a numerical approach to solve lane-emden-type equations ...fractional differential equations (fdes)...

A NUMERICAL APPROACH TO SOLVE LANE-EMDEN-TYPE EQUATIONSBY THE FRACTIONAL ORDER OF RATIONAL BERNOULLI FUNCTIONS

K. PARAND1,2, H. YOUSEFI2, M. DELKHOSH2

1Department of Cognitive Modelling, Institute for Cognitive and Brain Sciences,Shahid Beheshti University, G.C., Tehran, Iran

E-mail: k [email protected] of Computer Sciences, Shahid Beheshti University, G.C., Tehran, Iran

Received October 14, 2016

Abstract. In this paper, a numerical method based on the hybrid of the quasi-linearization method (QLM) and the collocation method is suggested for solving well-known nonlinear Lane-Emden-type equations as singular initial value problems, whichmodel many phenomena in mathematical physics and astrophysics. First, by using theQLM method, the nonlinear ordinary differential equation is converted into a sequenceof linear differential equations, and then the linear equations by the fractional order ofrational Bernoulli collocation (FRBC) method on the semi-infinite interval [0,∞) aresolved. This method reduces the solution of these problems to the solution of a sys-tem of algebraic equations. Computational results of several problems are presented todemonstrate the viability and powerfulness of the method. Further, the fractional orderof rational Bernoulli functions has also been used for the first time. The first zeros ofstandard Lane-Emden equation and the approximations of y(x) for Lane-Emden-typeequations are given with unprecedented accuracy.

Key words: Lane-Emden-type equations; Fractional order of rational Bernoullifunctions; Quasilinearization method; Collocation method; Isother-mal gas spheres; Astrophysics; Nonlinear ODE.

1. INTRODUCTION

Many problems arising in astrophysics, fluid dynamics, quantum mechanics,and other fields are defined on infinite or semi-infinite intervals. In general, mostof these problems are not solvable exactly and therefore should be investigated withthe help of semi-analytical or numerical approximation methods. These methodsinclude Adomian decomposition method [1, 2], Homotopy perturbation method [3,4], Variational iteration method [5–7], Exp-function method [8–10], Finite differenceapproximation method [11, 12], Finite element method (FEM) [12, 13], Meshfreemethods [14–18], Spectral methods [19–25], and so on.

Fractional calculus is used to model various different phenomena in nature andis an emerging field drawing much attention in both theoretical and applied disci-plines. Fractional differential equations (FDEs) are a branch of differential equa-tions that most of them do not have exact analytic solutions, hence researchers have

Romanian Journal of Physics 62, 104 (2017) v.2.0*2017.3.13#69455133

Article no. 104 K. Parand, H. Yousefi, M. Delkhosh 2

tried to find solutions of FDEs using approximate and numerical techniques. Severalmethods have been used to solve fractional differential equations, fractional partialdifferential equations, fractional integro-differential equations and dynamic systemscontaining fractional derivatives; see for example, Refs. [26]-[42]. In this paper,we have applied a numerical method based on fractional order of rational Bernoullifunctions to obtain approximate numerical solutions of Lane-Emden-type differentialequations.

In recent years, the studies of singular initial value problems in the ordinarydifferential equations (ODEs) have attracted the attention of many applied mathe-maticians and astrophysicists. One class of these equations are the Lane-Emden-typeequations that are second order nonlinear ordinary differential equations on semi-infinite interval [0,∞) and have a singularity at the origin.

MATHEMATICAL PRELIMINARIES ON LANE-EMDEN-TYPE EQUATIONS

The Lane-Emden equation describes a variety of phenomena in theoreticalphysics and astrophysics, including aspects of stellar structure, the thermal historyof a spherical cloud of gas, isothermal gas spheres, and thermionic currents.Let P (r) denote the total pressure at a distance r from the center of spherical gascloud. The total pressure is due to the usual gas pressure and a contribution fromradiation:

P =1

3ςT 4+

RT

v,

where ς , T , R and v are, respectively, the radiation constant, the absolute tempera-ture, the gas constant, and the specific volume [43]. Let M(r) be the mass within asphere of radius r and G the constant of gravitation. The equilibrium equations forthe configuration are

dP

dr=−GM(r)

r2, (1)

dM(r)

dr= 4πρr2,

where ρ is the density, at a distance r from the center of a spherical star.Eliminating M yields:

1

r2d

dr

(r2

ρ

dP

dr

)=−4πGρ.

Pressure and density ρ = v−1 vary with r and P =Kρ1+1m , where K and m

are constants.We can insert this relation into Eq. (1) for the hydrostatic equilibrium condition and

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3 A numerical approach to solve Lane-Emden-type equations Article no. 104

from this we rewrite the equation to:[K(m+1)

4πGλ

1m−1

]1

r2d

dr

(r2

dy

dr

)=−ym,

where λ represents the central density of the star and y is a dimensionless quantitythat are both related to ρ through the following relation:

ρ= λym.

Let:

r = ax,

a=

[K(m+1)

4πGλ

1m−1

] 12

.

Inserting these relations into our previous relations we obtain the Lane-Emden equa-tion [44]:

1

x2d

dx

(x2

dy

dx

)=−ym,

and simplifying the previous equation we have:

y′′(x)+2

xy′(x)+ym(x) = 0, x > 0,

with the boundary conditions:

y(0) = 1, y′(0) = 0.

It has been claimed in the literature that only for m = 0, 1, and 5 the solutionsof the Lane-Emden equation could be exact. For the other values of m, the Lane-Emden equation is to be integrated numerically. Thus we decided to present a newand efficient technique to solve it numerically. This problem has been studied bymany researchers and has been solved by different techniques; the main numericalmethods are listed in Table 1. The Lane-Emden problems have a long history andonly a small subset of numerical techniques is given in the bibliography table, seeTable 1.

In the present paper, we intend to extend the application of Bernoulli polynomi-als to solve Lane-Emden-type equations. First, using the quasilinearization method(QLM) these equations convert to a sequence of linear ordinary differential equationsto obtain the solution. Then, the equation will be solved on a semi-infinite domainby truncated fractional order of rational Bernoulli series defined as basis functionsfor the collocation method. The main characteristic of this technique is that it re-duces these problems to those of solving a system of linear algebraic equations, thusgreatly simplifying the problems. The main idea of this algorithm is to transform

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rational basis functions into fractional calculus and then apply the QLM method forthe solution of nonlinear singular initial value problems.

The outline of this paper is as follows: in Sec. 2, the basic definitions andproperties of the Bernoulli functions are expressed. In Sec. 3, the quasilinearizationmethod and the numerical scheme by using fractional order of rational Bernoullicollocation method are described briefly. In Sec. 4, we show that the numericalresults demonstrate the high accuracy and efficiency of the method. The conclusionsare given in the last Section.

2. PRELIMINARIES AND NOTATIONS

In this Section, some notations, definitions, and preliminary facts are presentedthat will be used further in this work.

2.1. BERNOULLI POLYNOMIALS

Bernoulli polynomials play an important role in various expansions and ap-proximation formulas, which are useful both in analytic theory of numbers and inclassical and numerical analysis. The Bernoulli polynomials and numbers have beengeneralized by Norlund [84] to the Bernoulli polynomials and numbers of higherorder. Also, Vandiver in [85] generalized the Bernoulli numbers. Analogous poly-nomials and sets of numbers have been defined from time to time, witness the Eu-ler polynomials and numbers and the so-called Bernoulli polynomials of the secondkind. These polynomials can be defined by various methods depending on the appli-cations [86]-[93].The classical Bernoulli polynomials and numbers are defined, respectively, by theexponential generating functions [94, 95]:

text

et−1=

∞∑n=0

tn

n!Bn(x), |t|< 2π, x ∈ R; (2)

t

et−1=

∞∑n=0

tn

n!Bn, |t|< 2π. (3)

It is known that there is an explicit formula

Bn(x) =

n∑k=0

(n

k

)Bkx

n−k, n= 0,1, ... (4)

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Table 1

Lane-Emden type equations bibliography

Year Author/Authors Comment

1977 Pascual [45] Approximate the equation with Pade series1977 Sharma [46] A non-elliptic solution of the equation for index m= 51980 Mohan & Al-Bayaty [47] Power-series solution of the equation1986 Horedt [48] The exact calculation equations to 7 decimal places.1989 Bender et al. [49] δ-perturbation expansion method1993 Shawagfeh [50] Calculating the equation with ADM1996 Liu [51] Providing an approximate analytic solution of the equation1999 Roxburgh & Stockman [52] Using the specific power series2001 Mandelzweig & Tabakin [53] Using the quasi-linearization method2001 Wazwaz [1] Adomian decomposition method with the modified structure2003 He [5] Ritz method2004 Parand & Razzaghi [54] The tau method together with rational Legendre functions2005 Ramos [55] Linearization techniques2006 Yousefi [56] Legendre wavelets method2006 Momoniat & Harley [57] Approximate implicit solution of the equation2007 Yildirim & Ozis [4] Homotopy perturbation method2007 Ramos [58] Piecewise-adaptive decomposition methods2008 Benko et al. [59] Nystrom methods2008 Parand & Taghavi [60] Generalized Laguerre polynomials collocation method2008 Khalique & Ntsime [61] Classifying the equation and applying the standard Lagrangian2008 Dehghan & Shakeri [6] The variational iteration method2009 Bataineh et al. [62] The homotopy analysis method2009 Parand et al. [63] Rational Legendre pseudospectral approach2009 Singh et al. [64] The modified homotopy analysis method2010 Adibi & Rismani [65] The modified Legendre-spectral method2010 Parand et al. [66] Rational Chebyshev collocation method2010 Parand et al. [67] Sinc collocation method2010 Karimi-Vanani & Aminataei [68] Using an integral operator and Pade series2011 Yuzbasi [69] A collocation method based on the Bessel polynomials2011 Boyd [70] Chebyshev Spectral Methods2011 Parand et al. [18] A numerical method based on radial basis functions2012 Bhrawy & Alofi [71] Jacobi-Gauss collocation method2012 Pandey & Kumar [72] Using a Bernstein operational matrix2012 Mutsa & Shateyi [73] A successive linearization method2013 Dehghan et al. [74] Based on a class of Birkhoff-type interpolation method2013 Doha et al. [75] Utilizing a second kind Chebyshev operational matrix2013 Parand et al. [76] Using Bessel orthogonal functions collocation method2013 Wazwaz et al. [77] The systematic Adomian decomposition method2014 Gurbuz & Sezer[78] Laguerre polynomial approach2014 Mall & Chakaraverty[79] Chebyshev Neural Network based model2014 Rach et al. [2] Adomian decomposition method2014 Doha et al. [80] Jacobi rational-Gauss collocation method2015 Nasab et al. [81] A numerical method based on hybrid of Chebyshev wavelets2015 Azarnavid et al. [82] Picard-reproducing kernel Hilbert space method2016 Parand & Khaleqi [83] Rational Chebyshev of second kind collocation method

where BK =BK(0) are the Bernoulli numbers for each k= 0,1, ...,n. Thus, the firstfour such polynomials are

B0(x) = 1, B1(x) = x− 1

2, B2(x) = x2−x+

1

6, B3(x) = x3− 3

2x2+

1

2x

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They satisfy to the following relations:the binomial relation

Bn(x+y) =

n∑k=0

(n

k

)Bk(x)y

n−k,

the differential equations

B′n(x) = nBn−1(x),

the reciprocal relations

Bn(1−x) = (−1)nBn(x),

as well as the difference equations

Bn(1+x)−Bn(x) = nxn−1

As a consequence, the relationship between Bernoulli and Euler polynomials is ob-tained from

Bn(x) =n∑

k=0

(n

k

)Bk(0)En−k(x).

2.2. FRACTIONAL ORDER OF RATIONAL BERNOULLI FUNCTIONS

Most important equations in physics, chemistry, mechanics, and mathemati-cal sciences occur in the semi-infinite or infinite intervals. To solve these equationsusing Spectral methods various strategies have been proposed [96]-[106]. Accord-ing to [107], Bernoulli polynomials form a complete basis over the interval [0,1]and Calvert et al. [108] applied the rational Bernoulli functions (Rn(x;L)) to solvedifferent problems on semi-infinite intervals. As discussed before, we can apply dif-ferent Spectral basis that are used to solve problems in the semi-infinite interval. Nowby using the algebraic mapping x→ xα; α > 0 on the rational Bernoulli functions,we define the fractional order of rational Bernoulli functions in the interval [0,∞).The new basis functions that are denoted by FRBn(x), are defined as follows:

FRBn(x;L) =Rn(xα;L) (5)

and an analytical form of FRBn(x;L) for n= 0,1, · · · is as follows:

FRBn(x;L) =

n∑k=0

(n

k

)Bk

(xα

xα+L

)n−k

, (6)

where the constant parameter L sets the length scale of the mapping and Bk(y) is theBernoulli polynomial of degree k.

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2.3. FUNCTION APPROXIMATION

We show how to approximate the unknown function by the rational functions.We determine w(x) = αLxα−1

(xα+L)2as a non-negative, integrable, and real-valued weight

function for the fractional order of rational Bernoulli functions over the semi-infiniteinterval. Now, we define Γ = {x|0≤ x≤∞} and

L2w(Γ) = {µ : Γ−→ R| µ is measurable and ∥µ∥w <∞}, (7)

where

∥µ∥w =

(∫ ∞

0|µ(x)|2w(x) dx

) 12

, w(x) =αLxα−1

(xα+L)2, (8)

is the norm induced by the inner product of the space L2(Γ),

⟨ν,µ⟩w =

∫ ∞

0ν(x)µ(x)w(x) dx. (9)

Now, we assume

Sn = {FRB0(x),FRB1(x), · · · ,FRBn(x)},

is a finite dimensional subspace, therefore Sn is a complete subspace of L2(Γ). Theinterpolating function of a smooth function ν on a semi-infinite interval is denotedby ϵnν. It is an element of Sn and

ϵnν =n∑

k=0

akFRBk(x),

Here ϵnν is the best projection of ν upon Sn with respect to the inner product Eq. (9)and the norm Eq. (8). Then, we have

⟨ξnν−ν,FRBi(x)⟩= 0 ∀ FRBi(x) ∈ Sn.

3. DESCRIPTION OF THE QLM-FRBC TECHNIQUE

A numerical method based on hybrid of quasilinearization method (QLM) andfractional order of rational Bernoulli collocation (FRBC) method is employed forsolving the Lane-Emden type equations.

3.1. THE QUASILINEARIZATION METHOD

Most algorithms for solving nonlinear equations or systems of equations arevariations of Newton’s method. When we can apply the collocation method to solvea nonlinear differential equation, the result is a nonlinear system of algebraic equa-tions. Nonlinearity is not a major complication for Spectral methods but for simplic-ity, we use the linearization algorithm called the quasilinearization method (QLM).

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The method, is constructed to solve a wide variety of nonlinear ordinary differentialequations or their systems arising in such different physics, engineering, and biologyproblems as orbit determination, detection of periodicities, radiative transfer, and car-diology [99, 109]. The aim of the quasilinearization method pioneered by Kalaba andBellman [110] is to solve a nonlinear nth order ordinary differential equation in Ndimensions as a limit of a sequence of linear differential equations. For the generalproperties of the method, particularly its uniform and quadratic convergence, whichoften is monotonic, see [53]. The QLM is easily understandable since there is no dif-ficult technique for obtaining the general solution of a nonlinear equation in terms ofa finite set of particular solutions [53]. Many researchers have used in various papersthis method, and it was extended, see Refs. [111]-[114].For simplicity, we limit our discussion to nonlinear ordinary differential equation inone variable on the interval [0, b], which could be infinite:

L(n)y(x) = f(y(x),y(1)(x), · · · ,y(n−1)(x),x), (10)

with n boundary conditions

gK = f(y(0),y(1)(0), · · · ,y(n−1)(0)) = 0, k = 1, · · · , l (11)

andgK = f(y(b),y(1)(b), · · · ,y(n−1)(b)) = 0, k = l+1, · · · ,n. (12)

Here L(n) is linear nth order ordinary differential operator and nth f and g1,g2, · · · ,gnare nonlinear functions of y(x) and their n+ 1 derivatives y(s)(x), s = 1, · · · ,n.The QLM prescription [112, 115] determines the (r+1)th iterative approximationyr+1(x) to the solution of Eq. (2.1) as a solution of the linear differential equation

L(n)yr+1(x) = f(yr(x),y(1)r (x), · · · ,y(n−1)

r (x),x)

+

n−1∑s=0

(y(s)r+1(x)−y(s)r (x)

)fy(s)(yr(x),y

(1)r (x), · · · ,y(n−1)

r (x),x), (13)

where y(0)r (x) = yr(x), with linearized two-point boundary conditions

n−1∑s=0

(y(s)r+1(0)−y(s)r (0)

)gky(s)(yr(0),y

(1)r (0), · · · ,y(n−1)

r (0),0), k = 1, · · · , l

andn−1∑s=0

(y(s)r+1(b)−y(s)r (b)

)gky(s)(yr(b),y

(1)r (b), · · · ,y(n−1)

r (b), b), k = l+1, · · · ,n.

Here the functions fy(s) =∂f

∂y(s)and gky(s) =

∂gk∂y(s)

, s= 1,2, · · · ,n−1 are functional

derivatives of the functionals f(yr(x),y(1)r (x), · · · ,y(n−1)

r (x),x) and

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gk(yr(x),y(1)r (x), · · · ,y(n−1)

r (x),x), respectively. The zero-th approximation y0(x)is chosen from mathematical or physical considerations.

3.2. METHOD OF SOLUTION

In this Section, a reliable algorithm that consists of two distinct approaches tohandle in a realistic and efficient way the Lane-Emden-type equations is introduced.The proposed approaches depend mainly on the QLM and we apply in transform-ing an ordinary nonlinear differential equation into a sequence of linear differentialequations.In general, the Lane-Emden-type equations are formulated as

y′′(x)+k

xy′(x)+f(x,y) = h(x), k,x⩾ 0, (14)

with the initial conditions

y(0) =A, y′(0) =B,

where k, A, and B are real constants and f(x,y) and h(x) are some given functions.By using the QLM , the the general solution of the Lane-Emden type equations de-termines the (r+1)th iterative approximation yr+1(x) as a solution of the lineardifferential equation:

y′′r+1(x)+

k

xy′r(x)+f(x,yr(x))−h(x)−

(yr+1(x)−yr(x))fy(x,y(x))+k

x

(y′r+1(x)−y′r(x)

)= 0, (15)


yr+1(0) =A, y′r+1(0) =B. (16)

In the other approaches, the fractional order of rational Bernoulli collocation (FRBC)method is employed for solving the linear differential equations at each iteration inEq. (15) with the boundary conditions Eq. (16).We suppose that y0(x) ≡ A. At the first step, the trial solution for the (r+1)-thiteration has been constructed as follows:

un,r(x) =n∑

i=0

ai,r FRBi(x;L), (17)

where n is a positive integer and our goal is to find the coefficients of ai,r.In the next step, the boundary conditions in Eq. (16) are satisfied.

yn,r+1(x) =A+Bx+x2 un,r(x). (18)

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By substituting Eq. (18) in Eq. (15), we form residual function at each iterationQLM as follows:

Resr(x) =y′′n,r+1(x)+

k

xy′n,r(x)+f(x,yn,r(x))−h(x)

− (yn,r+1(x)−yn,r(x))fy(x,y(x))+k

x(y′n,r+1(x)−y′n,r(x)).

(19)

The main goal of the collocation method is to minimize the residual function for cal-culating unknown coefficients . By putting the arbitrary collocation points {xj}nj=0

in Eq. (19), a system of n+1 linear equations at each iteration will be constructedthat can be solved by the Newton method for unknown coefficients of ai,r. Stabilityand convergence analysis of Spectral methods have proved in Ref. [21].The described method is summarized in the following algorithm:Algorithm of the methodBegin

1. Input: n, L, max iteration.2. Selection of an initial guess (According to the initial conditions

we suppose that y0(x)≡ A ).For r = 0,1,2, · · · ,max iteration do

3. Calculation a linear combination of trial functions4. Satisfy the boundary conditions by calculating function5. Calculation of the residual function Resr(x)6. Creating of the system of equations by putting the

collocation points {xj}nj=0 in Resr(x).7. Solving the system of linear equations.

End ForEnd

4. PROBLEM REPLACEMENT AND NUMERICAL RESULTS

In this Section, the approximate solutions of the Lane-Emden-type equationsusing the QLM-FREC method are presented. In what follows, several numericalproblems are given to illustrate the performance and reliability of the present methodsof solution. The results are tabulated and compared with accurate results of othermethods.In this study, the roots of the generalized fractional order of the Chebyshev functionsof the first kind on the interval [0,L]

xj = L(1− cos( (2j−1)π

(n+1)2 )

2

) 1α

j = 1,2, · · · ,n+1.

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with α = 1/2 have been used as collocation points [33] and all of the computationshave been done by software Maple 2015.

4.1. PROBLEM 1 (THE STANDARD LANE-EMDEN EQUATION)

For k = 2, f(x,y) = ym(x), h(x) = 0, A = 1, and B = 0, Eq. (14) has beendefined the standard Lane-Emden equation that was used to model the thermal be-havior of a spherical cloud of gas acting under the mutual attraction of its moleculesand subject to the classical laws of thermodynamics.

y′′(x)+2

xy′(x)+ym(x) = 0 x≥ 0, (20)

subject to the boundary conditions

y(0) = 1, y′(0) = 0,

where m is a constant. In Eq. (20) for m= 0, exact solution is y(x) = 1− 13!x

2, form= 1 is y(x) = sin(x)

x and for m= 5 is y(x) = 1√1+x2

3

.

In other cases, there is not any exact analytical solution.By using Eqs. (15) and (16), we have:

y′′r+1(x)+

2

xy′r+1(x)+m yr+1(x)y

(m−1)r (x)− (m−1)y(m)

r (x) = 0, (21)


yr+1(0) = 1, y′r+1(0) = 0, r = 0,1,2, · · ·max iteration. (22)

The QLM iteration requires an initialization or “initial guess” y0(x). We supposethat y0(x)≡ 1. According to the algorithm that was presented, the residual functionis formed at each iteration QLM is as follows:

Resr(x) = y′′n,r+1(x)+

2

xy′n,r+1(x)

+myn,r+1(x)y(m−1)n,r (x)− (m−1)y(m)

n,r (x). (23)

The results show that the method works with proposed basis functions effectivelyto solve the standard Lane-Emden equation. Accurate numerical results for integerand fractional values of the nonlinear exponent m to 21 and 15 decimal places arereported, respectively. Tables 2 and 3 show comparison of the first zeros of stan-dard Lane-Emden equations, for the QLM-FRBC and numerical results given byHordet [48], Boyd [70], Calvert et al. [108], Motsa & Shateyi [73], and Seidov[116], for m= 1.5,2,2.5,3,3.5, and 4 with N = 50, 10th iteration, and α= 0.5. Ta-bles 4 and 5 show approximations of y(x) for the standard Lane-Emden equations form = 1.5,2,2.5,3,3.5, and 4 with N = 50, 10-th iteration, and α = 0.5, respectively

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obtained by Horedt [48] and the methods proposed in this paper. Figure 1 representsthe graphs of the standard Lane-Emden type equations for m= 1.5,2,2.5,3,3.5,4.

Table 2

Comparison of the first zeros of standard Lane-Emden equations, for m= 2,3, 4, and 10-th iteration.

N m= 2 m= 3 m= 4

L= 4.35 L= 5.35 L= 14.9710 4.352881319803328205701 6.901388207013917025341 15.62762157118125791438820 4.352874595949825746840 6.896848606945193403930 14.97147168295803598149730 4.352874595946124712137 6.896848619377009479001 14.97154634821606931288440 4.352874595946124676976 6.896848619376960369021 14.97154634883850645258950 4.352874595946124676973 6.896848619376960375457 14.971546348838095068327

Ref.[70] 4.352874595946125 6.896848619376960 14.971546348838095[73] 4.352874595946125 6.896848619376960 14.971546348838095[116] 4.352874595946124 6.896848619376960 14.971546348838093[108] 4.35287460 6.89684862 14.9715463[48] 4.35287460 6.89684862 14.9715463

Table 3

Comparison of the first zeros of standard Lane-Emden equations, for m= 1.5,2.5,3.5,

and 10-th iteration.

N m= 1.5 m= 2.5 m= 3.5

L= 3.65 L= 5.35 L= 9.5310 3.653758092714 5.35487272539126 9.48865589602939220 3.653753754218 5.35527545804990 9.53580482400810030 3.653753737242 5.35527545899074 9.53580534425233640 3.653753736386 5.35527545900911 9.53580534424485850 3.653753736116 5.35527545901048 9.535805344244851

Ref.[70] 3.653753736219 5.355275459011 9.535805344245[73] 3.653753736219 5.355275459011 9.535805344245[108] ————— 5.3552754600 ————–[48] 3.65375374 5.35527546 9.53580534

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Table 4

The obtained values of y(x) for standard Lane-Emden equations m= 2,3 and 4 by the present method

for the Problem 1 (with N = 50, 10-th iteration and α= 0.5).

m x Ref. [83] Ref. [48] QLM-FRBC Res(x)

0.10 0.99833499 0.9983350 0.9983349985461484173818026 2.872e−170.50 0.95935270 0.9593527 0.9593527158033827008810774 8.404e−191.00 0.84865410 0.8486541 0.8486541114082496769115987 3.666e−20

2 3.00 0.24182408 0.2418241 0.2418240830523409167561426 3.902e−244.00 0.04884013 4.884015e−2 4.88401499759444464956266e−2 1.895e−254.30 0.00681093 6.810943e−3 6.81094327420583009083738e−3 2.083e−244.35 0.00036603 3.660302e−4 3.66030179361285732371457e−4 7.483e−24

0.10 0.99833582 0.9983358 0.998335829569169472981137 2.480e−130.50 0.95983906 0.9598391 0.959839069944851709269472 1.158e−141.00 0.85505756 0.8550576 0.855057568588626301786127 3.379e−15

3 5.00 0.11081983 0.1108198 0.110819835139625598858184 2.392e−216.00 0.04373798 4.373798e−2 4.3737983889708140055495e−2 2.680e−206.80 0.00416778 4.167789e−3 4.1677893654534600127584e−3 8.228e−216.896 0.00013365 3.601115e−5 3.6011145436709633317477e−5 1.276e−19

0.10 1.01585088 0.9.983367 0.99833665953955883440012 1.313e−100.20 0.99242503 0.9.933862 0.99338621353232638564112 8.595e−110.50 0.96031089 0.9.603109 0.96031090234223392518492 2.111e−121.00 0.86081381 .8.608138 0.86081381220840617132238 2.003e−11

4 5.00 0.23592273 0.2.359227 0.23592273104249156716087 2.761e−1410.0 0.05967377 5.967274e−2 5.967274158948779838014e−2 4.519e−1414.0 0.008.33293 8.330527e−4 8.330526695424819843717e−3 6.270e−1514.9 0.0005.7667 5.764189e−4 5.764188662135426830223e−4 6.752e−16

Fig. 1 – The obtained graphs of standard Lane-Emden equation solutions for several m for Problem 1.

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Table 5

The obtained values of y(x) for standard Lane-Emden m= 1.5, 2.5, and 3.5 by the present method for

Problem 1 (with N = 50, 10th iteration, and α= 0.5).

m x Ref. [83] Ref. [48] QLM-FRBC Res(x)

0.10 0.99833458 0.9983346 0.9983391507397410 5.760e−20.50 0.95910386 0.9591039 0.9591044903969852 1.097e−31.00 0.84516976 0.8451698 0.8451697099901000 4.398e−5

1.5 3.00 0.15885061 0.1588576 0.1588576084002391 1.729e−73.60 0.01107792 1.1090e−2 1.10909945425596e−2 1.607e−73.65 0.00076275 7.6392e−4 7.63924148072572e−4 1.315e−6

0.10 0.99820016 0.9983354 0.9983354163296818 2.157e−50.50 0.95959775 0.9595978 0.9595977541616153 4.102e−71.00 0.85194420 0.8519442 0.8519441989615764 8.833e−8

2.5 4.00 0.13768075 0.1376807 0.1376807330235731 2.419e−105.00 0.02901986 2.901919E−2 2.90191866495909e−2 1.519e−105.30 0.00425986 4.259544e−3 4.25954353378507e−3 3.688e−10

0.10 —– 0.9983362 0.9983362446858018 2.433e−100.50 —– 0.9600768 0.9600767558148295 9.164e−111.00 —– 0.8580096 0.8580095937895475 1.724e−11

3.5 5.00 —– 0.1786843 0.1786842657489721 6.540e−149.00 —– 1.180312e−2 1.18031215295922e−2 2.747e−149.50 —– 7.472341e−4 7.47234075333894e−4 1.430e−139.53 —– 1.207723e−4 1.20772344475770e−4 2.760e−13

4.2. PROBLEM 2 (THE ISOTHERMAL GAS SPHERES EQUATION)

According to Eq. (14), if k = 2, f(x,y) = ey(x), h(x) = 0 , A= 0, and B = 0,the isothermal gas spheres equation has been defined as follows:

y′′(x)+2

xy′(x)+ ey(x) = 0 x≥ 0, (24)


y(0) = 0, y′(0) = 0, (25)

Davis [117] and Van Gorder [118] have discussed about Eq. (24) that can beused to view the isothermal gas spheres, where the temperature remains constant.This equation has been solved by some researchers, for example Wazwaz [1] andChowdhury and Hashim [119] by using ADM and HPM, respectively, Parand et al.[76, 103] by using Hermite collocation method, and Bessel orthogonal functions col-location method, respectively. A series solution has been investigated by Wazwaz [1],Liao [120] and Singh et al. [64] by using ADM, ADM, and MHAM, respectively:

y(x)≃−1

6x2+

1

5!x4− 8

21.6!x6+

122

81.8!x8− 61.67

459.10!x10+ · · · . (26)

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By using Eqs. (15) and (16), we have:

y′′r+1(x)+

2

xy′r+1(x)+ eyr(x)

(yr+1(x)−yr(x)+1

)= 0 (27)


yr+1(0) = 0, y′r+1(0) = 0, r = 0,1,2, · · · . (28)

We suppose that y0(x)≡ 0, i.e. the initial guess is satisfying the boundary conditions.According to the algorithm that was presented, the residual function is formed at eachiteration QLM as follows:


2

xy′n,r+1(x)+eyn,r+1(x)

(yn,r+1(x)−yn,r(x)+1

)(29)

In this equation, the QLM-FRBC method with n = 40, 10th iteration, and L = 2.5are considered.Table 6 shows the comparison of y(x) obtained by the proposed method in this paperand those obtained by Wazwaz [1]. The resulting graph of the isothermal gas spheresequation in comparison to the presented method and the Log graph of the residualerror of approximate solution of the isothermal gas spheres equation are shown inFigs. 2 and 3.

Table 6

The obtained values of y(x) for the Lane-Emden equation by the present method for Problem 2.

x ADM Ref. [103] QLM-FRBC Res(x)

0.1 −0.0016658339 −0.0016664188 −0.0016658338620605875 7.705e−210.2 −0.0066533671 −0.0066539713 −0.0066533671004231439 5.738e−220.5 −0.0411539568 −0.0411545150 −0.0411539572927138819 6.117e−241.0 −0.1588273537 −0.1588281737 −0.1588276775243942141 5.155e−261.5 −0.3380131103 −0.3380198308 −0.3380194247607996012 1.092e−272.0 −0.5599626601 −0.5598233120 −0.5598230043355377849 1.401e−302.5 −0.8100196713 −0.8063410846 −0.8063408705983917220 7.758e−30

4.3. PROBLEM 3

According to Eq. (14), if f(x,y) = sinh(y(x)), A= 1 and B = 0 the equationhas been defined as follows:

y′′(x)+2

xy′(x)+sinh(y(x)) = 0 x≥ 0, (30)


y(0) = 1, y′(0) = 0,

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Fig. 2 – Graph of residual error for Problem 2.

Fig. 3 – Graph of solution for Problem 2.

A series solution that has been investigated by Wazwaz [1] by using Adomian De-composition Method (ADM) is

y(x)≃ 1− (e2+1)x2

12e+

1

480

(e4−1)x4

e2− 1

30240

(2e6+3e2−3e4−2)x6

e3

+1

26127360

(61e8+104e6−104e2−61)x8

e4+ · · · .

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By using the QLM, the solution of the equation determines the (r+1)th iterativeapproximation yr+1(x) as a solution of the linear differential equation:

y′′r+1(x)+

2

xy′r+1(x)+sinh(yr(x))+

(yr+1(x)−yr(x)

)sinh(yr(x)) = 0 (31)


yr+1(0) = 1, y′r+1(0) = 0, r = 0,1,2, · · · . (32)



2

xy′n,r+1(x)+sinh(yn,r(x))+

(yn,r+1(x)−yn,r(x)

)sinh(yn,r(x)) (33)

In this equation, the QLM-FRBC method with n= 40, 10th iteration, and dL= 2 isused.Table 7 shows the comparison of y(x) obtained by the proposed method in this paperand the results obtained by Wazwaz [1]. The resulting graph of the isothermal gasspheres equation in comparison to the presented method and the Log graph of theresidual error of approximate solution of the equation are shown in Figs. 4 and 5.

Table 7



0.1 0.9980428414 0.9980428414 0.998042841444807299 3.095e−230.2 0.9921894348 0.9921894347 0.992189434812197457 8.626e−250.5 0.9519611019 0.9519610925 0.951961092744912401 6.488e−271.0 0.8182516669 0.8182429282 0.818242928490522158 1.052e−321.5 0.6258916077 0.6254387632 0.625438763484943800 1.252e−342.0 0.4136691039 0.4066228877 0.406622887545649827 8.598e−38

4.4. PROBLEM 4

According to Eq. (14), if f(x,y) = sin(y(x)), A = 1 and B = 0 the equationhas been defined as follows:

y′′(x)+2

xy′(x)+sin(y(x)) = 0 x≥ 0, (34)


y(0) = 1, y′(0) = 0,

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The series solution investigated by Wazwaz [1] by using the Adomian DecompositionMethod (ADM) is

y(x)≃ 1− 1

6k1x

2+1

120k1k2x

4+k1( 1

3024k21−

1

5040k22)x6+k1k2

( 113

3265920k21

− 1

362880k22)x8+k1

( 1781

82892800k21k

22−

1

399168000k42−

19

23950080k41)x10+ · · · .

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By using the QLM, the solution of the equation determines the (r+1)th iterativeapproximation yr+1(x) as a solution of the linear differential equation:

y′′r+1(x)+

2

xy′r+1(x)+sin(yr(x))+cos(yr(x))

(yr+1(x)−yr(x)

)= 0 (35)


yr+1(0) = 1, y′r+1(0) = 0, r = 0,1,2, · · · ,max Iteration. (36)



2

xy′n,r+1(x)+sin(yn,r(x))+cos(yn,r(x))

(yn,r+1(x)−yn,r(x)

)(37)

In this equation, the QLM-FRBC method with n = 40, 10th iteration, and L = 2 isconsidered.


Table 8 shows the comparison of y(x) obtained by the proposed method in thispaper and the results obtained by Wazwaz [1]. The resulting graph of the equationin comparison to the presented method and the Log graph of the residual error ofapproximate solution of the isothermal gas spheres equation are shown in Figs. 6and 7.

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Table 8



0.1 0.9985979358 0.9986051425 0.998597927380765484 2.979e−220.2 0.9943962733 0.9944062706 0.994396264880561327 8.601e−240.5 0.9651777886 0.9651881683 0.965177780164444317 7.234e−261.0 0.8636811027 0.8636881301 0.863681125560823255 7.270e−281.5 0.7050419247 0.7050524103 0.705045233465384717 2.264e−332.0 0.5063720330 0.5064687568 0.506463627288849547 2.317e−36

5. CONCLUSIONS

The main purpose of this paper was to propose a powerful numerical methodfor solving Lane-Emden-type equations arising in astrophysics. The proposed me-thod is based on converting an ordinary nonlinear differential equation into a se-quence of linear differential equations through the quasilinearization method (QLM),which then are solved using the collocation method. The useful properties of QLMand fractional order of rational Bernoulli collocation (FRBC) method make it a com-putationally efficient method to approximate the solution on the semi-infinite interval[0,∞). The present method has several advantages, such as:

1. For the first time, to the best of our knowledge, the fractional order of rationalBernoulli functions (FRB) is introduced as a new basis for Spectral methods andthis basis can be used to develop a framework or theory in Spectral methods. One

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can see that some equations are solved more accurately using the fractional basis.

2. For the first time, to the best of our knowledge, the fractional basis was used forsolving an ordinary differential equation (nonlinear singular Lane-Emden-typedifferential equation) and it provided insight into an important issue.

3. The convergence of the obtained results is illustrated.

4. Accurate numerical results for the first root are achieved for the integer and frac-tional values of the nonlinear exponent m of the standard Lane-Emden equationsto 21 and 15 decimal places respectively, after only ten iterations. Moreover a verygood approximation solution of y(x) for Lane-Emden type equations is given.

5. This paper tells a quite complete history of different methods used in the literaturefor solving Lane-Emden-type equations.

6. The method is valid for large interval calculations and it was also shown that theaccuracy can be improved by increasing the number of collocation points.

7. The approximate solution obtained by the proposed method shows its superiorityon the other method.

8. The presented results show that this approach can be used for effectively solvingother initial and boundary value problems.

Acknowledgements. The authors are very grateful to reviewers for carefully reading the manuscriptand for their comments and suggestions that have improved the paper.

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(c) RJP 62(Nos. 1-2), id:104-1 (2017) v.2.0*2017.3.13#69455133

a numerical approach to solve lane-emden-type equations ...fractional differential equations (fdes)...

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