Approximate Analytical Solution of Fractional Lane-Emden Equation by Mittag-Le ﬄ er Function Method

The classical Lane-Emden di ﬀ erential equation, a nonlinear second-order di ﬀ erential equation, models the structure of an isothermal gas sphere in equilibrium under its own gravitation. In this paper, the Mittag-Le ﬄ er function expansion method is used to solve a class of fractional Lane-Emden di ﬀ erential equation. In the proposed di ﬀ erential equation, the polytropic term f ( y ( x )) = y m ( x ) (where m = 0 , 1 , 2 ,... is the polytropic index; 0 < x ≤ 1) is replaced with a linear combination f ( y ( x )) = a 0 + a 1 y ( x ) + a 2 y 2 ( x ) + ··· + a m y m ( x ) + ··· + a N y N ( x ) , 0 ≤ m ≤ N , N ∈ N 0 . Explicit solutions of the fractional equation, when f ( y ) are elementary functions are presented. In particular, we consider the special cases of the trigonometric, hyperbolic and exponential functions. Several examples are given to illustrate the method. Comparison of the Mittag-Le ﬄ er function method with other methods indicates that the method gives accurate and reliable approximate solutions of the fractional Lane-Emden di ﬀ erential equation.


Introduction
Fractional calculus is a generalization and extension of classical calculus to non-integer orders. In recent times, fractional calculus has attracted the attention of researchers in several areas including mathematics, physics, biology, chemistry, engineering, economics and psychology ( [1], [2], [3], [4], [5], [6], [7], [8], [9]). Several definitions of fractional calculus have been formulated by researchers. The most common and widely used definitions are the Caputo, Grünwald-Letnikov and Riemann-Liouville derivatives ( [5], [8], [10]). The Grünwald-Letnikov derivative is mostly limited to numerical algorithms. The Riemann-Liouville fractional derivative has certain limitations and so becomes unsuitable in modeling some real-life phenomena since it requires the definition of fractional order initial conditions which have no physical meaningful explanation yet ( [8]). The main advantage of the Caputo derivative is that it takes on the same form of initial conditions for the integer-order differential equations.
Recently, the Lane-Emden equation has been investigated by several researchers due to its significant applications in mathematical physics and astrophysics ( [11]). The classical Lane-Emden equation, first introduced in 1870 by Lane and studied in 1907 by Emden is of the form: with initial conditions Equation (1) with various forms of f (y(x)) has been used to model several phenomena such as the theory of stellar structure, thermal explosions, the thermal behavior of a spherical cloud of gas, isothermal gas spheres and thermionic currents ( [12], [13]). Various methods have been presented by several authors to obtain the solution of the initial value problem (1) - (2). The Adomian decomposition method was employed in [11] to investigate the initial value problem (1) - (2). Solutions of the Lane-Emden equation have also been obtained via the series method (see e.g., [14], [15]). The series solutions obtained in [15] were compared with the results obtained using the homotopy perturbation method. A numerical algorithm was developed by Vanani and Aminataei [16] for solving the Lane-Emden equation. Ogunniran et al. in [17] investigated the linear stabilities of some explicit members of Runge-Kutta methods in integrating the Lane-Emden equation. However, the standard Lane-Emden differential equation is the one given precisely by the initial value problem ( [14]) where m is the polytropic index and y = y(x) is the polytrope. Clearly, the equation (3) is linear when m = 0, 1 and as a result the analytical solutions of the corresponding equations are realisable in closed forms. By extension it is mentioned in [14] that a closed form solution is also possible for m = 5. For numerical solutions of second order ordinary differential equations, see [18] and [19]. As a result of the significant importance of fractional calculus in modeling real-life phenomena accurately, the fractional Lane-Emden equation has been formulated and studied by researchers in very recent times. By using the collocation method, a numerical solution of (3) was obtained in [20]. Some other researchers have also sought numerical solutions to the fractional Lane-Emden equation (see, e.g., [21], [22]). Approximate solutions based on orthonormal Bernoulli's polynomials method, Homotopy-Adomian decomposition method and the series expansion method were treated in [23], [24] and [25] respectively. Other treatise on the fractional Lane-Emden equation can be found in [26], [27], [28], [29] and [30]. In [28, Subsection 5.2.2], the authors considered the power series solutions of the fractional Lane-Emden equation with the polytropic term y m with the fractional derivative described in the Caputo sense, while Malik and Mohammed [25] presented approximate solutions of fractional Lane-Emden equation using conformable Homotopy-Adomian decomposition method and conformable residual power series method.
Arafa et al. in [31] used the Mittag-Leffler function method to solve a simple fractional differential equation of the form with the fractional derivative described in the Caputo sense. In this paper, the Mittag-Leffler function expansion method is employed to solve a class of fractional Lane-Emden initial value problem whose polytropic term f (y(x)) = y m (x) (where m = 0, 1, 2, . . . is the polytropic index; 0 < x ≤ 1) is replaced with a finite series f (y(x)) = a 0 + a 1 y(x) + a 2 y 2 (x) + · · · + a m y m (x) + · · · + a N y N (x), 0 ≤ m ≤ N, N ∈ N 0 , with the fractional derivative described in the Caputo sense. The analytical solutions of the corresponding fractional equations, when f (y) are elementary functions, are presented, from which several examples of fractional Lane-Emden equations are given. In particular, we consider the special cases where f (y) are trigonometric, hyperbolic and exponential functions. Comparison of the Mittag-Leffler function method with other methods shows that the method is reliably capable of solving analytically, nonlinear fractional Lane-Emden differential equations, and by extension, one can apply the method to solve several nonlinear fractional Lane-Emden equations when the functions f (y) are given by other special functions. The motivation for the forms of the nonlinear function f (y) considered in this paper arises from the need to address those situations where the expansion coefficients a 0 , a 1 , . . . , a N−1 are not identically zero.
Definition 2.1. For α > 0, the Caputo fractional derivative of order α is defined as follows (m ≥ 1) : where is the Riemann-Liouville fractional integral of order α. Here The following properties of the Caputo derivative hold ( [8], [10]). Lemma 2.1. Suppose m − 1 < α < m; n, m ∈ N; α ∈ R; λ, σ ∈ C. Let f (x) and g(x) be such that c D α f (x) and c D α g(x) exist. We have that

Fractional Lane-Emden Equations and Their Approximate Solutions
In this section, we apply the Mittag-Leffler function discussed in Section 3 to obtain approximate solutions to the fractional Lane-Emden initial value problem Here c D µ denotes the Caputo fractional differential operator of order µ, µ > 0. The resulting solution is given as the Mittag-Leffler function. The function f (y(x)) will be chosen in such a way that it can be expanded in a power series, in particular, the Maclaurin series.
Towards this end, let the function f (y(x)) be given by the power series where a m , m = 0, 1, 2, 3, . . . , are the expansion coefficients (constants). For computation reasons, one is interested in the approximation We proceed by giving the following result that will be needed in the sequel.
Proposition 4.1. For 1 < α ≤ 2, 0 < x ≤ 1, the function f N (y(x)) admits the power series expansion where the constants C α ,N are given by Here the numbers B α,N ,m , m = 3, 4, 5, . . . are given by the finite series with the Mittag-Leffler expansion coefficients given by In particular, the special cases B α,N ,m , m = 0, 1, 2, are given respectively by Proof. By definition, we use the Mittag-Leffler function in (12) to see that where we have written (m ≥ 1) Let A α k,N be given by Then clearly, Furthermore, the classical Cauchy product tells us that Also, in a similar way, we see that That is, one has where Similarly, by taking more products, we see that where Continuing in this way, we obtain where (m = 3, 4, 5, . . . ) Upon inserting (30) -(31) into (19), one gets where and we obtain the result as required.
With the Proposition 4.1 in place we now present the main result of this paper.
admits a series solution given by the Mittag-Leffler function where The coefficients C α ,N , ≥ 0, are as given in Proposition 4.1. Proof. The starting point is to substitute the Mittag-Leffler function (18) in the equation (34) to evaluate the Caputo fractional derivatives c D α y(x) and x β−α c D β y(x). Towards this end, using Corollary 2.1, we have that (38) and similarly one sees that Substituting the series (38), (39) in (34) and then applying Proposition 4.1 gives Comparing the coefficients of x α , = 0, 1, 2, · · · on both sides of (41) gives b +1 and as a result, where the coefficients C α ,N are as given in Proposition 4.1.
Remark 4.1. For computation purposes, in addition to the first two coefficients B α,N ,m , m = 1, 2, given in Propostion 4.1, we present explicitly the coefficients B α,N ,m , 3 ≤ m ≤ 20: One sees that the fractional differential equation under consideration now takes the interesting form which clearly is a generalization of the fractional Lane-Emden equation in the sense that the new function f (y(x)) is a linear combination of the classical polytropic term y m (x), 0 ≤ m ≤ N. Equation (45) is the one considered in [28, Subsection 5.2.2] using the power series method. To make our calculations explicit and computationally interesting we consider those elementary functions whose expansion coefficients are explicitly known. Such elementary functions to be examined with their explicit associated expansion coefficients a m , m = 0, 1, 2, 3, . . . , are the trigonometric, hyperbolic and exponential functions. These functions are enumerated as follows.
For extensive discussions of models involving these functions as well as the physical interpretations of the solutions, see [12], [13], [42].
We proceed with the computation of the approximate solution of the problem (10) according to whether the functions f (y) are trigonometric, hyperbolic or exponential functions.

Lane-Emden Equation Involving Trigonometric Functions
In this subsection, we treat the equation (10) as well as (34) for which the functions f (y(x)) are trigonometric functions, namely, the sine function and cosine function. The Maclaurin series representations for these trigonometric functions are employed.
In this case the fractional initial value problem (10) reduces to a special one In solving the initial value problem (46), we consider the series expansion of sin y(x), namely, In this case the problem (34) becomes By Theorem 4.1, the analytical solution of the reduced fractional problem (48) is then given by the Mittag-Leffler function expansion where the associated expansion coefficients are given explicitly by The coefficients C α ,N , ≥ 0, appearing in (51) take the formulation In order to see explicitly the values of the coefficients C α ,N , we consider the first values N ∈ N 0 and this is presented as follows.
(a) N = 0. In this case, equation (48) reduces to the initial value problem It follows from Theorem 4.1 that the solution of the initial value problem (53) is given by the Mittag-Leffler expansion where the expansion coefficients B 0 admit the explicit formulation ( ≥ 0) with the numbers C α ,0 , ≥ 1, given by For further explicit calculations we proceed by assigning special values to the parameters α, β and ω and this is done in the following way as examples.
Substituting these coefficients into (58) gives the solution of the initial value problem (57): One clearly sees from the series (54) that the solution of the initial value problem (63) gives where the expansion coefficients B 0 = B 0 3 2 , 1 2 ; 2 admit the Gamma function representation and The first Mittag-Leffler expansion coefficients B 0 3 2 , 1 2 ; 2 , = 1, 2, 3, 4, 5, 6, are computed as follows: Upon substituting these coefficients into (64) yields the solution Indeed it is understood here that equation (48) reduces to the initial value problem It is also seen here that the solution of the fractional problem (69) gives with the expansion coefficients B 1 = B 1 (α, β; ω) ( ≥ 0) given by Here the coefficients C α ,1 , ≥ 0, on the right of (71) are given by where the numbers B α,N ,m , m = 1, 3, are as illustrated in (16) and (27) respectively. whose solution admits the series representation From equations (71) and (72), we respectively have and It is straightforward to see that the first Mittag-Leffler expansion coefficients B 1 (2, 1; 2), = 1, 2, 3, 4, 5, 6, are calculated as follows.

The Special Case f (y(x)) = cos y(x)
It is seen here that the fractional Lane-Emden equation under consideration is Finding the approximate solution of the initial value problem (86) amounts to solving the problem where we have We now proceed to the explicit computation of the first expansion coefficients B N (α, β; ω). This is carried out by first considering the first values of N ∈ N 0 .
Upon substituting these coefficients in (110) yields the series solution

Lane-Emden Equation Involving Hyperbolic Functions
This subsection discusses the approximate solution of the initial value problems (10) and (34) for which the functions f (y(x)) are the hyperbolic sine function and the hyperbolic cosine function. We also make use of the Maclaurin series representations for these hyperbolic functions.

The Special Case f (y(x)) = sinh y(x)
We consider the fractional Lane-Emden initial value problem In solving the initial value problem (115), we are interested in the series expansion of sinh y(x), so that finding the approximate solution of the problem (115) amounts to solving the corresponding problem We proceed to obtaining the solutions of the initial value problem (117) for the first values of N ∈ N.
(a) N = 0. Clearly, in this case a 1 = 1 and as a result the resulting initial value problem coincides with the problem in (53).

The Special Case f (y(x)) = cosh y(x)
We consider the fractional Lane-Emden equation In order to solve the initial value problem (130), we consider the series expansion of cosh y(x): and as a result we have the associated problem For N = 0, we observe that the resulting initial value problem coincides with the problem in (93). It follows immediately that the solution to the problem (133) is given by Example 4.11. Consider the problem One easily sees that the solution to the problem (135) gives

The Case of Exponential Functions f (y) = exp(±y)
In this subsection, we compute the approximate solution of the fractional Lane-Emden problem (10) as well as the initial problem (34) for which the functions f (y(x)) are exponential functions. The Maclaurin series expansion for these exponential functions are applied.

Fractional Isothermal Gas Sphere Equation
The fractional isothermal gas sphere equation is given by Also in this case, we look for the approximate solution of the problem (137) by solving the complementary equation The problem y + 2 x y + e y = 0, y(0) = 0, y (0) = 0 (139) models the isothermal gas spheres where the temperature remains constant and the index m is infinite ( [42]). It is straightforward to see by the initial conditions in (140) that the expansion coefficients appearing in the series solution of (140) are given by B 0 0 (2, 1; 2) = 0, with C 2 ,0 = δ 0 . Thus one has the Mittag-Leffler expansion coefficient (142) Consequently we obtain the closed form solution Example 4.13. N = 1: It is seen here that The coefficients of the series solution of (144) has the formulation B 0 1 (2, 1; 2) = 0, Further simplification gives the following first expansion coefficients B 1 (2, 1; 2), = 1, 2, 3, 4, 5. Upon substituting we obtain the solution Example 4.14. N = 2 : Consider the initial value problem The series solution of (149) has the following expansion coefficients.

Concluding Remarks
In this paper, we have used the Mittag-Leffler function expansion method to find approximate solutions of a class of fractional Lane-Emden equation whose nonlinear forms of f (y) are expressible as f (y(x)) = a 0 + a 1 y(x) + a 2 y 2 (x) + · · · + a m y m (x) + · · · + a N y N (x), 0 ≤ m ≤ N, N ∈ N 0 ; the values of the expansion coefficients a m , 0 ≤ m ≤ N, were explicitly provided. We considered the cases for which the functions f (y(x)) are trigonometric, hyperbolic and exponential functions. In all these cases, the Lane-Emden equations for the special values N = 0; α = 2, β = 1, ω = 2, coincide with the classical and standard ones ((57), (97), (140)) and consequently the corresponding solutions are given in closed forms ((62), (101), (143)).
In the case where the functions f (y) are exponential functions, our results can be compared with those of [11, equations 278 (59) and (70)] (see also [25, equation (25)]). The other nonlinear forms of f (y) are the trigonometric and hyperbolic functions. Our approximate solutions in the case of trigonometric functions are comparable with the solutions [11, equations (81) and (83) ]; and in the case of hyperbolic functions, see [11, equations (89) and (91)] (see also [22]).
The method employed in this paper can be applied to other similar cases in applied sciences where the models are given as strongly nonlinear ordinary differential equations. By extension, the method, can therefore, be accurately and reliably used to compute approximate solutions of nonlinear fractional differential equations of Lane-Emden type where the nonlinear forms of f (y) involve several other special functions.