An efficient scheme for solving a system of fractional differential equations with boundary conditions
- Veysel Fuat Hatipoglu^{1},
- Sertan Alkan^{2} and
- Aydin Secer^{3}Email author
https://doi.org/10.1186/s13662-017-1260-9
© The Author(s) 2017
Received: 18 May 2017
Accepted: 30 June 2017
Published: 26 July 2017
Abstract
In this study, the sinc collocation method is used to find an approximate solution of a system of differential equations of fractional order described in the Caputo sense. Some theorems are presented to prove the applicability of the proposed method to the system of fractional order differential equations. Some numerical examples are given to test the performance of the method. Approximate solutions are compared with exact solutions by examples. Some graphs and tables are presented to show the performance of the proposed method.
Keywords
1 Introduction
2 Preliminaries
In this section, some preliminaries and notations related to fractional calculus and sinc basis functions are given. For more details we refer the reader to monographs [1–7, 33–35].
Definition 1
Theorem 1
Theorem 2
Definition 2
Definition 3
Definition 4
Definition 5
Definition 6
Theorem 3
Proof
See [18]. □
For the term of fractional in (1.1), the infinite quadrature rule must be truncated to a finite sum. The following theorem indicates the conditions under which an exponential convergence results.
Theorem 4
Proof
See [18]. □
Lemma 1
Proof
See [36]. □
3 The sinc-collocation method
Theorem 5
Similarly, the order α derivative of \(f(x)\) for \(0<\alpha<1\) is given by the following theorem.
Theorem 6
Proof
Theorem 7
4 Computational examples
In this section, two problems that have homogeneous boundary conditions will be tested by using the present method via Mathematica10 on a personal computer. In all the examples, we take \(d=\pi/2\), \(L=M=N\).
Example 1
Maximum absolute error for Example 1
N | Max. absolute error in u | Max. absolute error in v |
---|---|---|
5 | 9.642 × 10^{−3} | 3.145 × 10^{−3} |
10 | 9.843 × 10^{−4} | 4.176 × 10^{−4} |
20 | 5.077 × 10^{−5} | 8.190 × 10^{−5} |
Numerical results for Example 1 when \(\pmb{N=40}\)
x | Exact sol. in u | Exact sol. in v | Absolute error in u | Absolute error in v |
---|---|---|---|---|
0 | 0 | 0 | 0 | 0 |
0.1 | −0.0999 | 0.09 | 7.02 × 10^{−7} | 1.27 × 10^{−5} |
0.2 | −0.1984 | 0.16 | 2.10 × 10^{−6} | 2.07 × 10^{−5} |
0.3 | −0.2919 | 0.21 | 3.15 × 10^{−6} | 1.58 × 10^{−5} |
0.4 | −0.3744 | 0.24 | 3.71 × 10^{−6} | 1.28 × 10^{−5} |
0.5 | −0.4375 | 0.25 | 3.62 × 10^{−6} | 1.14 × 10^{−5} |
0.6 | −0.4704 | 0.24 | 3.30 × 10^{−6} | 1.11 × 10^{−5} |
0.7 | −0.4599 | 0.21 | 3.06 × 10^{−6} | 1.14 × 10^{−5} |
0.8 | −0.3904 | 0.16 | 2.22 × 10^{−6} | 1.01 × 10^{−5} |
0.9 | −0.2409 | 0.09 | 1.24 × 10^{−6} | 8.14 × 10^{−6} |
1 | 0 | 0 | 0 | 0 |
Example 2
Maximum absolute error for Example 2
N | Max. absolute error in u | Max. absolute error in v |
---|---|---|
5 | 4.274 × 10^{−3} | 3.387 × 10^{−3} |
10 | 3.737 × 10^{−4} | 9.082 × 10^{−4} |
20 | 2.282 × 10^{−5} | 2.793 × 10^{−4} |
Numerical results for Example 2 when \(\pmb{N=40}\)
x | Exact sol. in u | Exact sol. in v | Absolute error in u | Absolute error in v |
---|---|---|---|---|
0 | 0 | 0 | 0 | 0 |
0.1 | −0.009 | 0.09 | 7.43 × 10^{−6} | 8.46 × 10^{−5} |
0.2 | −0.032 | 0.16 | 8.57 × 10^{−6} | 1.37 × 10^{−4} |
0.3 | −0.063 | 0.21 | 8.29 × 10^{−6} | 1.23 × 10^{−4} |
0.4 | −0.096 | 0.24 | 8.64 × 10^{−6} | 1.15 × 10^{−4} |
0.5 | −0.125 | 0.25 | 9.17 × 10^{−6} | 1.09 × 10^{−4} |
0.6 | −0.144 | 0.24 | 9.58 × 10^{−6} | 1.03 × 10^{−4} |
0.7 | −0.147 | 0.21 | 9.58 × 10^{−6} | 9.36 × 10^{−5} |
0.8 | −0.128 | 0.16 | 8.70 × 10^{−6} | 7.32 × 10^{−5} |
0.9 | −0.081 | 0.09 | 5.90 × 10^{−6} | 4.53 × 10^{−5} |
1 | 0 | 0 | 0 | 0 |
5 Conclusion
This study focuses on the application of the sinc-collocation method to obtain the approximate solutions of the system of fractional order differential equations (1.1). The proposed method is applied to some special examples in order to illustrate the applicability and accuracy of the proposed method for equation (1.1). Obtained numerical solutions are compared with exact solutions and results are presented in tables and by graphics. Regarding the findings, it can be concluded that the sinc-collocation method is an effective and convenient method for obtaining the approximate solution of a system differential equations of fractional order.
Declarations
Acknowledgements
The authors express their sincere thanks to the referee(s) for the careful and detailed reading of the 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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