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An accurate method for solving a singular secondorder fractional EmdenFowler problem
Advances in Difference Equations volume 2018, Article number: 30 (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.
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).
In this paper, we discuss how to solve the following class of secondorder fractional EmdenFowler problems:
subject to
where a and b are constants. When \(\alpha =1\), \(\lambda =2\), and \(h(x)=1 \), Eq. (1.1) becomes the LaneEmden type equation. With some special forms of \(g(u(x))\), they describe many phenomena in mathematical physics and astrophysics such as the thermal behavior of a spherical cloud of gas, theory of stellar structure, theory of thermionic currents, and isothermal gas spheres. We use the RKHSM to solve it. We derive the reproducing kernel and the RKHSM. In addition, we prove the existence and uniform convergence of the approximate solution using this approach. Error estimation for our approximation is proved. Moreover, some numerical examples are presented in Section 6 to show the efficiency of the proposed method.
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.
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
The RiemannLiouville fractional integral operator \(I^{\alpha }\) of order \(\alpha >0\) on the usual Lebesgue space \(L_{1}[0,1]\) is given by
where \(\Gamma (\zeta )=\int_{0}^{\infty }t^{\zeta 1}e^{t}\,dt\) is the Euler gamma function, see [18, 45, 46].
For any \(f\in L_{1}[0,1]\), \(\alpha ,\beta \geqslant 0\), and \(\gamma >1\), \(I^{\alpha }\) exists for any \(x\in [ 0,1]\) and
Definition 2.2
The Caputo fractional derivative of order α is defined by
provided that the integral exists, where \(n= [ \alpha ] +1\), \([ \alpha ] \) is the integer part of the positive real number \(\alpha ,x>0\).
The Caputo fractional derivative satisfies the following properties for \(f\in L_{1}[0,1]\) and \(\alpha ,\beta \geqslant 0\):
and
The basic concept of this paper is Legendre polynomials. For this reason, we study some of their properties.
Definition 2.3
The Legendre polynomials \(\{\ell_{k}(x):k=0,1,2,\ldots \} \) are the eigenfunctions of the SturmLiouville problem
In order to use these polynomials on the interval \([0,1]\), we define the shifted Legendre polynomials by \(\ell S_{i}(x)=\ell_{i}(2x1)\). It is easy to see that
where \(\delta _{ij}= \bigl\{ \scriptsize{ \begin{array}{l@{\quad}l} 1 , & i=j \\ 0 , & i\neq j\end{array}}\). In this paper, we use orthonormal fractional shifted Legendre functions which are defined by \(S_{i}(x)=\sqrt{(2i+1)\alpha } S\ell_{i}(x^{\alpha })\), \(i=0,1,2,\ldots\) . Then
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
Let A be a nonempty abstract set. A function \(K:A\times A\rightarrow C\) is a reproducing kernel of the Hilbert space H if and only if

\(K(\cdot,x)\in H\) 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).
Construction of a reproducing kernel with fractional shifted Legendre functions form
Let
be the set of all orthonormal fractional shifted Legendre functions of degree \(\leq n\alpha \) with the following inner product:
and the norm
It is easy to see that Ω is a Hilbert space. In the next theorem, we show that Ω is a reproducing kernel space, and we present its reproducing kernel.
Theorem 3.1
Ω is a reproducing kernel space, and its reproducing kernel is
Proof
If \(u(x)\in \Omega \), then
Thus, for any \(y\in [ 0,1]\),
and
which completes the proof. □
Analysis of RKHSM for a secondorder fractional linear initial EmdenFowler problem
In this section, we discuss how to solve the following class of secondorder fractional linear initial EmdenFowler problems:
subject to
Let
Then L is a linear operator. Let \(\{ x_{1},x_{2},\ldots,x_{n} \} \) be n nodes of nodes in the interval \([0,1]\). In this paper, we choose
Let
for \(i=1,2,\ldots,n\), \(\psi _{n+1}(x)=S_{n+1}(x)\), and \(\psi _{n+2}(x)=S_{n+2}(x)\). Using the GramSchmidt orthonormalization to generate orthonormal basis \(\{ \overset{\_}{\psi }_{i}(x) \} _{i=1}^{n+2}\) of Ω, we have
where \(\alpha _{ij}\) are the coefficients of GramSchmidt orthonormalization, \(\alpha _{ii}>0\), \(i=1,2,\ldots,n+2\). In the next theorem, we show the existence of solution of Problem (4.1)(4.2).
Theorem 4.1
The approximate solution of Problem (4.1)(4.2) is given by
Proof
From Eq. (4.3) and from the fact that \(K_{n}(x,\eta )\) is a reproducing kernel and \(Lu(\eta )=f(\eta )\), we get
where \(\gamma _{1}=(u(x),\overset{\_}{\psi }_{n+1}(x))\) and \(\gamma _{2}=(u(x),\overset{\_}{\psi }_{n+2}(x))\). To find \(\gamma _{1}\) and \(\gamma _{2}\), we set
Next, we want to prove that \(u_{n}(x)\) is a solution to Eqs. (4.1)(4.2) at \(x_{i}\) for \(i=1,2,\ldots,n\). For any \(k\in \{1,2,\ldots,n\}\),
Then, for any \(m\in \{1,2,\ldots,n\}\),
where \(\delta _{ij}= \bigl\{\scriptsize{ \begin{array}{l@{\quad}l} 1, & i=j \\ 0, & i\neq j\end{array}} \bigr\} \). This implies that
for \(k=1,2,\ldots,n\). □
In the next theorem, we present the error estimate of our approximation.
Theorem 4.2
Let \(u_{n}(x)\) be the approximate solution of Eq. (4.1) in the space Ω and \(u(x)\) be the exact solution of Eq. (4.1). If \(\{ x_{i} \} _{i=1}^{n}\) is n nodes in \([0,1]\) and \(f(x)\in C^{4}[0,1]\), then
where C is constant and
Proof
Let
From Theorem 4.1, we see that
for \(i=1,2,\ldots,n\). Using Roll’s theorem, we get
and
Let \(l_{i}(x)\) be an interpolating polynomial of degree 1 of \(R_{n}^{\prime \prime }(x)\) at \(z_{1}\) and \(z_{i+1}\), where \(i\in \{1,2,\ldots,n2\}\). Then \(l\equiv 0\) on \([z_{i},z_{i+1}]\). Then there exists \(\zeta _{i}\in [ x_{i},z_{i+1}]\) and a constant \(\omega _{i}\) such that
Thus,
where \(\omega =\max \{\omega _{1},\omega _{2},\ldots,\omega _{n2}\}\). Hence,
which implies that
Moreover,
which implies that
Thus,
Now,
for some positive constant κ. Thus,
where \(\varrho = \Vert L^{1} \Vert _{\Omega } \kappa \). Then
In case that the initial condition is given by
we use the following change of variable:
Thus, our problem becomes
subject to
where \(g(x)=f(x)+\frac{b \lambda }{x^{\alpha }}\). Then we apply the procedure which is described in this section. □
Analysis of RKHSM for the secondorder fractional nonlinear initial EmdenFowler problem
In this section, we discuss how to solve the following class of secondorder fractional nonlinear initial EmdenFowler problems:
subject to
Let
We construct a homotopy as follows:
where \(\kappa \in [ 0,1]\) is an embedding parameter. If \(\kappa =0\), we get a linear equation
which can be solved by using RKHSM as we described in the pervious section. If \(\lambda =1\), it turns out to be Problem (4.1). Following the homotopy perturbation method [47], we expand the solution in terms of the homotopy parameter κ as
Substitute Eq. (5.4) into Eq. (5.3) and equate the coefficients of identical powers of λ to get the following system:
To solve the above equations, we use the RKHSM which is described in the previous section, and we obtain
where
From Eq. (5.4), it is easy to see that the solution to Problem (5.1)(5.2) is given by
We approximate the solution of Problem (5.1)(5.2) by
Results and discussion
In this section, we apply the fractional shifted Legendre functions reproducing kernel method outlined in the previous sections to solve numerically the following five examples. Note that the maximum number of terms in the series is \(M=15\). Let the absolute error be defined by
Example 6.1
Consider the following secondorder fractional EmdenFowler problem:
subject to
The graphs of the approximate solutions for different values of α are given in Figure 1. The absolute error obtained by the presented method is shown in Table 1.
Example 6.2
Consider the following secondorder fractional EmdenFowler problem:
subject to
The graphs of the approximate solutions for different values of α are given in Figure 2. The absolute error obtained by the presented method is shown in Table 2.
Example 6.3
Consider the following secondorder fractional EmdenFowler problem:
subject to
The graphs of the approximate solutions for different values of α are given in Figure 3. The absolute error obtained by the presented method is shown in Table 3.
Example 6.4
Consider the following secondorder fractional EmdenFowler problem:
subject to
The graphs of the approximate solutions for different values of α are given in Figure 4. The absolute error obtained by the presented method is shown in Table 4.
Example 6.5
Consider the following secondorder fractional EmdenFowler problem:
subject to
The graphs of the approximate solutions for different values of α are given in Figure 5. The absolute error obtained by the presented method is shown in Table 5.
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.
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DOI
MSC
 76A05
 76W05
 76Z99
 65L05
Keywords
 singular secondorder fractional EmdenFowler problem
 nonlinear initial value problem
 reproducing kernel Hilbert space method