A new operational matrix of Caputo fractional derivatives of Fermat polynomials: an application for solving the BagleyTorvik equation
 Youssri H Youssri^{1}Email author
https://doi.org/10.1186/s1366201711234
© The Author(s) 2017
Received: 18 November 2016
Accepted: 21 February 2017
Published: 4 March 2017
Abstract
Herein, an innovative operational matrix of fractionalorder derivatives (sensu Caputo) of Fermat polynomials is presented. This matrix is used for solving the fractional BagleyTorvik equation with the aid of tau spectral method. The basic approach of this algorithm depends on converting the fractional differential equation with its initial (boundary) conditions into a system of algebraic equations in the unknown expansion coefficients. The convergence and error analysis of the suggested expansion are carefully discussed in detail based on introducing some new inequalities, including the modified Bessel function of the first kind. The developed algorithm is tested via exhibiting some numerical examples with comparisons. The obtained numerical results ensure that the proposed approximate solutions are accurate and comparable to the analytical ones.
Keywords
MSC
1 Introduction
Spectral methods have prominent roles in treating various types of differential equations. Over the past four decades, the appeal of spectral methods for applications such as computational fluid dynamics has expanded. In fact, spectral methods are widely used in diverse applications such as wave propagation (for acoustic, elastic, seismic and electromagnetic waves), solid and structural analysis, marine engineering, biomechanics, astrophysics and even financial engineering. The main difference between these techniques is the specific choice of the trial and test functions. The main idea behind spectral methods is to assume spectral solutions of the form \(\sum a_{k} \phi_{k}(x)\). The expansion coefficients \(a_{k}\) can be determined if a suitable spectral method is applied. The collocation method requires enforcing the differential equation to be satisfied exactly at some nodes (collocation points). The tau method is a synonym for expanding the residual function as a series of orthogonal polynomials and then applying the boundary conditions as constraints. The Galerkin method principally depends on selecting some suitable combinations of orthogonal polynomials which satisfy the underlying boundary (initial) conditions which are called ‘basis functions’, and after that the residual is enforced to be orthogonal with the suggested basis functions. For an intensive study on spectral methods and their applications, see Canuto et al. [2], Hesthaven et al. [3], Boyd [4], Trefethen [5], Doha et al. [6] and Bhrawy et al. [7].
Many number and polynomial sequences can be generated by difference equations of order two. Fermat polynomials are among these polynomials. This class of polynomials is considered as a special case of the general class of \((p,q)\)Fibonacci polynomial sequence (see [8]). This class of polynomials is of fundamental interest in mathematics since it includes some polynomials which have numerous important applications in several fields such as combinatorics and number theory. There are many papers dealing with these kinds of polynomials from a theoretical point of view (see, for example, [8–10]); however, the numerical investigations concerning these polynomials are very rare. In this respect, recently, a collocation algorithm based on employing Fibonacci polynomials has been analyzed for solving VolterraFredholm integral equations in [11]. Moreover, a numerical approach with error estimation to solve general integrodifferentialdifference equations using Dickson polynomials has been developed in [12]. This gives us a motivation for utilizing these polynomials in several numerical applications.

To establish a new operational matrix of fractional derivatives of Fermat polynomials.

To analyze and present an algorithm for solving the BagleyTorvik equation based on applying the spectral tau method.
The outline of this paper is as follows. In Section 2, we introduce some necessary definitions of the fractional calculus. Moreover, in this section, an overview on Fermat polynomials is given including some properties and also some new formulae which are useful in the sequel. Section 3 is concerned with the construction of an operational matrix of fractional derivatives (OMFD) in the Caputo sense of Fermat polynomials. In Section 4, we analyze and present a spectral tau algorithm for solving the BagleyTorvik equation. We give a global error bound for the suggested Fermat expansion in Section 5. Some test problems with comparisons are displayed in Section 6. Finally, some concluding remarks are displayed in Section 7.
2 Preliminaries and useful formulae
This section is dedicated to presenting some fundamentals of the fractional calculus theory which will be useful throughout this article. Moreover, an overview on Fermat polynomials and some new formulae concerning these polynomials are presented.
2.1 Some fundamentals of fractional calculus
Definition 1
[21]
Definition 2
[21]
Definition 3
For survey on the fractional derivatives and integrals, one can be referred to [21, 22].
2.2 An overview on Fermat polynomials
Fermat polynomials can be obtained from these polynomials for the case corresponding to \(p(t)=3 t\) and \(q(t)=2\).
Note 1
It is worth mentioning here that \(\mathcal{F}_{k}(t)\) is a polynomial of degree \((k1)\) with integer coefficients.
Theorem 1
Proof
Theorem 2
Proof
3 Fermat operational matrix of the Caputo fractional derivative
3.1 Construction of Fermat OMFD
Theorem 3
Proof
4 Numerical treatment of the BagleyTorvik equation
In this section, we present a numerical spectral tau algorithm for solving the fractionalorder linear BagleyTorvik equation based on using the constructed Fermat operational matrix of derivatives.
5 Convergence and error analysis
This section gives a detailed study for the convergence and error analysis of the proposed Fermat expansion. To proceed in such a study, the following lemmas are useful in the sequel.
Lemma 1
Proof
Following procedures similar to those given in [27], the lemma can be obtained. □
Lemma 2
[28], p.375
Lemma 3
[29]
Lemma 4
Now, we are in a position to state and prove the following two theorems.
Theorem 4
Proof
Theorem 5
Proof
6 Test problems and comparisons
In this section, we use the Fermat tau operational matrix (FTM) method to solve numerically BagleyTorvik equations. Moreover, we compare our results with some techniques existing in the literature.
Example 1
[30]
Example 2
Example 3

Case i:$$f(t)=t^{5}t^{4}. $$

Case ii:$$f(t)=t^{3}. $$
Example 4
[35]
Example 5
Maximum pointwise error of Example 5
M  

5  10  15  
τ  15.92  131.78  321.52 
E  2.56⋅10^{−3}  7.95⋅10^{−9}  2.22⋅10^{−13} 
Example 6
Comparison between GTCM, FrTM and FTM for Example 6
t  GTCM [ 36 ]  FrTM [ 37 ]  FTM M = 15  Exact solution 

0.0  0  0  0  0 
0.1  0.036485547  0.036487480  0.036487479  0.036487479 
0.2  0.140634716  0.140639621  0.140639621  0.140639621 
0.3  0.307476229  0.307484627  0.307484627  0.307484627 
0.4  0.533271294  0.533284110  0.533284110  0.533284110 
0.5  0.814735609  0.814756949  0.814756950  0.814756950 
0.6  1.148805808  1.148837422  1.148837428  1.148837428 
0.7  1.532521264  1.532565426  1.532565443  1.532565443 
0.8  1.962974991  1.963029255  1.963029298  1.963029298 
0.9  2.437455982  2.437333971  2.437334072  2.437334072 
1.0  2.954070000  2.952583880  2.952584099  2.952584099 
Example 7
7 Conclusions
In this paper, we have developed a new operational matrix of fractional derivatives of Fermat polynomials. This matrix is established with the aid of introducing some new identities concerning Fermat polynomials. As far as we know, the introduced operational matrix is novel, and its application in handling fractional differential equations is also new. As an application, the BagleyTorvik equation is solved via a certain Fermat operational tau method. Some tested numerical examples including some comparisons are exhibited to demonstrate the features of the proposed method.
Declarations
Acknowledgements
The author would like to thank the anonymous referees for critically reading the manuscript and also for their constructive comments, which helped substantially to improve 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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