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A method for fractional Volterra integro-differential equations by Laguerre polynomials
- Dilek Varol Bayram^{1} and
- Ayşegül Daşcıoğlu^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s13662-018-1924-0
© The Author(s) 2018
- Received: 13 August 2018
- Accepted: 6 December 2018
- Published: 18 December 2018
Abstract
The main purpose of this study is to present an approximation method based on the Laguerre polynomials for fractional linear Volterra integro-differential equations. This method transforms the integro-differential equation to a system of linear algebraic equations by using the collocation points. In addition, the matrix relation for Caputo fractional derivatives of Laguerre polynomials is also obtained. Besides, some examples are presented to illustrate the accuracy of the method and the results are discussed.
Keywords
- Volterra integro-differential equations
- Laguerre polynomials
- Fractional integro-differential equations
1 Introduction
The fractional calculus represents a powerful tool in applied mathematics to study numerous problems from different fields of science and engineering such as mathematical physics, finance, hydrology, biophysics, thermodynamics, control theory, statistical mechanics, astrophysics, cosmology, and bioengineering [1]. Since the fractional calculus has attracted much more interest among mathematicians and other scientists, the solutions of the fractional differential and integro-differential equations have been studied frequently in recent years [2–10]. The methods that are used to find the solutions of the fractional Volterra integro-differential equations are given as Adomian decomposition [11], Bessel collocation [12, 13], CAS wavelets [14], Chebyshev pseudo-spectral [15], cubic B-spline wavelets [16], Euler wavelet [17], fractional differential transform [18], homotopy analysis [19], homotopy perturbation [20–23], Jacobi spectral-collocation [24, 25], Legendre collocation [26], Legendre wavelet [27], linear and quadratic interpolating polynomials [28], modification of hat functions [29], multi-domain pseudospectral [30], normalized systems functions [31], novel Legendre wavelet Petrov–Galerkin method [32], operational Tau [33], piecewise polynomial collocation [34], quadrature rules [35], reproducing kernel [36], second Chebyshev wavelet [37], second kind Chebyshev polynomials [38], sinc-collocation [39, 40], spline collocation [41], Taylor expansion [27], and variational iteration [20, 23].
Definition
([52])
Besides, the Caputo fractional derivative of a constant function is zero and the Caputo fractional differentiation operator is linear [53].
Besides, the main purpose of the solution method presented in this paper is to obtain the Caputo fractional derivative of the Laguerre polynomials in terms of the Laguerre polynomials and to give a matrix representation for this relation. The Caputo fractional derivative of the Laguerre polynomials is mentioned in Ref. [51, 55–57]. While these matrix relations have been given depending on approximate matrices, the relation proposed in this paper is new, exact, and simpler than the former ones.
This paper is organized as follows: In Sect. 2, the main matrix relations of the terms in Eq. (1) are established. In Sect. 3, the collocation method which is used to find the solution is introduced. In Sect. 4, some numerical examples are solved and their comparison with the existing results in the literature are presented to verify the accuracy and efficiency of the proposed method. The conclusion is given in Sect. 5.
2 Main matrix relations
Now, we will state a theorem that gives the Caputo fractional derivative of Laguerre polynomials in terms of Laguerre polynomials.
Theorem
Proof
2.1 Matrix relation for the differential part
2.2 Matrix relation for conditions
3 Method of solution
Finally, to obtain the solution of Eq. (1) under conditions (2), we replace or stack the n rows of the augmented matrix \([ \mathbf{W};\mathbf{G} ] \) with the rows of the augmented matrix \([ \mathbf{U}_{\mathrm{j}}; c_{j} ] \). In this way, the Laguerre coefficients are determined by solving the new linear algebraic system.
4 Numerical examples
In this section, we apply the proposed method to four examples existing in the literature and to a test example constructed for this method. We have performed all of the numerical computations using Mathcad 15. We also use the collocation points by using the formula \(x_{s} = { [ 1- \cos ( \frac{ ( s+1 ) \pi }{N+1} ) ] } / {2}\), \(s=0,1,\dots, N\).
Example 1
Using the homotopy analysis method, this problem was also solved by Awawdeh et al. [19]. They found the approximate solution for \(N=5\), but they did not state the numerical results of the errors of their method. Besides, Sahu et al. [32] found the approximate solution with the maximum absolute error \(4.2 \times 10^{-15}\) by the Legendre wavelet Petrov–Galerkin method for \(N=6\). If the results are compared, it can be said that the proposed method is better than the other methods since the exact solution is found for \(N=2\).
Example 2
By solving this system, we get \(a_{0} =1\), \(a_{1} =-1\). When we substitute the determined coefficients into Eq. (3), we get the exact solution.
This problem was also solved by Awawdeh et al. [19] with the homotopy analysis method. They found the approximate solution for \(N=5\), but they did not state the numerical results of the errors of their method. Besides, Sahu et al. [32] found the approximate solution with the maximum absolute error \(1.1 \times 10^{-16}\) by the Legendre wavelet Petrov–Galerkin method for \(N=6\). If the results are compared, it can be said that the proposed method is better than the other methods since the exact solution is found for \(N=1\).
Example 3
This problem was also solved by Awawdeh et al. [19] and they found the approximate solution by the homotopy analysis method for \(N=5\). By the proposed method, we have found the exact solution of the problem for \(N=2\). Apparently, our method is better than the other method.
Example 4
Comparison of the absolute errors of Example 4 for different methods
x | Linear scheme | Quadratic scheme | Linear-quadratic scheme | Our method |
---|---|---|---|---|
0.2 | 9.8 × 10^{−3} | 9.8 × 10^{−3} | 9.8 × 10^{−3} | 2.9 × 10^{−4} |
0.4 | 1.1 × 10^{−2} | 4.8 × 10^{−3} | 4.9 × 10^{−3} | 7.3 × 10^{−4} |
0.6 | 1.2 × 10^{−2} | 2.9 × 10^{−3} | 3.2 × 10^{−3} | 9.5 × 10^{−4} |
0.8 | 1.4 × 10^{−2} | 2.6 × 10^{−3} | 3.5 × 10^{−3} | 8.3 × 10^{−4} |
1 | 1.9 × 10^{−2} | 3.3 × 10^{−3} | 5.5 × 10^{−3} | 4.5 × 10^{−4} |
Example 5
Maximum errors of Example 5 for different N values
N | 2 | 4 | 6 | 8 | 10 |
---|---|---|---|---|---|
Maximum errors | 1.3 × 10^{−2} | 1.7 × 10^{−3} | 5.5 × 10^{−4} | 2.4 × 10^{−4} | 1.3 × 10^{−4} |
5 Conclusion
In this study, a collocation method based on Laguerre polynomials has been developed for solving the fractional linear Volterra integro-differential equations. For this purpose, the matrix relation for the Caputo fractional derivative of the Laguerre polynomials has been obtained for the first time in the literature. Using these relations and suitable collocation points, the integro-differential equation has been transformed into a system of algebraic equations. The method is faster and simpler than the other methods in the literature, and better than the homotopy analysis and Legendre wavelet method.
Declarations
Acknowledgements
The authors would like to thank the reviewers for their constructive comments to improve the quality of this work.
Funding
This work is supported by the Scientific Research Project Coordination Unit of Pamukkale University with numbers 2018KRM002-227 and 2018KRM002-457.
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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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