Physics Letters A 365 (2007) 111–115

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Series solution to the Thomas–Fermi equation

Hina Khan a,b, Hang Xu a,∗

a School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200030, Chinab Department of Humanities and Sciences, National University of Computer and Emerging Sciences, Islamabad campus, Pakistan

Received 1 November 2006; received in revised form 7 December 2006; accepted 18 December 2006

Available online 8 January 2007

Communicated by A.R. Bishop

Abstract

Here an analytic technique, namely the homotopy analysis method (HAM), is employed to solve the non-linear Thomas–Fermi equation. A newkind of transformation is being used here which has improved the results in comparison with Liao’s work. We also present the comparison of thiswork with some well-known results and prove the importance of this transformation and the freedom of HAM.© 2007 Elsevier B.V. All rights reserved.

Keywords: Thomas–Fermi equation; Series solution; Homotopy analysis method

1. Introduction

Consider a non-linear differential equation with singularityat x = 0, named as the Thomas–Fermi equation used to calcu-late the electrostatic potential in the Thomas–Fermi atom model[1,2] given by

(1)u′′(x) =√

u3(x)

x,

with the boundary conditions

(2)u(0) = 1, u(+∞) = 0,

in the common case. This problem has already been solvedby different techniques. The famous numerical solution ofthe same problem is given by Kobayashi [3]. Adomian [4]has applied the decomposition method to the Thomas–Fermiequation. Bush and Caldwell [5] also obtained the numeri-cal solution of the same problem using differential analyzer.Plindov and Pogrebnya [6] gave the analytical solution tothis problem by considering a sequence of functions ϕN(x),

* Corresponding author.E-mail addresses: hinakhan@sjtu.edu.cn (H. Khan), hangxu@sjtu.edu.cn

(H. Xu).

0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2006.12.064

which converges to the solution of the Thomas–Fermi equa-tion and found that N = 30 works well to approximate ϕ(x)

and ϕ′(x).Some mathematical aspects of the Thomas–Fermi equation

were presented by Hille [7]. Cedillo [8] applied perturbationtechnique in order to get the initial slope of Thomas–Fermiproblem, he used some transformation in order to get the per-turbation parameter which is smaller then the correspondingparameters obtained by the previous works. Recently Oulne [9]studied the same problem using Ritz variational approach. Liao[13,14] obtained a series solution of the same problem by an an-alytic technique, namely the homotopy analysis method (HAM,[10–22]).

In order to improve the convergence of the solution given byLiao [13,14] here, we use the same technique as Liao [13,14],but with new and better transformation to reconsider this veryfamous Thomas–Fermi problem. The beauty of HAM is, that itprovides us with a free way to control the convergence of ap-proximation series and it can adjust the convergence regionswhen needed. Here we have freedom of selection of initialguess. Only by considering the framework of HAM we can finddifferent initial guesses and then can select the best of themwhich works well for the result. In current work we introduce aparameter γ in relation with slope u′(0), and get better resultsas compare to Liao’s [13,14] work.

112 H. Khan, H. Xu / Physics Letters A 365 (2007) 111–115

2. Mathematical formulations

Let

(3)u′(0) = 1

γ,

and we introduce the following transformations

(4)u(x) = 1

γg(ξ), ξ = 1 + λx,

where λ is a constant parameter to be determined later, and γ

is unknown constant whose value is defined in Eq. (3). Aftertransformation the non-linear equation (1) takes the form

(5)(ξ − 1)λ3γ[g′′(ξ)

]2 − [g(ξ)

]3 = 0,

with the boundary conditions

(6)g(1) = γ, g′(1) = 1

λ, g(∞) = 0.

Considering the boundary conditions in (6) and physicalmeaning of u(x) as discussed by Liao [13,14], we choose thegiven base function

(7){ξ−m,m � 1

},

and the solution can be written as

(8)g(ξ) =+∞∑m=1

amξ−m,

where am are coefficients. This provides us the solution expres-sion of g(ξ).

2.1. HAM deformation equation

According to the rule of solution expression and the bound-ary conditions in (6), it is straightforward to choose the initialsolution

(9)g0(ξ) =(

1

λ+ 2γ

)ξ−1 −

(1

λ+ γ

)ξ−2

and besides to choose the auxiliary linear operator

(10)L[φ(ξ ;q)

] = ξ

2

∂2φ

∂ξ2+ ∂φ

∂ξ,

which has the following property that can be proved easily

(11)L[C1ξ

−1 + C2] = 0,

where C1 and C2 are integral constants. From Eq. (5), we areled to define a non-linear operator

N[Φ(ξ ;q),Γ (q)

] = (ξ − 1)λ3Γ (q)

[∂2Φ(ξ ;q)

∂ξ2

]2

(12)− [Φ(ξ ;q)

]3.

We construct the HAM deformation equation, where h̄ denotea non-zero auxiliary parameter

(13)(1 − q)L[Φ(ξ ;q) − g0(ξ)

] = qh̄N[Φ(ξ ;q),Γ (q)

],

subject to the corresponding boundary conditions

(14)Φ(1;q) = γ, Φ(∞;q) = 0,∂Φ(ξ ;q)

∂ξ

∣∣∣∣ξ=1

= 1

λ,

where q ∈ [0,1] is an embedding parameter.When q = 0, we have, from Eqs. (13) and (14), the solution

(15)Φ(ξ ;0) = g0(ξ), Γ (0) = γ0,

where γ0 is the initial approximation of γ . Since h̄ �= 0,Eq. (13), when q = 1, is equivalent to Eq. (5), provided

(16)Φ(ξ ;1) = g(ξ), Γ (1) = γ.

Thus, as q increases from 0 to 1, Φ(ξ ;q) varies from the initialguess g0(ξ) to the solution g(ξ) of the original equations (5)and (6).

Using (15), we expand Φ(ξ ;q) and Γ (q) in the Taylor serieswith respect to q , i.e.

(17)Φ(ξ ;q) = g0(ξ) ++∞∑m=1

gm(ξ)qm,

(18)Γ (q) = γ0 ++∞∑m=1

γmqm,

where

(19)gm(ξ) = 1

m!∂mΦ(ξ ;q)

∂qm

∣∣∣∣q=0

, γm = 1

m!∂mΓ (q)

∂qm

∣∣∣∣q=0

.

Note that the above two series are dependent upon the auxiliaryparameters h̄, γ0 and λ. Assuming that h̄, γ0 and λ are properlychosen so that the above two series converge at q = 1, and using(16), we have the solution series

g(ξ) = g0(ξ) ++∞∑m=1

gm(ξ),

(20)γ = γ0 ++∞∑m=1

γm.

2.1.1. High-order deformation equationFor brevity, define the vectors

�gm = {g0(ξ), g1(ξ), g2(ξ), . . . , gm(ξ)

},

(21)�γm = {γ0, γ1, γ2, . . . , γm

}.

Differentiating the HAM deformation equation (13) m timeswith respect to q , then setting q = 0, and finally dividing bym!, we have the mth-order deformation equation

(22)L[gm(ξ) − χmgm−1(ξ)

] = h̄Rm(�gm−1, �γm−1),

subject to the boundary conditions

(23)gm(1) = γm, g′m(1) = 0, gm(∞) = 0,

where

(24)

Rm(�gm−1, �γm−1) = (ξ − 1)λ3m−1∑k=0

γkαm−1−k(ξ)

−m−1∑

gk(ξ)δm−1−k(ξ),

k=0

H. Khan, H. Xu / Physics Letters A 365 (2007) 111–115 113

and

(25)χm ={

0, m � 1,

1, m > 1,

in which

(26)αk =k∑

j=0

g′′j g′′

k−j , δk =k∑

j=0

gjgk−j .

Now Eq. (5) is converted into an infinite number of linear prob-lems as in (22). Here gm(ξ) for m � 1 is calculated by solvingthe linear equation (22) with linear boundary conditions (23),which can be easily solved by using symbolic computation soft-ware such as MATHEMATICA.

Let g∗m(ξ) represents a particular solution of equation

(27)L[g∗

m(ξ)] = h̄Rm(�gm−1, �γm−1),

then using property (11) and the boundary conditions (23), thegeneral solution of Eq. (22) is

(28)gm(ξ) = χmgm−1(ξ) + g∗m(ξ) + Cm

1 ξ−1 + Cm2 .

Here Cm1 , Cm

2 and γm are determined as follows:

(29)Cm

1 = g′ ∗m (1), Cm

2 = 0, γm = χmγm−1 + Cm1 + g∗(1).

3. Convergence theorem

If the series

(30)g0 ++∞∑m=1

gm

is convergent, where gm(ξ) is governed by Eqs. (22) and (23)under the definitions (10), (24) and (25), it must be an exactsolution of the Thomas–Fermi equation.

Proof. Write

s(ξ) = g0(ξ) ++∞∑m=1

gm(ξ),

(31)γ = γ0 ++∞∑m=1

γm.

If the series is convergent it must satisfy the following property

(32)limm+∞gm(ξ) = 0.

Due to (22), the definitions (10), (25) and above expression, onehas

(33)

h

+∞∑m=1

Rm(ξ) = limk+∞

k∑m=1

L[gm(ξ) − χmgm−1(ξ)

]

= L{

limk+∞

k∑m=1

[gm(ξ) − χmgm−1(ξ)

]}

= L[

limk+∞gk(ξ)

]= 0,

as h �= 0, so

(34)+∞∑m=1

Rm(ξ) = 0.

Then, due to the definition (24) the above expression can besatisfied as follow,

+∞∑m=1

Rm(ξ) =+∞∑m=1

[(ξ − 1)λ3

m−1∑k=0

γk

m−1−k∑j=0

g′′j g′′

m−1−k−j

]

−+∞∑m=1

[m−1∑k=0

gk

m−1−k∑j=0

gjgm−1−k−j

]

= (ξ − 1)λ3+∞∑k=0

γk

( +∞∑k=0

g′′k

)2

−( +∞∑

k=0

gk

)3

= (ξ − 1)λ3Γ (q)(s′′)2 − (s)3

(35)= 0.

Besides, due to (23) and definition (9), one gets

(36)s(1) = γ, s′(1) = 1

λ, s(+∞) = 0.

So s(ξ) satisfies the Thomas–Fermi equation (5) and the cor-responding boundary conditions (6) therefore it is an accuratesolution of it. This ends the proof.

4. Result analysis

Here the series contains three parameters h̄, γ0 and λ whichinfluence its convergence region, where γ is the parameter thatwe introduce in order to improve the convergence of the con-sidered problem as compare to previous work [13,14] with thesame technique. Firstly for λ, the residual error of the initiallyassumed solution g0(ξ) can be expressed by

(37)E0(λ) =∞∫

1

(R1[ �g0]

)2dξ.

Letting

(38)dE0

dλ= 0,

we can get an optimal value corresponding to the least square.Similarly for γ we do the similar process as for λ. Then thevalue of auxiliary parameter h̄ is selected through h̄-curve asdiscussed by Liao [14].

Kobayashi [3] gave the numerical result for the same prob-lem as u′(0) = −1.588071. Comparison of the current resultswith Liao [13,14] and Kobayashi [3] results is presented inTable 1, which shows that improvement of the results can be ob-tained by applying new transformation as present work resultsare better then Liao’s results [13,14]. Another comparison withKobayashi [3] result for the approximations of the initial slopeu′(0) obtained by the present work is given in Table 2 whenh̄ = −10/11, λ = 81/100 and γ0 = −4/5. Here we are usingthe homotopy-Padé approximations [14] in order to improve

114 H. Khan, H. Xu / Physics Letters A 365 (2007) 111–115

Table 1The [m,m] homotopy-Padé approximations comparison of u′(0)

m Liao [13] Liao [14] Present work

[10/10] −1.51508 −1.54600 −1.573824678[20/20] −1.58281 −1.56474 −1.582901807[30/30] −1.58606 −1.558032 −1.586494973

Kobayashi [3] result −1.588071

Table 2The [m,m] homotopy-Padé approximations of u′(0) for present work

m Present work Error (%)

[5/5] −1.542791808 2.85[10/10] −1.573824678 0.89[15/15] −1.579528916 0.54[20/20] −1.582901807 0.33[25/25] −1.584561180 0.22[30/30] −1.586494973 0.099

Kobayashi [3] result −1.588071

Table 3The approximations of u′′(0) for present work

m u′′(0)

10 −10.277720 −15.749730 −20.895840 −25.888350 −30.790260 −35.6318

Table 4The [m,m] homotopy-Padé approximations of u′′(0) for present work

m u′′(0)

[5/5] −131.1096635[10/10] −912.7566447[15/15] −3462.635582[20/20] −8619.110061[25/25] −20603.52249[30/30] −42515.36921

the approximations of u′(0). It is clear that the error decreasesas the order of approximation increases.

The convergent analytic approximations of u(x) for 0 � x <

∞, when h̄ = −10/11, λ = 81/100 and γ0 = −4/5 are listed inTable 5 and also in Fig. 1 for present work and results obtainedby Liao [14]. It is clear that the present results are better thanthose results given in Liao [14]. From Thomas–Fermi equa-tion (1) it is clear that at x = 0, the second derivative of u(x)

tends to infinity, Here we have shown the approximations ofu′′(0) when h̄ = −10/11, λ = 81/100 and γ0 = −4/5 in Ta-ble 3 and [m,m] homotopy-Padé approximations of the samefunction are given in Table 4 which shows divergence of u′′(0)

clearly.

5. Conclusion

This work proves the power and liberty of HAM that wehave freedom to try different new transformations to improve

Table 5The approximations of u(x) for present work and Liao [14]

x u(x) present work u(x) Liao [14]

0.10 0.913577 –0.20 0.818094 –0.25 0.776191 0.7552020.5 0.615917 0.6069870.75 0.50538 0.5023471.00 0.423772 0.4240081.25 0.362935 0.3632021.50 0.31449 0.3147781.75 0.275154 0.2754512.00 0.242718 0.2430092.25 0.21563 0.2158952.50 0.192795 0.1929842.75 0.173364 0.1734413.00 0.156719 0.1566333.25 0.142371 0.1420703.50 0.129937 0.1293703.75 0.119108 0.1182294.00 0.109632 0.1084044.25 0.101303 0.09969794.50 0.0939504 0.09194824.75 0.087432 0.08502185.00 0.0816296 0.07880786.00 0.0638162 0.05942307.00 0.0518005 0.04609788.00 0.0432859 0.03658739.00 0.0370023 0.029590910.0 0.0322081 0.024314315.0 0.0191843 0.010805420.0 0.0134937 0.0057849425.0 0.010357 0.0034737550.0 0.00473089 0.00063225575.0 0.00305246 0.000218210100.0 0.002251 0.0001002431000.0 0.000214641 0.000000135

Fig. 1. Comparison of our 60th and 80th order analytic approximations withLiao’s results [14]. Circles: 60th order homotopy analysis results; solid line:80th order analytic approximations; dashed line: Liao’s results.

H. Khan, H. Xu / Physics Letters A 365 (2007) 111–115 115

the results just by taking care of framework of HAM. As inLiao’s work [13,14] where he also has applied the same ho-motopy analysis method but the results obtained presently withnew transformation are more closer to the results obtained byKobayashi [3]. The similar technique as presented in this workcan be used in order to handle other non-linear problems andone may observe better results.

Acknowledgement

We express our sincere thanks to the anonymous reviewersfor their valuable suggestions.

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