numerical solution of ordinary differential equations refer, at least, to the time of Lanczos [28]. Moreover, the origin of some current spectral method, such as the Galerkin, tau, and pseudospectral methods can be found in the ‘weighted residual method’ of Finlayson and Scriven [29]. Nowadays, spectral methods are efficient techniques for solving a different kind of fractional differential and integral equations accurately [15, 17, 30, 31]. The main advantage of spectral methods lies in their accuracy for a given number of unknowns. For smooth problems in simple geometries, they offer exponential rates of convergence (spectral accuracy). By using the operational matrices for basis functions, spectral methods reduce the solution of fractional differential and integral equations into a solution of systems of algebraic equations which produce highly accurate solutions for these equations [22, 23, 30, 32].
This paper is structured as follows: In Section 2 some basic preliminaries of the fractional calculus are presented. The FSLPs and their properties are introduced in Section 3. Section 4 is devoted to an operational matrix of fractional integration for the FSLPs. Application of the FSLPs for solving the Bagley-Torvik equation is considered in Section 5. Convergence and an error estimate for the FSLPs expansion are given in Section 6. The efficiency and superiority of the proposed method is demonstrated by considering some numerical examples in Section 7. Finally, a conclusion is given in Section 8.
2 Preliminaries
In this section we review some basic definitions and preliminaries of the fractional calculus which are used in the next sections.
2.1 Fractional calculus
Fractional-order calculus is a branch of calculus which deals with integration and differentiation operators of non-integer order. Among the several formulations of the generalized derivative, the Riemann-Liouville and Caputo definition are most commonly used, which can be described as follows [3].
Definition 1
A real function \(f (t)\), \(t>0\), is said to be in the space \(C_{\mu}\), \(\mu \in\mathbb{R}\) if there exist a real number \(p >\mu\) and a function \(f_{1}(t)\in C[0,\infty)\) such that \(f (t)=t^{p} f_{1}(t)\), and it is said to be in the space \(C_{\mu}^{n}\), \(n \in\mathbb{N}\) if \(f ^{(n)} \in C_{\mu}\).
Definition 2
Definition 3
For more details of fractional calculus and their applications please refer to [1–3].
3 The FSLPs and their properties
4 Operational matrix of fractional integration of FSLPs
In recent years various operational matrices for the polynomials have been developed to cover the numerical solution of differential, integral and integro-differential equations. The main advantage of these operational matrices is that they replace differential and integral operators with some matrices. Consequently, they reduce such problems to those of solving a system of algebraic equations, greatly simplifying the problem [33–38]. In this section the operational matrix of fractional integration for FSLPs will be derived.
Theorem 4.1
Proof
5 Numerical solution of Bagley-Torvik equations
6 Error analysis
In this section, in order to demonstrate the efficiency of the proposed FSLPs method, we have given some theorems on convergence and error estimation. The next theorem gives an upper bound for the error function of the truncated FSLPs series.
Theorem 6.1
Proof
7 Numerical examples
In this section, the efficiency and superiority of the proposed method is demonstrated by some illustrative examples. All algorithms are performed by Maple 17.
Example 1
Example 2
Example 3
Example 4
Example 5
t | Exact | FSLPs ( α = 0.5) | FSLPs ( α = 1.0) | Ref. [ 27 ] | Ref. [ 23 ] |
---|---|---|---|---|---|
1.40625 | 4.85696 | 4.80915 | 4.85715 | 4.95531 | 4.67105 |
2.03125 | 6.83165 | 6.78579 | 6.85062 | 6.93440 | 6.48436 |
2.96875 | 7.67925 | 7.64470 | 7.67261 | 7.80605 | 7.21918 |
3.59375 | 6.97278 | 6.94967 | 6.98356 | 7.09830 | 6.51938 |
4.21875 | 5.48313 | 5.47278 | 5.48883 | 5.59310 | 5.09093 |
5.46875 | 1.28657 | 1.29947 | 1.28343 | 1.33675 | 1.11881 |
7.96875 | −4.53369 | −4.50974 | −4.53926 | −4.59731 | −4.30082 |
9.53125 | −3.64404 | −3.63542 | −3.64279 | −3.71142 | −3.40603 |
11.7188 | 0.59143 | 0.57883 | 0.59421 | 0.58569 | 0.61398 |
13.5938 | 2.64127 | 2.62760 | 2.63996 | 2.67926 | 2.51628 |
15.4688 | 1.72175 | 1.71945 | 1.72207 | 1.75636 | 1.60585 |
16.4063 | 0.63025 | 0.63383 | 0.62882 | 0.64944 | 0.56273 |
17.3438 | −0.44428 | −0.43668 | −0.44270 | −0.44298 | −0.45529 |
18.9063 | −1.50186 | −1.49344 | −1.49966 | −1.52298 | −1.44138 |
19.8438 | −1.52304 | −1.51713 | −1.518921 | −1.54859 | −1.44734 |
8 Discussion and conclusion
A new type of orthonormal fractional-order Legendre polynomials is defined. The operational matrix of fractional integration for this fractional-order basis is derived. By using this fractional operational matrix and collocation method a numerical method is proposed for solving the fractional Bagley-Torvik equations. A comparison is made between numerical results derived by the presented collocation method and other existing numerical method. According to the numerical results, we can conclude that the presented method is more accurate and effective for a numerical solution of the fractional Bagley-Torvik equations.
Declarations
Acknowledgements
We express our sincere thanks to the anonymous referees for valuable suggestions that improved the final manuscript.
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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