Corresponding Author: Dr. Mehmet Merdan, Department of Geomatics Engineering, Gümüshane University, 29100,Gümüshane, Turkey
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Approximate Analiytical Solutions of Lane-Emden Type Equation by Differential Transformation Method
1Mehmet Merdan, 2Ahmet Gökdogan and 3Ahmet Yildirim
1Department of Geomatics Engineering, Gümüshane University, 29100, Gümüshane, Turkey2Department of Software Engineering, Gümüshane University, 29100, Gümüshane, Turkey
3Department of Mathematics, Ege University, 35000, Izmir, Turkey
Abstract: In this article Differential Transformation Method (DTMs) is considered to find exact and approximate solutions of singular Initial Value Problems (IVPs) of the Lane-Emden type in second-orderOrdinary Differential Equations (ODEs). The method can simply be applied to many linear and nonlinear problems and is capable of reducing the size of computational work while still providing the series solution with fast convergence rate. Exact solutions can also be obtained from the known forms of the series solutions. The results show that the method is effective, suitable, easy, practical and accurate.
Key words: Differential transformation method • Lane-Emden type equation equations
INTRODUCTION
Many phenomena in mathematical physics andastrophysics can be corresponding to Lane-Emdentype equation. Lane-Emden type equation is a nonlinear differential equation which defines the equilibriumdensity distribution in self-gravitating sphere ofpolytrophic isothermal gas, has a singularity at theorigin and is of fundamental importance in the field of stellar structure, radioactive cooling and modeling of clusters of galaxies. Lane-Emden type equations are described following as:
(1)
with initial conditions
(2)
where the first denotes the differentiation withrespect to x, a, b are constant, f (x, y) is a nonlinear function of x and y. It is well known that an analytic solution of Lane-Emden type equation (1) is always possible [1] in the neighborhood of thesingular point x = 0 for the above initial conditions. It is named after the astrophysicists Jonathan H. Lane and Robert Emden [2], as it was first studied by them.
Bender et al. [4] handled the solution of Lane-Emden equation as well as those of a variety ofnonlinear problems in quantum mechanics andastrophysics by means of perturbation methods based on the existence of a small parameter. Shawagfeh [5] and Wazwaz [6, 7] presented to approximatedanalytical solutions of the above equation usingAdomian Decomposition Method (ADM). Nouh [8]accelerated the convergence of power series solution of Lane-Emden equation by using Euler-Abeltransformation and Padé approximation. Mandelzweig and Tabakin [9] applied Bellman and Kalaba’squasilinearization method and Ramos [10] usedpiecewise linearization technique based on thepiecewise linearization of Lane-Emden equation.Bozkhov and Martins [11] applied the Lie Groupmethod successfully to generalized Lane-Emdenequations of the first kind. Exact solutions ofgeneralized Lane-Emden solutions of the first kind are investigated by Goenner and Havas [12]. Approximated or exact solutions of Lane-Emden type equations are given by applying homotopy analysis method [13],modified homotopy analysis method [14], homotopy perturbation method [15-17] and variational iteration method [18, 19]. Parand et al. [20] presented solution of the nonlinear Lane-Emden type equations using hermite functions collocation method. Adibi and Rismani [21] applied a modified Legendre-spectral method forsolving singular IVPs of Lane-Emden type.
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The goal of this paper is to extend the differential transform method to approximate or exact solutions of Lane-Emden type equation.
DIFFERENTIALTRANSFORMATION METHOD
As in [22-32], the basic definition of thedifferential transformation method is given as follows:
Definition 2.1: If y(t) is analytic in the domain T, then it will be differentiated continuously with respect to time t,
(3)
for t = ti, then ϕ(t,k) = ϕ(ti,k), where k belongs to set of nonnegative integers, denoted as the K-domain.Consequently, Eq. (3) can be rewritten as
(4)
where Y(k) is called the spectrum of y(t) at t = t i,
Definition 2.2: If y(t) can be described by Taylor’s series, then y(t) can be shown as
(5)
Eq. (5) is called the inverse of y(t), with the symbol D denoting the differential transformation process.Upon combining (4) and (5), we attain
(6)
Using the differential transformation, a differential equation in the domain of interest can be transformed to an algebraic equation in the K-domain and the y(t) can be obtained by finite-term Taylor’s series plus a remainder, as
(7)
From the definitions (4) and (6), it is easy to obtain the following mathematical operations:
Table 1: Operations of the one dimensional differential transform
Original function Transformed function
Theorem 2.1: if
then
APPLICATIONS
In this section, we will apply the DTM to Lane-Emden type equations which have been widelydiscussed in the literature.
Lane-Emden type equationsExample 3.1.1: We first consider followinghomogeneous Lane-Emden equation
(8)
with initial condition,
(9)
applying the differential transform of (8) and (9), then
(10)
(11)
Substituting (11) in (10), the series followingsolution form can be obtained
World Appl. Sci. J., 13 (Special Issue of Applied Math): 55-61, 2011
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(12)
this series has the closed solution form 2xe , which is the exact solution of homogeneous Lane-Emden equation.
Example 3.1.2: Consider the time-dependent Lane-Emden equation
(13)
with initial condition,
(14)
applying the differential transform of (23) and (24), then
(15)
(16)
Substituting (16) in (15), we obtain the closed form solution as
(17)
which is the exact solution of the time-dependent Lane-Emden equation.
Example 3.1.3: Consider the time-dependent Lane-Emden equation
(18)with initial condition
(19)
applying the differential transform of (18) and (19), then
(20)
(21)
Substituting (21) in (20), we obtain the closed form solution as
(22)
which is the exact solution of the time-dependent Lane-Emden equation.
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Example 3.1.4: Consider nonlinear Lane-Emden equation
(23)with initial condition,
(24)
applying the differential transform of (23) and (24), then
(25)
(26)
and exponential nonlinearity [22]: ƒ(y) = eay
From the definition of transform,
(27)
Now, taking differentiation ƒ(y) = eay with respect to x, we get:
(28)
Application of the differential transform to Eq. (28) gives:
(29)
Replacing k+1 by k gives
(30)
Thus, the recursive relation for calculating the T-function of ƒ(y) = eay is:
(31)
Where, H(k) and G(k) are the T-function of eay and given by Eq.(31):
(32)
For each k, substituting Eq. (25) into (26) and (32) and by recursive method,
(33)
Substituting (33) in (5), we obtain the closed form solution as
(34)
which is the exact solution.
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Example 3.1.5: Consider nonlinear Lane-Emden type equation
(35)with initial condition,
(36)
applying the differential transform of (35) and (36), then
(37)
(38)
and logarithmic nonlinearity [22]:
By the definition of transform,
(39)
Further, differentiating
with respect to x, we get:
(40)or equivalently,
(41)
taking the differential transform to Eq. (41), we obtain
(42)
Replacing k+1 by k gives
(43)
Substitute k = 1 into Eq. (43) to get:
(44)For k≥2, Eq. (44) can be rewritten as:
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(45)
Thus, the recursive relation for calculating the T-function of ƒ(y) = In(ay+b) is:
(46)
Where H(k) is the T-function of In(y) and given by Eq. (46):
(47)
For each k, substituting Eq. (37) into (38) and (47) and by recursive method,
, (48)
Substitute all Y(k) into Eq. (5), we obtain the closed form solution as
(49)which is the exact solution.
CONCLUSIONS
In this paper, we used the DifferentialTransformation Method (DTM) for find to exact and approximate solutions of differential equations of theLane-Emden type as singular initial value problems .DTM can be applied to many complicated linear and strongly nonlinear ordinary or partial differentialequations and systems of partial differential equations and does not require linearization, discretization orperturbation. The obtained results show that thismethod is powerful and meaningful for solving theLane-Emden type equation.
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