Solution of Lane-Emden Equation by Differential Transform Method 1 ...

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Jan 4, 2012 - Keywords: Lane-Emden equation; differential transform method; polytropic index; analytic solution; non- linear differential equation.
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.12(2011) No.4,pp.478-484

Solution of Lane-Emden Equation by Differential Transform Method Supriya Mukherjee1 ∗ , Banamali Roy2 , Pratik Kumar Chaterjee3 1

3

Department of Mathematics, Swami Vivekananda Institute of Science and Technology South Gobindapur, P.S- Sonarpur, Kolkata-700145, West Bengal India. 2 Department of Mathematics, Bangabasi Evening College, 19 Rajkumar Chakraborty Sarani, Kolkata -700009, West Bengal, India. Student of Computer Science and Engineering , Swami Vivekananda Institute of Science and Technology, South Gobindapur, P.S- Sonarpur, Kolkata-700145, West Bengal, India. (Received 16 February 2011, accepted 26 September 2011)

Abstract: In this paper, we have tried to derive solutions of one of the widely studied and challenging equations in nonlinear dynamics the Lane-Emden y ′′ (x) + x2 y ′ (x) + y n = 0 equation for n =0, 1, 2, 3, 4 and 5, by a relatively new exact series method of solution known as the differential transform method (DTM). The Lane-Emden equation describes a variety of phenomena in theoretical physics and astrophysics. Keywords: Lane-Emden equation; differential transform method; polytropic index; analytic solution; nonlinear differential equation

1

Introduction

Since the beginning of stellar astrophysics, the investigation of stellar structure has been a central problem. There have been continuous efforts to deduce the radial profiles of pressure, density and mass of a star, and one of the key results that came out of these efforts are the Lane-Emden equation. First published by Jonathan Homer Lane in 1870 [1], the Lane-Emden equation describes the density profile of a gaseous star. Mathematically, the Lane-Emden equation is a second-order ODE with an arbitrary index, known as the polytropic index, involved in one of its terms. In astrophysics, the LaneEmden equation is essentially a Poisson equation for the gravitational potential of a self-gravitating, spherically symmetric polytropic fluid [1][2]. The Lane-Emden equation describes a variety of phenomena in theoretical physics and astrophysics, including aspects of stellar structure, the thermal history of a spherical cloud of gas, isothermal gas spheres, and thermionic currents [3].Since then the equation has been a center of attraction. Various authors have derived the solution to this equation both numerically and analytically for different values of the polytropic index. For example, Ramos used multistep methods [4] in obtaining analytical solutions of the Lane-Emden equation, Abbasbandy [5] showed that the results given by the Homotopy Perturbation Method are divergent in the case of strong nonlinearity when large physical parameters are involved and Liao [6] solved the above system by Homotopy analysis Method (HAM) for the special case of a = 1. Gorder et.al [7] have also solved the same problem by HAM for the case a > 0.Thus it is evident that the Lane Emden equations have received attention from both mathematicians and physicists for the past few years. This paper attempts to present the solutions of the Lane-Emden equation for the following values of polytropic indices, 0, 1, 2, 3, 4 and 5 by the Differential Transform method . This technique doesnt require any discretization, linearization or small perturbations and therefore it reduces significantly the numerical computation. The concept of differential transform was first introduced by Zhou [8] in solving linear and nonlinear initial value problems in electrical circuit analysis. The traditional Taylor series method takes a long time for computation of higher order derivatives. Instead, DTM is an iterative procedure for obtaining analytic Taylor series solution of differential equations and is much easier. In our previous work we have seen that the DTM provides the solution of the Duffing-Van der Pol oscillator equation in a rapidly convergent series [9] and that, it is in good agreement with the solution obtained by Chandrasekar et.al. [10]. ∗ Corresponding

author.

E-mail address: supriya [email protected] c Copyright⃝World Academic Press, World Academic Union IJNS.2011.12.31/565

S. Mukherjee, B. Roy, P. K. Chaterjee: Solution of Lane-Emden Equation by Differential Transform Method

2

479

Formulation of the problem

The equation of motion for a stars radius for spherical symmetry [3] is given by ρ

d2 r Gρ(r)M (r) dP (r) =− − dt2 r2 dr

(1)

where M (r), ρ(r) and P (r) describes the mass included within a radius r, the density at radius r, and the pressure at radius r, respectively. Now, assuming that the star is at hydrostatic equilibrium, LHS of the equation vanishes and we obtain, dP (r) Gρ(r)M (r) =− . (2) dr r2 Now, we have a situation with three unknowns (P, ρ, M ) and one relation. The other relation is a trivial description of the mass profile, given by dM (r) = 4πr2 ρ(r) (3) dr and the polytropic assumption is given by P (r) = Kρ(r)γ , (4) where, K is a positive constant and γ is the ratio of specific heat at constant pressure to that at constant volume. Although this assumption does not have any theoretical background, it has been experimentally verified that certain stars, such as white dwarfs, can be described quite well with polytropic models [11]. This assumption is a key to the derivation of the Lane-Emden equation. Differentiating equation (2) with respect to r and substituting equation (4) we obtain, 1−n dρ(r) n+1 k d 2 (r ρ(r) n ) = −4πGρ(r), 2 n r dr dr

where γ =

n+1 n

(5)

and n is the polytropic index. Introducing dimensionless variables x and y given by ρ(r) = ρc γ n (r)

and

(6)

1−n

Kρc n 1 r = ((n + 1) ) 2 x, 4πG where ρc is the density at the center of the star, the above equation (5) reduces to 1 d 2 dy(x) {x } + y n (x) = 0. x2 dx dx

(7)

(8)

On simplification of equation (8) we obtain the known form of the Lane-Emden nonlinear equation given by d2 y 2 dy + + y n = 0. 2 dx x dx

(9)

In this paper we have solved the above equation (9) for n = 0, 1, 2, 3, 4 and 5 using DTM with the initial conditions y(0) = a, y ′ (0) = 0.

(10)

The following section describes the Differential transform method.

3

Differential transform method

The differential transformation technique is one of the semi numerical analytical methods for ordinary and partial differential equations that use the form of polynomials as approximations of the exact solutions that are sufficiently differentiable. The basic definition and the fundamental theorems of the DTM and its applicability for various kinds of differential equations are given in [12] - [17].The method is an analytical and numerical method for solving a wide variety of differential

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International Journal of Nonlinear Science, Vol.12(2011), No.4, pp. 478-484

equations and usually gets the solution in a series form. Differential transform of a function f (x) is defined as follows: F (k) =

1 dk f (x) ]x=0 . [ k! dxk

(11)

In (11), f (x) is the original function and F (k) is the transformed function. The Taylor series expansion of the function f (x) about a point x = 0 is given as, ∞ ∑ xk dk f (x) f (x) = [ ]x=0 . (12) k! dxk k=0

Replacing

k 1 d f (x) k! [ dxk ]x=0

by F (k), we have, f (x) =

∞ ∑

xk F (k),

(13)

k=0

which may be defined as the inverse differential transform. From (11) and (12) it is easy to obtain the following mathematical operations: (i) If f (x) = g(x) ± h(x), then F (k) = G(k) ± H(k). (ii) If f (x) = cg(x), then F (k) = cG(k), where c is a constant. n (k+n)! (iii) If f (x) = d dxg(x) G(k + n). n , then F (k) = k! ∑k (iv) If f (x) = g(x)h(x), then F (k) = l=0 G(k)H(k − l). (v) If f (x) = xn , then F (k) = δ(k − n), where δ is the Kronecker delta. ∫x (vi) If f (x) = 0 g(t)dt, then F (k) = G(k−1) , where k ≥ 1. k (vii) If f (x) = u(x)v(x)w(x)...p(x), then ∑k ∑k−r ∑k−r−s...m U (r)V (s)W (m)...P (k−r−s−...m), where F (k), G(k), H(k), U (k), V (k), W (k), P (k) F (k) = r=0 s=0 ... m=0 are the differential transform of the functions f (x), g(x), h(x), u(x), v(x), w(x), p(x) respectively.

4

Solution of Lane-Emden equation by DTM

Recently, many analytical methods have been used to solve Lane-Emden equations. Most techniques in use for handling this equation are based on either series solutions or perturbation techniques. To recall a few, Bender et al. [18], proposed a new perturbation technique based on an artificial parameter δ, the method is often called δ-method whereas Shawagfeh [19] applied a nonperturbative approximate analytical solution for the Lane-Emden equation using the Adomian decomposition method. Mandelzweig et al. [20]used quasilinearization approach to solve the same. Parand et al. [21], [22] and [23] presented some numerical techniques to solve higher ordinary differential equations such as Lane-Emden. Their approach was based on a rational Chebyshev and rational Legendre tau method.Yousefi [24] presented a numerical method for solving the Lane-Emden equations as singular initial value problems. Recently, Dehghan and Shakeri [25] first applied anexponential transformation to this equation. Thus, there are various approaches towards solving the Lane-Emden equations. Here we present a rather easier method to solve this equation for the values of the polytropic index n = 0, 1, 2, 3, 4 and 5. The Lane-Emden equation as given by d2 y 2 dy n equation (9) is dx = 0 with the initial conditions y(0) = a, y ′ (0) = 0. (i) For the case n = 0, the equation 2 + x dx + y (9) takes the form, d2 y 2 dy + + 1 = 0. dx2 x dx Applying Differential Transform (DT) to equation (14) we have, dy d2 y DT [x dx 2 + 2 dx + x = 0], i.e. k ∑

δ(l − 1)(k + 2 − l)T (k + 2 − l) + 2(k + 1)T (k + 1) + δ(k − 1) = 0.

(14)

(15)

l=0

From the initial conditions (10) and using (12) we have, T (0) = a, T (1) = 0.

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(16)

S. Mukherjee, B. Roy, P. K. Chaterjee: Solution of Lane-Emden Equation by Differential Transform Method

481

For k = 1, 2, 3, ...10, from equation (15, 16) we have, 1 T (2) = − , 6

T (3) = 0 , T (4) = 0 = ... = T (10).

(17)

Thus, using equation (12) we have, y(x) = a −

x2 . 6

(18)

(ii) For the case n = 1, the equation (9) takes the form, d2 y 2 dy + + y = 0. dx2 x dx

(19)

Applying DTM we have, k ∑

δ(l − 1)(k + 2 − l)T (k + 2 − l) + 2(k + 1)T (k + 1) +

l=0

k ∑

δ(l − 1)T (k − l) = 0.

(20)

l=0

Using the initial conditions and equation (12) we get, a T (2) = − , 6

T (4) =

Thus, using equation (12) we have, y(x) = a −

a a , T (6) = − , 120 5040 ax2 6

+

ax4 120



y(x) = a

ax6 5040

T (3) = T (5)... = 0.

(21)

+ ... which gives the solution in the form,

sinx . x

(22)

(iii) For the case n = 2, the equation (9) takes the form, d2 y 2 dy + + y 2 = 0. dx2 x dx

(23)

Applying DTM we have, k ∑

δ(l − 1)(k + 2 − l)T (k + 2 − l) + 2(k + 1)T (k + 1) +

k k−s ∑ ∑

δ(s − 1)T (m)T (k − s − m) = 0.

(24)

l=0 m=0

l=0

Using the initial conditions and equation (12) we get, 2

T (2) = − a6 , T (4) =

a3 60 ,

4

T (6) = − 11a 7560 , T (8) =

a5 8505 ,

6

6929876a T (10) = − 85056944162400 ,

T (3) = T (5) = T (7) = T (9) = ... = 0.

(25)

Thus, using equation (12) we have, the solution in the form, y(x) = a −

a2 x2 a3 x4 a4 x6 a5 x8 6929874a6 x10 + − + − + ... . 6 60 7560 8505 85056944162400

(26)

(iv) For the case n = 3, the equation (9) takes the form, d2 y 2 dy + + y 3 = 0. 2 dx x dx

(27)

Applying DTM we have, k ∑

δ(l − 1)(k + 2 − l)T (k + 2 − l) + 2(k + 1)T (k + 1)

l=0

+

k k−r ∑ ∑ k−r−s ∑

(28) δ(r − 1)T (s)T (m)T (k − r − s − m) = 0.

r=0 s=0 m=0

Using the initial conditions and equation (12) we get,

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International Journal of Nonlinear Science, Vol.12(2011), No.4, pp. 478-484 3

T (2) = − a6 , T (4) =

a5 40 ,

7

T (6) = − 19a 5040 , T (8) =

619a9 1088640 ,

11

17117a T (10) = − 199584000 ,

T (3) = T (5) = T (7) = T (9) = ... = 0.

(29)

Thus, using equation (12) we have, the solution in the form, y(x) = a −

a3 x2 a5 x 4 19a7 x6 619a9 x8 17117a11 x10 + − + − + ... 6 40 5040 1088640 199584000

(30)

(v) For the case n = 4, the equation (9) takes the form, d2 y 2 dy + y 4 = 0. + dx2 x dx

(31)

Applying DTM we have, k ∑

δ(l − 1)(k + 2 − l)T (k + 2 − l) + 2(k + 1)T (k + 1)

l=0

+

k k−r ∑ ∑ k−r−s ∑ k−r−s−m ∑ r=0 s=0 m=0

(32) δ(r − 1)T (s)T (m)T (p)T (k − r − s − m − p) = 0.

p=0

Using the initial conditions and equation (12) we get, a4 a7 a10 43a13 26641a16 , T (4) = , T (6) = − , T (8) = , T (10) = − , 6 30 140 27216 74844000 T (3) = T (5) = T (7) = T (9) = ... = 0.

T (2) = −

(33)

Thus, using equation (12) we have, the solution in the form, y(x) = a −

a4 x2 a7 x4 a10 x8 43a13 x8 26641a16 x10 + − + − + ... 6 30 140 27216 7844000

(34)

(vi) For the case n = 5, the equation (9) takes the form, d2 y 2 dy + + y 5 = 0. dx2 x dx

(35)

Applying DTM we have, k ∑

δ(l − 1)(k + 2 − l)T (k + 2 − l) + 2(k + 1)T (k + 1)

l=0

+

k k−r ∑ ∑ k−r−s ∑ k−r−s−m ∑ k−r−s−m−p ∑ r=0 s=0 m=0

p=0

(36)

δ(r − 1)T (s)T (m)T (p)T (q)T (k − r − s − m − p − q) = 0.

q=0

Using the initial conditions and equation (12) we get, T (2) = −

a9 5a13 35a17 7a21 a5 , T (4) = , T (6) = − , T (8) = , T (10) = − , 6 24 432 10368 6912 T (3) = T (5) = T (7) = T (9) = ... = 0.

(37)

Thus, using equation (12) we have, y(x) = a −

a9 x4 5a13 x8 35a17 x8 7a21 x10 a5 x 2 + − + − + ... 6 24 432 10368 6912

and thus the solution in the form, y(x) = √

a 1+

(a2 x)2 3

.

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(38)

S. Mukherjee, B. Roy, P. K. Chaterjee: Solution of Lane-Emden Equation by Differential Transform Method

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483

Results

Figure 1(a) gives the plot of the normalized density y(x) against dimensionless radius x , for the following set of values of polytropic index, n = 0, 1, 5 with initial condition y(0) = a = 1 and dy(0) = 0 . It has been shown that for the dx polytropic index n = 0, y(x) becomes zero at x ≈ 2.4 and for the value n = 1, y(x) cut the x- axis at x ≈ 3.1 . But for the polytropic index n = 5 , we see from the Figure 1, that y(x) never falls below zero. It should be noted that the first root of the equation y(x) = 0 gives the radius of the polytrope and can be useful in the understanding of the stellar structure in stellar astrophysics. This implies that the case of n = 5 is unimportant in stellar astrophysics. Figure 1(b) gives the plot of normalized density y(x) against dimensionless radius x , for the following set of values of polytropic index, n = 2, 3, 4 with initial condition y(0) = a = 1 and dy(0) dx = 0. It is interesting to note that, in this case the solution of Lane Emden equation is an infinite series solution, rather than a closed form as obtained when n = 0, 1, 5. From this figure, it is clear that as polytropic index increases from n = 2 to n = 4 , the radius of the polytrope also increases. It also shows that for n = 2, 3 and 4 the normalized density y(x) decreases with x till x ≈ 5.5 . The radius of the polytrope in the case n = 2, 3, 4, are 2.7, 2.75, 2.78 respectively.

(a) Plot of the normalized density y(x) against dimensionless radius (b) Plot of the normalized density y(x) against dimensionless radius for different values of n(0, 1, 5) with initial condition y(0) = 1 and for different values of n(= 2, 3, 4) with initial condition y(0) = 1 dy(0) dy(0) = 0. and dx = 0. dx

Figure 1:

6

Conclusion

In this paper our primary aim was to investigate the Lane Emden equations and we have successfully arrived at the solutions of the same by the Differential Transform Method which is a very fast convergent, precise and time saving method. The Lane Emden equations are solved for all the values of the polytropic index from 0 to 5. The solution of the problem for the case n = 0, 1, 5 comes out in a closed form, whereas for the case of n = 2, 3, 4 the solution is obtained as an infinite series. The graphical representation of these results gives us the radius of the polytrope for different values of the polytropic index which may be useful in the study of the behavior of stellar structures in astrophysics. It is also significant that the DTM works successfully and accurately in deriving the solutions of the Lane Emden equations. Thus DTM is an important tool in handling highly nonlinear differential equations with a minimum size of computations and a wide interval of convergence. This emphasizes the fact that this method is applicable to many other systems of nonlinear equations and it is reliable and promising when compared with existing methods.

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