Solution of Modified Equations of Emden-Type by Differential Transform Method

In this paper the Modified Equations of Emden type (MEE 0 ), 3 x xx x        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.


Introduction
The modified equation of Emden type (MEE), also called the modified Painleve-Ince equation, where over dot denotes differentiation with respect to time and  and  are arbitrary parameters, have received attention from both mathematicians and physicists for more than a century [2][3][4][5][6].The above differential equation appears in a number of mathematical problems such as univalued functions defined by second order differential equations [7] and the Riccati equation [8].Physicists 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][10][11][12], in spherically symmetric expansion 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][16][17][18][19][20][21][22][23][24][25][26].This equation have been found to possess an explicit general solution for the following parametric choices, 0 The concept of differential transform was first introduced by Zhou [27] in solving linear and nonlinear initial value problems in electrical circuit analysis.The traditional Taylor series method takes a long time for compu-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 equation in a rapidly convergent series [28] and that, it is in good agreement with the solution obtained by Chandrasekar et al. [29].

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 particular 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][3][4][5][6][15][16][17][18][19][20][21][22][23][24][25][26]30].It has also been noted that the MEE possess the Painleve property for certain values of 20,22].
In [1] the authors have identified the first integrals of Equation (1) separately for each of the three ranges 1)    .The Hamiltonians are obtained from these integrals and are given by For the case 2 under the canonical transformation The general solution thus obtained by integrating the new Hamiltonian (4) and by using the canonical equations and where, 0 is an arbitrary constant of integration and erf is the error function [31].

t
In our present work we have solved the modified equation 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.

The Differential Transform Method
Differential transform of a function is defined as follows 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 which may be defined as the inverse differential transform.From ( 7) and ( 8) it is easy to obtain the following mathematical operations: where  is Kronecker delta.

Method
The equation of the modified Emden type is given as The initial conditions are ( 0 The inverse differential transform of T(k) is defined as Using the initial conditions (0) 0 have, and For e ab on, we have For k = 2, we have , For k = 3, 4, 5, 6, 7, we have         For the case 2 8    , from Equa have, tions (11) to (19) we       2 2 13 20 60 T 5 30         10 0 T  .

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    1, that the solution obt a 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-

References
[1] V. K. Chandrasekar, M. Senthilvelan and M. Lakshmanan, "On the General Solution for the Modified Emden-Type Equation ," Journal of Phys- general solution of Equation (1) for arbitrary values of  and  was explored for the first time by Chandrasekhar et al. [1].They have constructed the time-independent Hamiltonians from the time-independent integrals of Equation (1) and by the suitable use of canonical transformations, have converted these Hamiltonians 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].
of  .The graphical represe tion of the solution DTM in this paper is in good agreem ta nd thereby illustrate the reliability and the performance of the differential transform method.

Figure 2
gives us a comparison of the solution for MEE obtained by DTM with the so tained by classical Runge-Kutta 4 Method.

Figure 2 .
Figure 2. Plot of solution of Modified Equations of Emden type for the case α 2 = 8β taking α = 4 using DTM [Solid line] and RK4 method [Dotted line].

Table 1
gives the estimate of absolute error between the DTM-solu-

Table 1 . Comparison of the DTM-solutions with RK4 solu- tions and calculation of Absolute error.
(30)s more satisfactory r small times which are evident from Figure1r results by adding more terms on Equation(30)for longer time in-ns.