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 Open Access
An accurate method for solving a singular secondorder fractional EmdenFowler problem
 Muhammed I Syam^{1}Email author,
 HM Jaradat^{2},
 Marwan Alquran^{3} and
 Safwan AlShara’^{2}
https://doi.org/10.1186/s1366201814692
© The Author(s) 2018
 Received: 24 August 2017
 Accepted: 31 December 2017
 Published: 22 January 2018
Abstract
In this paper, we study a singular secondorder fractional EmdenFowler problem. The reproducing kernel Hilbert space method (RKHSM) is employed to compute an approximation to the proposed problem. The construction of the reproducing kernel based on orthonormal shifted Legendre polynomials is presented. The validity of the RKHSM is ascertained by presenting several examples. We prove the existence of solution of the singular secondorder fractional EmdenFowler problem. The convergence of the approximate solution using the proposed method is investigated. The uniform convergence of the approximate solution to the exact solution is presented. Error estimation to the proposed method is proven. The results reveal that the proposed analytical method can achieve excellent results in predicting the solutions of such problems.
Keywords
 singular secondorder fractional EmdenFowler problem
 nonlinear initial value problem
 reproducing kernel Hilbert space method
MSC
 76A05
 76W05
 76Z99
 65L05
1 Introduction
Recently, fractional initial and boundary value problems have been studied extensively. Several forms of them have been proposed in standard models, and there has been a significant interest in developing numerical schemes for their solutions. Several numerical techniques, such as Laplace and Fourier transforms [1, 2], Adomian decomposition and variational iteration methods [3, 4], eigenvector expansion [5], differential transform and finite differences methods [6, 7], power series method [8], residual power series method [9–12], collocation method [13], operational matrix of fractional integration method [14, 15], and wavelet method [16, 17], are used to solve such problems. Many applications of fractional calculus to various branches of science such as engineering, physics and economics can be found in [18, 19]. Considerable attention has been given to the theory of fractional ordinary differential equations and integral equations [20, 21]. Additionally, the existence of solutions of ordinary and fractional boundary value problems using monotone iterative sequences has been investigated by several authors [22–25]. Some applications are discussed by Singh et al. [26]. They analyze a new fractional model of chemical kinetics system related to a newly discovered AtanganaBaleanu derivative with fractional order having nonsingular and nonlocal kernel. Also, Singh et al. [27] presented a reliable algorithm based on the local fractional homotopy perturbation. Moreover, Singh et al. [28] proposed a new numerical algorithm, namely qhomotopy analysis Sumudu transform method (qHASTM), to obtain the approximate solution for the nonlinear fractional dynamical model of interpersonal and romantic relationships.
A reproducing kernel Hilbert space method (RKHSM) is a useful framework for constructing approximate solutions of linear and nonlinear boundary value problems. In recent years, a lot of attention has been devoted to the study of RKHSM to investigate various scientific models. The RKHSM, which accurately computes the series solution, is of great interest to applied sciences. This technique gives the solution in a rapidly convergent series with components that can be easily computed. We use the RKHSM to solve this problem since it is easy to implement and it gives very accurate results. This method is used for the investigation of several scientific applications, see [29–32] and [33–35]. Kumar et al. [36] presented a new numerical scheme based on a combination of qhomotopy analysis approach and Laplace transform approach to examine the FitzhughNagumo (FN) equation of fractional order. Kumar et al. [37] constituted a numerical algorithm based on the fractional homotopy analysis transform method to study the fractional model of Lienard’s equations. Baleanu et al. [38–44] considered the exact solutions of wave equations by the help of the local fractional Laplace variation iteration method (LFLVIM). They developed an iterative scheme for the exact solutions of local fractional wave equations (LFWEs).
We organize this paper as follows. In Section 2, we present some preliminaries which we use in this paper. In Section 3, we construct the reproducing kernel with fractional shifted Legendre polynomial form. In Section 4, we present the RKHSM for solving a secondorder fractional EmdenFowler problem. Convergence and error estimate of the proposed method are presented in this section. In Section 5, we present an iterative method to solve the secondorder fractional nonlinear EmdenFowler problem. This iterative method is a combination of the homotopy perturbation method and the RKHSM. Some numerical results are presented in Section 6 to illustrate the efficiency of the presented method. Finally, we conclude with some comments and conclusions in Section 7.
2 Preliminaries
In this section, we review the definition and some preliminary results of Caputo fractional derivatives as well as the definition of fractional shifted Legendre functions and their properties.
Definition 2.1
Definition 2.2
The basic concept of this paper is Legendre polynomials. For this reason, we study some of their properties.
Definition 2.3
The reproducing kernel Hilbert space method is a useful numerical technique to solve nonlinear problems. The reproducing kernel is given by the following definition.
Definition 2.4

\(K(\cdot,x)\in H\) for all \(x\in A\),

\((\phi (\cdot),K(\cdot,x))=\phi (x)\) for all \(x\in A\) and \(\phi \in H\).
The second condition is called the reproducing property, and a Hilbert space which possesses a reproducing kernel is called a reproducing kernel Hilbert space (RKHS).
3 Construction of a reproducing kernel with fractional shifted Legendre functions form
Theorem 3.1
Proof
4 Analysis of RKHSM for a secondorder fractional linear initial EmdenFowler problem
Theorem 4.1
Proof
In the next theorem, we present the error estimate of our approximation.
Theorem 4.2
Proof
5 Analysis of RKHSM for the secondorder fractional nonlinear initial EmdenFowler problem
6 Results and discussion
Example 6.1
Absolute error of Example 6.1
α  \(e_{\alpha }\) 

0.55  2.2∗10^{−16} 
0.65  2.4∗10^{−16} 
0.75  2.7∗10^{−17} 
0.85  2.3∗10^{−16} 
0.95  5.6∗10^{−17} 
1  0 
Example 6.2
Absolute error of Example 6.2
α  \(e_{\alpha }\) 

0.55  2.2∗10^{−16} 
0.65  6.5∗10^{−19} 
0.75  8.7∗10^{−19} 
0.85  4.3∗10^{−19} 
0.95  0 
1  8.7∗10^{−19} 
Example 6.3
Absolute error of Example 6.3
α  \(e_{\alpha }\) 

0.55  3.1∗10^{−17} 
0.65  1.0∗10^{−17} 
0.75  6.9∗10^{−18} 
0.85  0 
0.95  4.1∗10^{−18} 
1  3.5∗10^{−18} 
Example 6.4
Absolute error of Example 6.4
α  \(e_{\alpha }\) 

0.55  2.8∗10^{−14} 
0.65  0 
0.75  0 
0.85  3.5∗10^{−15} 
0.95  0 
1  0 
Example 6.5
Absolute error of Example 6.5
α  \(e_{\alpha }\) 

0.55  1.8∗10^{−12} 
0.65  2.3∗10^{−13} 
0.75  1.1∗10^{−13} 
0.85  0 
0.95  8.5∗10^{−14} 
1  0 
7 Conclusions
In this paper, we study the secondorder fractional EmdenFowler problem. The reproducing kernel Hilbert space method (RKHSM) is employed to compute an approximate solution to the proposed problem. The construction of the reproducing kernel based on the orthonormal shifted Legendre polynomials is presented. The validity of the RKHSM is ascertained by presenting five of our examples. It is worth mentioning that we get the same results as those in Wazwaz [48] for Examples 6.36.5 when \(\alpha =1\). We prove the existence of solution of the secondorder fractional EmdenFowler problem. The convergence of the approximate solution using the proposed method is investigated. The uniform convergence of the approximate solution to the exact solution is presented. Error estimation to the proposed method is proven. The results reveal that the proposed analytical method can achieve excellent results in predicting the solutions of such problems.
Declarations
Acknowledgements
The authors would like to thank the reviewers for the valuable comments.
Availability of data and materials
Not Applicable
Funding
Not Applicable
Authors’ contributions
The contribution of the authors is equal. 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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