Abstract

In this work we apply the differential transformation method or DTM for solving some classes of Lane-Emden type equations as a model for the dimensionless density distribution in an isothermal gas sphere and as a study of the gravitational potential of (white-dwarf) stars, which are nonlinear ordinary differential equations on the semi-infinite domain [1] [2] . The efficiency of the DTM is illustrated by investigating the convergence results for this type of the Lane-Emden equations. The numerical results show the reliability and accuracy of this method.

Keywords: Differential Transformation, Lane-Emden, Isothermal Gas Sphere, White-Dwarfs, Iterative Method

1. Introduction

Other classical nonlinear equation, which has been the object of much study, is Lane-Emden’s equation. This equation has the form

(1)

with and the subject to initial conditions

(2)

where and are constants and is a real-valued continuous function where and are constants and is a real-valued continuous function. The Equation (1) was used to model various problems, including the isothermal gas spheres, theory of thermionic currents and the gravitational potential of stars [1] among others.

Let us consider a spherical cloud of gas (see Figure 1) and denote its hydrostatic pressure at a distance from the centre by. Let be the mass of the spheres of radius the gravitational potential of the gas and the acceleration of gravity.

Then, we have the following equation

(3)

where is the gravitational constant.

Thus, three conditions are assumed for the determination of and

(4)

where is the density of the gas.

(5)

and

(6)

where and are arbitrary constants.

Now, solving (4) and (6) with when we have

(7)

or

(8)

where and. If this value of is replaced into Equation (5), we obtain

(9)

where.

Now, since, by integration, that is,. If is the central densitythen must be zero, a change from the condition in the previous case where was zero only at the boundary of the sphere.

Poisson’s equation is now replaced by

(10)

where, equation which is known as Liouville’s equation. If we assume symmetry as before, Equation (1) in polar coordinates reduces to the following

(11)

which replaces Equation (9).

If we let and, then (11) becomes

(12)

which is to solved subject to the boundary conditions and. The counterpart [2] of the Equation (12) in which is replaced by appears in Richardson’s theory of thermionic currents when one seeks to determine the density and electric force of an electron gas in the neighborhood of a hot body in thermal equilibrium.

Finally, now consider, then Equation (1) is turned to the white-dwarf equation, which introduced by [2] in his study of gravitational potential of the degenerate stars. This Equation is defined in the form

With and subject to initial conditions and. For instance if, we have Lane-Emden equation of index [3] .

The Differential Transformation Method is a semi-numerical-analytic method for solving ordinary and partial differential equations. Zhou first introduced the concept of DTM in 1986 [4] . This technique constructs an analytical solution in the form of a polynomial. DTM is an alternative procedure for obtaining analytical Taylor series solution of the differential equations. The series often coincides with the Taylor expansion of the true solution at point, in the value case, although the series can be rapidly convergent in a very small region.

Many numerical methods were developed for this type of nonlinear ordinary differential equations, specifically on Lane-Emden type equations such as the Adomian Decomposition Method (ADM) [5] , the Homotopy Perturbation Method (HPM) [6] [7] , the Homotopy Analysis Method (HAM) [8] and Berstein Operational Matrix of Integration [9] . In this paper, we show superiority of DTM by applying them on the some type LaneEmden type equations. The power series solution of the reduced equation transforms into an approximate implicit solution of the original equation. A spectral method (Legendre-Spectral method) was proposed to solve white-dwarf equation; this spectral method provides the most convenient computer implementation [10] .

2. Description of DTM

Differential transformation method of the function is defined as follows

(13)

In (13), is the original function and is the transformed function and the inverse differential transformation is defined by

(14)

In real applications, function is expressed by a finite series and Equation (14) can be written as

(15)

Equation (15) implies that

The following theorems can be deduced from Equations (13) and (15).

Theorem 1.

Theorem 2.

Theorem 3.

Theorem 4.

Theorem 5

Theorem 6 (Cárdenas)

The proofs of Theorems are available in [11] .

3. Test Problems

To illustrate the ability of DTM for the Lane-Emden type equation, three examples are provided. The results reveal that this method is very effective.

Example 1 Consider the nonlinear initial-value problem subject to. Multiplying both sides by we obtain

(16)

Applying theorems 1-6 to Equation (16)

(17)

where

(18)

(19)

(20)

for all.

Now, from the initial conditions we can obtain

(21)

Substituting Equation (21) into Equation (17) and by recursive method, the results are listed as follows.

.

and then,. For we have:

Now, as and, then

For we have:

In this case as and, then.

The lector can see that

For we have:

Now, we can see:

and then

For we have:

Here,

and

and so,. Consequently,.

For we have:

Here

and

Consequently,. Finally,

Therefore using (15), the closed form of the solution can be easily written as:

A series solution obtained by Wazwaz [5] and series expansion respectively is

(22)

Table 1 shows the comparison of obtained by the DTM (method proposed in this work) and those obtained by Wazwaz. The resulting graph of the isothermal gas spheres equation in comparison to the present method and those obtained by Wazwaz is shown in Figure 2.

Example 2 Consider the following problem subject to. Multiplying both sides by

(23)

As before, using theorems 1-6 we obtain

(24)

where and are as (18), (19) and (20) respectively for all. Now, from the initial conditions we have

(25)

Substituting Equation (25) into Equation (24) and by recursive method, the results are listed as follows.

For, we have respectively

So on, we can use (15) and the closed form of the solution can be easily written as

A solution obtained by Yahya [12] by using the power series method is

We can see Figure 3 and compare with [13] , the results are very good.

As final example and to illustrate the ability of DTM for white-dwarf equation, the next problem is provided for.

Example 3 Consider the problem subject to. Multiplying both sides by we obtain

(27)

Here, is easy to verify that the function has a series expansion

(28)

where. Therefore, Equation (27) takes the form

(29)

Using in (29) the above theorems we have the following

(30)

or

(31)

where

and successively. Also,

(32)

(33)

(34)

for all. Now, from the initial conditions we have

(35)

Substituting (35) into Equation (31) and by recursive method, the results are listed as follows.

For. or and then and then. For and so. For therefore. Using (15), the closed form of the soluyion can be easily written as

A series solution obtained by Chandrasekhar [2] using series expansion was

(36)

Table 2 shows the comparison of obtained by the DTM and those obtained by Parand [14] . The resulting graph of the white-dwarfs equation in comparison to the present method and the obtained by [14] is show in Figure 4.

4. On Convergence of DTM

We can write the DTM as

(37)

where increase function depends on its arguments through the function. The method (37) means steps, needed for early values to calculate. It is therefore necessary to have bootstrap values.

The method (37) is said to be convergent if for all IVP has to

Remark. The condition on the bootstrap values is equivalent to asking that for. Here, we are asking that bootstrap values approximate well and the initial data; if this is not, then no reason to expect that numerical solution closely matches the theoretical.

Now let us consider the following form of the Equation (1)

(38)

Here is a nonlinear differential operator, which encloses the linear and nonlinear term of the Lane-Emden type equation. Now, the linear term is always invertible and the nonlinear term is Therefore (38) may be written as

(39)

or

(40)

Applying DTM in (40) we can obtain

(41)

Remember that differential transformation of and are computed by using theorems 1 - 6.

Let us consider the Equation (38) in the following form

(42)

Here, is a nonlinear operator. It is noted that Equation (15) is equivalent to the sequence

(43)

This sequence is determined using the iterative scheme

(44)

and associated with.

The following theorem guarantees that the scheme of DTM converges to the solution of Lane-Emden Equation (1).

Theorem 7 Let be a nonlinear operator from a Banach space and be the solution (exact) of Equation (42). The series solution (14) converges to, if there exists a constant such that for.

Proof. We prove that the sequence is a Cauchy sequence in. Therefore,

Thus, for any

so implying that he sequence is Cauchy, i.e. since then , therefore there exists such that, i.e. converges.

Now, we can say too that Equation (42) is similar to solve, therefore this implies that if is continuous then

i.e. T is a solution of and this completes the proof.

Figure 5 shows the maximum point-wise error between the numerical solution obtained by using DTM and the Chandrasekhar solution. It is observed that both schemes are almost the same accuracy.

5. Conclusion

In this work, we presented the definition and handling of one-dimensional differential transformation method. Using DTM, the Lane-Emden equations were transformed into algebraic equations (iterative equations). The new scheme obtained by using DTM yields an analytical solution in the form of a rapidly convergent series. This method makes the solution procedure much more attractive. The figures and tables clearly show the high efficiency of DTM and the convergence of the method for three examples in investigated.

Acknowledgements

Foremost, I would like to express my sincere gratitude to Jean-Christophe Nave (Department of Mathematics and Statics McGill University) for the support of my research and the support of the Department of Mathematics of the Universidad Tecnológica de Pereira (Colombia) and the group GEDNOL.

References