Research Article Nonlinearities Distribution Homotopy...
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Research Article Nonlinearities Distribution Homotopy Perturbation Method Applied to Solve Nonlinear Problems: Thomas-Fermi Equation as a Case Study U. Filobello-Nino, 1 H. Vazquez-Leal, 1 K. Boubaker, 2 A. Sarmiento-Reyes, 3 A. Perez-Sesma, 1 A. Diaz-Sanchez, 3 V. M. Jimenez-Fernandez, 1 J. Cervantes-Perez, 1 J. Sanchez-Orea, 1 J. Huerta-Chua, 4 L. J. Morales-Mendoza, 5 M. Gonzalez-Lee, 5 C. Hernandez-Mejia, 3 and F. J. Gonzalez-Martinez 1 1 Electronic Instrumentation and Atmospheric Sciences School, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltr´ an S/N, 91000 Xalapa, VER, Mexico 2 Equipe de Physique des Dispositifs ` a Semiconducteurs, Facult´ e des Sciences de Tunis, Tunis El Manar University, 2092 Tunis, Tunisia 3 National Institute for Astrophysics, Optics and Electronics, Luis Enrique Erro No. 1, Santa Mar´ ıa Tonantzintla, 72840 Puebla, PUE, Mexico 4 Civil Engineering School, Universidad Veracruzana, Venustiano Carranza S/N, Colonia Revolucion, 93390 PozaRica, VER, Mexico 5 Department of Electronics Engineering, Universidad Veracruzana, Venustiano Carranza S/N, Colonia Revolucion, 93390 Poza Rica, VER, Mexico Correspondence should be addressed to H. Vazquez-Leal; [email protected] Received 3 November 2014; Revised 23 December 2014; Accepted 27 December 2014 Academic Editor: Shiping Lu Copyright © 2015 U. Filobello-Nino et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We propose an approximate solution of T-F equation, obtained by using the nonlinearities distribution homotopy perturbation method (NDHPM). Besides, we show a table of comparison, between this proposed approximate solution and a numerical of T-F, by establishing the accuracy of the results. 1. Introduction e numerical simulation for the charge distribution and the electric field inside an atom is a very difficult task, especially for complex atoms. For these cases, the omas-Fermi (T-F) equation can be employed in order to obtain highly accurate approximate results. T-F method is based on the fact that most of complex atoms, with a large number of electrons, have relatively large quantum numbers, and therefore the semiclassical approach can be employed. en, it is possible to apply the concept of cells in the phase space to the states of individual electrons [1]. e omas-Fermi equation describes mathematically an infinite atom, without border; however the model is not applicable to both distances from the nucleus: too small and too large. More precisely, its application domain is limited to a range of distances : large relative to the quantity 1/ and small compared to 1, where is the atomic number of the atom. However, for complex atoms, most of the electrons are in that interval, for which the study of T-F equation becomes important. One of the important consequences of the omas- Fermi theory follows from the above. e atom has an outer boundary for values of ≅1; thus the theory predicts that the dimensions of the atom are independent of , and for the same reason the ionization energy of the outer electrons is also independent of the atomic number [1]. Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2015, Article ID 405108, 9 pages http://dx.doi.org/10.1155/2015/405108
Transcript of Research Article Nonlinearities Distribution Homotopy...
Research ArticleNonlinearities Distribution HomotopyPerturbation Method Applied to Solve Nonlinear ProblemsThomas-Fermi Equation as a Case Study
U Filobello-Nino1 H Vazquez-Leal1 K Boubaker2 A Sarmiento-Reyes3
A Perez-Sesma1 A Diaz-Sanchez3 V M Jimenez-Fernandez1 J Cervantes-Perez1
J Sanchez-Orea1 J Huerta-Chua4 L J Morales-Mendoza5 M Gonzalez-Lee5
C Hernandez-Mejia3 and F J Gonzalez-Martinez1
1Electronic Instrumentation and Atmospheric Sciences School Universidad VeracruzanaCircuito Gonzalo Aguirre Beltran SN 91000 Xalapa VER Mexico2Equipe de Physique des Dispositifs a Semiconducteurs Faculte des Sciences de TunisTunis El Manar University 2092 Tunis Tunisia3National Institute for Astrophysics Optics and Electronics Luis Enrique Erro No 1 Santa Marıa Tonantzintla72840 Puebla PUE Mexico4Civil Engineering School Universidad Veracruzana Venustiano Carranza SN Colonia Revolucion93390 PozaRica VER Mexico5Department of Electronics Engineering Universidad Veracruzana Venustiano Carranza SNColonia Revolucion 93390 Poza Rica VER Mexico
Correspondence should be addressed to H Vazquez-Leal hvazquezuvmx
Received 3 November 2014 Revised 23 December 2014 Accepted 27 December 2014
Academic Editor Shiping Lu
Copyright copy 2015 U Filobello-Nino et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We propose an approximate solution of T-F equation obtained by using the nonlinearities distribution homotopy perturbationmethod (NDHPM) Besides we show a table of comparison between this proposed approximate solution and a numerical of T-Fby establishing the accuracy of the results
1 Introduction
The numerical simulation for the charge distribution and theelectric field inside an atom is a very difficult task especiallyfor complex atoms For these cases the Thomas-Fermi (T-F)equation can be employed in order to obtain highly accurateapproximate results T-F method is based on the fact thatmost of complex atoms with a large number of electronshave relatively large quantum numbers and therefore thesemiclassical approach can be employed Then it is possibleto apply the concept of cells in the phase space to the states ofindividual electrons [1]
The Thomas-Fermi equation describes mathematicallyan infinite atom without border however the model is not
applicable to both distances from the nucleus too small andtoo large More precisely its application domain is limited toa range of distances 119877 large relative to the quantity 1119885 andsmall compared to 1 where 119885 is the atomic number of theatom However for complex atoms most of the electrons arein that interval for which the study of T-F equation becomesimportant
One of the important consequences of the Thomas-Fermi theory follows from the above The atom has an outerboundary for values of 119877 cong 1 thus the theory predicts thatthe dimensions of the atom are independent of119885 and for thesame reason the ionization energy of the outer electrons isalso independent of the atomic number [1]
Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2015 Article ID 405108 9 pageshttpdxdoiorg1011552015405108
2 Journal of Applied Mathematics
Problems like the one mentioned give rise to the searchof solutions to nonlinear differential equations but unfortu-nately solving this kind of equations is a difficult task As amatter of fact most of the time only an approximate solutionto such problems can be got In order to approach varioustypes of nonlinear differential equations several methodshave been proposed such as those based on variationalapproaches [2ndash4] tanh method [5] exp-function [6] Ado-mianrsquos decomposition method [7ndash12] parameter expansion[13] homotopy perturbation method [14ndash27] homotopyanalysis method [25 28ndash32] and perturbation method [2833 34] among others Also a few exact solutions to nonlineardifferential equations have been reported occasionally [35]
This study proposes a variation of the homotopy per-turbation method (HPM) by using nonlinearities distribu-tions [17] which allows finding an approximate solution ofThomas-Fermi equation [23 36ndash41] On the other hand itwill be seen that it is convenient to introduce an adjustingparameter in order to enlarge the domain of convergence ofour approach
This paper is organized as follows In Section 2 we pro-vide the basic concept of nonlinearities distributions homo-topy perturbation method (NDHPM) Section 3 presents theapplication of NDHPM to find an approximate solution ofThomas-Fermi equation Section 4 discusses the main resultsobtained Finally a brief conclusion is given in Section 5
2 Basic Idea of NDHPM
The standard homotopy perturbation method (HPM) wasproposed by Ji Huan He It was introduced like a powerfultool to approach various kinds of nonlinear problems Thehomotopy perturbation method can be considered as acombination of the classical perturbation technique and thehomotopy (whose origin is in the topology) and it is notrestricted to small parameters as traditional perturbationmethods For example HPM method requires neither smallparameter nor linearization but only few iterations to obtainaccurate solutions
To figure out howHPMmethodworks consider a generalnonlinear equation in the form
119860(119906) minus 119891(119903) = 0 119903 isin Ω (1)
with the following boundary conditions
119861(119906120597119906
120597119899) = 0 119903 isin Γ (2)
where 119860 is a general differential operator 119861 is a boundaryoperator 119891(119903) a known analytical function and Γ is thedomain boundary forΩ119860 can be divided into two operators119871 and 119873 where 119871 is linear and 119873 nonlinear from this laststatement (1) can be rewritten as
119871 (119906) + 119873 (119906) minus 119891 (119903) = 0 (3)
Generally a homotopy can be constructed in the form
119867(V 119901) = (1 minus 119901)[119871(V) minus 119871(1199060)]
+ 119901[119871(V) + 119873(V) minus 119891(119903)] = 0
119901 isin [0 1] 119903 isin Ω
(4)
Or
119867(V 119901) = 119871 (V) minus 119871 (1199060) + 119901[119871(1199060) + 119873(V) minus 119891(119903)] = 0
119901 isin [0 1] 119903 isin Ω
(5)
where 119901 is a homotopy parameter whose values are withinrange of 0 and 1 119906
0is the first approximation for the solution
of (3) that satisfies the boundary conditionsAssuming that solution for (4) or (5) can be written as a
power series of 119901
V = V0+ V1119901 + V21199012+ sdot sdot sdot (6)
Substituting (6) into (5) and equating identical powersof 119901 terms there can be found values for the sequenceV0 V1 V2
When 119901 rarr 1 it yields the approximate solution for (1)in the form
V = V0+ V1+ V2+ V3+ sdot sdot sdot (7)
Another way to build a homotopy which is relevant for thispaper it is by considering the following general equation
119871 (V) + 119873 (V) = 0 (8)
where 119871(V) and 119873(V) are the linear and nonlinear operatorsrespectively It is desired that solution for 119871(V) = 0 describesaccurately the original nonlinear system
By the homotopy technique a formulation is constructedas follows [16]
(1 minus 119901) 119871 (V) + 119901 [119871 (V) + 119873 (V)] = 0 (9)
Again it is assumed that solution for (9) can be written inthe form (6) thus taking the limit when 119901 rarr 1 results inthe approximate solution of (8)The convergence of solutionsobtained by HPMmethod is discussed in [20 21 25 26]
A recent report [17] introduced a modified version ofhomotopy perturbation method which eases the solutionssearching process for (3) As first step a homotopy of thefollowing form is introduced
119867(V 119901) = (1 minus 119901)[119871(V) minus 119871(1199060)]
+ 119901[119871(V) + 119873(V 119901) minus 119891(119903 119901)] = 0(10)
It can be noticed that the homotopy function (10) is essentiallythe same as (4) except for the nonlinear operator119873 and thenonhomogeneous function 119891 which contain embedded thehomotopy parameter 119901 We propose first to subdivide119873 and119891 into a series of terms andmultiply 119901119894 by the most nonlinearterms where 119894 is an integer number greater or equal to zero
Journal of Applied Mathematics 3
The 119894 power is selected according to how much displacementis desired in the interactions for the corresponding nonlinearterm of 119873 or 119891 The introduction of that parameter withinthe differential equation is a strategy to redistribute thenonlinearities between the successive iterations of the HPMmethod and thus it increases the probabilities of finding thesought solutionThe rest of the method is exactly the same asthe standard procedure for the HPMTherefore we establishthat
V = V0+ V1119901 + V21199012+ sdot sdot sdot (11)
When 119901 rarr 1 it results in the approximate solution for (3)in the form
V = V0+ V1+ V2+ V3+ sdot sdot sdot (12)
An advantage of this procedure is that given the distributionof nonlinearities from the differential equation over thesuccessive iterations of (11) less complex analytic approxima-tions may be obtained than those generated by the originalstandard HPM
Finally convergence of solutions obtained by NDHPMmethod is discussed in [17]
3 Approximate Solution ofThomas-Fermi Equation
In order to facilitate understanding of the NDHPM methodwe will solve approximately the equation
11991010158401015840=11991032
11990912 119910 (0) = 1 119910
1015840(0) = minus119871 (13)
where prime denotes differentiation with respect to 119909119910(119909) is a function related to the electrostatic potential
inside the atom 119909 is a variable proportional to distance fromthe nucleus (119903) and the most accurate value of 1199101015840(0) is 119871 =
1588071022611375313 [36]The above equation is a modified version of the original
problem of Thomas-Fermi where (13) is subject to theboundary conditions 119910(0) = 1 119910(infin) = 0 [28 41] althoughthere are other conditions that can be applied [40]
Following references [16 24] instead of defining a linearand nonlinear part in the above equation we add and subtract1205721199101015840+120573119910 as it is shown in order to add a term to linear operator
to facilitate the search for convergent solutions thus (13) canbe rewritten as
11991010158401015840+ 1205721199101015840+ 120573119910 minus [
11991032
11990912+ 1205721199101015840+ 120573119910] = 0 (14)
where 120572 and 120573 are constant parameters to determineThe linear part is identified as
119871(119910) = 11991010158401015840+ 1205721199101015840+ 120573119910 (15)
and the nonlinear is
119873(119910) = minus[11991032
11990912+ 1205721199101015840+ 120573119910] (16)
In order to simplify the calculations we will redistribute thenonlinear term by using NDHPMmethod starting from (9)in the form
11991010158401015840+ 1205721199101015840+ 120573119910 minus 119901[
11991032
11990912+ 119901120572119910
1015840+ 119901120573119910] = 0 (17)
by substituting
119910 = 1199100+ 1199101119901 + 11991021199012+ sdot sdot sdot (18)
into (17) and equating terms having identical powers of 119901weobtain
11991010158401015840
0+ 1205721199101015840
0+ 1205731199100= 0 119910
0(0) = 1 119910
1015840
0(0) = minus119871 (19)
11991010158401015840
1+ 1205721199101015840
1+ 1205731199101minus11991032
0
11990912= 0 119910
1 (0) = 0 1199101015840
1(0) = 0
(20)
In this paper we study the first order approximation (19)-(20) We will see that the nonlinear term of 119873(119910) (see (16))contains the sufficient information to obtain a good analyticalapproximation for (13)
The solution of (19) that satisfies the initial conditions isgiven by
1199100=119861 minus 119871
119861 minus 119860119890minus119860119909
+119871 minus 119860
119861 minus 119860119890minus119861119909
(21)
where 119860 and 119861 are constants related to 120572 and 120573By substituting (21) into (20) we obtain
11991010158401015840
1+ 1205721199101015840
1+ 1205731199101minus
1
11990912(119861 minus 119871
119861 minus 119860119890minus119860119909
+119871 minus 119860
119861 minus 119860119890minus119861119909
)
32
= 0
(22)
In order to simplify (22) and obtain a handy approximationfor (13) we will assume that there is an adequate choice of 119860and 119861 (119860 ≪ 119861) such that 119890minus119860119909 ≫ 119890
minus119861119909 valid for 0 lt 119909 lt infinin such a way we can rewrite (22) approximately as
11991010158401015840
1+ 1205721199101015840
1+ 1205731199101=11989632
1119890minus31198601199092
11990912(1 +
31198962
21198961
119890minus(119861minus119860)119909
) (23)
where we have defined
1198961=119861 minus 119871
119861 minus 119860 119896
2=119871 minus 119860
119861 minus 119860 (24)
This assumption is justified laterTo solve (23) we employ the method of variation of
parameters [42] which requires evaluating the followingintegrals
1199061= minusint
119891(119909)119890minus119861119909
119889119909
119882 119906
2= int
119891(119909)119890minus119860119909
119889119909
119882 (25)
4 Journal of Applied Mathematics
where 1199101198671
= 119890minus119860119909 and 119910
1198672= 119890minus119861119909 are the solutions of the
homogeneous differential equation
11991010158401015840
1+ 1205721199101015840
1+ 1205731199101= 0 (26)
119882 is theWronskian of these two functions which is given by
119882(1199101 1199102) = (119860 minus 119861)119890
minus(119860+119861)119909(119860 = 119861) (27)
and 119891(119909) is the right-hand side of (23)Substituting 119891(119909) and (27) into (25) leads to
1199061= minus
1
119860 minus 119861(11989632
1lim120576rarr0
int
119909
120576
119890minus1198601199102
119889119910
radic119910
+3
2119896211989612
1lim120576rarr0
int
119909
120576
119890minus(119861minus1198602)119910
119889119910
radic119910)
1199062=
1
119860 minus 119861(11989632
1lim120576rarr0
int
119909
120576
119890minus(31198602minus119861)119910
119889119910
radic119910
+3
2119896211989612
1lim120576rarr0
int
119909
120576
119890minus1198601199102
119889119910
radic119910)
(28)
By substituting 119910 = 1199062 the above equations take the form
1199061= minus
1
119860 minus 119861(211989632
1lim120576rarr0
int
radic119909
120576
119890minus11990621198602
119889119906
+ 3119896211989612
1lim120576rarr0
int
radic119909
120576
119890minus(119861minus1198602)119906
2
119889119906)
(29)
1199062=
1
119860 minus 119861(211989632
1lim120576rarr0
int
radic119909
120576
119890(119861minus31198602)119906
2
119889119906
+ 3119896211989612
1lim120576rarr0
int
radic119909
120576
119890minus11986011990622119889119906)
(30)
Therefore the solution of (23) is written according to themethod of variation of parameters as [42]
1199101= 119891119890minus119860119909
+ 119892119890minus119861119909
+ 1199061119890minus119860119909
+ 1199062119890minus119861119909
(31)
where 119891 and 119892 are constants and 1199061and 119906
2are given by (29)
and (30) respectivelyApplying the initial condition 119910
1(0) = 0 to (31) leads to
1199101(0) = 119891 + 119892 minus
1
119860 minus 119861(211989632
1lim120576rarr0119909rarr0
int
radic119909
120576
119890minus11986011990622119889119906
+ 3119896211989612
1lim120576rarr0119909rarr0
int
radic119909
120576
119890minus(119861minus1198602)119906
2
119889119906)
+1
119860 minus 119861(211989632
1lim120576rarr0119909rarr0
int
radic119909
120576
119890(119861minus31198602)119906
2
119889119906
+ 3119896211989612
1lim120576rarr0119909rarr0
int
radic119909
120576
119890minus11986011990622119889119906) = 0
(32)
Equation (32) is simplified to
119891 + 119892 = 0 (33)
On the other hand to apply the condition 1199101015840
1(0) = 0 we
differentiate (31) to obtain
1199101015840
1= minus119860119891119890
minus119860119909minus 119861119892119890
minus119861119909
minus119890minus119860119909
119860 minus 119861(11989632
1
119890minus1198601199092
radic119909+3119896211989612
1119890minus(119861minus1198602)119909
2radic119909)
+119890minus119861119909
119860 minus 119861(11989632
1
119890minus(31198602minus119861)119909
radic119909+3119896211989612
1119890minus1198602119909
2radic119909)
minus 119860119890minus119860119909
1199061minus 119861119890minus119861119909
1199062
(34)
After performing algebraic simplifications to (34) we obtain
1199101015840
1= minus119860119891119890
minus119860119909minus 119861119892119890
minus119861119909minus 119860119890minus119860119909
1199061minus 119861119890minus119861119909
1199062 (35)
Applying the condition 1199101015840
1(0) = 0 to (35) the following is
obtained
119860119891 + 119861119892 = 0 (36)
since lim119909rarr0
1199061= 0 and lim
119909rarr01199062= 0 (see (29) and (30))
From (33) and (36) we obtain 119891 = 0 and 119892 = 0 since119860 = 119861 therefore (31) becomes
1199101= 1199061119890minus119860119909
+ 1199062119890minus119861119909
(37)
By substituting (21) and (37) into (12) we obtain the followingfirst order approximation for the solution of (13)
119910 = 1198961119890minus119860119909
+ 1198962119890minus119861119909
+ 1199061119890minus119860119909
+ 1199062119890minus119861119909
(38)
We will show later that it is possible to achieve a good fit andenlarge the domain of convergence for (38) (keeping at thesame time the handy character of the proposed solution) byintroducing in (38) an adjusting parameter (consistent withthe initial conditions) as it is shown in
119910 = 1198961119890minus119860119909
+ 1198962119890minus119861119909
+ 1205751199061119890minus119860119909
+ 1205751199062119890minus119861119909
(39)
The inclusion of parameter 120575 is motivated because thesolution 119910
0(119909) for the homogeneous equation (19) is undeter-
mined by a constant factor 120572 (without considering the initialvalues) and from the following argument
Let 1198840(119909) = 120572119910
0(119909) since 119884
0(0) = 120572 1199101015840
0(0) = minus119871120572 the
first order solution for a function say 119884(119909) which definesa slightly different problem from the original is given by119884(119909) = 119884
0(119909) + 119884
1(119909) or 119884(119909) = 120572119910
0(119909) + 119884
1(119909) where
1199100(0) = 1 1199101015840
0(0) = minus119871 119884
1(0) = 0 and 119884
1015840
1(0) = 0 Thus
after dividing by 120572 119910(119909) = 1199100(119909) + 120575119910
1(119909) is obtained (where
120575 = 119891(120572)120572 119910(119909) = 119884(119909)120572 and we have supposed thatfor some problems 119884
1(119909) = 119891(120572)119910
1(119909) As a matter of fact
following (21)ndash(30)we conclude that for this case119891(120572) = 12057232therefore 120575 = 12057212)
Journal of Applied Mathematics 5
Substituting (29) and (30) into (39) we obtain
119910 = 1198961119890minus119860119909
+ 1198962119890minus119861119909
minus120575119890minus119860119909
119860 minus 119861(211989632
1lim120576rarr0
int
radic119909
120576
119890minus11986011990622119889119906
+ 3119896211989612
1lim120576rarr0
int
radic119909
120576
119890minus(119861minus1198602)119906
2
119889119906)
+120575119890minus119861119909
119860 minus 119861(211989632
1lim120576rarr0
int
radic119909
120576
119890(119861minus31198602)119906
2
119889119906
+ 3119896211989612
1lim120576rarr0
int
radic119909
120576
119890minus11986011990622119889119906)
(40)
In [14] a high accurate approximation of normal distributionintegral was reported which let us write the above integralsas
lim120576rarr0
int
radic119909
120576
119890minus11986011990622119889119906
=1
2
radic2120587
119860tanh(39
2
radic119860119909
2120587minus111
2tanminus1( 35
111
radic119860119909
2120587))
lim120576rarr0
int
radic119909
120576
119890minus(119861minus1198602)119906
2
119889119906
=1
2
radic2120587
2119861 minus 119860tanh(39
2
radic(2119861 minus 119860)119909
2120587
minus111
2tanminus1( 35
111
radic(2119861 minus 119860)119909
2120587))
(41)
Note that
int
radic119909
0
119890(119861minus31198602)119906
2
119889119906 (42)
is not a Gaussian integral on the assumption that 119890minus119860119909 ≫ 119890minus119861119909
for 0 lt 119909 lt infin since it implies that 119860 ≪ 119861
Equations (41) allow us to write (40) as
119910 = 1198961119890minus119860119909
+ 1198962119890minus119861119909
minus120575119890minus119860119909
119860 minus 119861
sdot (11989632
1radic2120587
119860tanh(39
2
radic119860119909
2120587
minus111
2tanminus1( 35
111
radic119860119909
2120587))
+3
2119896211989612
1radic
2120587
2119861 minus 119860
sdot tanh(39
2
radic(2119861 minus 119860)119909
2120587
minus111
2tanminus1( 35
111
radic(2119861 minus 119860)119909
2120587)))
+120575119890minus119861119909
119860 minus 119861(211989632
1lim120576rarr0
int
radic119909
120576
119890(119861minus31198602)119906
2
119889119906
+3
2119896211989612
1radic2120587
119860
sdot tanh(392
radic119860119909
2120587minus111
2tanminus1( 35
111
radic119860119909
2120587)))
(43)
To transform (43) into an analytical expression we evaluate(42) keeping terms up to the ninth power to obtain thefollowing compact fractional power function
int
radic119909
0
119890(119861minus31198602)119906
2
119889119906
= radic119909 +1
3(119861 minus 3119860)119909
32+
1
10(119861 minus 3119860)
211990952
+1
42(119861 minus 3119860)
311990972
+1
216(119861 minus 3119860)
411990992
(44)
where the main contribution of this approximation to (43) isin the range of 119909 isin (0 1] because it is multiplied by a negativeexponential (see (43) and Section 4)
6 Journal of Applied Mathematics
Therefore substituting (44) into (43) the following isobtained
119910 = 1198961119890minus119860119909
+ 1198962119890minus119861119909
minus120575119890minus119860119909
119860 minus 119861
sdot (11989632
1radic2120587
119860tanh(39
2
radic119860119909
2120587
minus111
2tanminus1( 35
111
radic119860119909
2120587))
+3
2119896211989612
1radic
2120587
2119861 minus 119860
sdot tanh(39
2
radic(2119861 minus 119860)119909
2120587
minus111
2tanminus1( 35
111
radic(2119861 minus 119860) 119909
2120587)))
+120575119890minus119861119909
119860 minus 119861(211989632
1(radic119909 +
(119861 minus 3119860)11990932
3+(119861 minus 3119860)
211990952
10
+(119861 minus 3119860)
311990972
42+(119861 minus 3119860)
411990992
216)
+3
2119896211989612
1radic2120587
119860
sdot tanh(392
radic119860119909
2120587minus111
2tanminus1( 35
111
radic119860119909
2120587)))
(45)
In order to obtain a good approximation from (45) weoptimize the values of the aforementioned parameters asfollows 119860 = 0398521712348839 119861 = 6 and 120575 =
minus0623239813 using the procedure reported in [14 17 27](note that with these values the assumption 119860 ≪ 119861 issatisfied besides the above results allow to know 120572 and 120573
whose values are not needed in the search for the solutionof the system (19) and (20))
Figure 1 and Table 1 show the comparison betweenapproximate solution (45) and the exact solution The accu-racy of (45) is clear as an approximate solution for (13)
4 Discussion
Table 1 shows the comparison between numerical solutiongiven in [39] and approximations [15 38] and (45) It isclear that our approximation (45) is competitive consideringthat it is analytic and handy unlike other approximationswhich are expressed in terms of power series In [37 3841] T-F equation was solved by homotopy analysis method(HAM) The approximation [38] corresponds to an 80thorder analytical approximation and therefore is impracticalOn the other hand in [37] the results are given in terms of
1
08
06
04
02
0
y(x)
0 05 1 15 2 25 3 35 4 45 5
x
[39](45)
Figure 1 Comparison of Thomas-Fermirsquos numerical solution [39]and approximation (45)
infinite series and show particularly the calculation of theinitial slope 1199101015840(0) Moreover in [41] an accurate approximatesolution was obtained but it corresponds to the 40th orderapproximation In [23] (13) was solved by HPM and Padeapproximants in this case a solution of 26 power series termsis provided which is also not handy in practice Reference[15] presents an analytical approach but it is not accurateIn this study we used homotopy perturbation method withnonlinearities distributions to find an accurate solution for(13) (see Figure 1 and Table 1) Although (43) is just almostanalytic (due to the presence of a non-Gaussian integral) itcan be estimated that contribution of (42) to the solution(43) is well represented by (44) assuming the values are 119860 =
0398521712348839 119861 = 6 and 120575 = minus0623239813 thus weobtained the analytical approximation (45)We just kept onlyfive terms in (44) in order to obtain a handy approximationBesides the right-hand side of (44) adequately represents theindicated non-Gaussian integral in the range 119909 isin [0 1] Itimplies that the best contribution of the term proportional to119890minus119861119909
intradic119909
0119890(119861minus31198602)119906
2
119889119906 in (43) is just beginning of the intervalconsidered that is in a part of the domain where the T-F equation describes correctly the atom (see Section 1) Forvalues 119909 gt 1 the above-mentioned term is dominated forthe exponential 119890minus119861119909 In any case the value of the integralsof (40) can be evaluated by using numerical methods Aninteresting fact is that if more terms had been consideredin the development of (12) we would have obtained asa solution of (13) a series containing terms of hyperbolictangent functions to the form (41)
The strategy to redistribute the nonlinearities of T-Fequation between the successive iterations of the HPMmethod was important to obtain an approximate solution of(13) because it allows us to simplify the solution procedurefor the successive iterations of the HPM method Besideswe obtained a soluble coupled system of linear differentialequations (19) and (20) In this study we introduced anadjusting parameter in order to enlarge the domain ofconvergence of our approach with good results by using onlythe first order approximation (see Table 1 and Figure 1)
Journal of Applied Mathematics 7
Table 1 Numerical comparison of Thomas-Fermirsquos numerical solution [39] and approximations [15 38] and (45)
119909 Numerical [39] [38] (45) [15]000 mdash 1000000000 1000000000 1000000000025 0755880759 0776191000 0708502932 0680650028050 0606700008 0615917000 0570491745 0459455663075 0502964042 0505380000 0485018011 0307042537100 0424333179 0423772000 0421167227 0202655526125 0363227937 0362935000 0369542163 0131667603150 0314660642 0314490000 0326298526 0083800106175 0275233848 0275154000 0289315779 0051853222200 0242678587 0242718000 0257253699 0030801947225 0215439334 0215630000 0229210094 0017153465250 0192406328 0192795000 0204540381 0008491277275 0172758691 0173364000 0182755264 0003152503300 0155871862 0156719000 0163464474 0325 0141260504 0142371000 0146346052 minus0001738122
350 0128541381 0129937000 0131128799 minus0002581206
375 0117408054 0119108000 0117581337 minus0002874924
400 0107612958 0109632000 0105504623 minus000284628
425 0098954329 0101303000 0094726427 minus0002641801
450 0091266456 0093950400 0085097045 minus0002353931
475 0084412289 0087432000 0076485875 minus0002039168
500 0078277758 0081629600 0068778618 minus0001730443
A relevant fact is that NDHPM requires little iterationsto obtain accurate solutions if we have a first approximationcontaining as much information as possible for a nonlineardifferential equation For instance in our case (21) containsthe correct asymptotic character of the exact equation
5 Conclusion
In this work NDHPM was used to obtain an approximateanalytical solution for T-F equation Our solution is novelbecause it is expressed in terms of exponential and fractionalpower functions and above all in terms of normal distributionintegrals It is important to obtain an analytical expressionthat provides a good description of the solution for thenonlinear differential equations like (13) For instance thecharge distribution and the electrostatic potential inside anatom are adequately described by (43) and (45) It is worthmentioning that if the initial guess is suitably chosen it ispossible to obtain by this method an accurate approximationlike (43) even using the first terms of (12) Finally in contrastto Runge-Kutta numerical solution NDHPMmethod allowsto analyze quantitatively and qualitatively the solution
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the financial supportfrom the National Council for Science and Technology ofMexico (CONACyT) through Grant CB-2010-01 157024The authors would like to thank Rogelio-Alejandro Callejas-Molina and Roberto Ruiz-Gomez for their contribution tothis project
References
[1] L Landau and E M LifshitzMecanica Cuantica No RelativistaEditorial Reverte S A 2nd edition 1967
[2] LM B Assas ldquoApproximate solutions for the generalizedKdVndashBurgersrsquo equation by Hersquos variational iteration methodrdquo PhysicaScripta vol 76 no 2 pp 161ndash164 2007
[3] M Kazemnia S A Zahedi M Vaezi and N Tolou ldquoArseniccontamination in New Orleans soil temporal changes associ-ated with floodingrdquo Journal of Applied Sciences vol 8 no 22pp 4192ndash4197 2008
[4] R Noorzad A T Poor and M Omidvar ldquoVariational iterationmethod and homotopy-perturbation method for solving Burg-ers equation in fluid dynamicsrdquo Journal of Applied Sciences vol8 no 2 pp 369ndash373 2008
[5] D J Evans and K R Raslan ldquoThe tanh function method forsolving some important non-linear partial differential equa-tionsrdquo International Journal of Computer Mathematics vol 82no 7 pp 897ndash905 2005
[6] J Mahmoudi N Tolou I Khatami A Barari and D D GanjildquoExplicit solution of nonlinear ZK-BBM wave equation using
8 Journal of Applied Mathematics
Exp-function methodrdquo Journal of Applied Sciences vol 8 no 2pp 358ndash363 2008
[7] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988
[8] E Babolian and J Biazar ldquoOn the order of convergence ofAdomian methodrdquo Applied Mathematics and Computation vol130 no 2-3 pp 383ndash387 2002
[9] A Kooch and M Abadyan ldquoEfficiency of modified Ado-mian decomposition for simulating the instability of nano-electromechanical switches comparison with the conventionaldecomposition methodrdquo Trends in Applied Sciences Researchvol 7 no 1 pp 57ndash67 2012
[10] Y Khan H Vazquez-Leal and N Faraz ldquoAn auxiliary param-eter method using Adomian polynomials and Laplace transfor-mation for nonlinear differential equationsrdquoAppliedMathemat-ical Modelling vol 37 no 5 pp 2702ndash2708 2013
[11] S K Vanani S Heidari and M Avaji ldquoA low-cost numericalalgorithm for the solution of nonlinear delay boundary integralequationsrdquo Journal of Applied Sciences vol 11 no 20 pp 3504ndash3509 2011
[12] S H Chowdhury ldquoA comparison between the modifiedhomotopy perturbation method and adomian decompositionmethod for solving nonlinear heat transfer equationsrdquo Journalof Applied Sciences vol 11 no 7 pp 1416ndash1420 2011
[13] L-N Zhang and L Xu ldquoDetermination of the limit cycle byHersquos parameter expansion for oscillators in a u31 + u2 potentialrdquoZeitschrift fur NaturforschungmdashSection A Journal of PhysicalSciences vol 62 no 7-8 pp 396ndash398 2007
[14] H Vazquez-Leal R Castaneda-Sheissa U Filobello-Nino ASarmiento-Reyes and J Sanchez Orea ldquoHigh accurate simpleapproximation of normal distribution integralrdquo MathematicalProblems in Engineering vol 2012 Article ID 124029 22 pages2012
[15] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006
[16] A Fereidoon Y Rostamiyan M Akbarzade and D D GanjildquoApplication of Hersquos homotopy perturbation method to nonlin-ear shock damper dynamicsrdquo Archive of Applied Mechanics vol80 no 6 pp 641ndash649 2010
[17] H Vazquez-Leal U Filobello-Nino R Castaneda-SheissaL Hernandez-Martınez and A Sarmiento-Reyes ldquoModifiedHPMs inspired by homotopy continuation methodsrdquo Mathe-matical Problems in Engineering vol 2012 Article ID 30912319 pages 2012
[18] Y Khan H Vazquez-Leal and Q Wu ldquoAn efficient iteratedmethod for mathematical biology modelrdquo Neural Computingand Applications vol 23 no 3-4 pp 677ndash682 2013
[19] Y Khan H Vazquez-Leal and N Faraz ldquoAn efficient newiterative method for oscillator differential equationrdquo ScientiaIranica vol 19 no 6 pp 1473ndash1477 2012
[20] J Biazar and H Aminikhah ldquoStudy of convergence of homo-topy perturbation method for systems of partial differentialequationsrdquoComputersampMathematics with Applications vol 58no 11-12 pp 2221ndash2230 2009
[21] J Biazar and H Ghazvini ldquoConvergence of the homotopy per-turbation method for partial differential equationsrdquo NonlinearAnalysis Real World Applications vol 10 no 5 pp 2633ndash26402009
[22] Y Khan and Q Wu ldquoHomotopy perturbation transformmethod for nonlinear equations using Hersquos polynomialsrdquo Com-puters and Mathematics with Applications vol 61 no 8 pp1963ndash1967 2011
[23] M A Noor and S T Mohyud-Din ldquoHomotopy perturbationmethod for solvingThomas-Fermi equation using pade approx-imantsrdquo International Journal of Nonlinear Science vol 8 no 1pp 27ndash31 2009
[24] DDGanji A R Sahouli andM Famouri ldquoAnewmodificationofHersquos homotopy perturbationmethod for rapid convergence ofnonlinear undamped oscillatorsrdquo Journal of Applied Mathemat-ics and Computing vol 30 no 1-2 pp 181ndash192 2009
[25] M Turkyilmazoglu ldquoSome issues on HPM and HAMmethodsa convergence schemerdquoMathematical and Computer Modellingvol 53 no 9-10 pp 1929ndash1936 2011
[26] M Turkyilmazoglu ldquoConvergence of the homotopy perturba-tion methodrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 12 no 1ndash8 pp 9ndash14 2011
[27] U Filobello-Nino H Vazquez-Leal R Castaneda-Sheissa et alldquoAn approximate solution of Blasius equation by using HPMmethodrdquo Asian Journal of Mathematics and Statistics vol 5 no2 pp 50ndash59 2012
[28] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineering Springer Berlin Germany 1st edition 2011
[29] T Patel M N Mehta and V H Pradhan ldquoThe numericalsolution of Burgerrsquos equation arising into the irradiation oftumour tissue in biological diffusing system by homotopyanalysis methodrdquo Asian Journal of Applied Sciences vol 5 no1 pp 60ndash66 2012
[30] S Liao Advances in the Homotopy Analysis Method WorldScientific Hackensack NJ USA 2014
[31] M Turkyilmazoglu ldquoAnalytic approximate solutions of rotatingdisk boundary layer flow subject to a uniform suction orinjectionrdquo International Journal of Mechanical Sciences vol 52no 12 pp 1735ndash1744 2010
[32] M Turkyilmazoglu ldquoPurely analytic solutions of the compress-ible boundary layer flow due to a porous rotating disk with heattransferrdquo Physics of Fluids vol 21 no 10 2009
[33] U Filobello-Nino H Vazquez-Leal B Benhammouda et alldquoA handy approximation for a mediated bioelectrocatalysisprocess related to Michaelis-Menten equationrdquo SpringerPlusvol 3 article 162 2014
[34] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoUsing per-turbation methods and Laplace-Pade approximation to solvenonlinear problemsrdquo Miskolc Mathematical Notes vol 14 no1 pp 89ndash101 2013
[35] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoA handyexact solution for flow due to a stretching boundary with partialsliprdquo Revista Mexicana de Fısica E vol 59 no 1 pp 51ndash55 2013
[36] FM Fernandez ldquoRational approximation to theThomas-Fermiequationsrdquo Applied Mathematics and Computation vol 217 no13 pp 6433ndash6436 2011
[37] B Yao ldquoA series solution to the Thomas-Fermi equationrdquoApplied Mathematics and Computation vol 203 no 1 pp 396ndash401 2008
[38] H Khan and H Xu ldquoSeries solution to the Thomas-Fermiequationrdquo Physics Letters A vol 365 no 1-2 pp 111ndash115 2007
[39] K Parand andM Shahini ldquoRational Chebyshev pseudospectralapproach for solving Thomas-Fermi equationrdquo Physics LettersA vol 373 no 2 pp 210ndash213 2009
Journal of Applied Mathematics 9
[40] E Hille ldquoOn the Thomas-Fermi equationrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 62 pp 7ndash10 1969
[41] M Turkyilmazoglu ldquoSolution of the Thomas-Fermi equationwith a convergent approachrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 11 pp 4097ndash41032012
[42] D G ZillA First Course in Differential Equations withModelingApplications Brooks Cole 10th edition 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Applied Mathematics
Problems like the one mentioned give rise to the searchof solutions to nonlinear differential equations but unfortu-nately solving this kind of equations is a difficult task As amatter of fact most of the time only an approximate solutionto such problems can be got In order to approach varioustypes of nonlinear differential equations several methodshave been proposed such as those based on variationalapproaches [2ndash4] tanh method [5] exp-function [6] Ado-mianrsquos decomposition method [7ndash12] parameter expansion[13] homotopy perturbation method [14ndash27] homotopyanalysis method [25 28ndash32] and perturbation method [2833 34] among others Also a few exact solutions to nonlineardifferential equations have been reported occasionally [35]
This study proposes a variation of the homotopy per-turbation method (HPM) by using nonlinearities distribu-tions [17] which allows finding an approximate solution ofThomas-Fermi equation [23 36ndash41] On the other hand itwill be seen that it is convenient to introduce an adjustingparameter in order to enlarge the domain of convergence ofour approach
This paper is organized as follows In Section 2 we pro-vide the basic concept of nonlinearities distributions homo-topy perturbation method (NDHPM) Section 3 presents theapplication of NDHPM to find an approximate solution ofThomas-Fermi equation Section 4 discusses the main resultsobtained Finally a brief conclusion is given in Section 5
2 Basic Idea of NDHPM
The standard homotopy perturbation method (HPM) wasproposed by Ji Huan He It was introduced like a powerfultool to approach various kinds of nonlinear problems Thehomotopy perturbation method can be considered as acombination of the classical perturbation technique and thehomotopy (whose origin is in the topology) and it is notrestricted to small parameters as traditional perturbationmethods For example HPM method requires neither smallparameter nor linearization but only few iterations to obtainaccurate solutions
To figure out howHPMmethodworks consider a generalnonlinear equation in the form
119860(119906) minus 119891(119903) = 0 119903 isin Ω (1)
with the following boundary conditions
119861(119906120597119906
120597119899) = 0 119903 isin Γ (2)
where 119860 is a general differential operator 119861 is a boundaryoperator 119891(119903) a known analytical function and Γ is thedomain boundary forΩ119860 can be divided into two operators119871 and 119873 where 119871 is linear and 119873 nonlinear from this laststatement (1) can be rewritten as
119871 (119906) + 119873 (119906) minus 119891 (119903) = 0 (3)
Generally a homotopy can be constructed in the form
119867(V 119901) = (1 minus 119901)[119871(V) minus 119871(1199060)]
+ 119901[119871(V) + 119873(V) minus 119891(119903)] = 0
119901 isin [0 1] 119903 isin Ω
(4)
Or
119867(V 119901) = 119871 (V) minus 119871 (1199060) + 119901[119871(1199060) + 119873(V) minus 119891(119903)] = 0
119901 isin [0 1] 119903 isin Ω
(5)
where 119901 is a homotopy parameter whose values are withinrange of 0 and 1 119906
0is the first approximation for the solution
of (3) that satisfies the boundary conditionsAssuming that solution for (4) or (5) can be written as a
power series of 119901
V = V0+ V1119901 + V21199012+ sdot sdot sdot (6)
Substituting (6) into (5) and equating identical powersof 119901 terms there can be found values for the sequenceV0 V1 V2
When 119901 rarr 1 it yields the approximate solution for (1)in the form
V = V0+ V1+ V2+ V3+ sdot sdot sdot (7)
Another way to build a homotopy which is relevant for thispaper it is by considering the following general equation
119871 (V) + 119873 (V) = 0 (8)
where 119871(V) and 119873(V) are the linear and nonlinear operatorsrespectively It is desired that solution for 119871(V) = 0 describesaccurately the original nonlinear system
By the homotopy technique a formulation is constructedas follows [16]
(1 minus 119901) 119871 (V) + 119901 [119871 (V) + 119873 (V)] = 0 (9)
Again it is assumed that solution for (9) can be written inthe form (6) thus taking the limit when 119901 rarr 1 results inthe approximate solution of (8)The convergence of solutionsobtained by HPMmethod is discussed in [20 21 25 26]
A recent report [17] introduced a modified version ofhomotopy perturbation method which eases the solutionssearching process for (3) As first step a homotopy of thefollowing form is introduced
119867(V 119901) = (1 minus 119901)[119871(V) minus 119871(1199060)]
+ 119901[119871(V) + 119873(V 119901) minus 119891(119903 119901)] = 0(10)
It can be noticed that the homotopy function (10) is essentiallythe same as (4) except for the nonlinear operator119873 and thenonhomogeneous function 119891 which contain embedded thehomotopy parameter 119901 We propose first to subdivide119873 and119891 into a series of terms andmultiply 119901119894 by the most nonlinearterms where 119894 is an integer number greater or equal to zero
Journal of Applied Mathematics 3
The 119894 power is selected according to how much displacementis desired in the interactions for the corresponding nonlinearterm of 119873 or 119891 The introduction of that parameter withinthe differential equation is a strategy to redistribute thenonlinearities between the successive iterations of the HPMmethod and thus it increases the probabilities of finding thesought solutionThe rest of the method is exactly the same asthe standard procedure for the HPMTherefore we establishthat
V = V0+ V1119901 + V21199012+ sdot sdot sdot (11)
When 119901 rarr 1 it results in the approximate solution for (3)in the form
V = V0+ V1+ V2+ V3+ sdot sdot sdot (12)
An advantage of this procedure is that given the distributionof nonlinearities from the differential equation over thesuccessive iterations of (11) less complex analytic approxima-tions may be obtained than those generated by the originalstandard HPM
Finally convergence of solutions obtained by NDHPMmethod is discussed in [17]
3 Approximate Solution ofThomas-Fermi Equation
In order to facilitate understanding of the NDHPM methodwe will solve approximately the equation
11991010158401015840=11991032
11990912 119910 (0) = 1 119910
1015840(0) = minus119871 (13)
where prime denotes differentiation with respect to 119909119910(119909) is a function related to the electrostatic potential
inside the atom 119909 is a variable proportional to distance fromthe nucleus (119903) and the most accurate value of 1199101015840(0) is 119871 =
1588071022611375313 [36]The above equation is a modified version of the original
problem of Thomas-Fermi where (13) is subject to theboundary conditions 119910(0) = 1 119910(infin) = 0 [28 41] althoughthere are other conditions that can be applied [40]
Following references [16 24] instead of defining a linearand nonlinear part in the above equation we add and subtract1205721199101015840+120573119910 as it is shown in order to add a term to linear operator
to facilitate the search for convergent solutions thus (13) canbe rewritten as
11991010158401015840+ 1205721199101015840+ 120573119910 minus [
11991032
11990912+ 1205721199101015840+ 120573119910] = 0 (14)
where 120572 and 120573 are constant parameters to determineThe linear part is identified as
119871(119910) = 11991010158401015840+ 1205721199101015840+ 120573119910 (15)
and the nonlinear is
119873(119910) = minus[11991032
11990912+ 1205721199101015840+ 120573119910] (16)
In order to simplify the calculations we will redistribute thenonlinear term by using NDHPMmethod starting from (9)in the form
11991010158401015840+ 1205721199101015840+ 120573119910 minus 119901[
11991032
11990912+ 119901120572119910
1015840+ 119901120573119910] = 0 (17)
by substituting
119910 = 1199100+ 1199101119901 + 11991021199012+ sdot sdot sdot (18)
into (17) and equating terms having identical powers of 119901weobtain
11991010158401015840
0+ 1205721199101015840
0+ 1205731199100= 0 119910
0(0) = 1 119910
1015840
0(0) = minus119871 (19)
11991010158401015840
1+ 1205721199101015840
1+ 1205731199101minus11991032
0
11990912= 0 119910
1 (0) = 0 1199101015840
1(0) = 0
(20)
In this paper we study the first order approximation (19)-(20) We will see that the nonlinear term of 119873(119910) (see (16))contains the sufficient information to obtain a good analyticalapproximation for (13)
The solution of (19) that satisfies the initial conditions isgiven by
1199100=119861 minus 119871
119861 minus 119860119890minus119860119909
+119871 minus 119860
119861 minus 119860119890minus119861119909
(21)
where 119860 and 119861 are constants related to 120572 and 120573By substituting (21) into (20) we obtain
11991010158401015840
1+ 1205721199101015840
1+ 1205731199101minus
1
11990912(119861 minus 119871
119861 minus 119860119890minus119860119909
+119871 minus 119860
119861 minus 119860119890minus119861119909
)
32
= 0
(22)
In order to simplify (22) and obtain a handy approximationfor (13) we will assume that there is an adequate choice of 119860and 119861 (119860 ≪ 119861) such that 119890minus119860119909 ≫ 119890
minus119861119909 valid for 0 lt 119909 lt infinin such a way we can rewrite (22) approximately as
11991010158401015840
1+ 1205721199101015840
1+ 1205731199101=11989632
1119890minus31198601199092
11990912(1 +
31198962
21198961
119890minus(119861minus119860)119909
) (23)
where we have defined
1198961=119861 minus 119871
119861 minus 119860 119896
2=119871 minus 119860
119861 minus 119860 (24)
This assumption is justified laterTo solve (23) we employ the method of variation of
parameters [42] which requires evaluating the followingintegrals
1199061= minusint
119891(119909)119890minus119861119909
119889119909
119882 119906
2= int
119891(119909)119890minus119860119909
119889119909
119882 (25)
4 Journal of Applied Mathematics
where 1199101198671
= 119890minus119860119909 and 119910
1198672= 119890minus119861119909 are the solutions of the
homogeneous differential equation
11991010158401015840
1+ 1205721199101015840
1+ 1205731199101= 0 (26)
119882 is theWronskian of these two functions which is given by
119882(1199101 1199102) = (119860 minus 119861)119890
minus(119860+119861)119909(119860 = 119861) (27)
and 119891(119909) is the right-hand side of (23)Substituting 119891(119909) and (27) into (25) leads to
1199061= minus
1
119860 minus 119861(11989632
1lim120576rarr0
int
119909
120576
119890minus1198601199102
119889119910
radic119910
+3
2119896211989612
1lim120576rarr0
int
119909
120576
119890minus(119861minus1198602)119910
119889119910
radic119910)
1199062=
1
119860 minus 119861(11989632
1lim120576rarr0
int
119909
120576
119890minus(31198602minus119861)119910
119889119910
radic119910
+3
2119896211989612
1lim120576rarr0
int
119909
120576
119890minus1198601199102
119889119910
radic119910)
(28)
By substituting 119910 = 1199062 the above equations take the form
1199061= minus
1
119860 minus 119861(211989632
1lim120576rarr0
int
radic119909
120576
119890minus11990621198602
119889119906
+ 3119896211989612
1lim120576rarr0
int
radic119909
120576
119890minus(119861minus1198602)119906
2
119889119906)
(29)
1199062=
1
119860 minus 119861(211989632
1lim120576rarr0
int
radic119909
120576
119890(119861minus31198602)119906
2
119889119906
+ 3119896211989612
1lim120576rarr0
int
radic119909
120576
119890minus11986011990622119889119906)
(30)
Therefore the solution of (23) is written according to themethod of variation of parameters as [42]
1199101= 119891119890minus119860119909
+ 119892119890minus119861119909
+ 1199061119890minus119860119909
+ 1199062119890minus119861119909
(31)
where 119891 and 119892 are constants and 1199061and 119906
2are given by (29)
and (30) respectivelyApplying the initial condition 119910
1(0) = 0 to (31) leads to
1199101(0) = 119891 + 119892 minus
1
119860 minus 119861(211989632
1lim120576rarr0119909rarr0
int
radic119909
120576
119890minus11986011990622119889119906
+ 3119896211989612
1lim120576rarr0119909rarr0
int
radic119909
120576
119890minus(119861minus1198602)119906
2
119889119906)
+1
119860 minus 119861(211989632
1lim120576rarr0119909rarr0
int
radic119909
120576
119890(119861minus31198602)119906
2
119889119906
+ 3119896211989612
1lim120576rarr0119909rarr0
int
radic119909
120576
119890minus11986011990622119889119906) = 0
(32)
Equation (32) is simplified to
119891 + 119892 = 0 (33)
On the other hand to apply the condition 1199101015840
1(0) = 0 we
differentiate (31) to obtain
1199101015840
1= minus119860119891119890
minus119860119909minus 119861119892119890
minus119861119909
minus119890minus119860119909
119860 minus 119861(11989632
1
119890minus1198601199092
radic119909+3119896211989612
1119890minus(119861minus1198602)119909
2radic119909)
+119890minus119861119909
119860 minus 119861(11989632
1
119890minus(31198602minus119861)119909
radic119909+3119896211989612
1119890minus1198602119909
2radic119909)
minus 119860119890minus119860119909
1199061minus 119861119890minus119861119909
1199062
(34)
After performing algebraic simplifications to (34) we obtain
1199101015840
1= minus119860119891119890
minus119860119909minus 119861119892119890
minus119861119909minus 119860119890minus119860119909
1199061minus 119861119890minus119861119909
1199062 (35)
Applying the condition 1199101015840
1(0) = 0 to (35) the following is
obtained
119860119891 + 119861119892 = 0 (36)
since lim119909rarr0
1199061= 0 and lim
119909rarr01199062= 0 (see (29) and (30))
From (33) and (36) we obtain 119891 = 0 and 119892 = 0 since119860 = 119861 therefore (31) becomes
1199101= 1199061119890minus119860119909
+ 1199062119890minus119861119909
(37)
By substituting (21) and (37) into (12) we obtain the followingfirst order approximation for the solution of (13)
119910 = 1198961119890minus119860119909
+ 1198962119890minus119861119909
+ 1199061119890minus119860119909
+ 1199062119890minus119861119909
(38)
We will show later that it is possible to achieve a good fit andenlarge the domain of convergence for (38) (keeping at thesame time the handy character of the proposed solution) byintroducing in (38) an adjusting parameter (consistent withthe initial conditions) as it is shown in
119910 = 1198961119890minus119860119909
+ 1198962119890minus119861119909
+ 1205751199061119890minus119860119909
+ 1205751199062119890minus119861119909
(39)
The inclusion of parameter 120575 is motivated because thesolution 119910
0(119909) for the homogeneous equation (19) is undeter-
mined by a constant factor 120572 (without considering the initialvalues) and from the following argument
Let 1198840(119909) = 120572119910
0(119909) since 119884
0(0) = 120572 1199101015840
0(0) = minus119871120572 the
first order solution for a function say 119884(119909) which definesa slightly different problem from the original is given by119884(119909) = 119884
0(119909) + 119884
1(119909) or 119884(119909) = 120572119910
0(119909) + 119884
1(119909) where
1199100(0) = 1 1199101015840
0(0) = minus119871 119884
1(0) = 0 and 119884
1015840
1(0) = 0 Thus
after dividing by 120572 119910(119909) = 1199100(119909) + 120575119910
1(119909) is obtained (where
120575 = 119891(120572)120572 119910(119909) = 119884(119909)120572 and we have supposed thatfor some problems 119884
1(119909) = 119891(120572)119910
1(119909) As a matter of fact
following (21)ndash(30)we conclude that for this case119891(120572) = 12057232therefore 120575 = 12057212)
Journal of Applied Mathematics 5
Substituting (29) and (30) into (39) we obtain
119910 = 1198961119890minus119860119909
+ 1198962119890minus119861119909
minus120575119890minus119860119909
119860 minus 119861(211989632
1lim120576rarr0
int
radic119909
120576
119890minus11986011990622119889119906
+ 3119896211989612
1lim120576rarr0
int
radic119909
120576
119890minus(119861minus1198602)119906
2
119889119906)
+120575119890minus119861119909
119860 minus 119861(211989632
1lim120576rarr0
int
radic119909
120576
119890(119861minus31198602)119906
2
119889119906
+ 3119896211989612
1lim120576rarr0
int
radic119909
120576
119890minus11986011990622119889119906)
(40)
In [14] a high accurate approximation of normal distributionintegral was reported which let us write the above integralsas
lim120576rarr0
int
radic119909
120576
119890minus11986011990622119889119906
=1
2
radic2120587
119860tanh(39
2
radic119860119909
2120587minus111
2tanminus1( 35
111
radic119860119909
2120587))
lim120576rarr0
int
radic119909
120576
119890minus(119861minus1198602)119906
2
119889119906
=1
2
radic2120587
2119861 minus 119860tanh(39
2
radic(2119861 minus 119860)119909
2120587
minus111
2tanminus1( 35
111
radic(2119861 minus 119860)119909
2120587))
(41)
Note that
int
radic119909
0
119890(119861minus31198602)119906
2
119889119906 (42)
is not a Gaussian integral on the assumption that 119890minus119860119909 ≫ 119890minus119861119909
for 0 lt 119909 lt infin since it implies that 119860 ≪ 119861
Equations (41) allow us to write (40) as
119910 = 1198961119890minus119860119909
+ 1198962119890minus119861119909
minus120575119890minus119860119909
119860 minus 119861
sdot (11989632
1radic2120587
119860tanh(39
2
radic119860119909
2120587
minus111
2tanminus1( 35
111
radic119860119909
2120587))
+3
2119896211989612
1radic
2120587
2119861 minus 119860
sdot tanh(39
2
radic(2119861 minus 119860)119909
2120587
minus111
2tanminus1( 35
111
radic(2119861 minus 119860)119909
2120587)))
+120575119890minus119861119909
119860 minus 119861(211989632
1lim120576rarr0
int
radic119909
120576
119890(119861minus31198602)119906
2
119889119906
+3
2119896211989612
1radic2120587
119860
sdot tanh(392
radic119860119909
2120587minus111
2tanminus1( 35
111
radic119860119909
2120587)))
(43)
To transform (43) into an analytical expression we evaluate(42) keeping terms up to the ninth power to obtain thefollowing compact fractional power function
int
radic119909
0
119890(119861minus31198602)119906
2
119889119906
= radic119909 +1
3(119861 minus 3119860)119909
32+
1
10(119861 minus 3119860)
211990952
+1
42(119861 minus 3119860)
311990972
+1
216(119861 minus 3119860)
411990992
(44)
where the main contribution of this approximation to (43) isin the range of 119909 isin (0 1] because it is multiplied by a negativeexponential (see (43) and Section 4)
6 Journal of Applied Mathematics
Therefore substituting (44) into (43) the following isobtained
119910 = 1198961119890minus119860119909
+ 1198962119890minus119861119909
minus120575119890minus119860119909
119860 minus 119861
sdot (11989632
1radic2120587
119860tanh(39
2
radic119860119909
2120587
minus111
2tanminus1( 35
111
radic119860119909
2120587))
+3
2119896211989612
1radic
2120587
2119861 minus 119860
sdot tanh(39
2
radic(2119861 minus 119860)119909
2120587
minus111
2tanminus1( 35
111
radic(2119861 minus 119860) 119909
2120587)))
+120575119890minus119861119909
119860 minus 119861(211989632
1(radic119909 +
(119861 minus 3119860)11990932
3+(119861 minus 3119860)
211990952
10
+(119861 minus 3119860)
311990972
42+(119861 minus 3119860)
411990992
216)
+3
2119896211989612
1radic2120587
119860
sdot tanh(392
radic119860119909
2120587minus111
2tanminus1( 35
111
radic119860119909
2120587)))
(45)
In order to obtain a good approximation from (45) weoptimize the values of the aforementioned parameters asfollows 119860 = 0398521712348839 119861 = 6 and 120575 =
minus0623239813 using the procedure reported in [14 17 27](note that with these values the assumption 119860 ≪ 119861 issatisfied besides the above results allow to know 120572 and 120573
whose values are not needed in the search for the solutionof the system (19) and (20))
Figure 1 and Table 1 show the comparison betweenapproximate solution (45) and the exact solution The accu-racy of (45) is clear as an approximate solution for (13)
4 Discussion
Table 1 shows the comparison between numerical solutiongiven in [39] and approximations [15 38] and (45) It isclear that our approximation (45) is competitive consideringthat it is analytic and handy unlike other approximationswhich are expressed in terms of power series In [37 3841] T-F equation was solved by homotopy analysis method(HAM) The approximation [38] corresponds to an 80thorder analytical approximation and therefore is impracticalOn the other hand in [37] the results are given in terms of
1
08
06
04
02
0
y(x)
0 05 1 15 2 25 3 35 4 45 5
x
[39](45)
Figure 1 Comparison of Thomas-Fermirsquos numerical solution [39]and approximation (45)
infinite series and show particularly the calculation of theinitial slope 1199101015840(0) Moreover in [41] an accurate approximatesolution was obtained but it corresponds to the 40th orderapproximation In [23] (13) was solved by HPM and Padeapproximants in this case a solution of 26 power series termsis provided which is also not handy in practice Reference[15] presents an analytical approach but it is not accurateIn this study we used homotopy perturbation method withnonlinearities distributions to find an accurate solution for(13) (see Figure 1 and Table 1) Although (43) is just almostanalytic (due to the presence of a non-Gaussian integral) itcan be estimated that contribution of (42) to the solution(43) is well represented by (44) assuming the values are 119860 =
0398521712348839 119861 = 6 and 120575 = minus0623239813 thus weobtained the analytical approximation (45)We just kept onlyfive terms in (44) in order to obtain a handy approximationBesides the right-hand side of (44) adequately represents theindicated non-Gaussian integral in the range 119909 isin [0 1] Itimplies that the best contribution of the term proportional to119890minus119861119909
intradic119909
0119890(119861minus31198602)119906
2
119889119906 in (43) is just beginning of the intervalconsidered that is in a part of the domain where the T-F equation describes correctly the atom (see Section 1) Forvalues 119909 gt 1 the above-mentioned term is dominated forthe exponential 119890minus119861119909 In any case the value of the integralsof (40) can be evaluated by using numerical methods Aninteresting fact is that if more terms had been consideredin the development of (12) we would have obtained asa solution of (13) a series containing terms of hyperbolictangent functions to the form (41)
The strategy to redistribute the nonlinearities of T-Fequation between the successive iterations of the HPMmethod was important to obtain an approximate solution of(13) because it allows us to simplify the solution procedurefor the successive iterations of the HPM method Besideswe obtained a soluble coupled system of linear differentialequations (19) and (20) In this study we introduced anadjusting parameter in order to enlarge the domain ofconvergence of our approach with good results by using onlythe first order approximation (see Table 1 and Figure 1)
Journal of Applied Mathematics 7
Table 1 Numerical comparison of Thomas-Fermirsquos numerical solution [39] and approximations [15 38] and (45)
119909 Numerical [39] [38] (45) [15]000 mdash 1000000000 1000000000 1000000000025 0755880759 0776191000 0708502932 0680650028050 0606700008 0615917000 0570491745 0459455663075 0502964042 0505380000 0485018011 0307042537100 0424333179 0423772000 0421167227 0202655526125 0363227937 0362935000 0369542163 0131667603150 0314660642 0314490000 0326298526 0083800106175 0275233848 0275154000 0289315779 0051853222200 0242678587 0242718000 0257253699 0030801947225 0215439334 0215630000 0229210094 0017153465250 0192406328 0192795000 0204540381 0008491277275 0172758691 0173364000 0182755264 0003152503300 0155871862 0156719000 0163464474 0325 0141260504 0142371000 0146346052 minus0001738122
350 0128541381 0129937000 0131128799 minus0002581206
375 0117408054 0119108000 0117581337 minus0002874924
400 0107612958 0109632000 0105504623 minus000284628
425 0098954329 0101303000 0094726427 minus0002641801
450 0091266456 0093950400 0085097045 minus0002353931
475 0084412289 0087432000 0076485875 minus0002039168
500 0078277758 0081629600 0068778618 minus0001730443
A relevant fact is that NDHPM requires little iterationsto obtain accurate solutions if we have a first approximationcontaining as much information as possible for a nonlineardifferential equation For instance in our case (21) containsthe correct asymptotic character of the exact equation
5 Conclusion
In this work NDHPM was used to obtain an approximateanalytical solution for T-F equation Our solution is novelbecause it is expressed in terms of exponential and fractionalpower functions and above all in terms of normal distributionintegrals It is important to obtain an analytical expressionthat provides a good description of the solution for thenonlinear differential equations like (13) For instance thecharge distribution and the electrostatic potential inside anatom are adequately described by (43) and (45) It is worthmentioning that if the initial guess is suitably chosen it ispossible to obtain by this method an accurate approximationlike (43) even using the first terms of (12) Finally in contrastto Runge-Kutta numerical solution NDHPMmethod allowsto analyze quantitatively and qualitatively the solution
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the financial supportfrom the National Council for Science and Technology ofMexico (CONACyT) through Grant CB-2010-01 157024The authors would like to thank Rogelio-Alejandro Callejas-Molina and Roberto Ruiz-Gomez for their contribution tothis project
References
[1] L Landau and E M LifshitzMecanica Cuantica No RelativistaEditorial Reverte S A 2nd edition 1967
[2] LM B Assas ldquoApproximate solutions for the generalizedKdVndashBurgersrsquo equation by Hersquos variational iteration methodrdquo PhysicaScripta vol 76 no 2 pp 161ndash164 2007
[3] M Kazemnia S A Zahedi M Vaezi and N Tolou ldquoArseniccontamination in New Orleans soil temporal changes associ-ated with floodingrdquo Journal of Applied Sciences vol 8 no 22pp 4192ndash4197 2008
[4] R Noorzad A T Poor and M Omidvar ldquoVariational iterationmethod and homotopy-perturbation method for solving Burg-ers equation in fluid dynamicsrdquo Journal of Applied Sciences vol8 no 2 pp 369ndash373 2008
[5] D J Evans and K R Raslan ldquoThe tanh function method forsolving some important non-linear partial differential equa-tionsrdquo International Journal of Computer Mathematics vol 82no 7 pp 897ndash905 2005
[6] J Mahmoudi N Tolou I Khatami A Barari and D D GanjildquoExplicit solution of nonlinear ZK-BBM wave equation using
8 Journal of Applied Mathematics
Exp-function methodrdquo Journal of Applied Sciences vol 8 no 2pp 358ndash363 2008
[7] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988
[8] E Babolian and J Biazar ldquoOn the order of convergence ofAdomian methodrdquo Applied Mathematics and Computation vol130 no 2-3 pp 383ndash387 2002
[9] A Kooch and M Abadyan ldquoEfficiency of modified Ado-mian decomposition for simulating the instability of nano-electromechanical switches comparison with the conventionaldecomposition methodrdquo Trends in Applied Sciences Researchvol 7 no 1 pp 57ndash67 2012
[10] Y Khan H Vazquez-Leal and N Faraz ldquoAn auxiliary param-eter method using Adomian polynomials and Laplace transfor-mation for nonlinear differential equationsrdquoAppliedMathemat-ical Modelling vol 37 no 5 pp 2702ndash2708 2013
[11] S K Vanani S Heidari and M Avaji ldquoA low-cost numericalalgorithm for the solution of nonlinear delay boundary integralequationsrdquo Journal of Applied Sciences vol 11 no 20 pp 3504ndash3509 2011
[12] S H Chowdhury ldquoA comparison between the modifiedhomotopy perturbation method and adomian decompositionmethod for solving nonlinear heat transfer equationsrdquo Journalof Applied Sciences vol 11 no 7 pp 1416ndash1420 2011
[13] L-N Zhang and L Xu ldquoDetermination of the limit cycle byHersquos parameter expansion for oscillators in a u31 + u2 potentialrdquoZeitschrift fur NaturforschungmdashSection A Journal of PhysicalSciences vol 62 no 7-8 pp 396ndash398 2007
[14] H Vazquez-Leal R Castaneda-Sheissa U Filobello-Nino ASarmiento-Reyes and J Sanchez Orea ldquoHigh accurate simpleapproximation of normal distribution integralrdquo MathematicalProblems in Engineering vol 2012 Article ID 124029 22 pages2012
[15] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006
[16] A Fereidoon Y Rostamiyan M Akbarzade and D D GanjildquoApplication of Hersquos homotopy perturbation method to nonlin-ear shock damper dynamicsrdquo Archive of Applied Mechanics vol80 no 6 pp 641ndash649 2010
[17] H Vazquez-Leal U Filobello-Nino R Castaneda-SheissaL Hernandez-Martınez and A Sarmiento-Reyes ldquoModifiedHPMs inspired by homotopy continuation methodsrdquo Mathe-matical Problems in Engineering vol 2012 Article ID 30912319 pages 2012
[18] Y Khan H Vazquez-Leal and Q Wu ldquoAn efficient iteratedmethod for mathematical biology modelrdquo Neural Computingand Applications vol 23 no 3-4 pp 677ndash682 2013
[19] Y Khan H Vazquez-Leal and N Faraz ldquoAn efficient newiterative method for oscillator differential equationrdquo ScientiaIranica vol 19 no 6 pp 1473ndash1477 2012
[20] J Biazar and H Aminikhah ldquoStudy of convergence of homo-topy perturbation method for systems of partial differentialequationsrdquoComputersampMathematics with Applications vol 58no 11-12 pp 2221ndash2230 2009
[21] J Biazar and H Ghazvini ldquoConvergence of the homotopy per-turbation method for partial differential equationsrdquo NonlinearAnalysis Real World Applications vol 10 no 5 pp 2633ndash26402009
[22] Y Khan and Q Wu ldquoHomotopy perturbation transformmethod for nonlinear equations using Hersquos polynomialsrdquo Com-puters and Mathematics with Applications vol 61 no 8 pp1963ndash1967 2011
[23] M A Noor and S T Mohyud-Din ldquoHomotopy perturbationmethod for solvingThomas-Fermi equation using pade approx-imantsrdquo International Journal of Nonlinear Science vol 8 no 1pp 27ndash31 2009
[24] DDGanji A R Sahouli andM Famouri ldquoAnewmodificationofHersquos homotopy perturbationmethod for rapid convergence ofnonlinear undamped oscillatorsrdquo Journal of Applied Mathemat-ics and Computing vol 30 no 1-2 pp 181ndash192 2009
[25] M Turkyilmazoglu ldquoSome issues on HPM and HAMmethodsa convergence schemerdquoMathematical and Computer Modellingvol 53 no 9-10 pp 1929ndash1936 2011
[26] M Turkyilmazoglu ldquoConvergence of the homotopy perturba-tion methodrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 12 no 1ndash8 pp 9ndash14 2011
[27] U Filobello-Nino H Vazquez-Leal R Castaneda-Sheissa et alldquoAn approximate solution of Blasius equation by using HPMmethodrdquo Asian Journal of Mathematics and Statistics vol 5 no2 pp 50ndash59 2012
[28] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineering Springer Berlin Germany 1st edition 2011
[29] T Patel M N Mehta and V H Pradhan ldquoThe numericalsolution of Burgerrsquos equation arising into the irradiation oftumour tissue in biological diffusing system by homotopyanalysis methodrdquo Asian Journal of Applied Sciences vol 5 no1 pp 60ndash66 2012
[30] S Liao Advances in the Homotopy Analysis Method WorldScientific Hackensack NJ USA 2014
[31] M Turkyilmazoglu ldquoAnalytic approximate solutions of rotatingdisk boundary layer flow subject to a uniform suction orinjectionrdquo International Journal of Mechanical Sciences vol 52no 12 pp 1735ndash1744 2010
[32] M Turkyilmazoglu ldquoPurely analytic solutions of the compress-ible boundary layer flow due to a porous rotating disk with heattransferrdquo Physics of Fluids vol 21 no 10 2009
[33] U Filobello-Nino H Vazquez-Leal B Benhammouda et alldquoA handy approximation for a mediated bioelectrocatalysisprocess related to Michaelis-Menten equationrdquo SpringerPlusvol 3 article 162 2014
[34] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoUsing per-turbation methods and Laplace-Pade approximation to solvenonlinear problemsrdquo Miskolc Mathematical Notes vol 14 no1 pp 89ndash101 2013
[35] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoA handyexact solution for flow due to a stretching boundary with partialsliprdquo Revista Mexicana de Fısica E vol 59 no 1 pp 51ndash55 2013
[36] FM Fernandez ldquoRational approximation to theThomas-Fermiequationsrdquo Applied Mathematics and Computation vol 217 no13 pp 6433ndash6436 2011
[37] B Yao ldquoA series solution to the Thomas-Fermi equationrdquoApplied Mathematics and Computation vol 203 no 1 pp 396ndash401 2008
[38] H Khan and H Xu ldquoSeries solution to the Thomas-Fermiequationrdquo Physics Letters A vol 365 no 1-2 pp 111ndash115 2007
[39] K Parand andM Shahini ldquoRational Chebyshev pseudospectralapproach for solving Thomas-Fermi equationrdquo Physics LettersA vol 373 no 2 pp 210ndash213 2009
Journal of Applied Mathematics 9
[40] E Hille ldquoOn the Thomas-Fermi equationrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 62 pp 7ndash10 1969
[41] M Turkyilmazoglu ldquoSolution of the Thomas-Fermi equationwith a convergent approachrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 11 pp 4097ndash41032012
[42] D G ZillA First Course in Differential Equations withModelingApplications Brooks Cole 10th edition 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 3
The 119894 power is selected according to how much displacementis desired in the interactions for the corresponding nonlinearterm of 119873 or 119891 The introduction of that parameter withinthe differential equation is a strategy to redistribute thenonlinearities between the successive iterations of the HPMmethod and thus it increases the probabilities of finding thesought solutionThe rest of the method is exactly the same asthe standard procedure for the HPMTherefore we establishthat
V = V0+ V1119901 + V21199012+ sdot sdot sdot (11)
When 119901 rarr 1 it results in the approximate solution for (3)in the form
V = V0+ V1+ V2+ V3+ sdot sdot sdot (12)
An advantage of this procedure is that given the distributionof nonlinearities from the differential equation over thesuccessive iterations of (11) less complex analytic approxima-tions may be obtained than those generated by the originalstandard HPM
Finally convergence of solutions obtained by NDHPMmethod is discussed in [17]
3 Approximate Solution ofThomas-Fermi Equation
In order to facilitate understanding of the NDHPM methodwe will solve approximately the equation
11991010158401015840=11991032
11990912 119910 (0) = 1 119910
1015840(0) = minus119871 (13)
where prime denotes differentiation with respect to 119909119910(119909) is a function related to the electrostatic potential
inside the atom 119909 is a variable proportional to distance fromthe nucleus (119903) and the most accurate value of 1199101015840(0) is 119871 =
1588071022611375313 [36]The above equation is a modified version of the original
problem of Thomas-Fermi where (13) is subject to theboundary conditions 119910(0) = 1 119910(infin) = 0 [28 41] althoughthere are other conditions that can be applied [40]
Following references [16 24] instead of defining a linearand nonlinear part in the above equation we add and subtract1205721199101015840+120573119910 as it is shown in order to add a term to linear operator
to facilitate the search for convergent solutions thus (13) canbe rewritten as
11991010158401015840+ 1205721199101015840+ 120573119910 minus [
11991032
11990912+ 1205721199101015840+ 120573119910] = 0 (14)
where 120572 and 120573 are constant parameters to determineThe linear part is identified as
119871(119910) = 11991010158401015840+ 1205721199101015840+ 120573119910 (15)
and the nonlinear is
119873(119910) = minus[11991032
11990912+ 1205721199101015840+ 120573119910] (16)
In order to simplify the calculations we will redistribute thenonlinear term by using NDHPMmethod starting from (9)in the form
11991010158401015840+ 1205721199101015840+ 120573119910 minus 119901[
11991032
11990912+ 119901120572119910
1015840+ 119901120573119910] = 0 (17)
by substituting
119910 = 1199100+ 1199101119901 + 11991021199012+ sdot sdot sdot (18)
into (17) and equating terms having identical powers of 119901weobtain
11991010158401015840
0+ 1205721199101015840
0+ 1205731199100= 0 119910
0(0) = 1 119910
1015840
0(0) = minus119871 (19)
11991010158401015840
1+ 1205721199101015840
1+ 1205731199101minus11991032
0
11990912= 0 119910
1 (0) = 0 1199101015840
1(0) = 0
(20)
In this paper we study the first order approximation (19)-(20) We will see that the nonlinear term of 119873(119910) (see (16))contains the sufficient information to obtain a good analyticalapproximation for (13)
The solution of (19) that satisfies the initial conditions isgiven by
1199100=119861 minus 119871
119861 minus 119860119890minus119860119909
+119871 minus 119860
119861 minus 119860119890minus119861119909
(21)
where 119860 and 119861 are constants related to 120572 and 120573By substituting (21) into (20) we obtain
11991010158401015840
1+ 1205721199101015840
1+ 1205731199101minus
1
11990912(119861 minus 119871
119861 minus 119860119890minus119860119909
+119871 minus 119860
119861 minus 119860119890minus119861119909
)
32
= 0
(22)
In order to simplify (22) and obtain a handy approximationfor (13) we will assume that there is an adequate choice of 119860and 119861 (119860 ≪ 119861) such that 119890minus119860119909 ≫ 119890
minus119861119909 valid for 0 lt 119909 lt infinin such a way we can rewrite (22) approximately as
11991010158401015840
1+ 1205721199101015840
1+ 1205731199101=11989632
1119890minus31198601199092
11990912(1 +
31198962
21198961
119890minus(119861minus119860)119909
) (23)
where we have defined
1198961=119861 minus 119871
119861 minus 119860 119896
2=119871 minus 119860
119861 minus 119860 (24)
This assumption is justified laterTo solve (23) we employ the method of variation of
parameters [42] which requires evaluating the followingintegrals
1199061= minusint
119891(119909)119890minus119861119909
119889119909
119882 119906
2= int
119891(119909)119890minus119860119909
119889119909
119882 (25)
4 Journal of Applied Mathematics
where 1199101198671
= 119890minus119860119909 and 119910
1198672= 119890minus119861119909 are the solutions of the
homogeneous differential equation
11991010158401015840
1+ 1205721199101015840
1+ 1205731199101= 0 (26)
119882 is theWronskian of these two functions which is given by
119882(1199101 1199102) = (119860 minus 119861)119890
minus(119860+119861)119909(119860 = 119861) (27)
and 119891(119909) is the right-hand side of (23)Substituting 119891(119909) and (27) into (25) leads to
1199061= minus
1
119860 minus 119861(11989632
1lim120576rarr0
int
119909
120576
119890minus1198601199102
119889119910
radic119910
+3
2119896211989612
1lim120576rarr0
int
119909
120576
119890minus(119861minus1198602)119910
119889119910
radic119910)
1199062=
1
119860 minus 119861(11989632
1lim120576rarr0
int
119909
120576
119890minus(31198602minus119861)119910
119889119910
radic119910
+3
2119896211989612
1lim120576rarr0
int
119909
120576
119890minus1198601199102
119889119910
radic119910)
(28)
By substituting 119910 = 1199062 the above equations take the form
1199061= minus
1
119860 minus 119861(211989632
1lim120576rarr0
int
radic119909
120576
119890minus11990621198602
119889119906
+ 3119896211989612
1lim120576rarr0
int
radic119909
120576
119890minus(119861minus1198602)119906
2
119889119906)
(29)
1199062=
1
119860 minus 119861(211989632
1lim120576rarr0
int
radic119909
120576
119890(119861minus31198602)119906
2
119889119906
+ 3119896211989612
1lim120576rarr0
int
radic119909
120576
119890minus11986011990622119889119906)
(30)
Therefore the solution of (23) is written according to themethod of variation of parameters as [42]
1199101= 119891119890minus119860119909
+ 119892119890minus119861119909
+ 1199061119890minus119860119909
+ 1199062119890minus119861119909
(31)
where 119891 and 119892 are constants and 1199061and 119906
2are given by (29)
and (30) respectivelyApplying the initial condition 119910
1(0) = 0 to (31) leads to
1199101(0) = 119891 + 119892 minus
1
119860 minus 119861(211989632
1lim120576rarr0119909rarr0
int
radic119909
120576
119890minus11986011990622119889119906
+ 3119896211989612
1lim120576rarr0119909rarr0
int
radic119909
120576
119890minus(119861minus1198602)119906
2
119889119906)
+1
119860 minus 119861(211989632
1lim120576rarr0119909rarr0
int
radic119909
120576
119890(119861minus31198602)119906
2
119889119906
+ 3119896211989612
1lim120576rarr0119909rarr0
int
radic119909
120576
119890minus11986011990622119889119906) = 0
(32)
Equation (32) is simplified to
119891 + 119892 = 0 (33)
On the other hand to apply the condition 1199101015840
1(0) = 0 we
differentiate (31) to obtain
1199101015840
1= minus119860119891119890
minus119860119909minus 119861119892119890
minus119861119909
minus119890minus119860119909
119860 minus 119861(11989632
1
119890minus1198601199092
radic119909+3119896211989612
1119890minus(119861minus1198602)119909
2radic119909)
+119890minus119861119909
119860 minus 119861(11989632
1
119890minus(31198602minus119861)119909
radic119909+3119896211989612
1119890minus1198602119909
2radic119909)
minus 119860119890minus119860119909
1199061minus 119861119890minus119861119909
1199062
(34)
After performing algebraic simplifications to (34) we obtain
1199101015840
1= minus119860119891119890
minus119860119909minus 119861119892119890
minus119861119909minus 119860119890minus119860119909
1199061minus 119861119890minus119861119909
1199062 (35)
Applying the condition 1199101015840
1(0) = 0 to (35) the following is
obtained
119860119891 + 119861119892 = 0 (36)
since lim119909rarr0
1199061= 0 and lim
119909rarr01199062= 0 (see (29) and (30))
From (33) and (36) we obtain 119891 = 0 and 119892 = 0 since119860 = 119861 therefore (31) becomes
1199101= 1199061119890minus119860119909
+ 1199062119890minus119861119909
(37)
By substituting (21) and (37) into (12) we obtain the followingfirst order approximation for the solution of (13)
119910 = 1198961119890minus119860119909
+ 1198962119890minus119861119909
+ 1199061119890minus119860119909
+ 1199062119890minus119861119909
(38)
We will show later that it is possible to achieve a good fit andenlarge the domain of convergence for (38) (keeping at thesame time the handy character of the proposed solution) byintroducing in (38) an adjusting parameter (consistent withthe initial conditions) as it is shown in
119910 = 1198961119890minus119860119909
+ 1198962119890minus119861119909
+ 1205751199061119890minus119860119909
+ 1205751199062119890minus119861119909
(39)
The inclusion of parameter 120575 is motivated because thesolution 119910
0(119909) for the homogeneous equation (19) is undeter-
mined by a constant factor 120572 (without considering the initialvalues) and from the following argument
Let 1198840(119909) = 120572119910
0(119909) since 119884
0(0) = 120572 1199101015840
0(0) = minus119871120572 the
first order solution for a function say 119884(119909) which definesa slightly different problem from the original is given by119884(119909) = 119884
0(119909) + 119884
1(119909) or 119884(119909) = 120572119910
0(119909) + 119884
1(119909) where
1199100(0) = 1 1199101015840
0(0) = minus119871 119884
1(0) = 0 and 119884
1015840
1(0) = 0 Thus
after dividing by 120572 119910(119909) = 1199100(119909) + 120575119910
1(119909) is obtained (where
120575 = 119891(120572)120572 119910(119909) = 119884(119909)120572 and we have supposed thatfor some problems 119884
1(119909) = 119891(120572)119910
1(119909) As a matter of fact
following (21)ndash(30)we conclude that for this case119891(120572) = 12057232therefore 120575 = 12057212)
Journal of Applied Mathematics 5
Substituting (29) and (30) into (39) we obtain
119910 = 1198961119890minus119860119909
+ 1198962119890minus119861119909
minus120575119890minus119860119909
119860 minus 119861(211989632
1lim120576rarr0
int
radic119909
120576
119890minus11986011990622119889119906
+ 3119896211989612
1lim120576rarr0
int
radic119909
120576
119890minus(119861minus1198602)119906
2
119889119906)
+120575119890minus119861119909
119860 minus 119861(211989632
1lim120576rarr0
int
radic119909
120576
119890(119861minus31198602)119906
2
119889119906
+ 3119896211989612
1lim120576rarr0
int
radic119909
120576
119890minus11986011990622119889119906)
(40)
In [14] a high accurate approximation of normal distributionintegral was reported which let us write the above integralsas
lim120576rarr0
int
radic119909
120576
119890minus11986011990622119889119906
=1
2
radic2120587
119860tanh(39
2
radic119860119909
2120587minus111
2tanminus1( 35
111
radic119860119909
2120587))
lim120576rarr0
int
radic119909
120576
119890minus(119861minus1198602)119906
2
119889119906
=1
2
radic2120587
2119861 minus 119860tanh(39
2
radic(2119861 minus 119860)119909
2120587
minus111
2tanminus1( 35
111
radic(2119861 minus 119860)119909
2120587))
(41)
Note that
int
radic119909
0
119890(119861minus31198602)119906
2
119889119906 (42)
is not a Gaussian integral on the assumption that 119890minus119860119909 ≫ 119890minus119861119909
for 0 lt 119909 lt infin since it implies that 119860 ≪ 119861
Equations (41) allow us to write (40) as
119910 = 1198961119890minus119860119909
+ 1198962119890minus119861119909
minus120575119890minus119860119909
119860 minus 119861
sdot (11989632
1radic2120587
119860tanh(39
2
radic119860119909
2120587
minus111
2tanminus1( 35
111
radic119860119909
2120587))
+3
2119896211989612
1radic
2120587
2119861 minus 119860
sdot tanh(39
2
radic(2119861 minus 119860)119909
2120587
minus111
2tanminus1( 35
111
radic(2119861 minus 119860)119909
2120587)))
+120575119890minus119861119909
119860 minus 119861(211989632
1lim120576rarr0
int
radic119909
120576
119890(119861minus31198602)119906
2
119889119906
+3
2119896211989612
1radic2120587
119860
sdot tanh(392
radic119860119909
2120587minus111
2tanminus1( 35
111
radic119860119909
2120587)))
(43)
To transform (43) into an analytical expression we evaluate(42) keeping terms up to the ninth power to obtain thefollowing compact fractional power function
int
radic119909
0
119890(119861minus31198602)119906
2
119889119906
= radic119909 +1
3(119861 minus 3119860)119909
32+
1
10(119861 minus 3119860)
211990952
+1
42(119861 minus 3119860)
311990972
+1
216(119861 minus 3119860)
411990992
(44)
where the main contribution of this approximation to (43) isin the range of 119909 isin (0 1] because it is multiplied by a negativeexponential (see (43) and Section 4)
6 Journal of Applied Mathematics
Therefore substituting (44) into (43) the following isobtained
119910 = 1198961119890minus119860119909
+ 1198962119890minus119861119909
minus120575119890minus119860119909
119860 minus 119861
sdot (11989632
1radic2120587
119860tanh(39
2
radic119860119909
2120587
minus111
2tanminus1( 35
111
radic119860119909
2120587))
+3
2119896211989612
1radic
2120587
2119861 minus 119860
sdot tanh(39
2
radic(2119861 minus 119860)119909
2120587
minus111
2tanminus1( 35
111
radic(2119861 minus 119860) 119909
2120587)))
+120575119890minus119861119909
119860 minus 119861(211989632
1(radic119909 +
(119861 minus 3119860)11990932
3+(119861 minus 3119860)
211990952
10
+(119861 minus 3119860)
311990972
42+(119861 minus 3119860)
411990992
216)
+3
2119896211989612
1radic2120587
119860
sdot tanh(392
radic119860119909
2120587minus111
2tanminus1( 35
111
radic119860119909
2120587)))
(45)
In order to obtain a good approximation from (45) weoptimize the values of the aforementioned parameters asfollows 119860 = 0398521712348839 119861 = 6 and 120575 =
minus0623239813 using the procedure reported in [14 17 27](note that with these values the assumption 119860 ≪ 119861 issatisfied besides the above results allow to know 120572 and 120573
whose values are not needed in the search for the solutionof the system (19) and (20))
Figure 1 and Table 1 show the comparison betweenapproximate solution (45) and the exact solution The accu-racy of (45) is clear as an approximate solution for (13)
4 Discussion
Table 1 shows the comparison between numerical solutiongiven in [39] and approximations [15 38] and (45) It isclear that our approximation (45) is competitive consideringthat it is analytic and handy unlike other approximationswhich are expressed in terms of power series In [37 3841] T-F equation was solved by homotopy analysis method(HAM) The approximation [38] corresponds to an 80thorder analytical approximation and therefore is impracticalOn the other hand in [37] the results are given in terms of
1
08
06
04
02
0
y(x)
0 05 1 15 2 25 3 35 4 45 5
x
[39](45)
Figure 1 Comparison of Thomas-Fermirsquos numerical solution [39]and approximation (45)
infinite series and show particularly the calculation of theinitial slope 1199101015840(0) Moreover in [41] an accurate approximatesolution was obtained but it corresponds to the 40th orderapproximation In [23] (13) was solved by HPM and Padeapproximants in this case a solution of 26 power series termsis provided which is also not handy in practice Reference[15] presents an analytical approach but it is not accurateIn this study we used homotopy perturbation method withnonlinearities distributions to find an accurate solution for(13) (see Figure 1 and Table 1) Although (43) is just almostanalytic (due to the presence of a non-Gaussian integral) itcan be estimated that contribution of (42) to the solution(43) is well represented by (44) assuming the values are 119860 =
0398521712348839 119861 = 6 and 120575 = minus0623239813 thus weobtained the analytical approximation (45)We just kept onlyfive terms in (44) in order to obtain a handy approximationBesides the right-hand side of (44) adequately represents theindicated non-Gaussian integral in the range 119909 isin [0 1] Itimplies that the best contribution of the term proportional to119890minus119861119909
intradic119909
0119890(119861minus31198602)119906
2
119889119906 in (43) is just beginning of the intervalconsidered that is in a part of the domain where the T-F equation describes correctly the atom (see Section 1) Forvalues 119909 gt 1 the above-mentioned term is dominated forthe exponential 119890minus119861119909 In any case the value of the integralsof (40) can be evaluated by using numerical methods Aninteresting fact is that if more terms had been consideredin the development of (12) we would have obtained asa solution of (13) a series containing terms of hyperbolictangent functions to the form (41)
The strategy to redistribute the nonlinearities of T-Fequation between the successive iterations of the HPMmethod was important to obtain an approximate solution of(13) because it allows us to simplify the solution procedurefor the successive iterations of the HPM method Besideswe obtained a soluble coupled system of linear differentialequations (19) and (20) In this study we introduced anadjusting parameter in order to enlarge the domain ofconvergence of our approach with good results by using onlythe first order approximation (see Table 1 and Figure 1)
Journal of Applied Mathematics 7
Table 1 Numerical comparison of Thomas-Fermirsquos numerical solution [39] and approximations [15 38] and (45)
119909 Numerical [39] [38] (45) [15]000 mdash 1000000000 1000000000 1000000000025 0755880759 0776191000 0708502932 0680650028050 0606700008 0615917000 0570491745 0459455663075 0502964042 0505380000 0485018011 0307042537100 0424333179 0423772000 0421167227 0202655526125 0363227937 0362935000 0369542163 0131667603150 0314660642 0314490000 0326298526 0083800106175 0275233848 0275154000 0289315779 0051853222200 0242678587 0242718000 0257253699 0030801947225 0215439334 0215630000 0229210094 0017153465250 0192406328 0192795000 0204540381 0008491277275 0172758691 0173364000 0182755264 0003152503300 0155871862 0156719000 0163464474 0325 0141260504 0142371000 0146346052 minus0001738122
350 0128541381 0129937000 0131128799 minus0002581206
375 0117408054 0119108000 0117581337 minus0002874924
400 0107612958 0109632000 0105504623 minus000284628
425 0098954329 0101303000 0094726427 minus0002641801
450 0091266456 0093950400 0085097045 minus0002353931
475 0084412289 0087432000 0076485875 minus0002039168
500 0078277758 0081629600 0068778618 minus0001730443
A relevant fact is that NDHPM requires little iterationsto obtain accurate solutions if we have a first approximationcontaining as much information as possible for a nonlineardifferential equation For instance in our case (21) containsthe correct asymptotic character of the exact equation
5 Conclusion
In this work NDHPM was used to obtain an approximateanalytical solution for T-F equation Our solution is novelbecause it is expressed in terms of exponential and fractionalpower functions and above all in terms of normal distributionintegrals It is important to obtain an analytical expressionthat provides a good description of the solution for thenonlinear differential equations like (13) For instance thecharge distribution and the electrostatic potential inside anatom are adequately described by (43) and (45) It is worthmentioning that if the initial guess is suitably chosen it ispossible to obtain by this method an accurate approximationlike (43) even using the first terms of (12) Finally in contrastto Runge-Kutta numerical solution NDHPMmethod allowsto analyze quantitatively and qualitatively the solution
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the financial supportfrom the National Council for Science and Technology ofMexico (CONACyT) through Grant CB-2010-01 157024The authors would like to thank Rogelio-Alejandro Callejas-Molina and Roberto Ruiz-Gomez for their contribution tothis project
References
[1] L Landau and E M LifshitzMecanica Cuantica No RelativistaEditorial Reverte S A 2nd edition 1967
[2] LM B Assas ldquoApproximate solutions for the generalizedKdVndashBurgersrsquo equation by Hersquos variational iteration methodrdquo PhysicaScripta vol 76 no 2 pp 161ndash164 2007
[3] M Kazemnia S A Zahedi M Vaezi and N Tolou ldquoArseniccontamination in New Orleans soil temporal changes associ-ated with floodingrdquo Journal of Applied Sciences vol 8 no 22pp 4192ndash4197 2008
[4] R Noorzad A T Poor and M Omidvar ldquoVariational iterationmethod and homotopy-perturbation method for solving Burg-ers equation in fluid dynamicsrdquo Journal of Applied Sciences vol8 no 2 pp 369ndash373 2008
[5] D J Evans and K R Raslan ldquoThe tanh function method forsolving some important non-linear partial differential equa-tionsrdquo International Journal of Computer Mathematics vol 82no 7 pp 897ndash905 2005
[6] J Mahmoudi N Tolou I Khatami A Barari and D D GanjildquoExplicit solution of nonlinear ZK-BBM wave equation using
8 Journal of Applied Mathematics
Exp-function methodrdquo Journal of Applied Sciences vol 8 no 2pp 358ndash363 2008
[7] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988
[8] E Babolian and J Biazar ldquoOn the order of convergence ofAdomian methodrdquo Applied Mathematics and Computation vol130 no 2-3 pp 383ndash387 2002
[9] A Kooch and M Abadyan ldquoEfficiency of modified Ado-mian decomposition for simulating the instability of nano-electromechanical switches comparison with the conventionaldecomposition methodrdquo Trends in Applied Sciences Researchvol 7 no 1 pp 57ndash67 2012
[10] Y Khan H Vazquez-Leal and N Faraz ldquoAn auxiliary param-eter method using Adomian polynomials and Laplace transfor-mation for nonlinear differential equationsrdquoAppliedMathemat-ical Modelling vol 37 no 5 pp 2702ndash2708 2013
[11] S K Vanani S Heidari and M Avaji ldquoA low-cost numericalalgorithm for the solution of nonlinear delay boundary integralequationsrdquo Journal of Applied Sciences vol 11 no 20 pp 3504ndash3509 2011
[12] S H Chowdhury ldquoA comparison between the modifiedhomotopy perturbation method and adomian decompositionmethod for solving nonlinear heat transfer equationsrdquo Journalof Applied Sciences vol 11 no 7 pp 1416ndash1420 2011
[13] L-N Zhang and L Xu ldquoDetermination of the limit cycle byHersquos parameter expansion for oscillators in a u31 + u2 potentialrdquoZeitschrift fur NaturforschungmdashSection A Journal of PhysicalSciences vol 62 no 7-8 pp 396ndash398 2007
[14] H Vazquez-Leal R Castaneda-Sheissa U Filobello-Nino ASarmiento-Reyes and J Sanchez Orea ldquoHigh accurate simpleapproximation of normal distribution integralrdquo MathematicalProblems in Engineering vol 2012 Article ID 124029 22 pages2012
[15] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006
[16] A Fereidoon Y Rostamiyan M Akbarzade and D D GanjildquoApplication of Hersquos homotopy perturbation method to nonlin-ear shock damper dynamicsrdquo Archive of Applied Mechanics vol80 no 6 pp 641ndash649 2010
[17] H Vazquez-Leal U Filobello-Nino R Castaneda-SheissaL Hernandez-Martınez and A Sarmiento-Reyes ldquoModifiedHPMs inspired by homotopy continuation methodsrdquo Mathe-matical Problems in Engineering vol 2012 Article ID 30912319 pages 2012
[18] Y Khan H Vazquez-Leal and Q Wu ldquoAn efficient iteratedmethod for mathematical biology modelrdquo Neural Computingand Applications vol 23 no 3-4 pp 677ndash682 2013
[19] Y Khan H Vazquez-Leal and N Faraz ldquoAn efficient newiterative method for oscillator differential equationrdquo ScientiaIranica vol 19 no 6 pp 1473ndash1477 2012
[20] J Biazar and H Aminikhah ldquoStudy of convergence of homo-topy perturbation method for systems of partial differentialequationsrdquoComputersampMathematics with Applications vol 58no 11-12 pp 2221ndash2230 2009
[21] J Biazar and H Ghazvini ldquoConvergence of the homotopy per-turbation method for partial differential equationsrdquo NonlinearAnalysis Real World Applications vol 10 no 5 pp 2633ndash26402009
[22] Y Khan and Q Wu ldquoHomotopy perturbation transformmethod for nonlinear equations using Hersquos polynomialsrdquo Com-puters and Mathematics with Applications vol 61 no 8 pp1963ndash1967 2011
[23] M A Noor and S T Mohyud-Din ldquoHomotopy perturbationmethod for solvingThomas-Fermi equation using pade approx-imantsrdquo International Journal of Nonlinear Science vol 8 no 1pp 27ndash31 2009
[24] DDGanji A R Sahouli andM Famouri ldquoAnewmodificationofHersquos homotopy perturbationmethod for rapid convergence ofnonlinear undamped oscillatorsrdquo Journal of Applied Mathemat-ics and Computing vol 30 no 1-2 pp 181ndash192 2009
[25] M Turkyilmazoglu ldquoSome issues on HPM and HAMmethodsa convergence schemerdquoMathematical and Computer Modellingvol 53 no 9-10 pp 1929ndash1936 2011
[26] M Turkyilmazoglu ldquoConvergence of the homotopy perturba-tion methodrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 12 no 1ndash8 pp 9ndash14 2011
[27] U Filobello-Nino H Vazquez-Leal R Castaneda-Sheissa et alldquoAn approximate solution of Blasius equation by using HPMmethodrdquo Asian Journal of Mathematics and Statistics vol 5 no2 pp 50ndash59 2012
[28] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineering Springer Berlin Germany 1st edition 2011
[29] T Patel M N Mehta and V H Pradhan ldquoThe numericalsolution of Burgerrsquos equation arising into the irradiation oftumour tissue in biological diffusing system by homotopyanalysis methodrdquo Asian Journal of Applied Sciences vol 5 no1 pp 60ndash66 2012
[30] S Liao Advances in the Homotopy Analysis Method WorldScientific Hackensack NJ USA 2014
[31] M Turkyilmazoglu ldquoAnalytic approximate solutions of rotatingdisk boundary layer flow subject to a uniform suction orinjectionrdquo International Journal of Mechanical Sciences vol 52no 12 pp 1735ndash1744 2010
[32] M Turkyilmazoglu ldquoPurely analytic solutions of the compress-ible boundary layer flow due to a porous rotating disk with heattransferrdquo Physics of Fluids vol 21 no 10 2009
[33] U Filobello-Nino H Vazquez-Leal B Benhammouda et alldquoA handy approximation for a mediated bioelectrocatalysisprocess related to Michaelis-Menten equationrdquo SpringerPlusvol 3 article 162 2014
[34] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoUsing per-turbation methods and Laplace-Pade approximation to solvenonlinear problemsrdquo Miskolc Mathematical Notes vol 14 no1 pp 89ndash101 2013
[35] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoA handyexact solution for flow due to a stretching boundary with partialsliprdquo Revista Mexicana de Fısica E vol 59 no 1 pp 51ndash55 2013
[36] FM Fernandez ldquoRational approximation to theThomas-Fermiequationsrdquo Applied Mathematics and Computation vol 217 no13 pp 6433ndash6436 2011
[37] B Yao ldquoA series solution to the Thomas-Fermi equationrdquoApplied Mathematics and Computation vol 203 no 1 pp 396ndash401 2008
[38] H Khan and H Xu ldquoSeries solution to the Thomas-Fermiequationrdquo Physics Letters A vol 365 no 1-2 pp 111ndash115 2007
[39] K Parand andM Shahini ldquoRational Chebyshev pseudospectralapproach for solving Thomas-Fermi equationrdquo Physics LettersA vol 373 no 2 pp 210ndash213 2009
Journal of Applied Mathematics 9
[40] E Hille ldquoOn the Thomas-Fermi equationrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 62 pp 7ndash10 1969
[41] M Turkyilmazoglu ldquoSolution of the Thomas-Fermi equationwith a convergent approachrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 11 pp 4097ndash41032012
[42] D G ZillA First Course in Differential Equations withModelingApplications Brooks Cole 10th edition 2012
Submit your manuscripts athttpwwwhindawicom
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Stochastic AnalysisInternational Journal of
4 Journal of Applied Mathematics
where 1199101198671
= 119890minus119860119909 and 119910
1198672= 119890minus119861119909 are the solutions of the
homogeneous differential equation
11991010158401015840
1+ 1205721199101015840
1+ 1205731199101= 0 (26)
119882 is theWronskian of these two functions which is given by
119882(1199101 1199102) = (119860 minus 119861)119890
minus(119860+119861)119909(119860 = 119861) (27)
and 119891(119909) is the right-hand side of (23)Substituting 119891(119909) and (27) into (25) leads to
1199061= minus
1
119860 minus 119861(11989632
1lim120576rarr0
int
119909
120576
119890minus1198601199102
119889119910
radic119910
+3
2119896211989612
1lim120576rarr0
int
119909
120576
119890minus(119861minus1198602)119910
119889119910
radic119910)
1199062=
1
119860 minus 119861(11989632
1lim120576rarr0
int
119909
120576
119890minus(31198602minus119861)119910
119889119910
radic119910
+3
2119896211989612
1lim120576rarr0
int
119909
120576
119890minus1198601199102
119889119910
radic119910)
(28)
By substituting 119910 = 1199062 the above equations take the form
1199061= minus
1
119860 minus 119861(211989632
1lim120576rarr0
int
radic119909
120576
119890minus11990621198602
119889119906
+ 3119896211989612
1lim120576rarr0
int
radic119909
120576
119890minus(119861minus1198602)119906
2
119889119906)
(29)
1199062=
1
119860 minus 119861(211989632
1lim120576rarr0
int
radic119909
120576
119890(119861minus31198602)119906
2
119889119906
+ 3119896211989612
1lim120576rarr0
int
radic119909
120576
119890minus11986011990622119889119906)
(30)
Therefore the solution of (23) is written according to themethod of variation of parameters as [42]
1199101= 119891119890minus119860119909
+ 119892119890minus119861119909
+ 1199061119890minus119860119909
+ 1199062119890minus119861119909
(31)
where 119891 and 119892 are constants and 1199061and 119906
2are given by (29)
and (30) respectivelyApplying the initial condition 119910
1(0) = 0 to (31) leads to
1199101(0) = 119891 + 119892 minus
1
119860 minus 119861(211989632
1lim120576rarr0119909rarr0
int
radic119909
120576
119890minus11986011990622119889119906
+ 3119896211989612
1lim120576rarr0119909rarr0
int
radic119909
120576
119890minus(119861minus1198602)119906
2
119889119906)
+1
119860 minus 119861(211989632
1lim120576rarr0119909rarr0
int
radic119909
120576
119890(119861minus31198602)119906
2
119889119906
+ 3119896211989612
1lim120576rarr0119909rarr0
int
radic119909
120576
119890minus11986011990622119889119906) = 0
(32)
Equation (32) is simplified to
119891 + 119892 = 0 (33)
On the other hand to apply the condition 1199101015840
1(0) = 0 we
differentiate (31) to obtain
1199101015840
1= minus119860119891119890
minus119860119909minus 119861119892119890
minus119861119909
minus119890minus119860119909
119860 minus 119861(11989632
1
119890minus1198601199092
radic119909+3119896211989612
1119890minus(119861minus1198602)119909
2radic119909)
+119890minus119861119909
119860 minus 119861(11989632
1
119890minus(31198602minus119861)119909
radic119909+3119896211989612
1119890minus1198602119909
2radic119909)
minus 119860119890minus119860119909
1199061minus 119861119890minus119861119909
1199062
(34)
After performing algebraic simplifications to (34) we obtain
1199101015840
1= minus119860119891119890
minus119860119909minus 119861119892119890
minus119861119909minus 119860119890minus119860119909
1199061minus 119861119890minus119861119909
1199062 (35)
Applying the condition 1199101015840
1(0) = 0 to (35) the following is
obtained
119860119891 + 119861119892 = 0 (36)
since lim119909rarr0
1199061= 0 and lim
119909rarr01199062= 0 (see (29) and (30))
From (33) and (36) we obtain 119891 = 0 and 119892 = 0 since119860 = 119861 therefore (31) becomes
1199101= 1199061119890minus119860119909
+ 1199062119890minus119861119909
(37)
By substituting (21) and (37) into (12) we obtain the followingfirst order approximation for the solution of (13)
119910 = 1198961119890minus119860119909
+ 1198962119890minus119861119909
+ 1199061119890minus119860119909
+ 1199062119890minus119861119909
(38)
We will show later that it is possible to achieve a good fit andenlarge the domain of convergence for (38) (keeping at thesame time the handy character of the proposed solution) byintroducing in (38) an adjusting parameter (consistent withthe initial conditions) as it is shown in
119910 = 1198961119890minus119860119909
+ 1198962119890minus119861119909
+ 1205751199061119890minus119860119909
+ 1205751199062119890minus119861119909
(39)
The inclusion of parameter 120575 is motivated because thesolution 119910
0(119909) for the homogeneous equation (19) is undeter-
mined by a constant factor 120572 (without considering the initialvalues) and from the following argument
Let 1198840(119909) = 120572119910
0(119909) since 119884
0(0) = 120572 1199101015840
0(0) = minus119871120572 the
first order solution for a function say 119884(119909) which definesa slightly different problem from the original is given by119884(119909) = 119884
0(119909) + 119884
1(119909) or 119884(119909) = 120572119910
0(119909) + 119884
1(119909) where
1199100(0) = 1 1199101015840
0(0) = minus119871 119884
1(0) = 0 and 119884
1015840
1(0) = 0 Thus
after dividing by 120572 119910(119909) = 1199100(119909) + 120575119910
1(119909) is obtained (where
120575 = 119891(120572)120572 119910(119909) = 119884(119909)120572 and we have supposed thatfor some problems 119884
1(119909) = 119891(120572)119910
1(119909) As a matter of fact
following (21)ndash(30)we conclude that for this case119891(120572) = 12057232therefore 120575 = 12057212)
Journal of Applied Mathematics 5
Substituting (29) and (30) into (39) we obtain
119910 = 1198961119890minus119860119909
+ 1198962119890minus119861119909
minus120575119890minus119860119909
119860 minus 119861(211989632
1lim120576rarr0
int
radic119909
120576
119890minus11986011990622119889119906
+ 3119896211989612
1lim120576rarr0
int
radic119909
120576
119890minus(119861minus1198602)119906
2
119889119906)
+120575119890minus119861119909
119860 minus 119861(211989632
1lim120576rarr0
int
radic119909
120576
119890(119861minus31198602)119906
2
119889119906
+ 3119896211989612
1lim120576rarr0
int
radic119909
120576
119890minus11986011990622119889119906)
(40)
In [14] a high accurate approximation of normal distributionintegral was reported which let us write the above integralsas
lim120576rarr0
int
radic119909
120576
119890minus11986011990622119889119906
=1
2
radic2120587
119860tanh(39
2
radic119860119909
2120587minus111
2tanminus1( 35
111
radic119860119909
2120587))
lim120576rarr0
int
radic119909
120576
119890minus(119861minus1198602)119906
2
119889119906
=1
2
radic2120587
2119861 minus 119860tanh(39
2
radic(2119861 minus 119860)119909
2120587
minus111
2tanminus1( 35
111
radic(2119861 minus 119860)119909
2120587))
(41)
Note that
int
radic119909
0
119890(119861minus31198602)119906
2
119889119906 (42)
is not a Gaussian integral on the assumption that 119890minus119860119909 ≫ 119890minus119861119909
for 0 lt 119909 lt infin since it implies that 119860 ≪ 119861
Equations (41) allow us to write (40) as
119910 = 1198961119890minus119860119909
+ 1198962119890minus119861119909
minus120575119890minus119860119909
119860 minus 119861
sdot (11989632
1radic2120587
119860tanh(39
2
radic119860119909
2120587
minus111
2tanminus1( 35
111
radic119860119909
2120587))
+3
2119896211989612
1radic
2120587
2119861 minus 119860
sdot tanh(39
2
radic(2119861 minus 119860)119909
2120587
minus111
2tanminus1( 35
111
radic(2119861 minus 119860)119909
2120587)))
+120575119890minus119861119909
119860 minus 119861(211989632
1lim120576rarr0
int
radic119909
120576
119890(119861minus31198602)119906
2
119889119906
+3
2119896211989612
1radic2120587
119860
sdot tanh(392
radic119860119909
2120587minus111
2tanminus1( 35
111
radic119860119909
2120587)))
(43)
To transform (43) into an analytical expression we evaluate(42) keeping terms up to the ninth power to obtain thefollowing compact fractional power function
int
radic119909
0
119890(119861minus31198602)119906
2
119889119906
= radic119909 +1
3(119861 minus 3119860)119909
32+
1
10(119861 minus 3119860)
211990952
+1
42(119861 minus 3119860)
311990972
+1
216(119861 minus 3119860)
411990992
(44)
where the main contribution of this approximation to (43) isin the range of 119909 isin (0 1] because it is multiplied by a negativeexponential (see (43) and Section 4)
6 Journal of Applied Mathematics
Therefore substituting (44) into (43) the following isobtained
119910 = 1198961119890minus119860119909
+ 1198962119890minus119861119909
minus120575119890minus119860119909
119860 minus 119861
sdot (11989632
1radic2120587
119860tanh(39
2
radic119860119909
2120587
minus111
2tanminus1( 35
111
radic119860119909
2120587))
+3
2119896211989612
1radic
2120587
2119861 minus 119860
sdot tanh(39
2
radic(2119861 minus 119860)119909
2120587
minus111
2tanminus1( 35
111
radic(2119861 minus 119860) 119909
2120587)))
+120575119890minus119861119909
119860 minus 119861(211989632
1(radic119909 +
(119861 minus 3119860)11990932
3+(119861 minus 3119860)
211990952
10
+(119861 minus 3119860)
311990972
42+(119861 minus 3119860)
411990992
216)
+3
2119896211989612
1radic2120587
119860
sdot tanh(392
radic119860119909
2120587minus111
2tanminus1( 35
111
radic119860119909
2120587)))
(45)
In order to obtain a good approximation from (45) weoptimize the values of the aforementioned parameters asfollows 119860 = 0398521712348839 119861 = 6 and 120575 =
minus0623239813 using the procedure reported in [14 17 27](note that with these values the assumption 119860 ≪ 119861 issatisfied besides the above results allow to know 120572 and 120573
whose values are not needed in the search for the solutionof the system (19) and (20))
Figure 1 and Table 1 show the comparison betweenapproximate solution (45) and the exact solution The accu-racy of (45) is clear as an approximate solution for (13)
4 Discussion
Table 1 shows the comparison between numerical solutiongiven in [39] and approximations [15 38] and (45) It isclear that our approximation (45) is competitive consideringthat it is analytic and handy unlike other approximationswhich are expressed in terms of power series In [37 3841] T-F equation was solved by homotopy analysis method(HAM) The approximation [38] corresponds to an 80thorder analytical approximation and therefore is impracticalOn the other hand in [37] the results are given in terms of
1
08
06
04
02
0
y(x)
0 05 1 15 2 25 3 35 4 45 5
x
[39](45)
Figure 1 Comparison of Thomas-Fermirsquos numerical solution [39]and approximation (45)
infinite series and show particularly the calculation of theinitial slope 1199101015840(0) Moreover in [41] an accurate approximatesolution was obtained but it corresponds to the 40th orderapproximation In [23] (13) was solved by HPM and Padeapproximants in this case a solution of 26 power series termsis provided which is also not handy in practice Reference[15] presents an analytical approach but it is not accurateIn this study we used homotopy perturbation method withnonlinearities distributions to find an accurate solution for(13) (see Figure 1 and Table 1) Although (43) is just almostanalytic (due to the presence of a non-Gaussian integral) itcan be estimated that contribution of (42) to the solution(43) is well represented by (44) assuming the values are 119860 =
0398521712348839 119861 = 6 and 120575 = minus0623239813 thus weobtained the analytical approximation (45)We just kept onlyfive terms in (44) in order to obtain a handy approximationBesides the right-hand side of (44) adequately represents theindicated non-Gaussian integral in the range 119909 isin [0 1] Itimplies that the best contribution of the term proportional to119890minus119861119909
intradic119909
0119890(119861minus31198602)119906
2
119889119906 in (43) is just beginning of the intervalconsidered that is in a part of the domain where the T-F equation describes correctly the atom (see Section 1) Forvalues 119909 gt 1 the above-mentioned term is dominated forthe exponential 119890minus119861119909 In any case the value of the integralsof (40) can be evaluated by using numerical methods Aninteresting fact is that if more terms had been consideredin the development of (12) we would have obtained asa solution of (13) a series containing terms of hyperbolictangent functions to the form (41)
The strategy to redistribute the nonlinearities of T-Fequation between the successive iterations of the HPMmethod was important to obtain an approximate solution of(13) because it allows us to simplify the solution procedurefor the successive iterations of the HPM method Besideswe obtained a soluble coupled system of linear differentialequations (19) and (20) In this study we introduced anadjusting parameter in order to enlarge the domain ofconvergence of our approach with good results by using onlythe first order approximation (see Table 1 and Figure 1)
Journal of Applied Mathematics 7
Table 1 Numerical comparison of Thomas-Fermirsquos numerical solution [39] and approximations [15 38] and (45)
119909 Numerical [39] [38] (45) [15]000 mdash 1000000000 1000000000 1000000000025 0755880759 0776191000 0708502932 0680650028050 0606700008 0615917000 0570491745 0459455663075 0502964042 0505380000 0485018011 0307042537100 0424333179 0423772000 0421167227 0202655526125 0363227937 0362935000 0369542163 0131667603150 0314660642 0314490000 0326298526 0083800106175 0275233848 0275154000 0289315779 0051853222200 0242678587 0242718000 0257253699 0030801947225 0215439334 0215630000 0229210094 0017153465250 0192406328 0192795000 0204540381 0008491277275 0172758691 0173364000 0182755264 0003152503300 0155871862 0156719000 0163464474 0325 0141260504 0142371000 0146346052 minus0001738122
350 0128541381 0129937000 0131128799 minus0002581206
375 0117408054 0119108000 0117581337 minus0002874924
400 0107612958 0109632000 0105504623 minus000284628
425 0098954329 0101303000 0094726427 minus0002641801
450 0091266456 0093950400 0085097045 minus0002353931
475 0084412289 0087432000 0076485875 minus0002039168
500 0078277758 0081629600 0068778618 minus0001730443
A relevant fact is that NDHPM requires little iterationsto obtain accurate solutions if we have a first approximationcontaining as much information as possible for a nonlineardifferential equation For instance in our case (21) containsthe correct asymptotic character of the exact equation
5 Conclusion
In this work NDHPM was used to obtain an approximateanalytical solution for T-F equation Our solution is novelbecause it is expressed in terms of exponential and fractionalpower functions and above all in terms of normal distributionintegrals It is important to obtain an analytical expressionthat provides a good description of the solution for thenonlinear differential equations like (13) For instance thecharge distribution and the electrostatic potential inside anatom are adequately described by (43) and (45) It is worthmentioning that if the initial guess is suitably chosen it ispossible to obtain by this method an accurate approximationlike (43) even using the first terms of (12) Finally in contrastto Runge-Kutta numerical solution NDHPMmethod allowsto analyze quantitatively and qualitatively the solution
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the financial supportfrom the National Council for Science and Technology ofMexico (CONACyT) through Grant CB-2010-01 157024The authors would like to thank Rogelio-Alejandro Callejas-Molina and Roberto Ruiz-Gomez for their contribution tothis project
References
[1] L Landau and E M LifshitzMecanica Cuantica No RelativistaEditorial Reverte S A 2nd edition 1967
[2] LM B Assas ldquoApproximate solutions for the generalizedKdVndashBurgersrsquo equation by Hersquos variational iteration methodrdquo PhysicaScripta vol 76 no 2 pp 161ndash164 2007
[3] M Kazemnia S A Zahedi M Vaezi and N Tolou ldquoArseniccontamination in New Orleans soil temporal changes associ-ated with floodingrdquo Journal of Applied Sciences vol 8 no 22pp 4192ndash4197 2008
[4] R Noorzad A T Poor and M Omidvar ldquoVariational iterationmethod and homotopy-perturbation method for solving Burg-ers equation in fluid dynamicsrdquo Journal of Applied Sciences vol8 no 2 pp 369ndash373 2008
[5] D J Evans and K R Raslan ldquoThe tanh function method forsolving some important non-linear partial differential equa-tionsrdquo International Journal of Computer Mathematics vol 82no 7 pp 897ndash905 2005
[6] J Mahmoudi N Tolou I Khatami A Barari and D D GanjildquoExplicit solution of nonlinear ZK-BBM wave equation using
8 Journal of Applied Mathematics
Exp-function methodrdquo Journal of Applied Sciences vol 8 no 2pp 358ndash363 2008
[7] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988
[8] E Babolian and J Biazar ldquoOn the order of convergence ofAdomian methodrdquo Applied Mathematics and Computation vol130 no 2-3 pp 383ndash387 2002
[9] A Kooch and M Abadyan ldquoEfficiency of modified Ado-mian decomposition for simulating the instability of nano-electromechanical switches comparison with the conventionaldecomposition methodrdquo Trends in Applied Sciences Researchvol 7 no 1 pp 57ndash67 2012
[10] Y Khan H Vazquez-Leal and N Faraz ldquoAn auxiliary param-eter method using Adomian polynomials and Laplace transfor-mation for nonlinear differential equationsrdquoAppliedMathemat-ical Modelling vol 37 no 5 pp 2702ndash2708 2013
[11] S K Vanani S Heidari and M Avaji ldquoA low-cost numericalalgorithm for the solution of nonlinear delay boundary integralequationsrdquo Journal of Applied Sciences vol 11 no 20 pp 3504ndash3509 2011
[12] S H Chowdhury ldquoA comparison between the modifiedhomotopy perturbation method and adomian decompositionmethod for solving nonlinear heat transfer equationsrdquo Journalof Applied Sciences vol 11 no 7 pp 1416ndash1420 2011
[13] L-N Zhang and L Xu ldquoDetermination of the limit cycle byHersquos parameter expansion for oscillators in a u31 + u2 potentialrdquoZeitschrift fur NaturforschungmdashSection A Journal of PhysicalSciences vol 62 no 7-8 pp 396ndash398 2007
[14] H Vazquez-Leal R Castaneda-Sheissa U Filobello-Nino ASarmiento-Reyes and J Sanchez Orea ldquoHigh accurate simpleapproximation of normal distribution integralrdquo MathematicalProblems in Engineering vol 2012 Article ID 124029 22 pages2012
[15] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006
[16] A Fereidoon Y Rostamiyan M Akbarzade and D D GanjildquoApplication of Hersquos homotopy perturbation method to nonlin-ear shock damper dynamicsrdquo Archive of Applied Mechanics vol80 no 6 pp 641ndash649 2010
[17] H Vazquez-Leal U Filobello-Nino R Castaneda-SheissaL Hernandez-Martınez and A Sarmiento-Reyes ldquoModifiedHPMs inspired by homotopy continuation methodsrdquo Mathe-matical Problems in Engineering vol 2012 Article ID 30912319 pages 2012
[18] Y Khan H Vazquez-Leal and Q Wu ldquoAn efficient iteratedmethod for mathematical biology modelrdquo Neural Computingand Applications vol 23 no 3-4 pp 677ndash682 2013
[19] Y Khan H Vazquez-Leal and N Faraz ldquoAn efficient newiterative method for oscillator differential equationrdquo ScientiaIranica vol 19 no 6 pp 1473ndash1477 2012
[20] J Biazar and H Aminikhah ldquoStudy of convergence of homo-topy perturbation method for systems of partial differentialequationsrdquoComputersampMathematics with Applications vol 58no 11-12 pp 2221ndash2230 2009
[21] J Biazar and H Ghazvini ldquoConvergence of the homotopy per-turbation method for partial differential equationsrdquo NonlinearAnalysis Real World Applications vol 10 no 5 pp 2633ndash26402009
[22] Y Khan and Q Wu ldquoHomotopy perturbation transformmethod for nonlinear equations using Hersquos polynomialsrdquo Com-puters and Mathematics with Applications vol 61 no 8 pp1963ndash1967 2011
[23] M A Noor and S T Mohyud-Din ldquoHomotopy perturbationmethod for solvingThomas-Fermi equation using pade approx-imantsrdquo International Journal of Nonlinear Science vol 8 no 1pp 27ndash31 2009
[24] DDGanji A R Sahouli andM Famouri ldquoAnewmodificationofHersquos homotopy perturbationmethod for rapid convergence ofnonlinear undamped oscillatorsrdquo Journal of Applied Mathemat-ics and Computing vol 30 no 1-2 pp 181ndash192 2009
[25] M Turkyilmazoglu ldquoSome issues on HPM and HAMmethodsa convergence schemerdquoMathematical and Computer Modellingvol 53 no 9-10 pp 1929ndash1936 2011
[26] M Turkyilmazoglu ldquoConvergence of the homotopy perturba-tion methodrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 12 no 1ndash8 pp 9ndash14 2011
[27] U Filobello-Nino H Vazquez-Leal R Castaneda-Sheissa et alldquoAn approximate solution of Blasius equation by using HPMmethodrdquo Asian Journal of Mathematics and Statistics vol 5 no2 pp 50ndash59 2012
[28] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineering Springer Berlin Germany 1st edition 2011
[29] T Patel M N Mehta and V H Pradhan ldquoThe numericalsolution of Burgerrsquos equation arising into the irradiation oftumour tissue in biological diffusing system by homotopyanalysis methodrdquo Asian Journal of Applied Sciences vol 5 no1 pp 60ndash66 2012
[30] S Liao Advances in the Homotopy Analysis Method WorldScientific Hackensack NJ USA 2014
[31] M Turkyilmazoglu ldquoAnalytic approximate solutions of rotatingdisk boundary layer flow subject to a uniform suction orinjectionrdquo International Journal of Mechanical Sciences vol 52no 12 pp 1735ndash1744 2010
[32] M Turkyilmazoglu ldquoPurely analytic solutions of the compress-ible boundary layer flow due to a porous rotating disk with heattransferrdquo Physics of Fluids vol 21 no 10 2009
[33] U Filobello-Nino H Vazquez-Leal B Benhammouda et alldquoA handy approximation for a mediated bioelectrocatalysisprocess related to Michaelis-Menten equationrdquo SpringerPlusvol 3 article 162 2014
[34] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoUsing per-turbation methods and Laplace-Pade approximation to solvenonlinear problemsrdquo Miskolc Mathematical Notes vol 14 no1 pp 89ndash101 2013
[35] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoA handyexact solution for flow due to a stretching boundary with partialsliprdquo Revista Mexicana de Fısica E vol 59 no 1 pp 51ndash55 2013
[36] FM Fernandez ldquoRational approximation to theThomas-Fermiequationsrdquo Applied Mathematics and Computation vol 217 no13 pp 6433ndash6436 2011
[37] B Yao ldquoA series solution to the Thomas-Fermi equationrdquoApplied Mathematics and Computation vol 203 no 1 pp 396ndash401 2008
[38] H Khan and H Xu ldquoSeries solution to the Thomas-Fermiequationrdquo Physics Letters A vol 365 no 1-2 pp 111ndash115 2007
[39] K Parand andM Shahini ldquoRational Chebyshev pseudospectralapproach for solving Thomas-Fermi equationrdquo Physics LettersA vol 373 no 2 pp 210ndash213 2009
Journal of Applied Mathematics 9
[40] E Hille ldquoOn the Thomas-Fermi equationrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 62 pp 7ndash10 1969
[41] M Turkyilmazoglu ldquoSolution of the Thomas-Fermi equationwith a convergent approachrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 11 pp 4097ndash41032012
[42] D G ZillA First Course in Differential Equations withModelingApplications Brooks Cole 10th edition 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 5
Substituting (29) and (30) into (39) we obtain
119910 = 1198961119890minus119860119909
+ 1198962119890minus119861119909
minus120575119890minus119860119909
119860 minus 119861(211989632
1lim120576rarr0
int
radic119909
120576
119890minus11986011990622119889119906
+ 3119896211989612
1lim120576rarr0
int
radic119909
120576
119890minus(119861minus1198602)119906
2
119889119906)
+120575119890minus119861119909
119860 minus 119861(211989632
1lim120576rarr0
int
radic119909
120576
119890(119861minus31198602)119906
2
119889119906
+ 3119896211989612
1lim120576rarr0
int
radic119909
120576
119890minus11986011990622119889119906)
(40)
In [14] a high accurate approximation of normal distributionintegral was reported which let us write the above integralsas
lim120576rarr0
int
radic119909
120576
119890minus11986011990622119889119906
=1
2
radic2120587
119860tanh(39
2
radic119860119909
2120587minus111
2tanminus1( 35
111
radic119860119909
2120587))
lim120576rarr0
int
radic119909
120576
119890minus(119861minus1198602)119906
2
119889119906
=1
2
radic2120587
2119861 minus 119860tanh(39
2
radic(2119861 minus 119860)119909
2120587
minus111
2tanminus1( 35
111
radic(2119861 minus 119860)119909
2120587))
(41)
Note that
int
radic119909
0
119890(119861minus31198602)119906
2
119889119906 (42)
is not a Gaussian integral on the assumption that 119890minus119860119909 ≫ 119890minus119861119909
for 0 lt 119909 lt infin since it implies that 119860 ≪ 119861
Equations (41) allow us to write (40) as
119910 = 1198961119890minus119860119909
+ 1198962119890minus119861119909
minus120575119890minus119860119909
119860 minus 119861
sdot (11989632
1radic2120587
119860tanh(39
2
radic119860119909
2120587
minus111
2tanminus1( 35
111
radic119860119909
2120587))
+3
2119896211989612
1radic
2120587
2119861 minus 119860
sdot tanh(39
2
radic(2119861 minus 119860)119909
2120587
minus111
2tanminus1( 35
111
radic(2119861 minus 119860)119909
2120587)))
+120575119890minus119861119909
119860 minus 119861(211989632
1lim120576rarr0
int
radic119909
120576
119890(119861minus31198602)119906
2
119889119906
+3
2119896211989612
1radic2120587
119860
sdot tanh(392
radic119860119909
2120587minus111
2tanminus1( 35
111
radic119860119909
2120587)))
(43)
To transform (43) into an analytical expression we evaluate(42) keeping terms up to the ninth power to obtain thefollowing compact fractional power function
int
radic119909
0
119890(119861minus31198602)119906
2
119889119906
= radic119909 +1
3(119861 minus 3119860)119909
32+
1
10(119861 minus 3119860)
211990952
+1
42(119861 minus 3119860)
311990972
+1
216(119861 minus 3119860)
411990992
(44)
where the main contribution of this approximation to (43) isin the range of 119909 isin (0 1] because it is multiplied by a negativeexponential (see (43) and Section 4)
6 Journal of Applied Mathematics
Therefore substituting (44) into (43) the following isobtained
119910 = 1198961119890minus119860119909
+ 1198962119890minus119861119909
minus120575119890minus119860119909
119860 minus 119861
sdot (11989632
1radic2120587
119860tanh(39
2
radic119860119909
2120587
minus111
2tanminus1( 35
111
radic119860119909
2120587))
+3
2119896211989612
1radic
2120587
2119861 minus 119860
sdot tanh(39
2
radic(2119861 minus 119860)119909
2120587
minus111
2tanminus1( 35
111
radic(2119861 minus 119860) 119909
2120587)))
+120575119890minus119861119909
119860 minus 119861(211989632
1(radic119909 +
(119861 minus 3119860)11990932
3+(119861 minus 3119860)
211990952
10
+(119861 minus 3119860)
311990972
42+(119861 minus 3119860)
411990992
216)
+3
2119896211989612
1radic2120587
119860
sdot tanh(392
radic119860119909
2120587minus111
2tanminus1( 35
111
radic119860119909
2120587)))
(45)
In order to obtain a good approximation from (45) weoptimize the values of the aforementioned parameters asfollows 119860 = 0398521712348839 119861 = 6 and 120575 =
minus0623239813 using the procedure reported in [14 17 27](note that with these values the assumption 119860 ≪ 119861 issatisfied besides the above results allow to know 120572 and 120573
whose values are not needed in the search for the solutionof the system (19) and (20))
Figure 1 and Table 1 show the comparison betweenapproximate solution (45) and the exact solution The accu-racy of (45) is clear as an approximate solution for (13)
4 Discussion
Table 1 shows the comparison between numerical solutiongiven in [39] and approximations [15 38] and (45) It isclear that our approximation (45) is competitive consideringthat it is analytic and handy unlike other approximationswhich are expressed in terms of power series In [37 3841] T-F equation was solved by homotopy analysis method(HAM) The approximation [38] corresponds to an 80thorder analytical approximation and therefore is impracticalOn the other hand in [37] the results are given in terms of
1
08
06
04
02
0
y(x)
0 05 1 15 2 25 3 35 4 45 5
x
[39](45)
Figure 1 Comparison of Thomas-Fermirsquos numerical solution [39]and approximation (45)
infinite series and show particularly the calculation of theinitial slope 1199101015840(0) Moreover in [41] an accurate approximatesolution was obtained but it corresponds to the 40th orderapproximation In [23] (13) was solved by HPM and Padeapproximants in this case a solution of 26 power series termsis provided which is also not handy in practice Reference[15] presents an analytical approach but it is not accurateIn this study we used homotopy perturbation method withnonlinearities distributions to find an accurate solution for(13) (see Figure 1 and Table 1) Although (43) is just almostanalytic (due to the presence of a non-Gaussian integral) itcan be estimated that contribution of (42) to the solution(43) is well represented by (44) assuming the values are 119860 =
0398521712348839 119861 = 6 and 120575 = minus0623239813 thus weobtained the analytical approximation (45)We just kept onlyfive terms in (44) in order to obtain a handy approximationBesides the right-hand side of (44) adequately represents theindicated non-Gaussian integral in the range 119909 isin [0 1] Itimplies that the best contribution of the term proportional to119890minus119861119909
intradic119909
0119890(119861minus31198602)119906
2
119889119906 in (43) is just beginning of the intervalconsidered that is in a part of the domain where the T-F equation describes correctly the atom (see Section 1) Forvalues 119909 gt 1 the above-mentioned term is dominated forthe exponential 119890minus119861119909 In any case the value of the integralsof (40) can be evaluated by using numerical methods Aninteresting fact is that if more terms had been consideredin the development of (12) we would have obtained asa solution of (13) a series containing terms of hyperbolictangent functions to the form (41)
The strategy to redistribute the nonlinearities of T-Fequation between the successive iterations of the HPMmethod was important to obtain an approximate solution of(13) because it allows us to simplify the solution procedurefor the successive iterations of the HPM method Besideswe obtained a soluble coupled system of linear differentialequations (19) and (20) In this study we introduced anadjusting parameter in order to enlarge the domain ofconvergence of our approach with good results by using onlythe first order approximation (see Table 1 and Figure 1)
Journal of Applied Mathematics 7
Table 1 Numerical comparison of Thomas-Fermirsquos numerical solution [39] and approximations [15 38] and (45)
119909 Numerical [39] [38] (45) [15]000 mdash 1000000000 1000000000 1000000000025 0755880759 0776191000 0708502932 0680650028050 0606700008 0615917000 0570491745 0459455663075 0502964042 0505380000 0485018011 0307042537100 0424333179 0423772000 0421167227 0202655526125 0363227937 0362935000 0369542163 0131667603150 0314660642 0314490000 0326298526 0083800106175 0275233848 0275154000 0289315779 0051853222200 0242678587 0242718000 0257253699 0030801947225 0215439334 0215630000 0229210094 0017153465250 0192406328 0192795000 0204540381 0008491277275 0172758691 0173364000 0182755264 0003152503300 0155871862 0156719000 0163464474 0325 0141260504 0142371000 0146346052 minus0001738122
350 0128541381 0129937000 0131128799 minus0002581206
375 0117408054 0119108000 0117581337 minus0002874924
400 0107612958 0109632000 0105504623 minus000284628
425 0098954329 0101303000 0094726427 minus0002641801
450 0091266456 0093950400 0085097045 minus0002353931
475 0084412289 0087432000 0076485875 minus0002039168
500 0078277758 0081629600 0068778618 minus0001730443
A relevant fact is that NDHPM requires little iterationsto obtain accurate solutions if we have a first approximationcontaining as much information as possible for a nonlineardifferential equation For instance in our case (21) containsthe correct asymptotic character of the exact equation
5 Conclusion
In this work NDHPM was used to obtain an approximateanalytical solution for T-F equation Our solution is novelbecause it is expressed in terms of exponential and fractionalpower functions and above all in terms of normal distributionintegrals It is important to obtain an analytical expressionthat provides a good description of the solution for thenonlinear differential equations like (13) For instance thecharge distribution and the electrostatic potential inside anatom are adequately described by (43) and (45) It is worthmentioning that if the initial guess is suitably chosen it ispossible to obtain by this method an accurate approximationlike (43) even using the first terms of (12) Finally in contrastto Runge-Kutta numerical solution NDHPMmethod allowsto analyze quantitatively and qualitatively the solution
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the financial supportfrom the National Council for Science and Technology ofMexico (CONACyT) through Grant CB-2010-01 157024The authors would like to thank Rogelio-Alejandro Callejas-Molina and Roberto Ruiz-Gomez for their contribution tothis project
References
[1] L Landau and E M LifshitzMecanica Cuantica No RelativistaEditorial Reverte S A 2nd edition 1967
[2] LM B Assas ldquoApproximate solutions for the generalizedKdVndashBurgersrsquo equation by Hersquos variational iteration methodrdquo PhysicaScripta vol 76 no 2 pp 161ndash164 2007
[3] M Kazemnia S A Zahedi M Vaezi and N Tolou ldquoArseniccontamination in New Orleans soil temporal changes associ-ated with floodingrdquo Journal of Applied Sciences vol 8 no 22pp 4192ndash4197 2008
[4] R Noorzad A T Poor and M Omidvar ldquoVariational iterationmethod and homotopy-perturbation method for solving Burg-ers equation in fluid dynamicsrdquo Journal of Applied Sciences vol8 no 2 pp 369ndash373 2008
[5] D J Evans and K R Raslan ldquoThe tanh function method forsolving some important non-linear partial differential equa-tionsrdquo International Journal of Computer Mathematics vol 82no 7 pp 897ndash905 2005
[6] J Mahmoudi N Tolou I Khatami A Barari and D D GanjildquoExplicit solution of nonlinear ZK-BBM wave equation using
8 Journal of Applied Mathematics
Exp-function methodrdquo Journal of Applied Sciences vol 8 no 2pp 358ndash363 2008
[7] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988
[8] E Babolian and J Biazar ldquoOn the order of convergence ofAdomian methodrdquo Applied Mathematics and Computation vol130 no 2-3 pp 383ndash387 2002
[9] A Kooch and M Abadyan ldquoEfficiency of modified Ado-mian decomposition for simulating the instability of nano-electromechanical switches comparison with the conventionaldecomposition methodrdquo Trends in Applied Sciences Researchvol 7 no 1 pp 57ndash67 2012
[10] Y Khan H Vazquez-Leal and N Faraz ldquoAn auxiliary param-eter method using Adomian polynomials and Laplace transfor-mation for nonlinear differential equationsrdquoAppliedMathemat-ical Modelling vol 37 no 5 pp 2702ndash2708 2013
[11] S K Vanani S Heidari and M Avaji ldquoA low-cost numericalalgorithm for the solution of nonlinear delay boundary integralequationsrdquo Journal of Applied Sciences vol 11 no 20 pp 3504ndash3509 2011
[12] S H Chowdhury ldquoA comparison between the modifiedhomotopy perturbation method and adomian decompositionmethod for solving nonlinear heat transfer equationsrdquo Journalof Applied Sciences vol 11 no 7 pp 1416ndash1420 2011
[13] L-N Zhang and L Xu ldquoDetermination of the limit cycle byHersquos parameter expansion for oscillators in a u31 + u2 potentialrdquoZeitschrift fur NaturforschungmdashSection A Journal of PhysicalSciences vol 62 no 7-8 pp 396ndash398 2007
[14] H Vazquez-Leal R Castaneda-Sheissa U Filobello-Nino ASarmiento-Reyes and J Sanchez Orea ldquoHigh accurate simpleapproximation of normal distribution integralrdquo MathematicalProblems in Engineering vol 2012 Article ID 124029 22 pages2012
[15] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006
[16] A Fereidoon Y Rostamiyan M Akbarzade and D D GanjildquoApplication of Hersquos homotopy perturbation method to nonlin-ear shock damper dynamicsrdquo Archive of Applied Mechanics vol80 no 6 pp 641ndash649 2010
[17] H Vazquez-Leal U Filobello-Nino R Castaneda-SheissaL Hernandez-Martınez and A Sarmiento-Reyes ldquoModifiedHPMs inspired by homotopy continuation methodsrdquo Mathe-matical Problems in Engineering vol 2012 Article ID 30912319 pages 2012
[18] Y Khan H Vazquez-Leal and Q Wu ldquoAn efficient iteratedmethod for mathematical biology modelrdquo Neural Computingand Applications vol 23 no 3-4 pp 677ndash682 2013
[19] Y Khan H Vazquez-Leal and N Faraz ldquoAn efficient newiterative method for oscillator differential equationrdquo ScientiaIranica vol 19 no 6 pp 1473ndash1477 2012
[20] J Biazar and H Aminikhah ldquoStudy of convergence of homo-topy perturbation method for systems of partial differentialequationsrdquoComputersampMathematics with Applications vol 58no 11-12 pp 2221ndash2230 2009
[21] J Biazar and H Ghazvini ldquoConvergence of the homotopy per-turbation method for partial differential equationsrdquo NonlinearAnalysis Real World Applications vol 10 no 5 pp 2633ndash26402009
[22] Y Khan and Q Wu ldquoHomotopy perturbation transformmethod for nonlinear equations using Hersquos polynomialsrdquo Com-puters and Mathematics with Applications vol 61 no 8 pp1963ndash1967 2011
[23] M A Noor and S T Mohyud-Din ldquoHomotopy perturbationmethod for solvingThomas-Fermi equation using pade approx-imantsrdquo International Journal of Nonlinear Science vol 8 no 1pp 27ndash31 2009
[24] DDGanji A R Sahouli andM Famouri ldquoAnewmodificationofHersquos homotopy perturbationmethod for rapid convergence ofnonlinear undamped oscillatorsrdquo Journal of Applied Mathemat-ics and Computing vol 30 no 1-2 pp 181ndash192 2009
[25] M Turkyilmazoglu ldquoSome issues on HPM and HAMmethodsa convergence schemerdquoMathematical and Computer Modellingvol 53 no 9-10 pp 1929ndash1936 2011
[26] M Turkyilmazoglu ldquoConvergence of the homotopy perturba-tion methodrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 12 no 1ndash8 pp 9ndash14 2011
[27] U Filobello-Nino H Vazquez-Leal R Castaneda-Sheissa et alldquoAn approximate solution of Blasius equation by using HPMmethodrdquo Asian Journal of Mathematics and Statistics vol 5 no2 pp 50ndash59 2012
[28] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineering Springer Berlin Germany 1st edition 2011
[29] T Patel M N Mehta and V H Pradhan ldquoThe numericalsolution of Burgerrsquos equation arising into the irradiation oftumour tissue in biological diffusing system by homotopyanalysis methodrdquo Asian Journal of Applied Sciences vol 5 no1 pp 60ndash66 2012
[30] S Liao Advances in the Homotopy Analysis Method WorldScientific Hackensack NJ USA 2014
[31] M Turkyilmazoglu ldquoAnalytic approximate solutions of rotatingdisk boundary layer flow subject to a uniform suction orinjectionrdquo International Journal of Mechanical Sciences vol 52no 12 pp 1735ndash1744 2010
[32] M Turkyilmazoglu ldquoPurely analytic solutions of the compress-ible boundary layer flow due to a porous rotating disk with heattransferrdquo Physics of Fluids vol 21 no 10 2009
[33] U Filobello-Nino H Vazquez-Leal B Benhammouda et alldquoA handy approximation for a mediated bioelectrocatalysisprocess related to Michaelis-Menten equationrdquo SpringerPlusvol 3 article 162 2014
[34] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoUsing per-turbation methods and Laplace-Pade approximation to solvenonlinear problemsrdquo Miskolc Mathematical Notes vol 14 no1 pp 89ndash101 2013
[35] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoA handyexact solution for flow due to a stretching boundary with partialsliprdquo Revista Mexicana de Fısica E vol 59 no 1 pp 51ndash55 2013
[36] FM Fernandez ldquoRational approximation to theThomas-Fermiequationsrdquo Applied Mathematics and Computation vol 217 no13 pp 6433ndash6436 2011
[37] B Yao ldquoA series solution to the Thomas-Fermi equationrdquoApplied Mathematics and Computation vol 203 no 1 pp 396ndash401 2008
[38] H Khan and H Xu ldquoSeries solution to the Thomas-Fermiequationrdquo Physics Letters A vol 365 no 1-2 pp 111ndash115 2007
[39] K Parand andM Shahini ldquoRational Chebyshev pseudospectralapproach for solving Thomas-Fermi equationrdquo Physics LettersA vol 373 no 2 pp 210ndash213 2009
Journal of Applied Mathematics 9
[40] E Hille ldquoOn the Thomas-Fermi equationrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 62 pp 7ndash10 1969
[41] M Turkyilmazoglu ldquoSolution of the Thomas-Fermi equationwith a convergent approachrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 11 pp 4097ndash41032012
[42] D G ZillA First Course in Differential Equations withModelingApplications Brooks Cole 10th edition 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Applied Mathematics
Therefore substituting (44) into (43) the following isobtained
119910 = 1198961119890minus119860119909
+ 1198962119890minus119861119909
minus120575119890minus119860119909
119860 minus 119861
sdot (11989632
1radic2120587
119860tanh(39
2
radic119860119909
2120587
minus111
2tanminus1( 35
111
radic119860119909
2120587))
+3
2119896211989612
1radic
2120587
2119861 minus 119860
sdot tanh(39
2
radic(2119861 minus 119860)119909
2120587
minus111
2tanminus1( 35
111
radic(2119861 minus 119860) 119909
2120587)))
+120575119890minus119861119909
119860 minus 119861(211989632
1(radic119909 +
(119861 minus 3119860)11990932
3+(119861 minus 3119860)
211990952
10
+(119861 minus 3119860)
311990972
42+(119861 minus 3119860)
411990992
216)
+3
2119896211989612
1radic2120587
119860
sdot tanh(392
radic119860119909
2120587minus111
2tanminus1( 35
111
radic119860119909
2120587)))
(45)
In order to obtain a good approximation from (45) weoptimize the values of the aforementioned parameters asfollows 119860 = 0398521712348839 119861 = 6 and 120575 =
minus0623239813 using the procedure reported in [14 17 27](note that with these values the assumption 119860 ≪ 119861 issatisfied besides the above results allow to know 120572 and 120573
whose values are not needed in the search for the solutionof the system (19) and (20))
Figure 1 and Table 1 show the comparison betweenapproximate solution (45) and the exact solution The accu-racy of (45) is clear as an approximate solution for (13)
4 Discussion
Table 1 shows the comparison between numerical solutiongiven in [39] and approximations [15 38] and (45) It isclear that our approximation (45) is competitive consideringthat it is analytic and handy unlike other approximationswhich are expressed in terms of power series In [37 3841] T-F equation was solved by homotopy analysis method(HAM) The approximation [38] corresponds to an 80thorder analytical approximation and therefore is impracticalOn the other hand in [37] the results are given in terms of
1
08
06
04
02
0
y(x)
0 05 1 15 2 25 3 35 4 45 5
x
[39](45)
Figure 1 Comparison of Thomas-Fermirsquos numerical solution [39]and approximation (45)
infinite series and show particularly the calculation of theinitial slope 1199101015840(0) Moreover in [41] an accurate approximatesolution was obtained but it corresponds to the 40th orderapproximation In [23] (13) was solved by HPM and Padeapproximants in this case a solution of 26 power series termsis provided which is also not handy in practice Reference[15] presents an analytical approach but it is not accurateIn this study we used homotopy perturbation method withnonlinearities distributions to find an accurate solution for(13) (see Figure 1 and Table 1) Although (43) is just almostanalytic (due to the presence of a non-Gaussian integral) itcan be estimated that contribution of (42) to the solution(43) is well represented by (44) assuming the values are 119860 =
0398521712348839 119861 = 6 and 120575 = minus0623239813 thus weobtained the analytical approximation (45)We just kept onlyfive terms in (44) in order to obtain a handy approximationBesides the right-hand side of (44) adequately represents theindicated non-Gaussian integral in the range 119909 isin [0 1] Itimplies that the best contribution of the term proportional to119890minus119861119909
intradic119909
0119890(119861minus31198602)119906
2
119889119906 in (43) is just beginning of the intervalconsidered that is in a part of the domain where the T-F equation describes correctly the atom (see Section 1) Forvalues 119909 gt 1 the above-mentioned term is dominated forthe exponential 119890minus119861119909 In any case the value of the integralsof (40) can be evaluated by using numerical methods Aninteresting fact is that if more terms had been consideredin the development of (12) we would have obtained asa solution of (13) a series containing terms of hyperbolictangent functions to the form (41)
The strategy to redistribute the nonlinearities of T-Fequation between the successive iterations of the HPMmethod was important to obtain an approximate solution of(13) because it allows us to simplify the solution procedurefor the successive iterations of the HPM method Besideswe obtained a soluble coupled system of linear differentialequations (19) and (20) In this study we introduced anadjusting parameter in order to enlarge the domain ofconvergence of our approach with good results by using onlythe first order approximation (see Table 1 and Figure 1)
Journal of Applied Mathematics 7
Table 1 Numerical comparison of Thomas-Fermirsquos numerical solution [39] and approximations [15 38] and (45)
119909 Numerical [39] [38] (45) [15]000 mdash 1000000000 1000000000 1000000000025 0755880759 0776191000 0708502932 0680650028050 0606700008 0615917000 0570491745 0459455663075 0502964042 0505380000 0485018011 0307042537100 0424333179 0423772000 0421167227 0202655526125 0363227937 0362935000 0369542163 0131667603150 0314660642 0314490000 0326298526 0083800106175 0275233848 0275154000 0289315779 0051853222200 0242678587 0242718000 0257253699 0030801947225 0215439334 0215630000 0229210094 0017153465250 0192406328 0192795000 0204540381 0008491277275 0172758691 0173364000 0182755264 0003152503300 0155871862 0156719000 0163464474 0325 0141260504 0142371000 0146346052 minus0001738122
350 0128541381 0129937000 0131128799 minus0002581206
375 0117408054 0119108000 0117581337 minus0002874924
400 0107612958 0109632000 0105504623 minus000284628
425 0098954329 0101303000 0094726427 minus0002641801
450 0091266456 0093950400 0085097045 minus0002353931
475 0084412289 0087432000 0076485875 minus0002039168
500 0078277758 0081629600 0068778618 minus0001730443
A relevant fact is that NDHPM requires little iterationsto obtain accurate solutions if we have a first approximationcontaining as much information as possible for a nonlineardifferential equation For instance in our case (21) containsthe correct asymptotic character of the exact equation
5 Conclusion
In this work NDHPM was used to obtain an approximateanalytical solution for T-F equation Our solution is novelbecause it is expressed in terms of exponential and fractionalpower functions and above all in terms of normal distributionintegrals It is important to obtain an analytical expressionthat provides a good description of the solution for thenonlinear differential equations like (13) For instance thecharge distribution and the electrostatic potential inside anatom are adequately described by (43) and (45) It is worthmentioning that if the initial guess is suitably chosen it ispossible to obtain by this method an accurate approximationlike (43) even using the first terms of (12) Finally in contrastto Runge-Kutta numerical solution NDHPMmethod allowsto analyze quantitatively and qualitatively the solution
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the financial supportfrom the National Council for Science and Technology ofMexico (CONACyT) through Grant CB-2010-01 157024The authors would like to thank Rogelio-Alejandro Callejas-Molina and Roberto Ruiz-Gomez for their contribution tothis project
References
[1] L Landau and E M LifshitzMecanica Cuantica No RelativistaEditorial Reverte S A 2nd edition 1967
[2] LM B Assas ldquoApproximate solutions for the generalizedKdVndashBurgersrsquo equation by Hersquos variational iteration methodrdquo PhysicaScripta vol 76 no 2 pp 161ndash164 2007
[3] M Kazemnia S A Zahedi M Vaezi and N Tolou ldquoArseniccontamination in New Orleans soil temporal changes associ-ated with floodingrdquo Journal of Applied Sciences vol 8 no 22pp 4192ndash4197 2008
[4] R Noorzad A T Poor and M Omidvar ldquoVariational iterationmethod and homotopy-perturbation method for solving Burg-ers equation in fluid dynamicsrdquo Journal of Applied Sciences vol8 no 2 pp 369ndash373 2008
[5] D J Evans and K R Raslan ldquoThe tanh function method forsolving some important non-linear partial differential equa-tionsrdquo International Journal of Computer Mathematics vol 82no 7 pp 897ndash905 2005
[6] J Mahmoudi N Tolou I Khatami A Barari and D D GanjildquoExplicit solution of nonlinear ZK-BBM wave equation using
8 Journal of Applied Mathematics
Exp-function methodrdquo Journal of Applied Sciences vol 8 no 2pp 358ndash363 2008
[7] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988
[8] E Babolian and J Biazar ldquoOn the order of convergence ofAdomian methodrdquo Applied Mathematics and Computation vol130 no 2-3 pp 383ndash387 2002
[9] A Kooch and M Abadyan ldquoEfficiency of modified Ado-mian decomposition for simulating the instability of nano-electromechanical switches comparison with the conventionaldecomposition methodrdquo Trends in Applied Sciences Researchvol 7 no 1 pp 57ndash67 2012
[10] Y Khan H Vazquez-Leal and N Faraz ldquoAn auxiliary param-eter method using Adomian polynomials and Laplace transfor-mation for nonlinear differential equationsrdquoAppliedMathemat-ical Modelling vol 37 no 5 pp 2702ndash2708 2013
[11] S K Vanani S Heidari and M Avaji ldquoA low-cost numericalalgorithm for the solution of nonlinear delay boundary integralequationsrdquo Journal of Applied Sciences vol 11 no 20 pp 3504ndash3509 2011
[12] S H Chowdhury ldquoA comparison between the modifiedhomotopy perturbation method and adomian decompositionmethod for solving nonlinear heat transfer equationsrdquo Journalof Applied Sciences vol 11 no 7 pp 1416ndash1420 2011
[13] L-N Zhang and L Xu ldquoDetermination of the limit cycle byHersquos parameter expansion for oscillators in a u31 + u2 potentialrdquoZeitschrift fur NaturforschungmdashSection A Journal of PhysicalSciences vol 62 no 7-8 pp 396ndash398 2007
[14] H Vazquez-Leal R Castaneda-Sheissa U Filobello-Nino ASarmiento-Reyes and J Sanchez Orea ldquoHigh accurate simpleapproximation of normal distribution integralrdquo MathematicalProblems in Engineering vol 2012 Article ID 124029 22 pages2012
[15] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006
[16] A Fereidoon Y Rostamiyan M Akbarzade and D D GanjildquoApplication of Hersquos homotopy perturbation method to nonlin-ear shock damper dynamicsrdquo Archive of Applied Mechanics vol80 no 6 pp 641ndash649 2010
[17] H Vazquez-Leal U Filobello-Nino R Castaneda-SheissaL Hernandez-Martınez and A Sarmiento-Reyes ldquoModifiedHPMs inspired by homotopy continuation methodsrdquo Mathe-matical Problems in Engineering vol 2012 Article ID 30912319 pages 2012
[18] Y Khan H Vazquez-Leal and Q Wu ldquoAn efficient iteratedmethod for mathematical biology modelrdquo Neural Computingand Applications vol 23 no 3-4 pp 677ndash682 2013
[19] Y Khan H Vazquez-Leal and N Faraz ldquoAn efficient newiterative method for oscillator differential equationrdquo ScientiaIranica vol 19 no 6 pp 1473ndash1477 2012
[20] J Biazar and H Aminikhah ldquoStudy of convergence of homo-topy perturbation method for systems of partial differentialequationsrdquoComputersampMathematics with Applications vol 58no 11-12 pp 2221ndash2230 2009
[21] J Biazar and H Ghazvini ldquoConvergence of the homotopy per-turbation method for partial differential equationsrdquo NonlinearAnalysis Real World Applications vol 10 no 5 pp 2633ndash26402009
[22] Y Khan and Q Wu ldquoHomotopy perturbation transformmethod for nonlinear equations using Hersquos polynomialsrdquo Com-puters and Mathematics with Applications vol 61 no 8 pp1963ndash1967 2011
[23] M A Noor and S T Mohyud-Din ldquoHomotopy perturbationmethod for solvingThomas-Fermi equation using pade approx-imantsrdquo International Journal of Nonlinear Science vol 8 no 1pp 27ndash31 2009
[24] DDGanji A R Sahouli andM Famouri ldquoAnewmodificationofHersquos homotopy perturbationmethod for rapid convergence ofnonlinear undamped oscillatorsrdquo Journal of Applied Mathemat-ics and Computing vol 30 no 1-2 pp 181ndash192 2009
[25] M Turkyilmazoglu ldquoSome issues on HPM and HAMmethodsa convergence schemerdquoMathematical and Computer Modellingvol 53 no 9-10 pp 1929ndash1936 2011
[26] M Turkyilmazoglu ldquoConvergence of the homotopy perturba-tion methodrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 12 no 1ndash8 pp 9ndash14 2011
[27] U Filobello-Nino H Vazquez-Leal R Castaneda-Sheissa et alldquoAn approximate solution of Blasius equation by using HPMmethodrdquo Asian Journal of Mathematics and Statistics vol 5 no2 pp 50ndash59 2012
[28] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineering Springer Berlin Germany 1st edition 2011
[29] T Patel M N Mehta and V H Pradhan ldquoThe numericalsolution of Burgerrsquos equation arising into the irradiation oftumour tissue in biological diffusing system by homotopyanalysis methodrdquo Asian Journal of Applied Sciences vol 5 no1 pp 60ndash66 2012
[30] S Liao Advances in the Homotopy Analysis Method WorldScientific Hackensack NJ USA 2014
[31] M Turkyilmazoglu ldquoAnalytic approximate solutions of rotatingdisk boundary layer flow subject to a uniform suction orinjectionrdquo International Journal of Mechanical Sciences vol 52no 12 pp 1735ndash1744 2010
[32] M Turkyilmazoglu ldquoPurely analytic solutions of the compress-ible boundary layer flow due to a porous rotating disk with heattransferrdquo Physics of Fluids vol 21 no 10 2009
[33] U Filobello-Nino H Vazquez-Leal B Benhammouda et alldquoA handy approximation for a mediated bioelectrocatalysisprocess related to Michaelis-Menten equationrdquo SpringerPlusvol 3 article 162 2014
[34] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoUsing per-turbation methods and Laplace-Pade approximation to solvenonlinear problemsrdquo Miskolc Mathematical Notes vol 14 no1 pp 89ndash101 2013
[35] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoA handyexact solution for flow due to a stretching boundary with partialsliprdquo Revista Mexicana de Fısica E vol 59 no 1 pp 51ndash55 2013
[36] FM Fernandez ldquoRational approximation to theThomas-Fermiequationsrdquo Applied Mathematics and Computation vol 217 no13 pp 6433ndash6436 2011
[37] B Yao ldquoA series solution to the Thomas-Fermi equationrdquoApplied Mathematics and Computation vol 203 no 1 pp 396ndash401 2008
[38] H Khan and H Xu ldquoSeries solution to the Thomas-Fermiequationrdquo Physics Letters A vol 365 no 1-2 pp 111ndash115 2007
[39] K Parand andM Shahini ldquoRational Chebyshev pseudospectralapproach for solving Thomas-Fermi equationrdquo Physics LettersA vol 373 no 2 pp 210ndash213 2009
Journal of Applied Mathematics 9
[40] E Hille ldquoOn the Thomas-Fermi equationrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 62 pp 7ndash10 1969
[41] M Turkyilmazoglu ldquoSolution of the Thomas-Fermi equationwith a convergent approachrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 11 pp 4097ndash41032012
[42] D G ZillA First Course in Differential Equations withModelingApplications Brooks Cole 10th edition 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 7
Table 1 Numerical comparison of Thomas-Fermirsquos numerical solution [39] and approximations [15 38] and (45)
119909 Numerical [39] [38] (45) [15]000 mdash 1000000000 1000000000 1000000000025 0755880759 0776191000 0708502932 0680650028050 0606700008 0615917000 0570491745 0459455663075 0502964042 0505380000 0485018011 0307042537100 0424333179 0423772000 0421167227 0202655526125 0363227937 0362935000 0369542163 0131667603150 0314660642 0314490000 0326298526 0083800106175 0275233848 0275154000 0289315779 0051853222200 0242678587 0242718000 0257253699 0030801947225 0215439334 0215630000 0229210094 0017153465250 0192406328 0192795000 0204540381 0008491277275 0172758691 0173364000 0182755264 0003152503300 0155871862 0156719000 0163464474 0325 0141260504 0142371000 0146346052 minus0001738122
350 0128541381 0129937000 0131128799 minus0002581206
375 0117408054 0119108000 0117581337 minus0002874924
400 0107612958 0109632000 0105504623 minus000284628
425 0098954329 0101303000 0094726427 minus0002641801
450 0091266456 0093950400 0085097045 minus0002353931
475 0084412289 0087432000 0076485875 minus0002039168
500 0078277758 0081629600 0068778618 minus0001730443
A relevant fact is that NDHPM requires little iterationsto obtain accurate solutions if we have a first approximationcontaining as much information as possible for a nonlineardifferential equation For instance in our case (21) containsthe correct asymptotic character of the exact equation
5 Conclusion
In this work NDHPM was used to obtain an approximateanalytical solution for T-F equation Our solution is novelbecause it is expressed in terms of exponential and fractionalpower functions and above all in terms of normal distributionintegrals It is important to obtain an analytical expressionthat provides a good description of the solution for thenonlinear differential equations like (13) For instance thecharge distribution and the electrostatic potential inside anatom are adequately described by (43) and (45) It is worthmentioning that if the initial guess is suitably chosen it ispossible to obtain by this method an accurate approximationlike (43) even using the first terms of (12) Finally in contrastto Runge-Kutta numerical solution NDHPMmethod allowsto analyze quantitatively and qualitatively the solution
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the financial supportfrom the National Council for Science and Technology ofMexico (CONACyT) through Grant CB-2010-01 157024The authors would like to thank Rogelio-Alejandro Callejas-Molina and Roberto Ruiz-Gomez for their contribution tothis project
References
[1] L Landau and E M LifshitzMecanica Cuantica No RelativistaEditorial Reverte S A 2nd edition 1967
[2] LM B Assas ldquoApproximate solutions for the generalizedKdVndashBurgersrsquo equation by Hersquos variational iteration methodrdquo PhysicaScripta vol 76 no 2 pp 161ndash164 2007
[3] M Kazemnia S A Zahedi M Vaezi and N Tolou ldquoArseniccontamination in New Orleans soil temporal changes associ-ated with floodingrdquo Journal of Applied Sciences vol 8 no 22pp 4192ndash4197 2008
[4] R Noorzad A T Poor and M Omidvar ldquoVariational iterationmethod and homotopy-perturbation method for solving Burg-ers equation in fluid dynamicsrdquo Journal of Applied Sciences vol8 no 2 pp 369ndash373 2008
[5] D J Evans and K R Raslan ldquoThe tanh function method forsolving some important non-linear partial differential equa-tionsrdquo International Journal of Computer Mathematics vol 82no 7 pp 897ndash905 2005
[6] J Mahmoudi N Tolou I Khatami A Barari and D D GanjildquoExplicit solution of nonlinear ZK-BBM wave equation using
8 Journal of Applied Mathematics
Exp-function methodrdquo Journal of Applied Sciences vol 8 no 2pp 358ndash363 2008
[7] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988
[8] E Babolian and J Biazar ldquoOn the order of convergence ofAdomian methodrdquo Applied Mathematics and Computation vol130 no 2-3 pp 383ndash387 2002
[9] A Kooch and M Abadyan ldquoEfficiency of modified Ado-mian decomposition for simulating the instability of nano-electromechanical switches comparison with the conventionaldecomposition methodrdquo Trends in Applied Sciences Researchvol 7 no 1 pp 57ndash67 2012
[10] Y Khan H Vazquez-Leal and N Faraz ldquoAn auxiliary param-eter method using Adomian polynomials and Laplace transfor-mation for nonlinear differential equationsrdquoAppliedMathemat-ical Modelling vol 37 no 5 pp 2702ndash2708 2013
[11] S K Vanani S Heidari and M Avaji ldquoA low-cost numericalalgorithm for the solution of nonlinear delay boundary integralequationsrdquo Journal of Applied Sciences vol 11 no 20 pp 3504ndash3509 2011
[12] S H Chowdhury ldquoA comparison between the modifiedhomotopy perturbation method and adomian decompositionmethod for solving nonlinear heat transfer equationsrdquo Journalof Applied Sciences vol 11 no 7 pp 1416ndash1420 2011
[13] L-N Zhang and L Xu ldquoDetermination of the limit cycle byHersquos parameter expansion for oscillators in a u31 + u2 potentialrdquoZeitschrift fur NaturforschungmdashSection A Journal of PhysicalSciences vol 62 no 7-8 pp 396ndash398 2007
[14] H Vazquez-Leal R Castaneda-Sheissa U Filobello-Nino ASarmiento-Reyes and J Sanchez Orea ldquoHigh accurate simpleapproximation of normal distribution integralrdquo MathematicalProblems in Engineering vol 2012 Article ID 124029 22 pages2012
[15] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006
[16] A Fereidoon Y Rostamiyan M Akbarzade and D D GanjildquoApplication of Hersquos homotopy perturbation method to nonlin-ear shock damper dynamicsrdquo Archive of Applied Mechanics vol80 no 6 pp 641ndash649 2010
[17] H Vazquez-Leal U Filobello-Nino R Castaneda-SheissaL Hernandez-Martınez and A Sarmiento-Reyes ldquoModifiedHPMs inspired by homotopy continuation methodsrdquo Mathe-matical Problems in Engineering vol 2012 Article ID 30912319 pages 2012
[18] Y Khan H Vazquez-Leal and Q Wu ldquoAn efficient iteratedmethod for mathematical biology modelrdquo Neural Computingand Applications vol 23 no 3-4 pp 677ndash682 2013
[19] Y Khan H Vazquez-Leal and N Faraz ldquoAn efficient newiterative method for oscillator differential equationrdquo ScientiaIranica vol 19 no 6 pp 1473ndash1477 2012
[20] J Biazar and H Aminikhah ldquoStudy of convergence of homo-topy perturbation method for systems of partial differentialequationsrdquoComputersampMathematics with Applications vol 58no 11-12 pp 2221ndash2230 2009
[21] J Biazar and H Ghazvini ldquoConvergence of the homotopy per-turbation method for partial differential equationsrdquo NonlinearAnalysis Real World Applications vol 10 no 5 pp 2633ndash26402009
[22] Y Khan and Q Wu ldquoHomotopy perturbation transformmethod for nonlinear equations using Hersquos polynomialsrdquo Com-puters and Mathematics with Applications vol 61 no 8 pp1963ndash1967 2011
[23] M A Noor and S T Mohyud-Din ldquoHomotopy perturbationmethod for solvingThomas-Fermi equation using pade approx-imantsrdquo International Journal of Nonlinear Science vol 8 no 1pp 27ndash31 2009
[24] DDGanji A R Sahouli andM Famouri ldquoAnewmodificationofHersquos homotopy perturbationmethod for rapid convergence ofnonlinear undamped oscillatorsrdquo Journal of Applied Mathemat-ics and Computing vol 30 no 1-2 pp 181ndash192 2009
[25] M Turkyilmazoglu ldquoSome issues on HPM and HAMmethodsa convergence schemerdquoMathematical and Computer Modellingvol 53 no 9-10 pp 1929ndash1936 2011
[26] M Turkyilmazoglu ldquoConvergence of the homotopy perturba-tion methodrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 12 no 1ndash8 pp 9ndash14 2011
[27] U Filobello-Nino H Vazquez-Leal R Castaneda-Sheissa et alldquoAn approximate solution of Blasius equation by using HPMmethodrdquo Asian Journal of Mathematics and Statistics vol 5 no2 pp 50ndash59 2012
[28] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineering Springer Berlin Germany 1st edition 2011
[29] T Patel M N Mehta and V H Pradhan ldquoThe numericalsolution of Burgerrsquos equation arising into the irradiation oftumour tissue in biological diffusing system by homotopyanalysis methodrdquo Asian Journal of Applied Sciences vol 5 no1 pp 60ndash66 2012
[30] S Liao Advances in the Homotopy Analysis Method WorldScientific Hackensack NJ USA 2014
[31] M Turkyilmazoglu ldquoAnalytic approximate solutions of rotatingdisk boundary layer flow subject to a uniform suction orinjectionrdquo International Journal of Mechanical Sciences vol 52no 12 pp 1735ndash1744 2010
[32] M Turkyilmazoglu ldquoPurely analytic solutions of the compress-ible boundary layer flow due to a porous rotating disk with heattransferrdquo Physics of Fluids vol 21 no 10 2009
[33] U Filobello-Nino H Vazquez-Leal B Benhammouda et alldquoA handy approximation for a mediated bioelectrocatalysisprocess related to Michaelis-Menten equationrdquo SpringerPlusvol 3 article 162 2014
[34] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoUsing per-turbation methods and Laplace-Pade approximation to solvenonlinear problemsrdquo Miskolc Mathematical Notes vol 14 no1 pp 89ndash101 2013
[35] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoA handyexact solution for flow due to a stretching boundary with partialsliprdquo Revista Mexicana de Fısica E vol 59 no 1 pp 51ndash55 2013
[36] FM Fernandez ldquoRational approximation to theThomas-Fermiequationsrdquo Applied Mathematics and Computation vol 217 no13 pp 6433ndash6436 2011
[37] B Yao ldquoA series solution to the Thomas-Fermi equationrdquoApplied Mathematics and Computation vol 203 no 1 pp 396ndash401 2008
[38] H Khan and H Xu ldquoSeries solution to the Thomas-Fermiequationrdquo Physics Letters A vol 365 no 1-2 pp 111ndash115 2007
[39] K Parand andM Shahini ldquoRational Chebyshev pseudospectralapproach for solving Thomas-Fermi equationrdquo Physics LettersA vol 373 no 2 pp 210ndash213 2009
Journal of Applied Mathematics 9
[40] E Hille ldquoOn the Thomas-Fermi equationrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 62 pp 7ndash10 1969
[41] M Turkyilmazoglu ldquoSolution of the Thomas-Fermi equationwith a convergent approachrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 11 pp 4097ndash41032012
[42] D G ZillA First Course in Differential Equations withModelingApplications Brooks Cole 10th edition 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Journal of Applied Mathematics
Exp-function methodrdquo Journal of Applied Sciences vol 8 no 2pp 358ndash363 2008
[7] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988
[8] E Babolian and J Biazar ldquoOn the order of convergence ofAdomian methodrdquo Applied Mathematics and Computation vol130 no 2-3 pp 383ndash387 2002
[9] A Kooch and M Abadyan ldquoEfficiency of modified Ado-mian decomposition for simulating the instability of nano-electromechanical switches comparison with the conventionaldecomposition methodrdquo Trends in Applied Sciences Researchvol 7 no 1 pp 57ndash67 2012
[10] Y Khan H Vazquez-Leal and N Faraz ldquoAn auxiliary param-eter method using Adomian polynomials and Laplace transfor-mation for nonlinear differential equationsrdquoAppliedMathemat-ical Modelling vol 37 no 5 pp 2702ndash2708 2013
[11] S K Vanani S Heidari and M Avaji ldquoA low-cost numericalalgorithm for the solution of nonlinear delay boundary integralequationsrdquo Journal of Applied Sciences vol 11 no 20 pp 3504ndash3509 2011
[12] S H Chowdhury ldquoA comparison between the modifiedhomotopy perturbation method and adomian decompositionmethod for solving nonlinear heat transfer equationsrdquo Journalof Applied Sciences vol 11 no 7 pp 1416ndash1420 2011
[13] L-N Zhang and L Xu ldquoDetermination of the limit cycle byHersquos parameter expansion for oscillators in a u31 + u2 potentialrdquoZeitschrift fur NaturforschungmdashSection A Journal of PhysicalSciences vol 62 no 7-8 pp 396ndash398 2007
[14] H Vazquez-Leal R Castaneda-Sheissa U Filobello-Nino ASarmiento-Reyes and J Sanchez Orea ldquoHigh accurate simpleapproximation of normal distribution integralrdquo MathematicalProblems in Engineering vol 2012 Article ID 124029 22 pages2012
[15] J-H He ldquoSome asymptotic methods for strongly nonlinearequationsrdquo International Journal of Modern Physics B vol 20no 10 pp 1141ndash1199 2006
[16] A Fereidoon Y Rostamiyan M Akbarzade and D D GanjildquoApplication of Hersquos homotopy perturbation method to nonlin-ear shock damper dynamicsrdquo Archive of Applied Mechanics vol80 no 6 pp 641ndash649 2010
[17] H Vazquez-Leal U Filobello-Nino R Castaneda-SheissaL Hernandez-Martınez and A Sarmiento-Reyes ldquoModifiedHPMs inspired by homotopy continuation methodsrdquo Mathe-matical Problems in Engineering vol 2012 Article ID 30912319 pages 2012
[18] Y Khan H Vazquez-Leal and Q Wu ldquoAn efficient iteratedmethod for mathematical biology modelrdquo Neural Computingand Applications vol 23 no 3-4 pp 677ndash682 2013
[19] Y Khan H Vazquez-Leal and N Faraz ldquoAn efficient newiterative method for oscillator differential equationrdquo ScientiaIranica vol 19 no 6 pp 1473ndash1477 2012
[20] J Biazar and H Aminikhah ldquoStudy of convergence of homo-topy perturbation method for systems of partial differentialequationsrdquoComputersampMathematics with Applications vol 58no 11-12 pp 2221ndash2230 2009
[21] J Biazar and H Ghazvini ldquoConvergence of the homotopy per-turbation method for partial differential equationsrdquo NonlinearAnalysis Real World Applications vol 10 no 5 pp 2633ndash26402009
[22] Y Khan and Q Wu ldquoHomotopy perturbation transformmethod for nonlinear equations using Hersquos polynomialsrdquo Com-puters and Mathematics with Applications vol 61 no 8 pp1963ndash1967 2011
[23] M A Noor and S T Mohyud-Din ldquoHomotopy perturbationmethod for solvingThomas-Fermi equation using pade approx-imantsrdquo International Journal of Nonlinear Science vol 8 no 1pp 27ndash31 2009
[24] DDGanji A R Sahouli andM Famouri ldquoAnewmodificationofHersquos homotopy perturbationmethod for rapid convergence ofnonlinear undamped oscillatorsrdquo Journal of Applied Mathemat-ics and Computing vol 30 no 1-2 pp 181ndash192 2009
[25] M Turkyilmazoglu ldquoSome issues on HPM and HAMmethodsa convergence schemerdquoMathematical and Computer Modellingvol 53 no 9-10 pp 1929ndash1936 2011
[26] M Turkyilmazoglu ldquoConvergence of the homotopy perturba-tion methodrdquo International Journal of Nonlinear Sciences andNumerical Simulation vol 12 no 1ndash8 pp 9ndash14 2011
[27] U Filobello-Nino H Vazquez-Leal R Castaneda-Sheissa et alldquoAn approximate solution of Blasius equation by using HPMmethodrdquo Asian Journal of Mathematics and Statistics vol 5 no2 pp 50ndash59 2012
[28] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineering Springer Berlin Germany 1st edition 2011
[29] T Patel M N Mehta and V H Pradhan ldquoThe numericalsolution of Burgerrsquos equation arising into the irradiation oftumour tissue in biological diffusing system by homotopyanalysis methodrdquo Asian Journal of Applied Sciences vol 5 no1 pp 60ndash66 2012
[30] S Liao Advances in the Homotopy Analysis Method WorldScientific Hackensack NJ USA 2014
[31] M Turkyilmazoglu ldquoAnalytic approximate solutions of rotatingdisk boundary layer flow subject to a uniform suction orinjectionrdquo International Journal of Mechanical Sciences vol 52no 12 pp 1735ndash1744 2010
[32] M Turkyilmazoglu ldquoPurely analytic solutions of the compress-ible boundary layer flow due to a porous rotating disk with heattransferrdquo Physics of Fluids vol 21 no 10 2009
[33] U Filobello-Nino H Vazquez-Leal B Benhammouda et alldquoA handy approximation for a mediated bioelectrocatalysisprocess related to Michaelis-Menten equationrdquo SpringerPlusvol 3 article 162 2014
[34] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoUsing per-turbation methods and Laplace-Pade approximation to solvenonlinear problemsrdquo Miskolc Mathematical Notes vol 14 no1 pp 89ndash101 2013
[35] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoA handyexact solution for flow due to a stretching boundary with partialsliprdquo Revista Mexicana de Fısica E vol 59 no 1 pp 51ndash55 2013
[36] FM Fernandez ldquoRational approximation to theThomas-Fermiequationsrdquo Applied Mathematics and Computation vol 217 no13 pp 6433ndash6436 2011
[37] B Yao ldquoA series solution to the Thomas-Fermi equationrdquoApplied Mathematics and Computation vol 203 no 1 pp 396ndash401 2008
[38] H Khan and H Xu ldquoSeries solution to the Thomas-Fermiequationrdquo Physics Letters A vol 365 no 1-2 pp 111ndash115 2007
[39] K Parand andM Shahini ldquoRational Chebyshev pseudospectralapproach for solving Thomas-Fermi equationrdquo Physics LettersA vol 373 no 2 pp 210ndash213 2009
Journal of Applied Mathematics 9
[40] E Hille ldquoOn the Thomas-Fermi equationrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 62 pp 7ndash10 1969
[41] M Turkyilmazoglu ldquoSolution of the Thomas-Fermi equationwith a convergent approachrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 11 pp 4097ndash41032012
[42] D G ZillA First Course in Differential Equations withModelingApplications Brooks Cole 10th edition 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 9
[40] E Hille ldquoOn the Thomas-Fermi equationrdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 62 pp 7ndash10 1969
[41] M Turkyilmazoglu ldquoSolution of the Thomas-Fermi equationwith a convergent approachrdquo Communications in NonlinearScience and Numerical Simulation vol 17 no 11 pp 4097ndash41032012
[42] D G ZillA First Course in Differential Equations withModelingApplications Brooks Cole 10th edition 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
