On The Convergence Of The Series Solution Of A Lane - Emden Equation Of The Second Kind

Journal of Science and Technology

3 (1), 2009, 82 - 89

©BEYKENT UNIVERSITY

ON THE CONVERGENCE OF THE SERiES

SOLUTION OF A LANE-EMDEN EQUATION OF

THE SECOND KIND

Afgan ASLANOV

Department of Mathematics and Computing, Beykent University, Istanbul afganaslanov@beykent.edu.tr

Received: 09.01.2009 Accepted: 02.02.2009

Abstract

In this paper we study the series solution of Lane-Emden equation describing the dimensionless density distribution in an isothermal gas sphere. We

establish a new result on the radius of convergence of this solution.

Keywords: Isothermal gas sphere, Lane-Emden equation, Approximate solution, Hydrodynamics.

1.Introduction.

The basic equation of this study is the Lane-Emden equation of the second kind [4,8]

2

y" + - y' + e

y

= 0, y(0)=y

,

(0)=0, (1)

x

where =d/dx describing the non-dimensional density distribution y in an isothermal gas sphere. The transformation y^ -y transforms (1) into

2

-y '' + - -y ' - e

- y

= 0,

y(0)= ^ y'(0)=0.

This is an equation derived by Bonnor (1956) to describe what is now commonly known as Bonnor-Ebert gas spheres. The interested reader is also referred to the books Emden [6], Binney and Tremaine [2] and Kopenhahn & Weigert [8].

The equation (2) can easily be derived from Poisson's equation and the equation for hydrostatic equilibrium given by

dP dM(r)p(r)G dM . . . .

— =

v

' , =4 nr

2

p(r),

dr r2 dr

where P is the pressure at radius r, M is the mass of the star at radius r, and p is the density at a distance r from the centre of the star. We can combine these equations to obtain

1 d

f

r

2

dP ^

2

_{dr p(r) dr}

= -4nGp. (3)

For an isothermal gas, we have

P=Kp+D, (4) where K and D are constants. The constants K and D depend on the

thermodynamic properties of the isothermal gas sphere. In Chavanis [5] these constants are related to physical properties of finite isothermal spheres. Substituting (4) into (3) and making the substitutions

p= pce-y, r =

r \ i/ 2

' K ^

x

(5)

4n

Gp

c

J

(3) reduces to (2). A lower bound on the ratio p/pc >1/32.1 from (5) is obtained

by Chavanis [5] for the stability of finite isothermal spheres (Critical points of free energy with density contrast pc /p >32.1 are unstable saddle points). Eq.

(2) can be written as

~T d

f

x* d f ] = .

(6)

x dx

f

dx J

The transformation x^ix, y ^ -y transforms (1) into

y ' '+(2/x)y '+ e"y =0, y(0)=y'(0)=0. ( 7 )

Eq. (7) is used in Richardson's theory of thermionic currents [10] which is related to the emission of electricity from hot bodies.

Wazwaz [11] has used the Adomian decomposition method to determine a power series solution of (1):

1 2 1 4 8 6 122 8 61-67 10

y = — x

2

+

— x

4

x + x x

lu

+...

(8)

6 5! 6!- 21 8!- 81 10!- 495

The corresponding transformations easily transforms (8) into a power series solutions admitted by (2) and (7), respectively.

Momoniat and Harley [9] were interested in obtaining approximate solutions admitted by (1) that have a larger radius of convergence than the power series solution admitted by (1). They used the Lie group method (see [3,7]) to reduce

Second Kind

In this paper we try to analyze the series solution (8) (in fact, the power solution admitted by (1) is exactly the Adomian decomposition solution obtained by Wazwaz, [11]) by using a new inequality on the components of an approximate solution. This inequality will demonstrate that the radius of convergence of the power series is greater than or equal to V 2 and therefore, the bounds on the ratio p/pc will be improved.

2.A convergence radius of the power series solution

Usually, authors supposed that the radius of convergence is one (see for example, [9]). Now we shall establish that the radius of convergence is at least V 2 . We established the next relationship between the components of Adomian decomposition solution (see also, [1]):

2 2 y0

=

y

( 0 )

=

0; y

( x )

=

; y

2

(x)

=

-

—

x

—

y ;

2

•

3 2

•

2

•

5

x

2 f v2 f 3 A

3 • 2 • 7

y 3 ( x ) ^ 2 + y r ; =

-v

2

y

x

4 • 2 • 9

_v

y 3 + y i y 2 + 4 " ₆

_y

^ x

2

r

1

2

1

2

1

4

^

y5 ( x ) = - 5• 2•1 11 y 4 + 2 y 2 + y ^ 3 + 2 y y 2 yI J - - ( 9 )

For detailed proof of these relations in more general conditions, see [1]. Let us now prove the inequality

2k

I y

k

(x)!< . do)

We use the mathematical induction method. For k = 4,5 we have

I

y41= 1 y5 !

=

4 • 2 • 9 V1890 6 120 6 6

3

J 1632960 2

4

4'

x

10

r

61 1 1 1 1 1 1 1 1 1

5• 2-11V1632960 2 120

2

6 1890 2 36 120 4! 6

4 10

= 2.8014-10

-6

x

lu

= 0.00025x

10

.

2

5

5

3

Now let (10) be true for k = n - 1 and we shall establish this inequality for

y„

( x ) = - _{[ y„_ı + ( y y„-2 + y2 y „ - 3 + • • • ) + ( yı y y „ - 3 + ^ 2 y y2 y „ - 4 + • • • ) +} 2M(2M +1)

+(Y y y y y - 4 + Y yı yı y

2

y„-

5

+ •••)+•••+

ı ı ry ın - 4 y s + -ı

ryr

3

y

2

+-2(n

-

5)! ı r y n . y i y 2

+

(n

-

4)! (n

-

3)! (n

-1)!

where ek =1 if k ^ n - k - 1 and =1/2! otherwise, similarly Sk =1, if in the

corresponding expressionymyryqall indices m, r, q are different, and dk =1/2!

if just two indices are equal, that is, for example m = q, and dk =1/3! if m = r = q. In like manner, ji=1, 1/2!, 1/3!, 1/(2!2!) or 1/4!. Let us estimate the

expressions in the right hand side of (11). Using induction we have

(ıı) I y ı y n - 2 + £2 y 2 y n - 3 + • • • < x2 x2 ( n - 2 ) 2 ( n - 3 ) x2Ml x2M2 6 2n - 2( n - 2 ) 22 • ( ı 2 0 / 4 ) 2n - 3( n - 3 )3 2 n - 2 f 1 1 1 ( n - ı ) / 2 J.. \ 2mi • 2M M M 1 2 n-ı ı ı 3(n - 2)3 30(n - 3)3 236(n - 4)3

J

du u (n-1-u)3

where M1=(n-1)/2 or (n-2)/2 and M2 = M1 or M1 +1. The substitution u = (n - 1 ) s i n21 gives

du

J -T

_{J Qir} dt

uz(n-ı -u)3 ( n - ı )5 J sin51 cos51 ( n - ı )5

2

f

3 3 M sin t + 6 l n +

-J

J Ç1 ı dt

J

J Qir dt

sin31 cos51 J sin51 cos31

ı (n - 1 )5 ^ 2 c o s21 2 s i n21 cos t 4 c o s41 4 s i n4 t . f (n -1)5 3 n - 1 3 n - 1 2 n -1 - u 2 u + 6ln u 1 " n -1 " 2 1 " n - 1 " 2 ^ 1 - u U n - 1 - u 4 u _J

Then easily we have (n-1)/2 du u (n-1-u)3 (n-1)5 3 n - 1 3 n-1 I 3 + 6ln, 2 n - 4 2 3 1 (n-1)2_ + 1 (n-1) 2 A n - 4 4(n - 4)2 (n-1)5 for n >5, therefore, (n-1)2 n - 1 n - 4 — + +3ln 36 2 3 -L75

l<

2 n - 2 / 1 y y ^ + y2 yn-3+•••< - ¿ n r ı (n-1)5 ı 0J76

""İT+

006(

" "

ı|!

J< J

ı 2 n-2 X n - 1 0^56

-0 1 7 6 3(n - 2)3 + 30(n - 3)3 + 236(n - 4)3 + (n-1)3 (12) 2

On The Convergence Of The Series Solution Of A Lane - Emden Equation Of The Second Kind

For the expression (¿1 y y yn-3 + 52 y1 y2 yn_ 4 + . . . ) in (11), grouping we

obtain

I ¿1 y y 1 y n - 3 + ¿ 2 y y 2 y n - 4 + ...|<l y11 ^ | \ y1 yn-31 +1 y2yn-41 + . . .j

+1 y 21 ( - 21 y2 y«-51 +1 y3y„-61 + - j + . . . +1 yKx y^2 y ^3 1 ,

where K1=(n-2)/3, (n-1)/3 or n/3, K1< K2< K3 and K3= K1 or K1+1. Using (12) we can estimate the parentheses in the right hand side of this expression:

1, ,

x

21 y y n - 31+1 y y u ! +••• < 2 2 n - 4 f 1 1 1 0.176 j _{1 1 1} 3 j n-2 6(n - 3)3 30(n - 4)3 236(n-5)3 (n - 2) xLn-4 0 . 5 6 - 1 / 6 x2 n - 4 0.4 < <• 2n - 2 (n - 3)3 2n - 2 (n - 3 )3' and similarly, 1 1 x 2 ( n - 3 ) 0 4

o1 yIyN-51+1 y^.y„-61 + - < ~1 y y - 41+1 yIyN-5 \ +... < '

1 x 2 n - 8 0 4 2 \ y 3 y n - 7 \ + \ y 4 y n - 8 \ + . . . < 2n - 4 n - 5) 3 ^ Thus we have x 2 n - 4 0 4 \ ¿ 1 y 1 y 1 y n - 3 + ¿ 2 y y 2 y n - 4 + . . . \<\ y 1 \ • ' 2n - 2 (n - 3) 3 x 2 n - 6 0 4 x 2 n - 8 0 4 + \ y2 \ 2n-3 n - 4 ) 3 + \ y3 \ 2n-4 („ - 5 )3 + • " + \ y^yK2yK % \ •2n-20 4 f 1 1 1 1 j

<-

x 0.4 2' in-3 ^ (w-3)3 30(n - 4)3 236(n-5)3 43(n - 6)3 j ... \ y K3 x2n-20.4 • 0 . 5 6 1 , ,

<

; 7. (13) 2n - 1 (n

-

3)3

Now grouping the expression yx y1 y1 y yn-4 + y2 y1 y1 y2 yn-5 + . . . in like manner we have

+

1

y2

!

V 6 y^y^y-7

1 1

y2y;yn-8

1

+... J+... <1 yJ V ^ w „ - 4

1

+

1 1

y y ^

1

+

, f 1, . 1. . j 0.4• 0.56• 0.56x

2n-2

,

x

+

\

y2

\

I

6 \

^ > ^ - 6

\

+ - I

+...

<

r^^

- 4) 3 ( 1 4 )

Thus (11), (12), (13) and (14) (and analogs of Eq. (14) for other terms) together imply

I yn ( x)|<

_{2n(2n +1)2}

2n n - 1

x

2( n-1)

0.56 x

2( n-1)

0.4 • 0.56 x

2( n-1)

(n-1)

3

( n - 2)

3

(n

-

3)

3

• +

...

<

2n(2n +1)2

n - 1

1

0.56 0.4 • 0.56 0.4 • 0.56

2

(n

-1)

3

(n

-

2)

3

(n

-

3)

3

(n

-

4)

3

- +

...

Consider the expression

hg 0.56

n-1-k

k

3

k=5

Since the terms of the series increases with respect to k > 5, we obtain

hh 0.56

n-2-k

< n - 5 < 1

h

k

3 <

(n-1)

3

< ( n - 1 )

2 , and therefore, \yn

(x)

\<-2n

_I

n n-1 nn-1

. 2

n -5 2

2

h — —

+ — — —

yn5y22

+

2n(2n + 1)2

n-1

(n -1)

2

2(n - 5 ) !

yn

-4

y3

o n - 1 o n - 1

y ^

y n- 1]

<-2n

(n

-

3)!'

2

n-1

(n-1)!-1

(n

-

4)!

2"

-1

1 1

n - 5

2n(2n+1)2

n-1

(n-1)

2

2(n - 5 ) ! 6

n

120'

1 1 2

n-1

1 1

~î"-1

( n - 4 ) ! 6

n-4

1890 (n-3)! 6

n-3

120 6

n - 1

(n- 1)!

J

It is easy to show that

2

n-1

1 1 1 2

n-1

1 1 1

<

<-2(n-5) ! 6

n-5

120

2

(n-1)

3

' ( n - 4 ) ! 6

n-4

1890 (n-1)

n - 1

_{1 1 2}

n-1

<

1

2(n-3)!6

n - 3

120 6

n - 1

(n-1)! (n-1)

3

'

for n >6, and therefore,

2n

!y

n

( x)|<

4n(n

+

0.5)2

n - 1

1 3

(n -1)

(n -1)

2n

<

_{3 •}

2

n

n

Second Kind

Now this is clear that the approximate solution

y (x)

=

y^x)

+

y

2

(x)

+...

+

y

n

(x)

+...

converges for all x2<2. Taking x = V 2 in (8) we have that the (alternating)

series converges and

0 > y ( V 2 ) > - 2 / 6 = - 1 / 3 .

Therefore, for the expression p / pc we have

p/p

c

> e

1/3

= 1.3956...

The ratio given in (15) is larger than the result in Momoniat and Harley [9] by 1.3956-1.18867426= 0.20693 and the series solution is valid on the

dimensional domain

K

4nGp

c

Momoniat and Harley [9] taking the first 10 terms in the series representation

r < V2

1/2

obtained the result

K

r <1.091

4nGp

c

1/2

Concluding remarks

We have investigated the convergence of the series solution of Lane-Emden equation of the second kind. It has been shown that the corresponding series solution converges for x < V 2 . This result is useful to overcome the difficulties in the study some other type of solutions. For example, in Momoniat and Harley [9] the implicit approximate solution diverges from the power solution for x>1. The main difficulty in their approach is that they were restricted by the condition x<1. For example, they used the approximation

(1 + x)n « 1 + nx which holds for x<<1. We do not think that the divergence

of the approximate implicit solution from the power solution is a good phenomenon. The main problem is to find a more proper algorithm with more suitable restrictions.

REFERENCES

[1]. Aslanov, A., A; generalization of Lane-Emden equation. Int. J. Comp. Math. 85, 1709-1725 (2008).

[2]. Binney, J., Tremain, S.; Galactic Dynamics. Princeton University Press, Princeton, NL (1987).

[3]. Bluman, G.W., Kumei, S.; Symmetries and Differential Equations. Springer, Berlin (1989).

[4]. Chandrasekhar, S.; An Introduction to the Study of Stellar Structure, Dover Publications, Inc., New York (1939).

[5]. Chavanis, P.-H.; Gravitational instability of finite isothermal spheres. arXiv:astro-ph/0103159 v2, 17 July 2001.

[6]. Emden, R.; Gaskugeln-Anwendungen der Mechan. Warmtheorie, Druck und Verlag Von B. G. Teubner, Leipzig and Berlin (1907).

[7]. Ibragimov, N.H.; Elementary Lie Group Analysis and Ordinary Differential Equations. Wiley, Chichester (1999).

[8]. Kippenhahn, R., Weigert, A.; Stellar Structure and Evolution. Ch. 19 Springer Verlag, Berlin (1990).

[9]. Momoniat, E., Harley, C.; Approximate implicit solution of a Lane-Emden equation, New Astron. 11, 520-526 (2006).

[10]. Richardson, O.U.; The Emission of Electricity from Hot Bodies, Longman's Green and Company, London (1921).

[11]. Wazwaz, A.M.; A new algoritm for solving differential equations of Lane-Emden type, Appl. Math. Comput. 118, 287, (2001).