Journal of Modern Physics, 2011, 2, 559-563
doi:10.4236/jmp.2011.26065 Published Online June 2011 (http://www.SciRP.org/journal/jmp)
Copyright © 2011 SciRes. JMP
Solution of Modified Equations of Emden-Type by
Differ ential Transform Method
Supriya Mukherjee1, Banamali Roy2, Pratik Kumar Chatterjee3
1Department of Mathematics Swami Vivekananda Institute of Science & Techno logy,
South Gobindapur, P.S-Sonarpur, Kolkata, West Bengal, India
2Department of Mat hem at i cs Ba ngabasi Evenin g C oll e ge, 19 Rajkumar Chakraborty Sarani,
Kolkata, West Bengal, India
3Department of Computer Science and Engineering Swami Vivekananda Institute of Science & Technology,
South Gobindapur, P.S-Sonarpur, Kolkat, West Bengal, India
E-mail: supriya_ju@yahoo.com, banamaliroy@yahoo.co.in, pratik_kumar_chatterjee@yahoo.in
Received February 2, 2011; revised March 24, 2011; accepted April 6, 2011
Abstract
In this paper the Modified Equations of Emden type (MEE0), 3
xxxx


  is solved numerically by
the differential transform method. This technique doesn’t require any discretization, linearization or small
perturbations and therefore it reduces significantly the numerical computation. The current results of this
paper are in excellent agreement with those provided by Chandrasekar et al. [1] and thereby illustrate the
reliability and the performance of the differential transform method. We have also compared the results with
the classical Runge-Kutta 4 (RK4) Method.
Keywords: Modified Equations of Emden Type, Differential Transforms Method, Runge-Kutta 4 (RK4)
Method
1. Introduction
The modified equation of Emden type (MEE), also called
the modified Painleve-Ince equation,
30xxxx


  (1)
where over dot denotes differentiation with respect to
time and
and
are arbitrary parameters, have re-
ceived attention from both mathematicians and physicists
for more than a century [2-6]. The above differential
equation appears in a number of mathematical problems
such as univalued functions defined by second order dif-
ferential equations [7] and the Riccati equation [8]. Phy-
sicists have found this equation in the study of equilibrium
configurations of a spherical gas cloud acting under the
mutual attraction of its molecules and subject to the laws
of thermodynamics [9-12], in spherically symmetric ex-
pansion or collapse of a relativistic gravitating mass [13]
and in the modeling of the fusion of pellets [14]. The
invariance and the integrability of this equation have been
a subject of study for the past two decades by a number of
authors [15-26]. This equation have been found to possess
an explicit general solution for the following parametric
choices,
0
, (2a)
0
, (2b)
2
9
(2c)
2
(2d)
However, the general solution of Equation (1) for arbi-
trary values of
and
was explored for the first time
by Chandrasekhar et al. [1]. They have constructed the
time-independent Hamiltonians from the time-inde-
pendent integrals of Equation (1) and by the suitable use
of canonical transformations, have converted these Ham-
iltonians to their standard forms. The general solutions are
then obtained by integrating these new Hamiltonians. We
present here a humble effort to arrive at the same by the
Differential Transform Method [DTM].
The concept of differential transform was first intro-
duced by Zhou [27] in solving linear and nonlinear initial
value problems in electrical circuit analysis. The tradi-
tional Taylor series method takes a long time for compu-
560 S. MUKHERJEE ET AL.
tation 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 equa-
tion in a rapidly convergent series [28] and that, it is in
good agreement with the solution obtained by Chandra-
sekar et al. [29].
2. The Modified Emden-Type Equations
As already mentioned, the modified equation of Emden
type cannot be integrated straightforwardly for arbitrary
values of
and
. The solution of MEE for the par-
ticular choice of parameters given by (2a) and (2b) can be
obtained by simple integration and for the choice (2c), the
equation is linearizable to a free particle equation. In the
fourth case the general solution can be expressed in terms
of the Weierstrass elliptic function [2-6,15-26,30]. It has
also been noted that the MEE possess the Painleve prop-
erty for certain values of
28
4
r




 [19,
20,22].
In [1] the authors have identified the first integrals of
Equation (1) separately for each of the three ranges 1)
28
, 2) 28
, and 3) 28
. The Hamilto-
nians are obtained from these integrals and are given by
22
2
22
1
42 222
4
log 8
4
11 8
22
1log sec8
244
r
r
px
p
rr
Hpxp
rr
xxpxp















(3)
For the case 28
the Hamiltonian
2
4
log 4
H
px
p




 reduces to the standard form
2
2
1
log
2
Hp U




32
, (4)
under the canonical transformation
4,
P
xU
2
8
U
p
 . (5)
The general solution thus obtained by integrating the
new Hamiltonian (4) and by using the canonical equa-
tions and
UP
2
PU
is given by,
  
2
11
81
exp 2
2
xterf zEerf z
i







(6)
where,
0
2exp
π
tit E
z
2
, 2
12log
2
EU U
,
0 is an arbitrary constant of integration and erf is the
error function [31].
t
In our present work we have solved the modified equ-
ation of Emden type by the Differential transform method
and we have compared the results with Equation (6) [1].
We have also compared the results with those obtained by
Runge-Kutta 4 Method.
3. The Differential Transform Method
Differential transform of a function is defined as
follows
)(xf
 
0
d
1
!d
k
k
x
fx
Fk kx
. (7)
In (7),
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 is given as 0x
 
00
d
!d
k
k
k
kx
fx
x
fx kx




.
Replacing

0
d
1
!d
k
k
x
fx
kx
by

F
k, we have

0
k
k
fx xFx
(8)
which may be defined as the inverse differential trans-
form.
From (7) and (8) it is easy to obtain the following ma-
thematical operations:
1) If
f
xgxhx, then
 
F
kGkHk
2) If
f
xcgx, then
 
F
kcGk, where c is
a constant.
3) If then

d
d
n
n
g
x
fx
x
, then


!
!
kn
F
kGk
k
n
.
4) If
f
xgxhx
 
0
k
l
, then

F
kGlHkl
.
5) If
fx x
n
, then
 
,
F
kkn
 where
is Kronecker delta.
 
0
d,
x
f
xgtt
then
 
1Gk
Fk k
,6) If where
1k.
Copyright © 2011 SciRes. JMP
S. MUKHERJEE ET AL.
561
7) If
 
f
xuxvxwx
kks
, then
 
00
sm
F
kUsVmWks


 m.
w
here

F
k,

Gk,

H
k,
Uk,

Vk,
Wk
are then differe the tial transform offunctions
f
x,

g
x,

hx ,
ux,

vx,

wx respectively.
4. Solu oMede
Equations Using Differential Transform
tionf the odifi Emdn-Type
Method
The equation of the modified Emden type is given as
2
ddxx
3
20
d
dxx
t
t

 (9)
The initial conditions are (0) 0x and (0)x1
(where prime denotes diff
time).
rm (DT) thave
erentiation with respect to
Applying Differential Transfoo (9), we
2
3
dd 0
xx
DTx x




2d
dt
t
or

2
3
2
dd 0
d
d
xx
DTDTxDT x
t
t

  

  

 ,
l
(10)
The inverse differential transform of T(k) is defined as
i.e.
 
  
0
00
21 211
0.
k
l
kks
sm
kkTk TlklTk
TsTmTks m
 




 
0
k
k
x
tTk
.t (11)
Using the initial conditions (0) 0x; d(0) 1
x
dt
we
have, and
For e abon, we have
(0) 0T
k = 0 in th
(1) 1T.
ove equati
 

200 2T T20 1000TTTTT

 .
(12)
For k = 1, we have
 

   

 
1
0
632021 1
0
m
TT

1100
3.
6
TTTTT
mTmTTT
T





(13)
For k = 2, we have
,
0,
 
   

23
0
22
00
12 433
20
40.
l
s
sm
TlTlTl
TsTmTs m
T

 
 

 (14)
For k = 3, 4, 5, 6, 7, we have

2
530 20
T

, (15)
60T
, (16)

3
17 2
72520 105
T


, (17)
80T
, (18)

42
1395 3792
91020600 30240
T3
1440


and
, (19)
10 0T. (20)
For the case 28
, from Equa
have,
tions (11) to (19) we
20T, (21)

36
T
, (22)
40T, (23)

22
13
20 60
T530

 , (24)
60T, (25)

33
17 2231
72520 10552920
T


, (26)
80T, (27)

4
3
4
1395 6025
91020600 907200092160
735
1000000
T




, (28)
10 0T.
From Equation (11) we have
(29)
 
0
l
l
x
ttTl

23
7
231
t
35
4
9
13
660 52920
735
1000000
xtt tt
t



(30)
Copyright © 2011 SciRes. JMP
S. MUKHERJEE ET AL.
562
5. Comparison of Results
The solution plot of the Modified Equations o
type using DTM is given in Figure 1 for the parametric
choice
f Emden
28
for different values of
. The graph-
ical represetion of the solution
DTM in this paper is in good agreem
ta nd thereby illustrate the
reliability and the performance of the differential trans-
form method. Figure 2 gives us a comparison of the so-
lution for MEE obtained by DTM with the so
tained by classical Runge-Kutta 4 Method. Table 1 gives
the estimate of absolute error between the DTM-solu-
lear from Figures 1, 2
ained by DTM is a bet-
nta (30) obtained by the
ent with those ob-
ined by Chandrasekar et al. [1] a
lution ob-
tions with RK4 solutions. It is c
nd Tab le 1, that the solution obta
ter approximation to the exact solution (as obtained in [1])
than the classical RK4 method. Therefore, the DTM is a
very efficient and accurate method that can be used to
provide analytical solution for nonlinear differential equ-
Figure 1. Plot of solution (30) of Modified Equations of
Emden type for the case α2 = 8β taking α = 4 and α = 5.
Figure 2. Plot of solution of Modified Equations of Emden
type for the case α2 = 8β taking α = 4 using DTM [Solid line]
and RK4 method [Dotted line].
Table 1. Comparison of the DTM- solutions with RK4 solu-
tions and calculation of Absolute error.
Time RK4 solutionDTM solution x(t) Absolute error
0 0 0 0
0.1 0.09934 0.42503 0.32569
0.2 0.1948 0.67042 0.47562
0.3 0.28299 0.80679 0.5238
0.4 0.36135 0.87702 0.51567
0.5 0.42833 0.90712 0.47879
0.6 0.48334 0.91287 0.42953
0.7 0.52667 0.90384 0.37717
0.8 0.55915 0.88584 0.32669
0.9 0.58201 0.8624 0.28039
1.4 0.59858 0.71514 0.11656
1 0.59662 0.83567 0.23905
1.1 0.60436 0.80693 0.20257
1.2 0.60652 0.77697 0.17045
1.3 0.60427 0.74628 0.14201
1.5 0.5903 0.68373 0.09343
1.6 0.
0.56852
5801 0.65215
0.62048
0.07205
0.05196 1.7
1.8 0.55602 0.58874 0.03272
1.9 0.54293 0.55697 0.01404
2 0.52952 0.52517 0.00435
atio
6. clus
The main co this pa to cons ap-
proate ansolution Modifieions
of Een tyave achis goal ing
DThe aof DTfact thaides
its withlytical mation.
con that posed dal transf
is afectivs of solious lineon-
lineuationsalso indat a
small numbeuments icient to an
accurate solupracticplicatios is
that, oly a limber of terms are requio be
ummed, and he ay be cd
owever the method gives more satisfactory
r small times which are evident from Figure 1
r results by adding more terms on
Equation (30) for longer time in-
ns.
Con ions
ncern ofper wastruct an
ximalytical for thed Equat
mdpe. We hhieved tby apply
M. Tdvantage M is the t it prov
user an anaapproxiThe results
firmthe proifferentiorm method
n efe meanving varar and n
ar differential eq. They icate th
r of argare suff provide
tion in e. An imn of thi
nited num
therefore t
red t
omputessolution m
rapidly. H
esults for
and we may get bette
he right hand side oft
tervals.
7. References
[1] V. K. Chandrasekar, M. Senthilvelan and M. Lakshma-
nan, “On the General Solution for the Modified Em-
den-Type Equation30xxx x

 ,Journal of Phys-
Copyright © 2011 SciRes. JMP
S. MUKHERJEE ET AL.
Copyright © 2011 SciRes. JMP
563
ics A, Vol. 40, No. 18, 2004, pp. 4717-4727.
doi:10.1088/1751-8113/40/18/003
[2] P. Painleve, “Sur les Equations Différentielles du Second
Ordre et d’Ordre Supérieure Dont L’Intégrale Générale
est Uniforme,”Acta Mathematica, Vol. 25, No. 1, 1902,
pp. 1-85.
[3] E. L. Ince, “Ordinary Differential Equations,” Dover,
New York, 1956.
[4] H. T. Davis, “Introduction to Nonlinear Differential and
Integral Equations,” Dover, New York, 1962.
[5] E. Kamke, “Differential Gleichungen Losungsmethoden
und
Losungen,” Teubner, Stuggart, 1983.
y, “Ordinary Differential Equations and
Nostrand, New York, 1960.
d A. K.
First-Order
ol. 20, No. 16, 1987,
[6] G. M. Murph
Their Solutions,” Van
[7] V. V. Golubev, “Lectures on Analytical Theory of Dif-
ferential Equations,” Gostekhizdat, Moscow, 1950.
[8] J.S. R.Chisholm an Common, “A Class Of Sec-
ond-Order Differential Equations and Related
Systems,” Journal of Physics A, V
pp. 5459-5472. doi:10.1088/0305-4470/20/16/020
[9] I. C. Moreira, “Lie Symmetries for the Reduced Three-
Wave,” Hadronic Journal, Vol. 7, 1984, p. 475.
[10] P. G. L. Leach, “First Integrals for the Modified Emden
Equation
0
n
qtqq

 ,Journal of Physics, Vol.
26, No. 10, 1985, p. 2510.
Significan
[11] S. Chandrasekhar, “An Introduction to the Study of Stel-
lar Structure,” Dover, New York, 1957.
[12] J. M. Dixon and J. A. Tuszynski, “Solutions of a Gener-
alized Emden Equation and Their Physicalce,”
Physical Review A, Vol. 41, No. 8, 1990, pp. 4166-4173.
doi:10.1103/PhysRevA.41.4166
[13] G. C. McVittie, “The Mass-Particle in an Expanding Uni-
verse,” Monthly Notices of the Royal Astronomical Soci-
ebra SL
1989,
arte and I. C. Moreira, “One
p. L701.
ety, Vol. 93, 1933, pp. 325-339.
[14] V. J. Erwin, W. F.Ames and E.Adams, “Wave Phe-
nomenon: Modern Theory and Applications,” In: C.
Rogers and J. B. Moodie, Eds., Wave Phenomenon: Modern
Theory and Applications, North-Holland, Amsterdam, 1984.
[15] F. M. Mahomed and P. G. L. Leach, “The Lie Alg
(3,R) and Linearization,” Quaestiones Mathematicae, Vol.
12, No. 2,pp.121-139.
[16] L. G. S.Duarte, S. E. S.Du
Dimensional Equations with the Maximum Number of
Symmetry Generators,” Journal of Physics A: Mathe-
matical General, Vol. 20, No. 11, 1987,
doi:10.1088/0305-4470/20/11/005
[17] S. E. Bouquet, M. R. Feix and P. G. L. Leach, “Properties
of Second Order Ordinary Differential Equations Invari-
ant under Time Translation and Self Similar Transforma-
tion,” Journal of Mathematical Physics, Vol. 32, No. 6,
1991, pp.1480-1490. doi:10.1063/1.529306
[18] W. Sarlet, F. M. Mahomed and P. G. L. Leach, “Symme-
tries of Nonlinear Differential Equations and Lineariza-
tion,” Journal of Physics A: Mathematical General, Vol.
20, No. 2, 1987, pp.277-292.
doi:10.1088/0305-4470/20/2/014
[19] P. G. L. Leach, M. R. Feix and S. Bouquet, “Analysis and
Solution of a Nonlinear Second-Order Equation through
Rescaling and through a Dynamical Point of View,”
al General, Vol. 26,
Journal of Mathematical Physics, Vol. 29, No. 12, 1988,
pp. 2563-2569.
[20] R. L. Lemmer and P. G. L. Leach, “The Painlev´e Test,
Hidden Symmetries and the Equationyyyky
 

Journal of Physics A: Mathematic
30,
1993. pp. 5017-5024.
doi:10.1088/0305-4470/26/19/030
[21] W.H. Steeb, “Invertible Point Transformations and Non-
linear Differential Equations” World Scientific, London,
1993.
[22] M. R. Feix, C. Geronimi, L. Cairo, P.G.L. Leach, R.L.
Lemmer and S. Bouquet, “On the Singularity Analysis of
Ordinary Differential Equation Invariant under Time
Translation and Rescaling,” Journ
matical General, Vol. 30, 1997, pp
al of Physics A: Mathe-
. 7437-7461.
doi:10.1088/0305-4470/30/21/017
[23] N. H. Ibragimov, “Elementary Lie Group Analysis and
Ordinary Differential Equations,” John Wiley & Sons,
New York, 1999.
[24] P. G. L. Leach, S. Cotsakis and G. P. Flessas, “Symme-
tries, Singularities and Integrability in Complex Dynam-
ics II: Rescalings and Time-Translatio Systems,”
Journal of Mathematical Analysis and Applications, Vol.
251, 2000, pp. 587-608
ns in 2D
. doi:10.1006/jmaa.2000.7033
lan and M. Lakshma-
ns,” Proceedings of the Royal Society, Vol. 461,
[25] V. K. Chandrasekar, M. Senthilve
nan, “On the Complete Integrability and Linearization of
Certain Second Order Nonlinear Ordinary Differential
Equatio
No. 2060, 2005, p. 2451. doi:10.1098/rspa.2005.1465
[26] V. K. Chandrasekar, S. N. Pandey, M. Senthilvelan and
M. Lakshmanan, “A Simple and Unified Approach to
Identify Integrable Nonlinear Oscillators and Systems,”
Journal of Mathematical Physics, Vol. 47, No.
p.023508.
2, 2006,
[27] J. K. Zhou, “Differential Transformation and Its Applica-
tion in Electrical Circuits,” Huazhong University Press,
Wuhan, 1986.
[28] S. Mukherjee, B. Roy and S. Dutta, “Solution of Duffing-
Van der Pol Oscillator Equation by a Differential Trans-
form Method,” Physica Scripta , Vol. 83, No. 1, 2010,
Article ID 015006. doi:10.1088/0031-8949/83/01/015006
[29] V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan
“New Aspects of Integrability of Force- Free Duffing-
Van der Pol Oscillator and Related Nonlinear System,”
Journal of Physics A, Vol. 37, No. 16, 2004, p. 4527.
,
doi:10.1088/0305-4470/37/16/004
[30] M. Euler, N. Euler and P. G. L. Leach, “The Riccati and
Ermakov-Pinney Hierarchies,” Report No. 8, Institut
Mittag-Leffler, Sweden, 2005/2006.
[31] M. Abramowitz and I. A. Stegun, “Handbook of Mathe-
matical Functions with Formulas, Graphs, and Mathe-
matical Tables,” Dover, New York, 1972.