Research ArticleClosed-Form Solutions of the Thomas-Fermi inHeavy Atoms and the Langmuir-Blodgett in CurrentFlow ODEs in Mathematical Physics

Efstathios E Theotokoglou Theodoros I Zarmpoutis and Ioannis H Stampouloglou

School of Applied Mathematical and Physical Sciences Department of Mechanics Laboratory of Testing and MaterialsNational Technical University of Athens Zographou Campus Theocaris Building 5 Heroes of Polytechniou Avenue157-73 Athens Greece

Correspondence should be addressed to Efstathios E Theotokoglou stathiscentralntuagr

Received 25 March 2015 Revised 2 July 2015 Accepted 7 July 2015

Academic Editor Hang Xu

Copyright copy 2015 Efstathios E Theotokoglou et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Twokinds of second-order nonlinear ordinary differential equations (ODEs) appearing inmathematical physics are analyzed in thispaperThe first one concerns theThomas-Fermi (TF) equation while the second concerns the Langmuir-Blodgett (LB) equation incurrent flow According to a mathematical methodology recently developed the exact analytic solutions of both TF and LB ODEsare proposed Both of these are nonlinear of the second order and by a series of admissible functional transformations are reducedto Abelrsquos equations of the second kind of the normal form The closed form solutions of the TF and LB equations in the phase andphysical plane are given Finally a new interesting result has been obtained related to the derivative of the TF function at the limit

The paper is dedicated to the memory of the authorsrsquo Professor in National Technical University of Athens Dr D EPanayotounakos who contributed a lot to the study of Thomas-Fermi and Langmuir-Blodgett equations

1 Introduction

An equation of considerable interest called the Langmuir-Blodgett (LB) equation [1 2] appeared in connectionwith thetheory of flow of a current from a hot cathode to a positivelycharged anode in a high vacuumThe cathode and anode arelong coaxial cylinders It was proved [2 page 409] that theLB equation has an analytic expansion in the neighborhoodof a fixed value of the independent variable which assumesarbitrarily given values of the dependent variable and itsderivative provided the zero value However the aboveequation does not admit exact analytic solution in terms ofknown tabulated functions [3 4] A lot of applications havebeen proposed for Langmuir equation Horvath et al [5] usedMartin-Synge algorithm or anti-Langmuir in liquid chro-matography Shang and Zheng [6] proposed the system of

Zakharov equations which involves the interaction betweenLangmuir and ion-acoustic waves in plasma Graham andCairns [7] posed the constraints on the formation andstructure of Langmuir eigenmodes in the solar wind and soforth

On the other hand the Thomas-Fermi (TF) equationappears in the problem of determining the effective nuclearcharge in heavy atoms [2 8 9] Numerous applications havealso been proposed for TF equation Tsurumi and Wadati[10] applied the TF approximation to obtain the dynamicsof magnetically trapped boson-fermion Liao [11] proposedan analytic technique for TF equation Abbasbandy andBervillier [12] used analytic continuation of Taylor series toconfront the TF equation Ourabah and Tribeche [13] pre-sented a TFmodel based on thermal nonextensive relativisticeffects and so forth

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 721637 8 pageshttpdxdoiorg1011552015721637

2 Mathematical Problems in Engineering

Both LB and TF equations do not admit exact analyticsolution in terms of known (tabulated) functions In thispaper using series of admissible functional transformationsat first both LB and TF nonlinear ODEs are reduced toequivalent Abelrsquos equations of the second kind of the normalform According to a mathematical methodology recentlydeveloped [14ndash16] we extract closed-form solution of theaboveAbel equations in both phase and physical planes underinitial data in accordance with the physical problems

The mathematical technique introduced in our study isgeneral and can be applied to a very large class of yet unsolv-able ODEs in nonlinear mechanics and generally in mathe-matical physicsThemethod employed is exact in accordancewith the boundary conditions of each problem under con-sideration and gives us the possibility not to consider the apriori approximate construction of the solutions in the formof power series some coefficient of which must be estimatedIn addition with our analysis the exact value of the first orderderivative of Thomas Fermi equation at 119909 = 0 is derivedin a different way from the methods already existing in theliterature [17ndash21]

2 Preliminaries Notations

A nonlinear ODE of considerable interest called the Lang-muir-Blodgett (LB) equation [1] is the following

311991011991010158401015840

119909119909+ 11991010158402

119909+ 4119910119910

1015840

119909+ 1199102minus 1 = 0 (1)

Here the notations 1199101015840119909= 119889119910119889119909 11991010158401015840

119909119909= 11988921199101198891199092 are used

for the total derivativesThis equation of constant coefficients appeared in con-

nection with the theory of flow of a current from a hot cath-ode to a positive charged anode in a high vacuum The cath-ode and anode are long coaxial cylinders and the independentvariable 119910 is defined by the equation 119910 = 119891(119903119903

0) where 119903 is

the radius of the anode enclosing a cathode of radius 1199030 The

independent variable 119909 is given by 119909 = ln(1199031199030)

Equation (1) is converted into a somewhat more tractableform bymeans of the transformation 119910 = 119890minus1199092119911(119909) proposedby Davis [2] and thus becomes

311991111991110158401015840

119909119909+ 11991110158402

119909minus 119890119909= 0 (2)

Numerical techniques concerning the solution of (1) areincluded in [6]

The TF nonlinear ODE

11991010158401015840

119909119909= 119909minus12

11991032 (3)

appears in the problem of determining the effective nuclearcharge in heavy atoms [8 9] The solution is defined for theboundary values

119910 (0) = 1

119910 (infin) = 0

(4)

Interesting approximate and numerical integrationsmethods have been proposed by Sommerfeld [22] Bush andCaldwell [17] Kobayashi et al [20] Feynman et al [18] andCoulson and March [19]

The differential equation (3) belongs to equations ofEmden-Fowler type [3 4] Taking into account the knownexact parametric solutions of this class of nonlinear ODEs [4pages 241ndash250] the above mentioned equation can not solveanalytically in terms of known functions

Many investigations have studied the TF equation obey-ing to the boundary conditions (4) (see [18ndash21]) and theycarried out the outward numerical integration (119909 = 0 119910 rarr

infin) for a sequence of values of initial slope converging step bystep to that curve which should approach the 119909-axis asymp-totically Finally for a semianalytical solution methodologywe must refer to Sommerfeld [22]

The Emden-Fowler nonlinear ODE of the normal form is

11991010158401015840

119909119909= 119860119909119899119910119898 (5)

where 119860 and 119899 119898 are arbitrary parameters We note thefollowing regarding the reduction of (5) (see [4])

For 119898 = 1 and 119898 = minus2119899 minus 3 the admissible functionaltransformations

120585 =

2119899 + 119898 + 3

119898 minus 1

119909(119899+2)(119898minus1)

119910

119906 = 119909(119899+2)(119898minus1)

(1199091199101015840

119909+

119899 + 2

119898 minus 1

119910)

(6)

reduce the nonlinear ODE (5) to the following Abel equationof the second kind of the normal form

1199061199061015840

120585minus 119906 = minus

(119899 + 2) (119899 + 119898 + 1)

(2119899 + 119898 + 3)2

120585

+ 119860(

119898 minus 1

2119899 + 119898 + 3

)

2

120585119898

(7)

In what follows by a series of admissible functionaltransformations we prove that both of (1) and (3) can beexactly reduced to Abelrsquos equations of the second kind of thenormal form 119910119910

1015840

119909minus 119910 = 119891(119909) According to the following

reduction procedure the exact analytic solutions of the aboveAbel equations are constructed avoiding the approximateexpression of the solutions in power series

3 The Reduction Procedure

31 The LB Equation The LB equation (1) is a second ordernonlinear ODE of constant coefficients Thus by the substi-tution

1199101015840

119909= 119901 (119910)

997904rArr 11991010158401015840

119909119909= 1199011015840

119910119901

(8)

it can be reduced to the followingAbel equation of the secondkind

31199101199011199011015840

119910= minus1199012minus 4119910119901 minus 119910

2+ 1 (9)

Furthermore the well-known transformation [4 page 50]

119908 (119910) = 119901119890int(13119910)119889119910

= 119901 exp (ln 10038161003816100381610038161199101003816100381610038161003816

13

) = 11991013119901

997904rArr 119901 = 119910minus13

119908 119910 = 0

(10)

Mathematical Problems in Engineering 3

giving the expressions

1199011015840

119910= 119910minus13

1199081015840

119910minus

1

3

119910minus43

119908

1199011199011015840

119910= 119910minus23

1199081199081015840

119910minus

1

3

119910minus53

1199082

(11)

transforms (9) into the following Abel equation of the secondkind

1199081199081015840

119910= minus

4

3

11991013119908 +

1

3

119910minus13

minus

1

3

11991053 (12)

Finally the substitution

119908 (119910) = 119899 (120585)

120585 =

4

3

int11991013119889119910

997904rArr 119910 = 12058534

(13)

reduces (12) to the followingAbel equation of the second kindof the normal form

1198991198991015840

120585+ 119899 =

1

4

120585minus12

minus

1

4

120585 (14)

or setting

119899 (120585) = minus119911 (120585) (15)

to the form

1199111199111015840

120585minus 119911 =

1

4

120585minus12

minus

1

4

120585 (16)

The solution of (16) and thus the solutions of (14) (12)and (9) constitutes the intermediate integral of the LB equa-tion (1) in the phase plane In other words when obtainingthe solution of the transformed equation (16) in the form119911 = 119911(120585

lowast

119862)lowast

119862 = first integration constant the solution of (14)becomes 119899(120585) = minus119911(120585) the solution of (12) is119908(119910) = minus119911(11991043)and finally the solution of (8) is 119901(119910) = minus119910

minus13119911(11991043) Thus

the solution to the original Langmuir equation (1) in thephysical plane can be obtained by the integration by parts ofthe following equation

119889119910

119901 (119910)

= 119889119909

997904rArr int

119889119910

11991013119911 (11991043)

= minus119909 + 119862

119862 = second integration constant

(17)

Note that the reduced Abel equation (16) does not admitan exact analytic solution in terms of known (tabulated)functions [4 pages 29ndash45]

32TheTFEquation Equation (3) is a typical Emden-Fowlernonlinear equation with 119860 = 1 119899 = minus12 and 119898 = 32A thorough examination concerning approximate expansion

solution of (3) is included in [20] as well as all the referencescited there By now with the aid of the transformations (6)one reduces the original TF equation (3) to the followingAbelequation of the second kind of the normal form

1199061199061015840

120585minus 119906 = minus

12

49

120585 +

1

49

12058532 (18)

where

120585 = 71199093119910 (19)

119906 = 1199093(1199091199101015840

119909+ 3119910) (20)

In what follows based on a mathematical constructionrecently developed in [14 15] concerning the closed-formanalytic solution of an Abel equation of the second kind ofthe normal form 119910119910

1015840

119909minus119910 = 119891(119909) we provide the closed-form

solutions of the reduced Abel equation (18) that is to say theconstruction of the intermediate integral of the original TFequation (3) in the phase plane as well as the final solutionin the physical plane in accordance with the given boundaryconditions

4 Closed-Form Solutions of the TF Equationin the Phase Plane-Final Solutions

We consider the reduced Abel equation (18) namely theequation

1199061199061015840

120585minus 119906 = 119860120585 + 119861120585

32 119860 = minus

12

49

119861 =

1

49

(21)

obeying (as the first of the boundary conditions (4) 119910(0) = 1and the well-known result 1199101015840

119909(0) =

lowast

119861 = 0 [2 8 17ndash20]) thetransformed (19) and (20) equivalent condition

at 120585 = 1205850= 0 997888rarr 119906 (0) = 119906

0= 0 (22)

It was recently proved [14 15] that an Abel equation ofthe second kind of the normal form 119910119910

1015840

119909minus 119910 = 119891(119909) admits

an exact analytic solution in terms of known (tabulated)functions Hence we perform the solution of (21) whichconstitutes the solution of the TF equation in the phase planeas follows

120585 = ln 10038161003816100381610038161003816120585 + 2

lowast

119862

10038161003816100381610038161003816

119906 (120585) =

1

2

120585 [119873 (120585) +

1

3

]

119873

1015840

120585(120585) =

4 [119866 (120585) + 2 (119860120585 + 120585

32

)]

120585 [119873 (120585) + 43]

(23a)

1

4

[(120585 sin 120585 + cos 120585)lowast

119860(120585) + cos2120585] (2120585lowast

119860(120585) + cos 120585)

(120585

lowast

119860(120585))

3

119890120585

= 4 [119866 + 2 (119860120585 + 119861120585

32

)]

(23b)

4 Mathematical Problems in Engineering

wherelowast

119860 =

lowast

119860(120585) = the cosine integral = ci (120585)

= C + ln 120585 +infin

sum

119906=1

(minus1)119906 120585

2119906

(2119906) (2119906)

C = Eulerrsquos number = 0 5572156649015325

(23c)

Herelowast

119862 is integration constant while the function 119873(120585) isgiven implicitly in terms of the subsidiary function 119866(120585)which can be determined from (23b) and the knownmemberof (21) as in the following three cases

Case 1 (119876(120585) lt 0 (119901 lt 0)) Consider

119873

(1)

(120585) = 2radicminus

119901

3

cos 1198863

119873

(2)

(120585) = minus2radicminus

119901

3

cos119886 minus 1205873

119873

(3)

(120585) = minus2radicminus

119901

3

cos 119886 + 1205873

cos 119886 = minus119902

2radicminus (1199013)3

0 lt 119886 lt 120587

(24)

Case 2 (119876(120585) gt 0) Consider

119873(120585) =3

radicminus

119902

2

+ radic119876 +3

radicminus

119902

2

minus radic119876 (25)

Case 3 (119876(120585) = 0) Consider

119873

(1)

(120585) = 23

radicminus

119902

2

119873

(2)

(120585) = 119873

(3)

(120585) = minus3

radicminus

119902

2

(26)

In all these formulae the quantities 119876 119901 and 119902 are givenby

119876(120585) =

1

27

1199013(120585) +

1

4

1199022(120585)

119901 (120585) = minus

1198862

3

+ 119887

119902 (120585) =

2

27

1198863minus

1

3

119886119887 + 119888

119886 = minus4

119887 =

3 + 4 [119866 (120585) + (119860120585 + 119861120585

32

)]

120585

119888 = minus

4 [119866 (120585) + 2 (119860120585 + 119861120585

32

)]

120585

(27)

In order to define the type of the function 119873

(119894)

(120585)

(119894 = 1 2 3) which must be selected among the closed-formsolutions ((23a) (23b) and (23c)) to (26) as well as the valueof the subsidiary function 119866(120585) at 120585 = 0 and the value ofthe integration constant

lowast

119862 we must combine the above Abelsolutions with the boundary data

We symbolize by 120585 = ln |2lowast

119862| the substitution while byL0

any function L(120585) at 1205850= ln |2

lowast

119862| Thus functions 119906(120585) and(119873

1015840

120585)(120585) given in (23a) perform

119906 (120585) =

1

2

120585 [119873

(119894)

(120585) +

1

3

]

119873

1015840

120585

(119894)

(120585) =

4 [119866 (120585) + 2 (minus (1249) (120585) + (149) 120585

32

)]

120585 [119873

(119894)

(120585) + 43]

(28)

1199060=

1

2

ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

(119873

(119894)

0+

1

3

)

119873

1015840(119894)

1205850

(120585)

=

4 [1198660+ 2 (minus (1249) ln 1003816100381610038161003816

10038162

lowast

119862

10038161003816100381610038161003816+ (149) (ln 1003816100381610038161003816

10038162

lowast

119862

10038161003816100381610038161003816)

32

)]

ln 100381610038161003816100381610038162

lowast

119862

10038161003816100381610038161003816[119873

(119894)

0+ 43]

(29)

On the other hand the Abel nonlinear ODE (21) at 1205850

becomes

1 +

minus (1249) ln 100381610038161003816100381610038162

lowast

119862

10038161003816100381610038161003816+ (149) (ln 1003816100381610038161003816

10038162

lowast

119862

10038161003816100381610038161003816)

32

1199060

= 1 +

minus (1249) ln 100381610038161003816100381610038162

lowast

119862

10038161003816100381610038161003816+ (149) (ln 1003816100381610038161003816

10038162

lowast

119862

10038161003816100381610038161003816)

32

ln 100381610038161003816100381610038162

lowast

119862

10038161003816100381610038161003816(119873

(119894)

0+ 43)

=

1199061015840

1205850

120585

1015840

1205850

(30)

Combination of the above equations results in the followingquadratic to119873

0equation

119873

(119894)

0

2

+ 119873

(119894)

0+

4

9

+[

[

[

1 minus

1

lowast

119862 ln 100381610038161003816100381610038162

lowast

119862

10038161003816100381610038161003816

]

]

]

sdot [minus21198660+ 2(minus

12

49

ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

+

1

49

(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)

32

)]

= 0

(31)

including the value of 1198660and the constant on integration

lowast

119862

Mathematical Problems in Engineering 5

Similarly by (23b) one gets

32 1198660+ [2(minus

12

49

ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

+

1

49

ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

32

)]

lowast

119860

3

0

= [ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

sin(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

) + cos(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)]

lowast

1198600

+ cos2 (ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)(2 ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

lowast

1198600

+ cos(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

))

(32)

that constitute a second equation including the value of 1198660

and the constant of integrationlowast

119862 Also in this equation bymeans of (23c) we have

lowast

1198600=

lowast

119860(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

) = ci(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)

= C + ln(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

) +

infin

sum

119896=1

(minus1)119896ln 100381610038161003816100381610038162

lowast

119862

10038161003816100381610038161003816

2119896

(2119896) (2119896)

C = Eulerrsquos number asymp 05572

(33)

Summarizing from (19) and (20) one concludes that since bythe first boundary conditions 119910(0) = 1 for 119909 rarr 0 rArr 119910 rarr 1

then for 119909 rarr 0 rArr 1199101015840

119909rarr 0 or 1199101015840

119909rarr

lowast

119861 wherelowast

119861 is a fixedvalue Thus combining (23a) (23b) and (23c) together withthe first of (28) and the Abel nonlinear ODE (21) one maywrite that for 119909 rarr 0 rArr 120585

0rarr 0 119906

0rarr 0 119873

0rarr 0

Therefore by making use of the first boundary conditiongiven in (4) (119909 rarr 0 119910 rarr 1) as well as of all the aboveobservations the solutions of (31) and (32) perform the valuesfor 119866

0and 2

lowast

119862 and they can be written in the followingalgebraic forms

minus1 plusmn radic1 minus 4F (1198660

lowast

119862)

2

= 0(34a)

H(

lowast

119862) = 1198660 (34b)

where

F(1198660

lowast

119862) =

4

9

+[

[

[

1 minus

1

lowast

119862 ln 100381610038161003816100381610038162

lowast

119862

10038161003816100381610038161003816

]

]

]

[minus21198660

+ 2(minus

12

4

ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

+

1

49

(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)

32

)]

(35a)

H(

lowast

119862)

=

1

32

[ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

sin(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

) + cos(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)]

sdot

lowast

11986000+ cos2 (ln

1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

) [2 ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

lowast

1198600

+ cos(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)]

lowast

1198600

lowast

119862 as in (33)

(35b)

Thus we are now able to estimate 1199010 1199020 and 119876

0given by

(27) in terms of the known valueslowast

119862 and 1198660 This estimation

furnishes the value of1198760and therefore the type of the solution

119873

(119894)

(120585) (119894 = 1 2 3) near the origin 1205850among the formulae (24)

to (26)From the above evaluation it is concluded that the closed

form solutions of the Abel nonlinear ODE (21) given as

119906(119894)(120585) =

1

2

120585 [119873

(119894)

(120585) +

1

3

]

120585 = ln1003816100381610038161003816100381610038161003816

120585 + 2

lowast

1198621

1003816100381610038161003816100381610038161003816

(119894 = 1 2 3)

(36)

have been completely determinedSince we obtained the above closed- form solutions of the

intermediate Abel nonlinear ODE (21) the solutions to theoriginal TF equation (3) can be determined as follows By (19)and (21) we perform

119909 =

1

713

(

120585

119910

)

13

1199101015840

119909=

1

1199094119906 minus 3

119910

119909

(37)

while taking the total differential in (19) we extract

119889120585 = 211199092119910119889119909 + 7119909

3119889119910

lArrrArr 1205851015840

119909=

119889120585

119889119909

= 71199092(1199091199101015840

119909+ 3119910) = 7

119906

119909

(38)

where 119906 = 119906(120585 + 2

lowast

1198621) denotes the known solution of the

Abel nonlinear ODE (18) Using (34a) (34b) and (37) onewrites

1199101015840

120585=

1199101015840

119909

1205851015840

119909

= 743(

119910

120585

) 11990643

minus 3119910713(

119910

120585

)

13

=

119906 (120585)

120585

119910 minus

3

71205854311991043

(39)

6 Mathematical Problems in Engineering

that is the Bernoulli equation

11991013

=

exp ((1120585) int (119906 (120585) 120585) 119889120585)

119862 + (17) int exp [((1120585) int (119906 (120585) 120585) 119889120585)] 119889120585

119909 =

1

713

(

120585

119910

)

13

119909 ge 0 119910 ge 0 120585 = parameter (120585 ge 0)

119862 = second constant of integration

(40)

The last step of the above analysis concerns the calculationof the second constant of integration 119862 Thus from the firstof the boundary conditions (4) and the parametric solution(40) one writes

1199091997888rarr 0

1199101997888rarr 1

1205851997888rarr 0

lim1205851rarr0

(11991013)

=

lim1205851rarr0

exp ((1120585) int (119906 (120585) 120585) 119889120585)

lim1205851rarr0

119862 + (17) int exp [((1120585) int (119906 (120585) 120585) 119889120585)] 119889120585

(41)

and estimates the constant of integration 119862 By the second ofthe boundary conditions (4) one writes 119909

2rarr +infin 119910

2rarr 0

and 120585 rarr 1205852 Again through the parametric solution (40) one

writes

lim120585rarr1205852

[exp [1120585

int

119906 (120585)

120585

119889120585]] (42a)

and evaluates the value of 1205852 The evaluation of the second

constant of integration can be obtained again through theboundary condition 1119909

1rarr 0 rArr 119910

1rarr infin that is

lim120585rarr1205851

119862 +

1

7

int exp [(minus1120585

int

119906 (120585)

120585

119889120585)

1

12058513119889120585]

= 0

(42b)

Equations (41) and (42a) (42b) complete the solutions of theproblem under consideration that is the construction of aclosed-form parametric solution of the TF equation (3)

It is worthwhile to remark that since the solution isgiven in a closed form the mathematical methodology beingdeveloped results in the ad hoc definition of the TF function119910in approximate power series with 1199101015840

119909(0) =

lowast

119861 (lowast

119861 = a factor thatmust be determined) and with boundary conditions given by1199100(0) = 119909

0= 1 119910

1rarr 0 as 119909 rarr infin The value 1199101015840

119909(0) =

lowast

119861

can be evaluated by way of the already constructed closed-form solution This is a very interesting result in accordancewith previous results presented by Bush and Caldwell [17]Feynman et al [18] Coulson and March [19] Kobayashi etal [20] and Kobayashi [21]

5 Closed-Form Solutions of the LBEquation in the Physical Plane

It was already proved (Section 3) that the LB nonlinear ODE(1) has been reduced to an Abel ODE of the second kindof the normal form (15) It was also proved that this Abelequation admits exact analytic solutions given as in case ofthe TF nonlinear ODE by (24) to (26) Thus the solutionof the LB equation (1) in the phase plane is given by theformulae (24) to (26) if instead of the parenthesis (119860120585 +11986112058532) the quantity (120585minus124minus1205854) is introducedThe solution

of the original equation (1) in the physical plane is giventhrough the combination of the above prescribed formulaein combination with (17) As in case of the TF equationthe whole problem includes two constants of integration

lowast

119862

and 119862 which are to be determined by using the convenientboundary data of the problem under consideration In orderto define simultaneously the type of the function 119873

(119894)

(120585)

(119894 = 1 2 3) which must be selected in accordance with themodified previously developed formulae (24) to (26) andthus define the modified function 119866(120585) and the constant ofintegration

lowast

119862 wemust combine the aboveAbel solutionswiththe prescribed convenient boundary data

Based on what was previously developed in Section 3for the original (LB) equation (1) the following boundaryconditions hold true

at 119909 = 1199090

997904rArr 119910 = 1199100= ℓ

1199101015840

119909= 1199101015840

1199090

= 119898

(43)

where ℓ and119898 are arbitrary given values provided119898 = 0Based on these boundary conditions the substitution

(8) furnishes 119901(1199100) = 119901

0= 119898 while transformation (10)

furnishes 119908(1199100) = 119908

0= 11991013

01199010= ℓ13119898 too Thus the sub-

stitutions (13) (15) and the Abel nonlinear ODE (16) performthe following relations

1199080= 119899 (120585

0) = 1198990= ℓ13119898

1205850= 11991043

0= ℓ43

1198990= minus119911 (120585

0) = minus119911

0

997904rArr 1199110= minusℓ13119898

1199111015840

1205850

= minus

(14) (ℓminus23

minus ℓ43)

ℓ13119898

+ 1 = 1 minus

1

4

ℓminus1minus ℓ

119898

(44)

We are now able to evaluate all the before mentionedquantities Indeed (23a) and (23b) for 120585

0= ℓ43 1199110= minusℓ13119898

Mathematical Problems in Engineering 7

1199111015840

1205850

= 1 minus (ℓminus1minus ℓ)4119898 furnish the following two equations

1198730= minus(

2ℓ13119898

ℓ43

+ 4

lowast

119862

+

1

3

)

81198660+ 2 (ℓ

43minus ℓ13)P ln |P| [ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 1003816100381610038161003816

10038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C]

4 [ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 100381610038161003816100381610038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C] minus cos (ln |P|)

=

1

4

[ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 100381610038161003816100381610038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C] [ln |P| sin (ln |P|) + cos (ln |P|)]

[ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 100381610038161003816100381610038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C]

2

(ln |P|)2

+

cos2 (ln |P|)

[ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 100381610038161003816100381610038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C]

2

(ln |P|)2

P = ℓ43

+ 2

lowast

119862 gt 0 2

lowast

119862 + 11989743

gt 0 2

lowast

119862 lt 1

(45)

where1198730and 119866

0are the values of119873(120585) and 119866(120585) at 120585 = 120585

0=

ℓ43 respectively On the other hand the formula for 1198731015840

120585in

(23a) at 120585 = 1205850results in

119873

1015840

1205850

=

1

(1198730+ 43)

4 [1198660+ (12) (ℓ

minus23minus ℓ43)]

(ℓ43

+ 4

lowast

119862)

2 (46)

while by the first of (23a) and (44) we have

1199111015840

1205850

=

1

2

119873

1015840

1205850

(1205850+ 4

lowast

119862) +

1

2

(1198730+

1

3

) (47)

or equivalently

1199111015840

1205850

=

2 [1198660+ (12) (ℓ

minus23minus ℓ43)]

(1198730+ 43) (ℓ

43+ 4

lowast

119862)

+

1

2

(1198730+

1

3

) (48)

The last relation (48) in combination together with (44)permits us to write

2 [1198660+ (12) (ℓ

minus23minus ℓ43)]

(1198730+ (43)) (ℓ

43+ 4

lowast

119862)

+

1

2

(1198730+

1

3

)

= 1 minus

1

4

ℓminus1minus ℓ

119898

(49)

The above three equations (46) to (49) constitute a nonlinear(transcendental) system for119873

01198660 and 120582 by means of which

one calculates these unknowns in terms of the parameters ℓand 119898 Thus the discriminant 119876

0(119876 at 120585 = 120585

0= ℓ43) given

in (27) is also known the fact that permits us to define thetype of the function119873(119894)(120585) for 120585 gt 120585

0= ℓ43 (24) to (26)This

completes the solution of the problem under consideration

which is the construction of the solution of the LB equationin the phase-plane (and thus in the physical plane) in themain interval [120585

0= ℓ43 120585] If this solution is not unique

inside [ℓ43 120585] then we follow step by step the methodologydeveloped in [15]

6 Conclusions

By a series of admissible functional transformations wereduce the nonlinear TF and LB equations to Abelrsquos equa-tions of the second kind of the normal form of (16) and(18) respectively These equations do not admit closed-formsolutions in terms of known (tabulated) functions Thisunsolvability is due to the fact that only very special formsof this kind of (16) and (18) can be solved in parametric form[4] Our goal is the development of the construction of theexact analytic solutions of the above equations based on amathematical technique leading to the derivation of closeform solutions for the Abel equation of the second kind of thenormal form [3 4] The proposed methodology constitutesthe intermediate integral of the TF and LB equations inthe phase and physical planes The reduction procedure inthe paper and the constructed solutions are very generaland can be applied to a large number of nonlinear ODEsin mathematical physics and nonlinear mechanics Also themethod employed is exact in accordance with the boundaryconditions of each problem under consideration and guidesus to the a priori approximate construction of the solutions inpower series some coefficients of which must be evaluated

With the proposed procedure the exact value of first orderderivative of TF equation is derived at 119909 = 0 In particularthe solution for the TF equation is finally constructed bycalculating 4 constants from the specific boundary conditions(4) (see (34a) (34b) (41) (42a) and (42b)) Equations(40) constitute the parametric solutions of the TF problemWith our method essentially a different way of calculating

8 Mathematical Problems in Engineering

1199101015840

119909(0) =

lowast

119861 (Section 4) is proposed with reference to themethods proposed in [17ndash21] where the value of

lowast

119861 has beencalculated having kept the solution from approximate seriesin which their coefficients had been calculated from theboundary conditions In conclusion in this paper an analyticmethod is developed that does not contradict previous resultsbut it is a different way of calculating them

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] I Langmuir and K B Blodgett ldquoCurrents limited by spacecharge between coaxial cylindersrdquo Physical Review vol 22 no4 pp 347ndash356 1923

[2] H T Davis Introduction to Nonlinear Differential and IntegralEquations Dover New York NY USA 1962

[3] E Kamke Differentialgleichungen I Gewohnliche Differential-gleichungen and II Partielle Differentialgleichungen Akademis-che Verlagsgesellschaft Leipzig Germany 1962

[4] A D Polyanin and V F Zaitsev Handbook of Exact Solutionsfor Ordinary Differential Equations CRCPress Boca Raton FlaUSA 1999

[5] K Horvath J N Fairchild K Kaczmarski and G Guio-chon ldquoMartin-Synge algorithm for the solution of equilibrium-dispersive model of liquid chromatographyrdquo Journal of Chro-matography A vol 1217 no 52 pp 8127ndash8135 2010

[6] Y Shang and X Zheng ldquoThe first-integral method and abun-dant explicit exact solutions to the zakharov equationsrdquo Journalof Applied Mathematics vol 2012 Article ID 818345 16 pages2012

[7] D B Graham and I H Cairns ldquoConstraints on the formationand structure of langmuir eigenmodes in the solar windrdquoPhysical Review Letters vol 111 no 12 Article ID 121101 2013

[8] L H Thomas ldquoThe calculation of atomic fieldsrdquoMathematicalProceedings of the Cambridge Philosophical Society vol 23 no5 pp 542ndash548 1927

[9] E Fermi ldquoStatistical method of investigating electrons inatomsrdquo Zeitschrift fur Physik vol 48 pp 73ndash79 1927

[10] T Tsurumi and MWadati ldquoDynamics of magnetically trappedboson-fermion mixturesrdquo Journal of the Physical Society ofJapan vol 69 no 1 pp 97ndash103 2000

[11] S Liao ldquoAn explicit analytic solution to the Thomas-Fermiequationrdquo Applied Mathematics and Computation vol 144 no2-3 pp 495ndash506 2003

[12] S Abbasbandy and C Bervillier ldquoAnalytic continuation ofTaylor series and the boundary value problems of some nonlin-ear ordinary differential equationsrdquo Applied Mathematics andComputation vol 218 no 5 pp 2178ndash2199 2011

[13] K Ourabah and M Tribeche ldquoRelativistic formulation of thegeneralized nonextensiveThomas-FermimodelrdquoPhysicaA vol393 pp 470ndash474 2014

[14] D E Panayotounakos ldquoExact analytic solutions of unsolvableclasses of first and second order nonlinear ODEs (part I Abelrsquosequations)rdquo Applied Mathematics Letters vol 18 no 2 pp 155ndash162 2005

[15] D E Panayotounakos N B Sotiropoulos A B Sotiropoulouand N D Panayotounakou ldquoExact analytic solutions of non-linear boundary value problems in fluid mechanics (Blasiusequations)rdquo Journal of Mathematical Physics vol 46 no 3Article ID 033101 2005

[16] D E Panayotounakos E ETheotokoglou and M P MarkakisldquoExact analytic solutions for the unforded damped duffingnonlinear oscillatorrdquo Comptes Rendus Mechanics Journal vol334 pp 311ndash316 2006

[17] V Bush and S H Caldwell ldquoThomas-Fermi equation solutionby the differential analyzerrdquo Physical Review vol 38 no 10 pp1898ndash1901 1931

[18] R P Feynman N Metropolis and E Teller ldquoEquations of stateof elements based on the generalized fermi-thomas theoryrdquoPhysical Review vol 75 no 10 pp 1561ndash1573 1949

[19] C A Coulson and N H March ldquoMomenta in atoms usingthe Thomas-Fermi methodrdquo Proceedings of the Physical SocietySection A vol 63 no 4 pp 367ndash374 1950

[20] S Kobayashi TMatsukuma S Nagai and K Umeda ldquoAccuratevalue of the initial slope of the ordinary TF functionrdquo Journal ofthe Physical Society of Japan vol 10 no 9 pp 759ndash762 1955

[21] S Kobayashi ldquoSome coefficients of the series expansion of theTFD functionrdquo Journal of the Physical Society of Japan vol 10no 9 pp 824ndash825 1955

[22] S A Sommerfeld ldquoIntegrazione Asintotica dellrsquoEquazione Dif-ferenziali diThomas-FermirdquoAccademia dei Lincei Atti-Rendic-onte vol 4 pp 85ndash110 1932

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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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

Journal of

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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

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Both LB and TF equations do not admit exact analyticsolution in terms of known (tabulated) functions In thispaper using series of admissible functional transformationsat first both LB and TF nonlinear ODEs are reduced toequivalent Abelrsquos equations of the second kind of the normalform According to a mathematical methodology recentlydeveloped [14ndash16] we extract closed-form solution of theaboveAbel equations in both phase and physical planes underinitial data in accordance with the physical problems

The mathematical technique introduced in our study isgeneral and can be applied to a very large class of yet unsolv-able ODEs in nonlinear mechanics and generally in mathe-matical physicsThemethod employed is exact in accordancewith the boundary conditions of each problem under con-sideration and gives us the possibility not to consider the apriori approximate construction of the solutions in the formof power series some coefficient of which must be estimatedIn addition with our analysis the exact value of the first orderderivative of Thomas Fermi equation at 119909 = 0 is derivedin a different way from the methods already existing in theliterature [17ndash21]

2 Preliminaries Notations

A nonlinear ODE of considerable interest called the Lang-muir-Blodgett (LB) equation [1] is the following

311991011991010158401015840

119909119909+ 11991010158402

119909+ 4119910119910

1015840

119909+ 1199102minus 1 = 0 (1)

Here the notations 1199101015840119909= 119889119910119889119909 11991010158401015840

119909119909= 11988921199101198891199092 are used

for the total derivativesThis equation of constant coefficients appeared in con-

nection with the theory of flow of a current from a hot cath-ode to a positive charged anode in a high vacuum The cath-ode and anode are long coaxial cylinders and the independentvariable 119910 is defined by the equation 119910 = 119891(119903119903

0) where 119903 is

the radius of the anode enclosing a cathode of radius 1199030 The

independent variable 119909 is given by 119909 = ln(1199031199030)

Equation (1) is converted into a somewhat more tractableform bymeans of the transformation 119910 = 119890minus1199092119911(119909) proposedby Davis [2] and thus becomes

311991111991110158401015840

119909119909+ 11991110158402

119909minus 119890119909= 0 (2)

Numerical techniques concerning the solution of (1) areincluded in [6]

The TF nonlinear ODE

11991010158401015840

119909119909= 119909minus12

11991032 (3)

appears in the problem of determining the effective nuclearcharge in heavy atoms [8 9] The solution is defined for theboundary values

119910 (0) = 1

119910 (infin) = 0

(4)

Interesting approximate and numerical integrationsmethods have been proposed by Sommerfeld [22] Bush andCaldwell [17] Kobayashi et al [20] Feynman et al [18] andCoulson and March [19]

The differential equation (3) belongs to equations ofEmden-Fowler type [3 4] Taking into account the knownexact parametric solutions of this class of nonlinear ODEs [4pages 241ndash250] the above mentioned equation can not solveanalytically in terms of known functions

Many investigations have studied the TF equation obey-ing to the boundary conditions (4) (see [18ndash21]) and theycarried out the outward numerical integration (119909 = 0 119910 rarr

infin) for a sequence of values of initial slope converging step bystep to that curve which should approach the 119909-axis asymp-totically Finally for a semianalytical solution methodologywe must refer to Sommerfeld [22]

The Emden-Fowler nonlinear ODE of the normal form is

11991010158401015840

119909119909= 119860119909119899119910119898 (5)

where 119860 and 119899 119898 are arbitrary parameters We note thefollowing regarding the reduction of (5) (see [4])

For 119898 = 1 and 119898 = minus2119899 minus 3 the admissible functionaltransformations

120585 =

2119899 + 119898 + 3

119898 minus 1

119909(119899+2)(119898minus1)

119910

119906 = 119909(119899+2)(119898minus1)

(1199091199101015840

119909+

119899 + 2

119898 minus 1

119910)

(6)

reduce the nonlinear ODE (5) to the following Abel equationof the second kind of the normal form

1199061199061015840

120585minus 119906 = minus

(119899 + 2) (119899 + 119898 + 1)

(2119899 + 119898 + 3)2

120585

+ 119860(

119898 minus 1

2119899 + 119898 + 3

)

2

120585119898

(7)

In what follows by a series of admissible functionaltransformations we prove that both of (1) and (3) can beexactly reduced to Abelrsquos equations of the second kind of thenormal form 119910119910

1015840

119909minus 119910 = 119891(119909) According to the following

reduction procedure the exact analytic solutions of the aboveAbel equations are constructed avoiding the approximateexpression of the solutions in power series

3 The Reduction Procedure

31 The LB Equation The LB equation (1) is a second ordernonlinear ODE of constant coefficients Thus by the substi-tution

1199101015840

119909= 119901 (119910)

997904rArr 11991010158401015840

119909119909= 1199011015840

119910119901

(8)

it can be reduced to the followingAbel equation of the secondkind

31199101199011199011015840

119910= minus1199012minus 4119910119901 minus 119910

2+ 1 (9)

Furthermore the well-known transformation [4 page 50]

119908 (119910) = 119901119890int(13119910)119889119910

= 119901 exp (ln 10038161003816100381610038161199101003816100381610038161003816

13

) = 11991013119901

997904rArr 119901 = 119910minus13

119908 119910 = 0

(10)

Mathematical Problems in Engineering 3

giving the expressions

1199011015840

119910= 119910minus13

1199081015840

119910minus

1

3

119910minus43

119908

1199011199011015840

119910= 119910minus23

1199081199081015840

119910minus

1

3

119910minus53

1199082

(11)

transforms (9) into the following Abel equation of the secondkind

1199081199081015840

119910= minus

4

3

11991013119908 +

1

3

119910minus13

minus

1

3

11991053 (12)

Finally the substitution

119908 (119910) = 119899 (120585)

120585 =

4

3

int11991013119889119910

997904rArr 119910 = 12058534

(13)

reduces (12) to the followingAbel equation of the second kindof the normal form

1198991198991015840

120585+ 119899 =

1

4

120585minus12

minus

1

4

120585 (14)

or setting

119899 (120585) = minus119911 (120585) (15)

to the form

1199111199111015840

120585minus 119911 =

1

4

120585minus12

minus

1

4

120585 (16)

The solution of (16) and thus the solutions of (14) (12)and (9) constitutes the intermediate integral of the LB equa-tion (1) in the phase plane In other words when obtainingthe solution of the transformed equation (16) in the form119911 = 119911(120585

lowast

119862)lowast

119862 = first integration constant the solution of (14)becomes 119899(120585) = minus119911(120585) the solution of (12) is119908(119910) = minus119911(11991043)and finally the solution of (8) is 119901(119910) = minus119910

minus13119911(11991043) Thus

the solution to the original Langmuir equation (1) in thephysical plane can be obtained by the integration by parts ofthe following equation

119889119910

119901 (119910)

= 119889119909

997904rArr int

119889119910

11991013119911 (11991043)

= minus119909 + 119862

119862 = second integration constant

(17)

Note that the reduced Abel equation (16) does not admitan exact analytic solution in terms of known (tabulated)functions [4 pages 29ndash45]

32TheTFEquation Equation (3) is a typical Emden-Fowlernonlinear equation with 119860 = 1 119899 = minus12 and 119898 = 32A thorough examination concerning approximate expansion

solution of (3) is included in [20] as well as all the referencescited there By now with the aid of the transformations (6)one reduces the original TF equation (3) to the followingAbelequation of the second kind of the normal form

1199061199061015840

120585minus 119906 = minus

12

49

120585 +

1

49

12058532 (18)

where

120585 = 71199093119910 (19)

119906 = 1199093(1199091199101015840

119909+ 3119910) (20)

In what follows based on a mathematical constructionrecently developed in [14 15] concerning the closed-formanalytic solution of an Abel equation of the second kind ofthe normal form 119910119910

1015840

119909minus119910 = 119891(119909) we provide the closed-form

solutions of the reduced Abel equation (18) that is to say theconstruction of the intermediate integral of the original TFequation (3) in the phase plane as well as the final solutionin the physical plane in accordance with the given boundaryconditions

4 Closed-Form Solutions of the TF Equationin the Phase Plane-Final Solutions

We consider the reduced Abel equation (18) namely theequation

1199061199061015840

120585minus 119906 = 119860120585 + 119861120585

32 119860 = minus

12

49

119861 =

1

49

(21)

obeying (as the first of the boundary conditions (4) 119910(0) = 1and the well-known result 1199101015840

119909(0) =

lowast

119861 = 0 [2 8 17ndash20]) thetransformed (19) and (20) equivalent condition

at 120585 = 1205850= 0 997888rarr 119906 (0) = 119906

0= 0 (22)

It was recently proved [14 15] that an Abel equation ofthe second kind of the normal form 119910119910

1015840

119909minus 119910 = 119891(119909) admits

an exact analytic solution in terms of known (tabulated)functions Hence we perform the solution of (21) whichconstitutes the solution of the TF equation in the phase planeas follows

120585 = ln 10038161003816100381610038161003816120585 + 2

lowast

119862

10038161003816100381610038161003816

119906 (120585) =

1

2

120585 [119873 (120585) +

1

3

]

119873

1015840

120585(120585) =

4 [119866 (120585) + 2 (119860120585 + 120585

32

)]

120585 [119873 (120585) + 43]

(23a)

1

4

[(120585 sin 120585 + cos 120585)lowast

119860(120585) + cos2120585] (2120585lowast

119860(120585) + cos 120585)

(120585

lowast

119860(120585))

3

119890120585

= 4 [119866 + 2 (119860120585 + 119861120585

32

)]

(23b)

4 Mathematical Problems in Engineering

wherelowast

119860 =

lowast

119860(120585) = the cosine integral = ci (120585)

= C + ln 120585 +infin

sum

119906=1

(minus1)119906 120585

2119906

(2119906) (2119906)

C = Eulerrsquos number = 0 5572156649015325

(23c)

Herelowast

119862 is integration constant while the function 119873(120585) isgiven implicitly in terms of the subsidiary function 119866(120585)which can be determined from (23b) and the knownmemberof (21) as in the following three cases

Case 1 (119876(120585) lt 0 (119901 lt 0)) Consider

119873

(1)

(120585) = 2radicminus

119901

3

cos 1198863

119873

(2)

(120585) = minus2radicminus

119901

3

cos119886 minus 1205873

119873

(3)

(120585) = minus2radicminus

119901

3

cos 119886 + 1205873

cos 119886 = minus119902

2radicminus (1199013)3

0 lt 119886 lt 120587

(24)

Case 2 (119876(120585) gt 0) Consider

119873(120585) =3

radicminus

119902

2

+ radic119876 +3

radicminus

119902

2

minus radic119876 (25)

Case 3 (119876(120585) = 0) Consider

119873

(1)

(120585) = 23

radicminus

119902

2

119873

(2)

(120585) = 119873

(3)

(120585) = minus3

radicminus

119902

2

(26)

In all these formulae the quantities 119876 119901 and 119902 are givenby

119876(120585) =

1

27

1199013(120585) +

1

4

1199022(120585)

119901 (120585) = minus

1198862

3

+ 119887

119902 (120585) =

2

27

1198863minus

1

3

119886119887 + 119888

119886 = minus4

119887 =

3 + 4 [119866 (120585) + (119860120585 + 119861120585

32

)]

120585

119888 = minus

4 [119866 (120585) + 2 (119860120585 + 119861120585

32

)]

120585

(27)

In order to define the type of the function 119873

(119894)

(120585)

(119894 = 1 2 3) which must be selected among the closed-formsolutions ((23a) (23b) and (23c)) to (26) as well as the valueof the subsidiary function 119866(120585) at 120585 = 0 and the value ofthe integration constant

lowast

119862 we must combine the above Abelsolutions with the boundary data

We symbolize by 120585 = ln |2lowast

119862| the substitution while byL0

any function L(120585) at 1205850= ln |2

lowast

119862| Thus functions 119906(120585) and(119873

1015840

120585)(120585) given in (23a) perform

119906 (120585) =

1

2

120585 [119873

(119894)

(120585) +

1

3

]

119873

1015840

120585

(119894)

(120585) =

4 [119866 (120585) + 2 (minus (1249) (120585) + (149) 120585

32

)]

120585 [119873

(119894)

(120585) + 43]

(28)

1199060=

1

2

ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

(119873

(119894)

0+

1

3

)

119873

1015840(119894)

1205850

(120585)

=

4 [1198660+ 2 (minus (1249) ln 1003816100381610038161003816

10038162

lowast

119862

10038161003816100381610038161003816+ (149) (ln 1003816100381610038161003816

10038162

lowast

119862

10038161003816100381610038161003816)

32

)]

ln 100381610038161003816100381610038162

lowast

119862

10038161003816100381610038161003816[119873

(119894)

0+ 43]

(29)

On the other hand the Abel nonlinear ODE (21) at 1205850

becomes

1 +

minus (1249) ln 100381610038161003816100381610038162

lowast

119862

10038161003816100381610038161003816+ (149) (ln 1003816100381610038161003816

10038162

lowast

119862

10038161003816100381610038161003816)

32

1199060

= 1 +

minus (1249) ln 100381610038161003816100381610038162

lowast

119862

10038161003816100381610038161003816+ (149) (ln 1003816100381610038161003816

10038162

lowast

119862

10038161003816100381610038161003816)

32

ln 100381610038161003816100381610038162

lowast

119862

10038161003816100381610038161003816(119873

(119894)

0+ 43)

=

1199061015840

1205850

120585

1015840

1205850

(30)

Combination of the above equations results in the followingquadratic to119873

0equation

119873

(119894)

0

2

+ 119873

(119894)

0+

4

9

+[

[

[

1 minus

1

lowast

119862 ln 100381610038161003816100381610038162

lowast

119862

10038161003816100381610038161003816

]

]

]

sdot [minus21198660+ 2(minus

12

49

ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

+

1

49

(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)

32

)]

= 0

(31)

including the value of 1198660and the constant on integration

lowast

119862

Mathematical Problems in Engineering 5

Similarly by (23b) one gets

32 1198660+ [2(minus

12

49

ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

+

1

49

ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

32

)]

lowast

119860

3

0

= [ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

sin(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

) + cos(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)]

lowast

1198600

+ cos2 (ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)(2 ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

lowast

1198600

+ cos(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

))

(32)

that constitute a second equation including the value of 1198660

and the constant of integrationlowast

119862 Also in this equation bymeans of (23c) we have

lowast

1198600=

lowast

119860(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

) = ci(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)

= C + ln(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

) +

infin

sum

119896=1

(minus1)119896ln 100381610038161003816100381610038162

lowast

119862

10038161003816100381610038161003816

2119896

(2119896) (2119896)

C = Eulerrsquos number asymp 05572

(33)

Summarizing from (19) and (20) one concludes that since bythe first boundary conditions 119910(0) = 1 for 119909 rarr 0 rArr 119910 rarr 1

then for 119909 rarr 0 rArr 1199101015840

119909rarr 0 or 1199101015840

119909rarr

lowast

119861 wherelowast

119861 is a fixedvalue Thus combining (23a) (23b) and (23c) together withthe first of (28) and the Abel nonlinear ODE (21) one maywrite that for 119909 rarr 0 rArr 120585

0rarr 0 119906

0rarr 0 119873

0rarr 0

Therefore by making use of the first boundary conditiongiven in (4) (119909 rarr 0 119910 rarr 1) as well as of all the aboveobservations the solutions of (31) and (32) perform the valuesfor 119866

0and 2

lowast

119862 and they can be written in the followingalgebraic forms

minus1 plusmn radic1 minus 4F (1198660

lowast

119862)

2

= 0(34a)

H(

lowast

119862) = 1198660 (34b)

where

F(1198660

lowast

119862) =

4

9

+[

[

[

1 minus

1

lowast

119862 ln 100381610038161003816100381610038162

lowast

119862

10038161003816100381610038161003816

]

]

]

[minus21198660

+ 2(minus

12

4

ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

+

1

49

(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)

32

)]

(35a)

H(

lowast

119862)

=

1

32

[ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

sin(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

) + cos(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)]

sdot

lowast

11986000+ cos2 (ln

1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

) [2 ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

lowast

1198600

+ cos(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)]

lowast

1198600

lowast

119862 as in (33)

(35b)

Thus we are now able to estimate 1199010 1199020 and 119876

0given by

(27) in terms of the known valueslowast

119862 and 1198660 This estimation

furnishes the value of1198760and therefore the type of the solution

119873

(119894)

(120585) (119894 = 1 2 3) near the origin 1205850among the formulae (24)

to (26)From the above evaluation it is concluded that the closed

form solutions of the Abel nonlinear ODE (21) given as

119906(119894)(120585) =

1

2

120585 [119873

(119894)

(120585) +

1

3

]

120585 = ln1003816100381610038161003816100381610038161003816

120585 + 2

lowast

1198621

1003816100381610038161003816100381610038161003816

(119894 = 1 2 3)

(36)

have been completely determinedSince we obtained the above closed- form solutions of the

intermediate Abel nonlinear ODE (21) the solutions to theoriginal TF equation (3) can be determined as follows By (19)and (21) we perform

119909 =

1

713

(

120585

119910

)

13

1199101015840

119909=

1

1199094119906 minus 3

119910

119909

(37)

while taking the total differential in (19) we extract

119889120585 = 211199092119910119889119909 + 7119909

3119889119910

lArrrArr 1205851015840

119909=

119889120585

119889119909

= 71199092(1199091199101015840

119909+ 3119910) = 7

119906

119909

(38)

where 119906 = 119906(120585 + 2

lowast

1198621) denotes the known solution of the

Abel nonlinear ODE (18) Using (34a) (34b) and (37) onewrites

1199101015840

120585=

1199101015840

119909

1205851015840

119909

= 743(

119910

120585

) 11990643

minus 3119910713(

119910

120585

)

13

=

119906 (120585)

120585

119910 minus

3

71205854311991043

(39)

6 Mathematical Problems in Engineering

that is the Bernoulli equation

11991013

=

exp ((1120585) int (119906 (120585) 120585) 119889120585)

119862 + (17) int exp [((1120585) int (119906 (120585) 120585) 119889120585)] 119889120585

119909 =

1

713

(

120585

119910

)

13

119909 ge 0 119910 ge 0 120585 = parameter (120585 ge 0)

119862 = second constant of integration

(40)

The last step of the above analysis concerns the calculationof the second constant of integration 119862 Thus from the firstof the boundary conditions (4) and the parametric solution(40) one writes

1199091997888rarr 0

1199101997888rarr 1

1205851997888rarr 0

lim1205851rarr0

(11991013)

=

lim1205851rarr0

exp ((1120585) int (119906 (120585) 120585) 119889120585)

lim1205851rarr0

119862 + (17) int exp [((1120585) int (119906 (120585) 120585) 119889120585)] 119889120585

(41)

and estimates the constant of integration 119862 By the second ofthe boundary conditions (4) one writes 119909

2rarr +infin 119910

2rarr 0

and 120585 rarr 1205852 Again through the parametric solution (40) one

writes

lim120585rarr1205852

[exp [1120585

int

119906 (120585)

120585

119889120585]] (42a)

and evaluates the value of 1205852 The evaluation of the second

constant of integration can be obtained again through theboundary condition 1119909

1rarr 0 rArr 119910

1rarr infin that is

lim120585rarr1205851

119862 +

1

7

int exp [(minus1120585

int

119906 (120585)

120585

119889120585)

1

12058513119889120585]

= 0

(42b)

Equations (41) and (42a) (42b) complete the solutions of theproblem under consideration that is the construction of aclosed-form parametric solution of the TF equation (3)

It is worthwhile to remark that since the solution isgiven in a closed form the mathematical methodology beingdeveloped results in the ad hoc definition of the TF function119910in approximate power series with 1199101015840

119909(0) =

lowast

119861 (lowast

119861 = a factor thatmust be determined) and with boundary conditions given by1199100(0) = 119909

0= 1 119910

1rarr 0 as 119909 rarr infin The value 1199101015840

119909(0) =

lowast

119861

can be evaluated by way of the already constructed closed-form solution This is a very interesting result in accordancewith previous results presented by Bush and Caldwell [17]Feynman et al [18] Coulson and March [19] Kobayashi etal [20] and Kobayashi [21]

5 Closed-Form Solutions of the LBEquation in the Physical Plane

It was already proved (Section 3) that the LB nonlinear ODE(1) has been reduced to an Abel ODE of the second kindof the normal form (15) It was also proved that this Abelequation admits exact analytic solutions given as in case ofthe TF nonlinear ODE by (24) to (26) Thus the solutionof the LB equation (1) in the phase plane is given by theformulae (24) to (26) if instead of the parenthesis (119860120585 +11986112058532) the quantity (120585minus124minus1205854) is introducedThe solution

of the original equation (1) in the physical plane is giventhrough the combination of the above prescribed formulaein combination with (17) As in case of the TF equationthe whole problem includes two constants of integration

lowast

119862

and 119862 which are to be determined by using the convenientboundary data of the problem under consideration In orderto define simultaneously the type of the function 119873

(119894)

(120585)

(119894 = 1 2 3) which must be selected in accordance with themodified previously developed formulae (24) to (26) andthus define the modified function 119866(120585) and the constant ofintegration

lowast

119862 wemust combine the aboveAbel solutionswiththe prescribed convenient boundary data

Based on what was previously developed in Section 3for the original (LB) equation (1) the following boundaryconditions hold true

at 119909 = 1199090

997904rArr 119910 = 1199100= ℓ

1199101015840

119909= 1199101015840

1199090

= 119898

(43)

where ℓ and119898 are arbitrary given values provided119898 = 0Based on these boundary conditions the substitution

(8) furnishes 119901(1199100) = 119901

0= 119898 while transformation (10)

furnishes 119908(1199100) = 119908

0= 11991013

01199010= ℓ13119898 too Thus the sub-

stitutions (13) (15) and the Abel nonlinear ODE (16) performthe following relations

1199080= 119899 (120585

0) = 1198990= ℓ13119898

1205850= 11991043

0= ℓ43

1198990= minus119911 (120585

0) = minus119911

0

997904rArr 1199110= minusℓ13119898

1199111015840

1205850

= minus

(14) (ℓminus23

minus ℓ43)

ℓ13119898

+ 1 = 1 minus

1

4

ℓminus1minus ℓ

119898

(44)

We are now able to evaluate all the before mentionedquantities Indeed (23a) and (23b) for 120585

0= ℓ43 1199110= minusℓ13119898

Mathematical Problems in Engineering 7

1199111015840

1205850

= 1 minus (ℓminus1minus ℓ)4119898 furnish the following two equations

1198730= minus(

2ℓ13119898

ℓ43

+ 4

lowast

119862

+

1

3

)

81198660+ 2 (ℓ

43minus ℓ13)P ln |P| [ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 1003816100381610038161003816

10038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C]

4 [ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 100381610038161003816100381610038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C] minus cos (ln |P|)

=

1

4

[ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 100381610038161003816100381610038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C] [ln |P| sin (ln |P|) + cos (ln |P|)]

[ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 100381610038161003816100381610038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C]

2

(ln |P|)2

+

cos2 (ln |P|)

[ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 100381610038161003816100381610038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C]

2

(ln |P|)2

P = ℓ43

+ 2

lowast

119862 gt 0 2

lowast

119862 + 11989743

gt 0 2

lowast

119862 lt 1

(45)

where1198730and 119866

0are the values of119873(120585) and 119866(120585) at 120585 = 120585

0=

ℓ43 respectively On the other hand the formula for 1198731015840

120585in

(23a) at 120585 = 1205850results in

119873

1015840

1205850

=

1

(1198730+ 43)

4 [1198660+ (12) (ℓ

minus23minus ℓ43)]

(ℓ43

+ 4

lowast

119862)

2 (46)

while by the first of (23a) and (44) we have

1199111015840

1205850

=

1

2

119873

1015840

1205850

(1205850+ 4

lowast

119862) +

1

2

(1198730+

1

3

) (47)

or equivalently

1199111015840

1205850

=

2 [1198660+ (12) (ℓ

minus23minus ℓ43)]

(1198730+ 43) (ℓ

43+ 4

lowast

119862)

+

1

2

(1198730+

1

3

) (48)

The last relation (48) in combination together with (44)permits us to write

2 [1198660+ (12) (ℓ

minus23minus ℓ43)]

(1198730+ (43)) (ℓ

43+ 4

lowast

119862)

+

1

2

(1198730+

1

3

)

= 1 minus

1

4

ℓminus1minus ℓ

119898

(49)

The above three equations (46) to (49) constitute a nonlinear(transcendental) system for119873

01198660 and 120582 by means of which

one calculates these unknowns in terms of the parameters ℓand 119898 Thus the discriminant 119876

0(119876 at 120585 = 120585

0= ℓ43) given

in (27) is also known the fact that permits us to define thetype of the function119873(119894)(120585) for 120585 gt 120585

0= ℓ43 (24) to (26)This

completes the solution of the problem under consideration

which is the construction of the solution of the LB equationin the phase-plane (and thus in the physical plane) in themain interval [120585

0= ℓ43 120585] If this solution is not unique

inside [ℓ43 120585] then we follow step by step the methodologydeveloped in [15]

6 Conclusions

By a series of admissible functional transformations wereduce the nonlinear TF and LB equations to Abelrsquos equa-tions of the second kind of the normal form of (16) and(18) respectively These equations do not admit closed-formsolutions in terms of known (tabulated) functions Thisunsolvability is due to the fact that only very special formsof this kind of (16) and (18) can be solved in parametric form[4] Our goal is the development of the construction of theexact analytic solutions of the above equations based on amathematical technique leading to the derivation of closeform solutions for the Abel equation of the second kind of thenormal form [3 4] The proposed methodology constitutesthe intermediate integral of the TF and LB equations inthe phase and physical planes The reduction procedure inthe paper and the constructed solutions are very generaland can be applied to a large number of nonlinear ODEsin mathematical physics and nonlinear mechanics Also themethod employed is exact in accordance with the boundaryconditions of each problem under consideration and guidesus to the a priori approximate construction of the solutions inpower series some coefficients of which must be evaluated

With the proposed procedure the exact value of first orderderivative of TF equation is derived at 119909 = 0 In particularthe solution for the TF equation is finally constructed bycalculating 4 constants from the specific boundary conditions(4) (see (34a) (34b) (41) (42a) and (42b)) Equations(40) constitute the parametric solutions of the TF problemWith our method essentially a different way of calculating

8 Mathematical Problems in Engineering

1199101015840

119909(0) =

lowast

119861 (Section 4) is proposed with reference to themethods proposed in [17ndash21] where the value of

lowast

119861 has beencalculated having kept the solution from approximate seriesin which their coefficients had been calculated from theboundary conditions In conclusion in this paper an analyticmethod is developed that does not contradict previous resultsbut it is a different way of calculating them

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] I Langmuir and K B Blodgett ldquoCurrents limited by spacecharge between coaxial cylindersrdquo Physical Review vol 22 no4 pp 347ndash356 1923

[2] H T Davis Introduction to Nonlinear Differential and IntegralEquations Dover New York NY USA 1962

[3] E Kamke Differentialgleichungen I Gewohnliche Differential-gleichungen and II Partielle Differentialgleichungen Akademis-che Verlagsgesellschaft Leipzig Germany 1962

[4] A D Polyanin and V F Zaitsev Handbook of Exact Solutionsfor Ordinary Differential Equations CRCPress Boca Raton FlaUSA 1999

[5] K Horvath J N Fairchild K Kaczmarski and G Guio-chon ldquoMartin-Synge algorithm for the solution of equilibrium-dispersive model of liquid chromatographyrdquo Journal of Chro-matography A vol 1217 no 52 pp 8127ndash8135 2010

[6] Y Shang and X Zheng ldquoThe first-integral method and abun-dant explicit exact solutions to the zakharov equationsrdquo Journalof Applied Mathematics vol 2012 Article ID 818345 16 pages2012

[7] D B Graham and I H Cairns ldquoConstraints on the formationand structure of langmuir eigenmodes in the solar windrdquoPhysical Review Letters vol 111 no 12 Article ID 121101 2013

[8] L H Thomas ldquoThe calculation of atomic fieldsrdquoMathematicalProceedings of the Cambridge Philosophical Society vol 23 no5 pp 542ndash548 1927

[9] E Fermi ldquoStatistical method of investigating electrons inatomsrdquo Zeitschrift fur Physik vol 48 pp 73ndash79 1927

[10] T Tsurumi and MWadati ldquoDynamics of magnetically trappedboson-fermion mixturesrdquo Journal of the Physical Society ofJapan vol 69 no 1 pp 97ndash103 2000

[11] S Liao ldquoAn explicit analytic solution to the Thomas-Fermiequationrdquo Applied Mathematics and Computation vol 144 no2-3 pp 495ndash506 2003

[12] S Abbasbandy and C Bervillier ldquoAnalytic continuation ofTaylor series and the boundary value problems of some nonlin-ear ordinary differential equationsrdquo Applied Mathematics andComputation vol 218 no 5 pp 2178ndash2199 2011

[13] K Ourabah and M Tribeche ldquoRelativistic formulation of thegeneralized nonextensiveThomas-FermimodelrdquoPhysicaA vol393 pp 470ndash474 2014

[14] D E Panayotounakos ldquoExact analytic solutions of unsolvableclasses of first and second order nonlinear ODEs (part I Abelrsquosequations)rdquo Applied Mathematics Letters vol 18 no 2 pp 155ndash162 2005

[15] D E Panayotounakos N B Sotiropoulos A B Sotiropoulouand N D Panayotounakou ldquoExact analytic solutions of non-linear boundary value problems in fluid mechanics (Blasiusequations)rdquo Journal of Mathematical Physics vol 46 no 3Article ID 033101 2005

[16] D E Panayotounakos E ETheotokoglou and M P MarkakisldquoExact analytic solutions for the unforded damped duffingnonlinear oscillatorrdquo Comptes Rendus Mechanics Journal vol334 pp 311ndash316 2006

[17] V Bush and S H Caldwell ldquoThomas-Fermi equation solutionby the differential analyzerrdquo Physical Review vol 38 no 10 pp1898ndash1901 1931

[18] R P Feynman N Metropolis and E Teller ldquoEquations of stateof elements based on the generalized fermi-thomas theoryrdquoPhysical Review vol 75 no 10 pp 1561ndash1573 1949

[19] C A Coulson and N H March ldquoMomenta in atoms usingthe Thomas-Fermi methodrdquo Proceedings of the Physical SocietySection A vol 63 no 4 pp 367ndash374 1950

[20] S Kobayashi TMatsukuma S Nagai and K Umeda ldquoAccuratevalue of the initial slope of the ordinary TF functionrdquo Journal ofthe Physical Society of Japan vol 10 no 9 pp 759ndash762 1955

[21] S Kobayashi ldquoSome coefficients of the series expansion of theTFD functionrdquo Journal of the Physical Society of Japan vol 10no 9 pp 824ndash825 1955

[22] S A Sommerfeld ldquoIntegrazione Asintotica dellrsquoEquazione Dif-ferenziali diThomas-FermirdquoAccademia dei Lincei Atti-Rendic-onte vol 4 pp 85ndash110 1932

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

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OptimizationJournal of

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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

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

giving the expressions

1199011015840

119910= 119910minus13

1199081015840

119910minus

1

3

119910minus43

119908

1199011199011015840

119910= 119910minus23

1199081199081015840

119910minus

1

3

119910minus53

1199082

(11)

transforms (9) into the following Abel equation of the secondkind

1199081199081015840

119910= minus

4

3

11991013119908 +

1

3

119910minus13

minus

1

3

11991053 (12)

Finally the substitution

119908 (119910) = 119899 (120585)

120585 =

4

3

int11991013119889119910

997904rArr 119910 = 12058534

(13)

reduces (12) to the followingAbel equation of the second kindof the normal form

1198991198991015840

120585+ 119899 =

1

4

120585minus12

minus

1

4

120585 (14)

or setting

119899 (120585) = minus119911 (120585) (15)

to the form

1199111199111015840

120585minus 119911 =

1

4

120585minus12

minus

1

4

120585 (16)

The solution of (16) and thus the solutions of (14) (12)and (9) constitutes the intermediate integral of the LB equa-tion (1) in the phase plane In other words when obtainingthe solution of the transformed equation (16) in the form119911 = 119911(120585

lowast

119862)lowast

119862 = first integration constant the solution of (14)becomes 119899(120585) = minus119911(120585) the solution of (12) is119908(119910) = minus119911(11991043)and finally the solution of (8) is 119901(119910) = minus119910

minus13119911(11991043) Thus

the solution to the original Langmuir equation (1) in thephysical plane can be obtained by the integration by parts ofthe following equation

119889119910

119901 (119910)

= 119889119909

997904rArr int

119889119910

11991013119911 (11991043)

= minus119909 + 119862

119862 = second integration constant

(17)

Note that the reduced Abel equation (16) does not admitan exact analytic solution in terms of known (tabulated)functions [4 pages 29ndash45]

32TheTFEquation Equation (3) is a typical Emden-Fowlernonlinear equation with 119860 = 1 119899 = minus12 and 119898 = 32A thorough examination concerning approximate expansion

solution of (3) is included in [20] as well as all the referencescited there By now with the aid of the transformations (6)one reduces the original TF equation (3) to the followingAbelequation of the second kind of the normal form

1199061199061015840

120585minus 119906 = minus

12

49

120585 +

1

49

12058532 (18)

where

120585 = 71199093119910 (19)

119906 = 1199093(1199091199101015840

119909+ 3119910) (20)

In what follows based on a mathematical constructionrecently developed in [14 15] concerning the closed-formanalytic solution of an Abel equation of the second kind ofthe normal form 119910119910

1015840

119909minus119910 = 119891(119909) we provide the closed-form

solutions of the reduced Abel equation (18) that is to say theconstruction of the intermediate integral of the original TFequation (3) in the phase plane as well as the final solutionin the physical plane in accordance with the given boundaryconditions

4 Closed-Form Solutions of the TF Equationin the Phase Plane-Final Solutions

We consider the reduced Abel equation (18) namely theequation

1199061199061015840

120585minus 119906 = 119860120585 + 119861120585

32 119860 = minus

12

49

119861 =

1

49

(21)

obeying (as the first of the boundary conditions (4) 119910(0) = 1and the well-known result 1199101015840

119909(0) =

lowast

119861 = 0 [2 8 17ndash20]) thetransformed (19) and (20) equivalent condition

at 120585 = 1205850= 0 997888rarr 119906 (0) = 119906

0= 0 (22)

It was recently proved [14 15] that an Abel equation ofthe second kind of the normal form 119910119910

1015840

119909minus 119910 = 119891(119909) admits

an exact analytic solution in terms of known (tabulated)functions Hence we perform the solution of (21) whichconstitutes the solution of the TF equation in the phase planeas follows

120585 = ln 10038161003816100381610038161003816120585 + 2

lowast

119862

10038161003816100381610038161003816

119906 (120585) =

1

2

120585 [119873 (120585) +

1

3

]

119873

1015840

120585(120585) =

4 [119866 (120585) + 2 (119860120585 + 120585

32

)]

120585 [119873 (120585) + 43]

(23a)

1

4

[(120585 sin 120585 + cos 120585)lowast

119860(120585) + cos2120585] (2120585lowast

119860(120585) + cos 120585)

(120585

lowast

119860(120585))

3

119890120585

= 4 [119866 + 2 (119860120585 + 119861120585

32

)]

(23b)

4 Mathematical Problems in Engineering

wherelowast

119860 =

lowast

119860(120585) = the cosine integral = ci (120585)

= C + ln 120585 +infin

sum

119906=1

(minus1)119906 120585

2119906

(2119906) (2119906)

C = Eulerrsquos number = 0 5572156649015325

(23c)

Herelowast

119862 is integration constant while the function 119873(120585) isgiven implicitly in terms of the subsidiary function 119866(120585)which can be determined from (23b) and the knownmemberof (21) as in the following three cases

Case 1 (119876(120585) lt 0 (119901 lt 0)) Consider

119873

(1)

(120585) = 2radicminus

119901

3

cos 1198863

119873

(2)

(120585) = minus2radicminus

119901

3

cos119886 minus 1205873

119873

(3)

(120585) = minus2radicminus

119901

3

cos 119886 + 1205873

cos 119886 = minus119902

2radicminus (1199013)3

0 lt 119886 lt 120587

(24)

Case 2 (119876(120585) gt 0) Consider

119873(120585) =3

radicminus

119902

2

+ radic119876 +3

radicminus

119902

2

minus radic119876 (25)

Case 3 (119876(120585) = 0) Consider

119873

(1)

(120585) = 23

radicminus

119902

2

119873

(2)

(120585) = 119873

(3)

(120585) = minus3

radicminus

119902

2

(26)

In all these formulae the quantities 119876 119901 and 119902 are givenby

119876(120585) =

1

27

1199013(120585) +

1

4

1199022(120585)

119901 (120585) = minus

1198862

3

+ 119887

119902 (120585) =

2

27

1198863minus

1

3

119886119887 + 119888

119886 = minus4

119887 =

3 + 4 [119866 (120585) + (119860120585 + 119861120585

32

)]

120585

119888 = minus

4 [119866 (120585) + 2 (119860120585 + 119861120585

32

)]

120585

(27)

In order to define the type of the function 119873

(119894)

(120585)

(119894 = 1 2 3) which must be selected among the closed-formsolutions ((23a) (23b) and (23c)) to (26) as well as the valueof the subsidiary function 119866(120585) at 120585 = 0 and the value ofthe integration constant

lowast

119862 we must combine the above Abelsolutions with the boundary data

We symbolize by 120585 = ln |2lowast

119862| the substitution while byL0

any function L(120585) at 1205850= ln |2

lowast

119862| Thus functions 119906(120585) and(119873

1015840

120585)(120585) given in (23a) perform

119906 (120585) =

1

2

120585 [119873

(119894)

(120585) +

1

3

]

119873

1015840

120585

(119894)

(120585) =

4 [119866 (120585) + 2 (minus (1249) (120585) + (149) 120585

32

)]

120585 [119873

(119894)

(120585) + 43]

(28)

1199060=

1

2

ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

(119873

(119894)

0+

1

3

)

119873

1015840(119894)

1205850

(120585)

=

4 [1198660+ 2 (minus (1249) ln 1003816100381610038161003816

10038162

lowast

119862

10038161003816100381610038161003816+ (149) (ln 1003816100381610038161003816

10038162

lowast

119862

10038161003816100381610038161003816)

32

)]

ln 100381610038161003816100381610038162

lowast

119862

10038161003816100381610038161003816[119873

(119894)

0+ 43]

(29)

On the other hand the Abel nonlinear ODE (21) at 1205850

becomes

1 +

minus (1249) ln 100381610038161003816100381610038162

lowast

119862

10038161003816100381610038161003816+ (149) (ln 1003816100381610038161003816

10038162

lowast

119862

10038161003816100381610038161003816)

32

1199060

= 1 +

minus (1249) ln 100381610038161003816100381610038162

lowast

119862

10038161003816100381610038161003816+ (149) (ln 1003816100381610038161003816

10038162

lowast

119862

10038161003816100381610038161003816)

32

ln 100381610038161003816100381610038162

lowast

119862

10038161003816100381610038161003816(119873

(119894)

0+ 43)

=

1199061015840

1205850

120585

1015840

1205850

(30)

Combination of the above equations results in the followingquadratic to119873

0equation

119873

(119894)

0

2

+ 119873

(119894)

0+

4

9

+[

[

[

1 minus

1

lowast

119862 ln 100381610038161003816100381610038162

lowast

119862

10038161003816100381610038161003816

]

]

]

sdot [minus21198660+ 2(minus

12

49

ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

+

1

49

(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)

32

)]

= 0

(31)

including the value of 1198660and the constant on integration

lowast

119862

Mathematical Problems in Engineering 5

Similarly by (23b) one gets

32 1198660+ [2(minus

12

49

ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

+

1

49

ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

32

)]

lowast

119860

3

0

= [ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

sin(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

) + cos(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)]

lowast

1198600

+ cos2 (ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)(2 ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

lowast

1198600

+ cos(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

))

(32)

that constitute a second equation including the value of 1198660

and the constant of integrationlowast

119862 Also in this equation bymeans of (23c) we have

lowast

1198600=

lowast

119860(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

) = ci(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)

= C + ln(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

) +

infin

sum

119896=1

(minus1)119896ln 100381610038161003816100381610038162

lowast

119862

10038161003816100381610038161003816

2119896

(2119896) (2119896)

C = Eulerrsquos number asymp 05572

(33)

Summarizing from (19) and (20) one concludes that since bythe first boundary conditions 119910(0) = 1 for 119909 rarr 0 rArr 119910 rarr 1

then for 119909 rarr 0 rArr 1199101015840

119909rarr 0 or 1199101015840

119909rarr

lowast

119861 wherelowast

119861 is a fixedvalue Thus combining (23a) (23b) and (23c) together withthe first of (28) and the Abel nonlinear ODE (21) one maywrite that for 119909 rarr 0 rArr 120585

0rarr 0 119906

0rarr 0 119873

0rarr 0

Therefore by making use of the first boundary conditiongiven in (4) (119909 rarr 0 119910 rarr 1) as well as of all the aboveobservations the solutions of (31) and (32) perform the valuesfor 119866

0and 2

lowast

119862 and they can be written in the followingalgebraic forms

minus1 plusmn radic1 minus 4F (1198660

lowast

119862)

2

= 0(34a)

H(

lowast

119862) = 1198660 (34b)

where

F(1198660

lowast

119862) =

4

9

+[

[

[

1 minus

1

lowast

119862 ln 100381610038161003816100381610038162

lowast

119862

10038161003816100381610038161003816

]

]

]

[minus21198660

+ 2(minus

12

4

ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

+

1

49

(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)

32

)]

(35a)

H(

lowast

119862)

=

1

32

[ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

sin(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

) + cos(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)]

sdot

lowast

11986000+ cos2 (ln

1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

) [2 ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

lowast

1198600

+ cos(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)]

lowast

1198600

lowast

119862 as in (33)

(35b)

Thus we are now able to estimate 1199010 1199020 and 119876

0given by

(27) in terms of the known valueslowast

119862 and 1198660 This estimation

furnishes the value of1198760and therefore the type of the solution

119873

(119894)

(120585) (119894 = 1 2 3) near the origin 1205850among the formulae (24)

to (26)From the above evaluation it is concluded that the closed

form solutions of the Abel nonlinear ODE (21) given as

119906(119894)(120585) =

1

2

120585 [119873

(119894)

(120585) +

1

3

]

120585 = ln1003816100381610038161003816100381610038161003816

120585 + 2

lowast

1198621

1003816100381610038161003816100381610038161003816

(119894 = 1 2 3)

(36)

have been completely determinedSince we obtained the above closed- form solutions of the

intermediate Abel nonlinear ODE (21) the solutions to theoriginal TF equation (3) can be determined as follows By (19)and (21) we perform

119909 =

1

713

(

120585

119910

)

13

1199101015840

119909=

1

1199094119906 minus 3

119910

119909

(37)

while taking the total differential in (19) we extract

119889120585 = 211199092119910119889119909 + 7119909

3119889119910

lArrrArr 1205851015840

119909=

119889120585

119889119909

= 71199092(1199091199101015840

119909+ 3119910) = 7

119906

119909

(38)

where 119906 = 119906(120585 + 2

lowast

1198621) denotes the known solution of the

Abel nonlinear ODE (18) Using (34a) (34b) and (37) onewrites

1199101015840

120585=

1199101015840

119909

1205851015840

119909

= 743(

119910

120585

) 11990643

minus 3119910713(

119910

120585

)

13

=

119906 (120585)

120585

119910 minus

3

71205854311991043

(39)

6 Mathematical Problems in Engineering

that is the Bernoulli equation

11991013

=

exp ((1120585) int (119906 (120585) 120585) 119889120585)

119862 + (17) int exp [((1120585) int (119906 (120585) 120585) 119889120585)] 119889120585

119909 =

1

713

(

120585

119910

)

13

119909 ge 0 119910 ge 0 120585 = parameter (120585 ge 0)

119862 = second constant of integration

(40)

The last step of the above analysis concerns the calculationof the second constant of integration 119862 Thus from the firstof the boundary conditions (4) and the parametric solution(40) one writes

1199091997888rarr 0

1199101997888rarr 1

1205851997888rarr 0

lim1205851rarr0

(11991013)

=

lim1205851rarr0

exp ((1120585) int (119906 (120585) 120585) 119889120585)

lim1205851rarr0

119862 + (17) int exp [((1120585) int (119906 (120585) 120585) 119889120585)] 119889120585

(41)

and estimates the constant of integration 119862 By the second ofthe boundary conditions (4) one writes 119909

2rarr +infin 119910

2rarr 0

and 120585 rarr 1205852 Again through the parametric solution (40) one

writes

lim120585rarr1205852

[exp [1120585

int

119906 (120585)

120585

119889120585]] (42a)

and evaluates the value of 1205852 The evaluation of the second

constant of integration can be obtained again through theboundary condition 1119909

1rarr 0 rArr 119910

1rarr infin that is

lim120585rarr1205851

119862 +

1

7

int exp [(minus1120585

int

119906 (120585)

120585

119889120585)

1

12058513119889120585]

= 0

(42b)

Equations (41) and (42a) (42b) complete the solutions of theproblem under consideration that is the construction of aclosed-form parametric solution of the TF equation (3)

It is worthwhile to remark that since the solution isgiven in a closed form the mathematical methodology beingdeveloped results in the ad hoc definition of the TF function119910in approximate power series with 1199101015840

119909(0) =

lowast

119861 (lowast

119861 = a factor thatmust be determined) and with boundary conditions given by1199100(0) = 119909

0= 1 119910

1rarr 0 as 119909 rarr infin The value 1199101015840

119909(0) =

lowast

119861

can be evaluated by way of the already constructed closed-form solution This is a very interesting result in accordancewith previous results presented by Bush and Caldwell [17]Feynman et al [18] Coulson and March [19] Kobayashi etal [20] and Kobayashi [21]

5 Closed-Form Solutions of the LBEquation in the Physical Plane

It was already proved (Section 3) that the LB nonlinear ODE(1) has been reduced to an Abel ODE of the second kindof the normal form (15) It was also proved that this Abelequation admits exact analytic solutions given as in case ofthe TF nonlinear ODE by (24) to (26) Thus the solutionof the LB equation (1) in the phase plane is given by theformulae (24) to (26) if instead of the parenthesis (119860120585 +11986112058532) the quantity (120585minus124minus1205854) is introducedThe solution

of the original equation (1) in the physical plane is giventhrough the combination of the above prescribed formulaein combination with (17) As in case of the TF equationthe whole problem includes two constants of integration

lowast

119862

and 119862 which are to be determined by using the convenientboundary data of the problem under consideration In orderto define simultaneously the type of the function 119873

(119894)

(120585)

(119894 = 1 2 3) which must be selected in accordance with themodified previously developed formulae (24) to (26) andthus define the modified function 119866(120585) and the constant ofintegration

lowast

119862 wemust combine the aboveAbel solutionswiththe prescribed convenient boundary data

Based on what was previously developed in Section 3for the original (LB) equation (1) the following boundaryconditions hold true

at 119909 = 1199090

997904rArr 119910 = 1199100= ℓ

1199101015840

119909= 1199101015840

1199090

= 119898

(43)

where ℓ and119898 are arbitrary given values provided119898 = 0Based on these boundary conditions the substitution

(8) furnishes 119901(1199100) = 119901

0= 119898 while transformation (10)

furnishes 119908(1199100) = 119908

0= 11991013

01199010= ℓ13119898 too Thus the sub-

stitutions (13) (15) and the Abel nonlinear ODE (16) performthe following relations

1199080= 119899 (120585

0) = 1198990= ℓ13119898

1205850= 11991043

0= ℓ43

1198990= minus119911 (120585

0) = minus119911

0

997904rArr 1199110= minusℓ13119898

1199111015840

1205850

= minus

(14) (ℓminus23

minus ℓ43)

ℓ13119898

+ 1 = 1 minus

1

4

ℓminus1minus ℓ

119898

(44)

We are now able to evaluate all the before mentionedquantities Indeed (23a) and (23b) for 120585

0= ℓ43 1199110= minusℓ13119898

Mathematical Problems in Engineering 7

1199111015840

1205850

= 1 minus (ℓminus1minus ℓ)4119898 furnish the following two equations

1198730= minus(

2ℓ13119898

ℓ43

+ 4

lowast

119862

+

1

3

)

81198660+ 2 (ℓ

43minus ℓ13)P ln |P| [ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 1003816100381610038161003816

10038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C]

4 [ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 100381610038161003816100381610038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C] minus cos (ln |P|)

=

1

4

[ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 100381610038161003816100381610038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C] [ln |P| sin (ln |P|) + cos (ln |P|)]

[ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 100381610038161003816100381610038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C]

2

(ln |P|)2

+

cos2 (ln |P|)

[ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 100381610038161003816100381610038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C]

2

(ln |P|)2

P = ℓ43

+ 2

lowast

119862 gt 0 2

lowast

119862 + 11989743

gt 0 2

lowast

119862 lt 1

(45)

where1198730and 119866

0are the values of119873(120585) and 119866(120585) at 120585 = 120585

0=

ℓ43 respectively On the other hand the formula for 1198731015840

120585in

(23a) at 120585 = 1205850results in

119873

1015840

1205850

=

1

(1198730+ 43)

4 [1198660+ (12) (ℓ

minus23minus ℓ43)]

(ℓ43

+ 4

lowast

119862)

2 (46)

while by the first of (23a) and (44) we have

1199111015840

1205850

=

1

2

119873

1015840

1205850

(1205850+ 4

lowast

119862) +

1

2

(1198730+

1

3

) (47)

or equivalently

1199111015840

1205850

=

2 [1198660+ (12) (ℓ

minus23minus ℓ43)]

(1198730+ 43) (ℓ

43+ 4

lowast

119862)

+

1

2

(1198730+

1

3

) (48)

The last relation (48) in combination together with (44)permits us to write

2 [1198660+ (12) (ℓ

minus23minus ℓ43)]

(1198730+ (43)) (ℓ

43+ 4

lowast

119862)

+

1

2

(1198730+

1

3

)

= 1 minus

1

4

ℓminus1minus ℓ

119898

(49)

The above three equations (46) to (49) constitute a nonlinear(transcendental) system for119873

01198660 and 120582 by means of which

one calculates these unknowns in terms of the parameters ℓand 119898 Thus the discriminant 119876

0(119876 at 120585 = 120585

0= ℓ43) given

in (27) is also known the fact that permits us to define thetype of the function119873(119894)(120585) for 120585 gt 120585

0= ℓ43 (24) to (26)This

completes the solution of the problem under consideration

which is the construction of the solution of the LB equationin the phase-plane (and thus in the physical plane) in themain interval [120585

0= ℓ43 120585] If this solution is not unique

inside [ℓ43 120585] then we follow step by step the methodologydeveloped in [15]

6 Conclusions

By a series of admissible functional transformations wereduce the nonlinear TF and LB equations to Abelrsquos equa-tions of the second kind of the normal form of (16) and(18) respectively These equations do not admit closed-formsolutions in terms of known (tabulated) functions Thisunsolvability is due to the fact that only very special formsof this kind of (16) and (18) can be solved in parametric form[4] Our goal is the development of the construction of theexact analytic solutions of the above equations based on amathematical technique leading to the derivation of closeform solutions for the Abel equation of the second kind of thenormal form [3 4] The proposed methodology constitutesthe intermediate integral of the TF and LB equations inthe phase and physical planes The reduction procedure inthe paper and the constructed solutions are very generaland can be applied to a large number of nonlinear ODEsin mathematical physics and nonlinear mechanics Also themethod employed is exact in accordance with the boundaryconditions of each problem under consideration and guidesus to the a priori approximate construction of the solutions inpower series some coefficients of which must be evaluated

With the proposed procedure the exact value of first orderderivative of TF equation is derived at 119909 = 0 In particularthe solution for the TF equation is finally constructed bycalculating 4 constants from the specific boundary conditions(4) (see (34a) (34b) (41) (42a) and (42b)) Equations(40) constitute the parametric solutions of the TF problemWith our method essentially a different way of calculating

8 Mathematical Problems in Engineering

1199101015840

119909(0) =

lowast

119861 (Section 4) is proposed with reference to themethods proposed in [17ndash21] where the value of

lowast

119861 has beencalculated having kept the solution from approximate seriesin which their coefficients had been calculated from theboundary conditions In conclusion in this paper an analyticmethod is developed that does not contradict previous resultsbut it is a different way of calculating them

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] I Langmuir and K B Blodgett ldquoCurrents limited by spacecharge between coaxial cylindersrdquo Physical Review vol 22 no4 pp 347ndash356 1923

[2] H T Davis Introduction to Nonlinear Differential and IntegralEquations Dover New York NY USA 1962

[3] E Kamke Differentialgleichungen I Gewohnliche Differential-gleichungen and II Partielle Differentialgleichungen Akademis-che Verlagsgesellschaft Leipzig Germany 1962

[4] A D Polyanin and V F Zaitsev Handbook of Exact Solutionsfor Ordinary Differential Equations CRCPress Boca Raton FlaUSA 1999

[5] K Horvath J N Fairchild K Kaczmarski and G Guio-chon ldquoMartin-Synge algorithm for the solution of equilibrium-dispersive model of liquid chromatographyrdquo Journal of Chro-matography A vol 1217 no 52 pp 8127ndash8135 2010

[6] Y Shang and X Zheng ldquoThe first-integral method and abun-dant explicit exact solutions to the zakharov equationsrdquo Journalof Applied Mathematics vol 2012 Article ID 818345 16 pages2012

[7] D B Graham and I H Cairns ldquoConstraints on the formationand structure of langmuir eigenmodes in the solar windrdquoPhysical Review Letters vol 111 no 12 Article ID 121101 2013

[8] L H Thomas ldquoThe calculation of atomic fieldsrdquoMathematicalProceedings of the Cambridge Philosophical Society vol 23 no5 pp 542ndash548 1927

[9] E Fermi ldquoStatistical method of investigating electrons inatomsrdquo Zeitschrift fur Physik vol 48 pp 73ndash79 1927

[10] T Tsurumi and MWadati ldquoDynamics of magnetically trappedboson-fermion mixturesrdquo Journal of the Physical Society ofJapan vol 69 no 1 pp 97ndash103 2000

[11] S Liao ldquoAn explicit analytic solution to the Thomas-Fermiequationrdquo Applied Mathematics and Computation vol 144 no2-3 pp 495ndash506 2003

[12] S Abbasbandy and C Bervillier ldquoAnalytic continuation ofTaylor series and the boundary value problems of some nonlin-ear ordinary differential equationsrdquo Applied Mathematics andComputation vol 218 no 5 pp 2178ndash2199 2011

[13] K Ourabah and M Tribeche ldquoRelativistic formulation of thegeneralized nonextensiveThomas-FermimodelrdquoPhysicaA vol393 pp 470ndash474 2014

[14] D E Panayotounakos ldquoExact analytic solutions of unsolvableclasses of first and second order nonlinear ODEs (part I Abelrsquosequations)rdquo Applied Mathematics Letters vol 18 no 2 pp 155ndash162 2005

[15] D E Panayotounakos N B Sotiropoulos A B Sotiropoulouand N D Panayotounakou ldquoExact analytic solutions of non-linear boundary value problems in fluid mechanics (Blasiusequations)rdquo Journal of Mathematical Physics vol 46 no 3Article ID 033101 2005

[16] D E Panayotounakos E ETheotokoglou and M P MarkakisldquoExact analytic solutions for the unforded damped duffingnonlinear oscillatorrdquo Comptes Rendus Mechanics Journal vol334 pp 311ndash316 2006

[17] V Bush and S H Caldwell ldquoThomas-Fermi equation solutionby the differential analyzerrdquo Physical Review vol 38 no 10 pp1898ndash1901 1931

[18] R P Feynman N Metropolis and E Teller ldquoEquations of stateof elements based on the generalized fermi-thomas theoryrdquoPhysical Review vol 75 no 10 pp 1561ndash1573 1949

[19] C A Coulson and N H March ldquoMomenta in atoms usingthe Thomas-Fermi methodrdquo Proceedings of the Physical SocietySection A vol 63 no 4 pp 367ndash374 1950

[20] S Kobayashi TMatsukuma S Nagai and K Umeda ldquoAccuratevalue of the initial slope of the ordinary TF functionrdquo Journal ofthe Physical Society of Japan vol 10 no 9 pp 759ndash762 1955

[21] S Kobayashi ldquoSome coefficients of the series expansion of theTFD functionrdquo Journal of the Physical Society of Japan vol 10no 9 pp 824ndash825 1955

[22] S A Sommerfeld ldquoIntegrazione Asintotica dellrsquoEquazione Dif-ferenziali diThomas-FermirdquoAccademia dei Lincei Atti-Rendic-onte vol 4 pp 85ndash110 1932

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

4 Mathematical Problems in Engineering

wherelowast

119860 =

lowast

119860(120585) = the cosine integral = ci (120585)

= C + ln 120585 +infin

sum

119906=1

(minus1)119906 120585

2119906

(2119906) (2119906)

C = Eulerrsquos number = 0 5572156649015325

(23c)

Herelowast

119862 is integration constant while the function 119873(120585) isgiven implicitly in terms of the subsidiary function 119866(120585)which can be determined from (23b) and the knownmemberof (21) as in the following three cases

Case 1 (119876(120585) lt 0 (119901 lt 0)) Consider

119873

(1)

(120585) = 2radicminus

119901

3

cos 1198863

119873

(2)

(120585) = minus2radicminus

119901

3

cos119886 minus 1205873

119873

(3)

(120585) = minus2radicminus

119901

3

cos 119886 + 1205873

cos 119886 = minus119902

2radicminus (1199013)3

0 lt 119886 lt 120587

(24)

Case 2 (119876(120585) gt 0) Consider

119873(120585) =3

radicminus

119902

2

+ radic119876 +3

radicminus

119902

2

minus radic119876 (25)

Case 3 (119876(120585) = 0) Consider

119873

(1)

(120585) = 23

radicminus

119902

2

119873

(2)

(120585) = 119873

(3)

(120585) = minus3

radicminus

119902

2

(26)

In all these formulae the quantities 119876 119901 and 119902 are givenby

119876(120585) =

1

27

1199013(120585) +

1

4

1199022(120585)

119901 (120585) = minus

1198862

3

+ 119887

119902 (120585) =

2

27

1198863minus

1

3

119886119887 + 119888

119886 = minus4

119887 =

3 + 4 [119866 (120585) + (119860120585 + 119861120585

32

)]

120585

119888 = minus

4 [119866 (120585) + 2 (119860120585 + 119861120585

32

)]

120585

(27)

In order to define the type of the function 119873

(119894)

(120585)

(119894 = 1 2 3) which must be selected among the closed-formsolutions ((23a) (23b) and (23c)) to (26) as well as the valueof the subsidiary function 119866(120585) at 120585 = 0 and the value ofthe integration constant

lowast

119862 we must combine the above Abelsolutions with the boundary data

We symbolize by 120585 = ln |2lowast

119862| the substitution while byL0

any function L(120585) at 1205850= ln |2

lowast

119862| Thus functions 119906(120585) and(119873

1015840

120585)(120585) given in (23a) perform

119906 (120585) =

1

2

120585 [119873

(119894)

(120585) +

1

3

]

119873

1015840

120585

(119894)

(120585) =

4 [119866 (120585) + 2 (minus (1249) (120585) + (149) 120585

32

)]

120585 [119873

(119894)

(120585) + 43]

(28)

1199060=

1

2

ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

(119873

(119894)

0+

1

3

)

119873

1015840(119894)

1205850

(120585)

=

4 [1198660+ 2 (minus (1249) ln 1003816100381610038161003816

10038162

lowast

119862

10038161003816100381610038161003816+ (149) (ln 1003816100381610038161003816

10038162

lowast

119862

10038161003816100381610038161003816)

32

)]

ln 100381610038161003816100381610038162

lowast

119862

10038161003816100381610038161003816[119873

(119894)

0+ 43]

(29)

On the other hand the Abel nonlinear ODE (21) at 1205850

becomes

1 +

minus (1249) ln 100381610038161003816100381610038162

lowast

119862

10038161003816100381610038161003816+ (149) (ln 1003816100381610038161003816

10038162

lowast

119862

10038161003816100381610038161003816)

32

1199060

= 1 +

minus (1249) ln 100381610038161003816100381610038162

lowast

119862

10038161003816100381610038161003816+ (149) (ln 1003816100381610038161003816

10038162

lowast

119862

10038161003816100381610038161003816)

32

ln 100381610038161003816100381610038162

lowast

119862

10038161003816100381610038161003816(119873

(119894)

0+ 43)

=

1199061015840

1205850

120585

1015840

1205850

(30)

Combination of the above equations results in the followingquadratic to119873

0equation

119873

(119894)

0

2

+ 119873

(119894)

0+

4

9

+[

[

[

1 minus

1

lowast

119862 ln 100381610038161003816100381610038162

lowast

119862

10038161003816100381610038161003816

]

]

]

sdot [minus21198660+ 2(minus

12

49

ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

+

1

49

(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)

32

)]

= 0

(31)

including the value of 1198660and the constant on integration

lowast

119862

Mathematical Problems in Engineering 5

Similarly by (23b) one gets

32 1198660+ [2(minus

12

49

ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

+

1

49

ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

32

)]

lowast

119860

3

0

= [ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

sin(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

) + cos(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)]

lowast

1198600

+ cos2 (ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)(2 ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

lowast

1198600

+ cos(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

))

(32)

that constitute a second equation including the value of 1198660

and the constant of integrationlowast

119862 Also in this equation bymeans of (23c) we have

lowast

1198600=

lowast

119860(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

) = ci(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)

= C + ln(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

) +

infin

sum

119896=1

(minus1)119896ln 100381610038161003816100381610038162

lowast

119862

10038161003816100381610038161003816

2119896

(2119896) (2119896)

C = Eulerrsquos number asymp 05572

(33)

Summarizing from (19) and (20) one concludes that since bythe first boundary conditions 119910(0) = 1 for 119909 rarr 0 rArr 119910 rarr 1

then for 119909 rarr 0 rArr 1199101015840

119909rarr 0 or 1199101015840

119909rarr

lowast

119861 wherelowast

119861 is a fixedvalue Thus combining (23a) (23b) and (23c) together withthe first of (28) and the Abel nonlinear ODE (21) one maywrite that for 119909 rarr 0 rArr 120585

0rarr 0 119906

0rarr 0 119873

0rarr 0

Therefore by making use of the first boundary conditiongiven in (4) (119909 rarr 0 119910 rarr 1) as well as of all the aboveobservations the solutions of (31) and (32) perform the valuesfor 119866

0and 2

lowast

119862 and they can be written in the followingalgebraic forms

minus1 plusmn radic1 minus 4F (1198660

lowast

119862)

2

= 0(34a)

H(

lowast

119862) = 1198660 (34b)

where

F(1198660

lowast

119862) =

4

9

+[

[

[

1 minus

1

lowast

119862 ln 100381610038161003816100381610038162

lowast

119862

10038161003816100381610038161003816

]

]

]

[minus21198660

+ 2(minus

12

4

ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

+

1

49

(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)

32

)]

(35a)

H(

lowast

119862)

=

1

32

[ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

sin(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

) + cos(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)]

sdot

lowast

11986000+ cos2 (ln

1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

) [2 ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

lowast

1198600

+ cos(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)]

lowast

1198600

lowast

119862 as in (33)

(35b)

Thus we are now able to estimate 1199010 1199020 and 119876

0given by

(27) in terms of the known valueslowast

119862 and 1198660 This estimation

furnishes the value of1198760and therefore the type of the solution

119873

(119894)

(120585) (119894 = 1 2 3) near the origin 1205850among the formulae (24)

to (26)From the above evaluation it is concluded that the closed

form solutions of the Abel nonlinear ODE (21) given as

119906(119894)(120585) =

1

2

120585 [119873

(119894)

(120585) +

1

3

]

120585 = ln1003816100381610038161003816100381610038161003816

120585 + 2

lowast

1198621

1003816100381610038161003816100381610038161003816

(119894 = 1 2 3)

(36)

have been completely determinedSince we obtained the above closed- form solutions of the

intermediate Abel nonlinear ODE (21) the solutions to theoriginal TF equation (3) can be determined as follows By (19)and (21) we perform

119909 =

1

713

(

120585

119910

)

13

1199101015840

119909=

1

1199094119906 minus 3

119910

119909

(37)

while taking the total differential in (19) we extract

119889120585 = 211199092119910119889119909 + 7119909

3119889119910

lArrrArr 1205851015840

119909=

119889120585

119889119909

= 71199092(1199091199101015840

119909+ 3119910) = 7

119906

119909

(38)

where 119906 = 119906(120585 + 2

lowast

1198621) denotes the known solution of the

Abel nonlinear ODE (18) Using (34a) (34b) and (37) onewrites

1199101015840

120585=

1199101015840

119909

1205851015840

119909

= 743(

119910

120585

) 11990643

minus 3119910713(

119910

120585

)

13

=

119906 (120585)

120585

119910 minus

3

71205854311991043

(39)

6 Mathematical Problems in Engineering

that is the Bernoulli equation

11991013

=

exp ((1120585) int (119906 (120585) 120585) 119889120585)

119862 + (17) int exp [((1120585) int (119906 (120585) 120585) 119889120585)] 119889120585

119909 =

1

713

(

120585

119910

)

13

119909 ge 0 119910 ge 0 120585 = parameter (120585 ge 0)

119862 = second constant of integration

(40)

The last step of the above analysis concerns the calculationof the second constant of integration 119862 Thus from the firstof the boundary conditions (4) and the parametric solution(40) one writes

1199091997888rarr 0

1199101997888rarr 1

1205851997888rarr 0

lim1205851rarr0

(11991013)

=

lim1205851rarr0

exp ((1120585) int (119906 (120585) 120585) 119889120585)

lim1205851rarr0

119862 + (17) int exp [((1120585) int (119906 (120585) 120585) 119889120585)] 119889120585

(41)

and estimates the constant of integration 119862 By the second ofthe boundary conditions (4) one writes 119909

2rarr +infin 119910

2rarr 0

and 120585 rarr 1205852 Again through the parametric solution (40) one

writes

lim120585rarr1205852

[exp [1120585

int

119906 (120585)

120585

119889120585]] (42a)

and evaluates the value of 1205852 The evaluation of the second

constant of integration can be obtained again through theboundary condition 1119909

1rarr 0 rArr 119910

1rarr infin that is

lim120585rarr1205851

119862 +

1

7

int exp [(minus1120585

int

119906 (120585)

120585

119889120585)

1

12058513119889120585]

= 0

(42b)

Equations (41) and (42a) (42b) complete the solutions of theproblem under consideration that is the construction of aclosed-form parametric solution of the TF equation (3)

It is worthwhile to remark that since the solution isgiven in a closed form the mathematical methodology beingdeveloped results in the ad hoc definition of the TF function119910in approximate power series with 1199101015840

119909(0) =

lowast

119861 (lowast

119861 = a factor thatmust be determined) and with boundary conditions given by1199100(0) = 119909

0= 1 119910

1rarr 0 as 119909 rarr infin The value 1199101015840

119909(0) =

lowast

119861

can be evaluated by way of the already constructed closed-form solution This is a very interesting result in accordancewith previous results presented by Bush and Caldwell [17]Feynman et al [18] Coulson and March [19] Kobayashi etal [20] and Kobayashi [21]

5 Closed-Form Solutions of the LBEquation in the Physical Plane

It was already proved (Section 3) that the LB nonlinear ODE(1) has been reduced to an Abel ODE of the second kindof the normal form (15) It was also proved that this Abelequation admits exact analytic solutions given as in case ofthe TF nonlinear ODE by (24) to (26) Thus the solutionof the LB equation (1) in the phase plane is given by theformulae (24) to (26) if instead of the parenthesis (119860120585 +11986112058532) the quantity (120585minus124minus1205854) is introducedThe solution

of the original equation (1) in the physical plane is giventhrough the combination of the above prescribed formulaein combination with (17) As in case of the TF equationthe whole problem includes two constants of integration

lowast

119862

and 119862 which are to be determined by using the convenientboundary data of the problem under consideration In orderto define simultaneously the type of the function 119873

(119894)

(120585)

(119894 = 1 2 3) which must be selected in accordance with themodified previously developed formulae (24) to (26) andthus define the modified function 119866(120585) and the constant ofintegration

lowast

119862 wemust combine the aboveAbel solutionswiththe prescribed convenient boundary data

Based on what was previously developed in Section 3for the original (LB) equation (1) the following boundaryconditions hold true

at 119909 = 1199090

997904rArr 119910 = 1199100= ℓ

1199101015840

119909= 1199101015840

1199090

= 119898

(43)

where ℓ and119898 are arbitrary given values provided119898 = 0Based on these boundary conditions the substitution

(8) furnishes 119901(1199100) = 119901

0= 119898 while transformation (10)

furnishes 119908(1199100) = 119908

0= 11991013

01199010= ℓ13119898 too Thus the sub-

stitutions (13) (15) and the Abel nonlinear ODE (16) performthe following relations

1199080= 119899 (120585

0) = 1198990= ℓ13119898

1205850= 11991043

0= ℓ43

1198990= minus119911 (120585

0) = minus119911

0

997904rArr 1199110= minusℓ13119898

1199111015840

1205850

= minus

(14) (ℓminus23

minus ℓ43)

ℓ13119898

+ 1 = 1 minus

1

4

ℓminus1minus ℓ

119898

(44)

We are now able to evaluate all the before mentionedquantities Indeed (23a) and (23b) for 120585

0= ℓ43 1199110= minusℓ13119898

Mathematical Problems in Engineering 7

1199111015840

1205850

= 1 minus (ℓminus1minus ℓ)4119898 furnish the following two equations

1198730= minus(

2ℓ13119898

ℓ43

+ 4

lowast

119862

+

1

3

)

81198660+ 2 (ℓ

43minus ℓ13)P ln |P| [ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 1003816100381610038161003816

10038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C]

4 [ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 100381610038161003816100381610038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C] minus cos (ln |P|)

=

1

4

[ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 100381610038161003816100381610038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C] [ln |P| sin (ln |P|) + cos (ln |P|)]

[ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 100381610038161003816100381610038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C]

2

(ln |P|)2

+

cos2 (ln |P|)

[ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 100381610038161003816100381610038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C]

2

(ln |P|)2

P = ℓ43

+ 2

lowast

119862 gt 0 2

lowast

119862 + 11989743

gt 0 2

lowast

119862 lt 1

(45)

where1198730and 119866

0are the values of119873(120585) and 119866(120585) at 120585 = 120585

0=

ℓ43 respectively On the other hand the formula for 1198731015840

120585in

(23a) at 120585 = 1205850results in

119873

1015840

1205850

=

1

(1198730+ 43)

4 [1198660+ (12) (ℓ

minus23minus ℓ43)]

(ℓ43

+ 4

lowast

119862)

2 (46)

while by the first of (23a) and (44) we have

1199111015840

1205850

=

1

2

119873

1015840

1205850

(1205850+ 4

lowast

119862) +

1

2

(1198730+

1

3

) (47)

or equivalently

1199111015840

1205850

=

2 [1198660+ (12) (ℓ

minus23minus ℓ43)]

(1198730+ 43) (ℓ

43+ 4

lowast

119862)

+

1

2

(1198730+

1

3

) (48)

The last relation (48) in combination together with (44)permits us to write

2 [1198660+ (12) (ℓ

minus23minus ℓ43)]

(1198730+ (43)) (ℓ

43+ 4

lowast

119862)

+

1

2

(1198730+

1

3

)

= 1 minus

1

4

ℓminus1minus ℓ

119898

(49)

The above three equations (46) to (49) constitute a nonlinear(transcendental) system for119873

01198660 and 120582 by means of which

one calculates these unknowns in terms of the parameters ℓand 119898 Thus the discriminant 119876

0(119876 at 120585 = 120585

0= ℓ43) given

in (27) is also known the fact that permits us to define thetype of the function119873(119894)(120585) for 120585 gt 120585

0= ℓ43 (24) to (26)This

completes the solution of the problem under consideration

which is the construction of the solution of the LB equationin the phase-plane (and thus in the physical plane) in themain interval [120585

0= ℓ43 120585] If this solution is not unique

inside [ℓ43 120585] then we follow step by step the methodologydeveloped in [15]

6 Conclusions

By a series of admissible functional transformations wereduce the nonlinear TF and LB equations to Abelrsquos equa-tions of the second kind of the normal form of (16) and(18) respectively These equations do not admit closed-formsolutions in terms of known (tabulated) functions Thisunsolvability is due to the fact that only very special formsof this kind of (16) and (18) can be solved in parametric form[4] Our goal is the development of the construction of theexact analytic solutions of the above equations based on amathematical technique leading to the derivation of closeform solutions for the Abel equation of the second kind of thenormal form [3 4] The proposed methodology constitutesthe intermediate integral of the TF and LB equations inthe phase and physical planes The reduction procedure inthe paper and the constructed solutions are very generaland can be applied to a large number of nonlinear ODEsin mathematical physics and nonlinear mechanics Also themethod employed is exact in accordance with the boundaryconditions of each problem under consideration and guidesus to the a priori approximate construction of the solutions inpower series some coefficients of which must be evaluated

With the proposed procedure the exact value of first orderderivative of TF equation is derived at 119909 = 0 In particularthe solution for the TF equation is finally constructed bycalculating 4 constants from the specific boundary conditions(4) (see (34a) (34b) (41) (42a) and (42b)) Equations(40) constitute the parametric solutions of the TF problemWith our method essentially a different way of calculating

8 Mathematical Problems in Engineering

1199101015840

119909(0) =

lowast

119861 (Section 4) is proposed with reference to themethods proposed in [17ndash21] where the value of

lowast

119861 has beencalculated having kept the solution from approximate seriesin which their coefficients had been calculated from theboundary conditions In conclusion in this paper an analyticmethod is developed that does not contradict previous resultsbut it is a different way of calculating them

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] I Langmuir and K B Blodgett ldquoCurrents limited by spacecharge between coaxial cylindersrdquo Physical Review vol 22 no4 pp 347ndash356 1923

[2] H T Davis Introduction to Nonlinear Differential and IntegralEquations Dover New York NY USA 1962

[3] E Kamke Differentialgleichungen I Gewohnliche Differential-gleichungen and II Partielle Differentialgleichungen Akademis-che Verlagsgesellschaft Leipzig Germany 1962

[4] A D Polyanin and V F Zaitsev Handbook of Exact Solutionsfor Ordinary Differential Equations CRCPress Boca Raton FlaUSA 1999

[5] K Horvath J N Fairchild K Kaczmarski and G Guio-chon ldquoMartin-Synge algorithm for the solution of equilibrium-dispersive model of liquid chromatographyrdquo Journal of Chro-matography A vol 1217 no 52 pp 8127ndash8135 2010

[6] Y Shang and X Zheng ldquoThe first-integral method and abun-dant explicit exact solutions to the zakharov equationsrdquo Journalof Applied Mathematics vol 2012 Article ID 818345 16 pages2012

[7] D B Graham and I H Cairns ldquoConstraints on the formationand structure of langmuir eigenmodes in the solar windrdquoPhysical Review Letters vol 111 no 12 Article ID 121101 2013

[8] L H Thomas ldquoThe calculation of atomic fieldsrdquoMathematicalProceedings of the Cambridge Philosophical Society vol 23 no5 pp 542ndash548 1927

[9] E Fermi ldquoStatistical method of investigating electrons inatomsrdquo Zeitschrift fur Physik vol 48 pp 73ndash79 1927

[10] T Tsurumi and MWadati ldquoDynamics of magnetically trappedboson-fermion mixturesrdquo Journal of the Physical Society ofJapan vol 69 no 1 pp 97ndash103 2000

[11] S Liao ldquoAn explicit analytic solution to the Thomas-Fermiequationrdquo Applied Mathematics and Computation vol 144 no2-3 pp 495ndash506 2003

[12] S Abbasbandy and C Bervillier ldquoAnalytic continuation ofTaylor series and the boundary value problems of some nonlin-ear ordinary differential equationsrdquo Applied Mathematics andComputation vol 218 no 5 pp 2178ndash2199 2011

[13] K Ourabah and M Tribeche ldquoRelativistic formulation of thegeneralized nonextensiveThomas-FermimodelrdquoPhysicaA vol393 pp 470ndash474 2014

[14] D E Panayotounakos ldquoExact analytic solutions of unsolvableclasses of first and second order nonlinear ODEs (part I Abelrsquosequations)rdquo Applied Mathematics Letters vol 18 no 2 pp 155ndash162 2005

[15] D E Panayotounakos N B Sotiropoulos A B Sotiropoulouand N D Panayotounakou ldquoExact analytic solutions of non-linear boundary value problems in fluid mechanics (Blasiusequations)rdquo Journal of Mathematical Physics vol 46 no 3Article ID 033101 2005

[16] D E Panayotounakos E ETheotokoglou and M P MarkakisldquoExact analytic solutions for the unforded damped duffingnonlinear oscillatorrdquo Comptes Rendus Mechanics Journal vol334 pp 311ndash316 2006

[17] V Bush and S H Caldwell ldquoThomas-Fermi equation solutionby the differential analyzerrdquo Physical Review vol 38 no 10 pp1898ndash1901 1931

[18] R P Feynman N Metropolis and E Teller ldquoEquations of stateof elements based on the generalized fermi-thomas theoryrdquoPhysical Review vol 75 no 10 pp 1561ndash1573 1949

[19] C A Coulson and N H March ldquoMomenta in atoms usingthe Thomas-Fermi methodrdquo Proceedings of the Physical SocietySection A vol 63 no 4 pp 367ndash374 1950

[20] S Kobayashi TMatsukuma S Nagai and K Umeda ldquoAccuratevalue of the initial slope of the ordinary TF functionrdquo Journal ofthe Physical Society of Japan vol 10 no 9 pp 759ndash762 1955

[21] S Kobayashi ldquoSome coefficients of the series expansion of theTFD functionrdquo Journal of the Physical Society of Japan vol 10no 9 pp 824ndash825 1955

[22] S A Sommerfeld ldquoIntegrazione Asintotica dellrsquoEquazione Dif-ferenziali diThomas-FermirdquoAccademia dei Lincei Atti-Rendic-onte vol 4 pp 85ndash110 1932

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

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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

Mathematical Problems in Engineering 5

Similarly by (23b) one gets

32 1198660+ [2(minus

12

49

ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

+

1

49

ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

32

)]

lowast

119860

3

0

= [ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

sin(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

) + cos(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)]

lowast

1198600

+ cos2 (ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)(2 ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

lowast

1198600

+ cos(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

))

(32)

that constitute a second equation including the value of 1198660

and the constant of integrationlowast

119862 Also in this equation bymeans of (23c) we have

lowast

1198600=

lowast

119860(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

) = ci(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)

= C + ln(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

) +

infin

sum

119896=1

(minus1)119896ln 100381610038161003816100381610038162

lowast

119862

10038161003816100381610038161003816

2119896

(2119896) (2119896)

C = Eulerrsquos number asymp 05572

(33)

Summarizing from (19) and (20) one concludes that since bythe first boundary conditions 119910(0) = 1 for 119909 rarr 0 rArr 119910 rarr 1

then for 119909 rarr 0 rArr 1199101015840

119909rarr 0 or 1199101015840

119909rarr

lowast

119861 wherelowast

119861 is a fixedvalue Thus combining (23a) (23b) and (23c) together withthe first of (28) and the Abel nonlinear ODE (21) one maywrite that for 119909 rarr 0 rArr 120585

0rarr 0 119906

0rarr 0 119873

0rarr 0

Therefore by making use of the first boundary conditiongiven in (4) (119909 rarr 0 119910 rarr 1) as well as of all the aboveobservations the solutions of (31) and (32) perform the valuesfor 119866

0and 2

lowast

119862 and they can be written in the followingalgebraic forms

minus1 plusmn radic1 minus 4F (1198660

lowast

119862)

2

= 0(34a)

H(

lowast

119862) = 1198660 (34b)

where

F(1198660

lowast

119862) =

4

9

+[

[

[

1 minus

1

lowast

119862 ln 100381610038161003816100381610038162

lowast

119862

10038161003816100381610038161003816

]

]

]

[minus21198660

+ 2(minus

12

4

ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

+

1

49

(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)

32

)]

(35a)

H(

lowast

119862)

=

1

32

[ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

sin(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

) + cos(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)]

sdot

lowast

11986000+ cos2 (ln

1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

) [2 ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

lowast

1198600

+ cos(ln1003816100381610038161003816100381610038161003816

2

lowast

119862

1003816100381610038161003816100381610038161003816

)]

lowast

1198600

lowast

119862 as in (33)

(35b)

Thus we are now able to estimate 1199010 1199020 and 119876

0given by

(27) in terms of the known valueslowast

119862 and 1198660 This estimation

furnishes the value of1198760and therefore the type of the solution

119873

(119894)

(120585) (119894 = 1 2 3) near the origin 1205850among the formulae (24)

to (26)From the above evaluation it is concluded that the closed

form solutions of the Abel nonlinear ODE (21) given as

119906(119894)(120585) =

1

2

120585 [119873

(119894)

(120585) +

1

3

]

120585 = ln1003816100381610038161003816100381610038161003816

120585 + 2

lowast

1198621

1003816100381610038161003816100381610038161003816

(119894 = 1 2 3)

(36)

have been completely determinedSince we obtained the above closed- form solutions of the

intermediate Abel nonlinear ODE (21) the solutions to theoriginal TF equation (3) can be determined as follows By (19)and (21) we perform

119909 =

1

713

(

120585

119910

)

13

1199101015840

119909=

1

1199094119906 minus 3

119910

119909

(37)

while taking the total differential in (19) we extract

119889120585 = 211199092119910119889119909 + 7119909

3119889119910

lArrrArr 1205851015840

119909=

119889120585

119889119909

= 71199092(1199091199101015840

119909+ 3119910) = 7

119906

119909

(38)

where 119906 = 119906(120585 + 2

lowast

1198621) denotes the known solution of the

Abel nonlinear ODE (18) Using (34a) (34b) and (37) onewrites

1199101015840

120585=

1199101015840

119909

1205851015840

119909

= 743(

119910

120585

) 11990643

minus 3119910713(

119910

120585

)

13

=

119906 (120585)

120585

119910 minus

3

71205854311991043

(39)

6 Mathematical Problems in Engineering

that is the Bernoulli equation

11991013

=

exp ((1120585) int (119906 (120585) 120585) 119889120585)

119862 + (17) int exp [((1120585) int (119906 (120585) 120585) 119889120585)] 119889120585

119909 =

1

713

(

120585

119910

)

13

119909 ge 0 119910 ge 0 120585 = parameter (120585 ge 0)

119862 = second constant of integration

(40)

The last step of the above analysis concerns the calculationof the second constant of integration 119862 Thus from the firstof the boundary conditions (4) and the parametric solution(40) one writes

1199091997888rarr 0

1199101997888rarr 1

1205851997888rarr 0

lim1205851rarr0

(11991013)

=

lim1205851rarr0

exp ((1120585) int (119906 (120585) 120585) 119889120585)

lim1205851rarr0

119862 + (17) int exp [((1120585) int (119906 (120585) 120585) 119889120585)] 119889120585

(41)

and estimates the constant of integration 119862 By the second ofthe boundary conditions (4) one writes 119909

2rarr +infin 119910

2rarr 0

and 120585 rarr 1205852 Again through the parametric solution (40) one

writes

lim120585rarr1205852

[exp [1120585

int

119906 (120585)

120585

119889120585]] (42a)

and evaluates the value of 1205852 The evaluation of the second

constant of integration can be obtained again through theboundary condition 1119909

1rarr 0 rArr 119910

1rarr infin that is

lim120585rarr1205851

119862 +

1

7

int exp [(minus1120585

int

119906 (120585)

120585

119889120585)

1

12058513119889120585]

= 0

(42b)

Equations (41) and (42a) (42b) complete the solutions of theproblem under consideration that is the construction of aclosed-form parametric solution of the TF equation (3)

It is worthwhile to remark that since the solution isgiven in a closed form the mathematical methodology beingdeveloped results in the ad hoc definition of the TF function119910in approximate power series with 1199101015840

119909(0) =

lowast

119861 (lowast

119861 = a factor thatmust be determined) and with boundary conditions given by1199100(0) = 119909

0= 1 119910

1rarr 0 as 119909 rarr infin The value 1199101015840

119909(0) =

lowast

119861

can be evaluated by way of the already constructed closed-form solution This is a very interesting result in accordancewith previous results presented by Bush and Caldwell [17]Feynman et al [18] Coulson and March [19] Kobayashi etal [20] and Kobayashi [21]

5 Closed-Form Solutions of the LBEquation in the Physical Plane

It was already proved (Section 3) that the LB nonlinear ODE(1) has been reduced to an Abel ODE of the second kindof the normal form (15) It was also proved that this Abelequation admits exact analytic solutions given as in case ofthe TF nonlinear ODE by (24) to (26) Thus the solutionof the LB equation (1) in the phase plane is given by theformulae (24) to (26) if instead of the parenthesis (119860120585 +11986112058532) the quantity (120585minus124minus1205854) is introducedThe solution

of the original equation (1) in the physical plane is giventhrough the combination of the above prescribed formulaein combination with (17) As in case of the TF equationthe whole problem includes two constants of integration

lowast

119862

and 119862 which are to be determined by using the convenientboundary data of the problem under consideration In orderto define simultaneously the type of the function 119873

(119894)

(120585)

(119894 = 1 2 3) which must be selected in accordance with themodified previously developed formulae (24) to (26) andthus define the modified function 119866(120585) and the constant ofintegration

lowast

119862 wemust combine the aboveAbel solutionswiththe prescribed convenient boundary data

Based on what was previously developed in Section 3for the original (LB) equation (1) the following boundaryconditions hold true

at 119909 = 1199090

997904rArr 119910 = 1199100= ℓ

1199101015840

119909= 1199101015840

1199090

= 119898

(43)

where ℓ and119898 are arbitrary given values provided119898 = 0Based on these boundary conditions the substitution

(8) furnishes 119901(1199100) = 119901

0= 119898 while transformation (10)

furnishes 119908(1199100) = 119908

0= 11991013

01199010= ℓ13119898 too Thus the sub-

stitutions (13) (15) and the Abel nonlinear ODE (16) performthe following relations

1199080= 119899 (120585

0) = 1198990= ℓ13119898

1205850= 11991043

0= ℓ43

1198990= minus119911 (120585

0) = minus119911

0

997904rArr 1199110= minusℓ13119898

1199111015840

1205850

= minus

(14) (ℓminus23

minus ℓ43)

ℓ13119898

+ 1 = 1 minus

1

4

ℓminus1minus ℓ

119898

(44)

We are now able to evaluate all the before mentionedquantities Indeed (23a) and (23b) for 120585

0= ℓ43 1199110= minusℓ13119898

Mathematical Problems in Engineering 7

1199111015840

1205850

= 1 minus (ℓminus1minus ℓ)4119898 furnish the following two equations

1198730= minus(

2ℓ13119898

ℓ43

+ 4

lowast

119862

+

1

3

)

81198660+ 2 (ℓ

43minus ℓ13)P ln |P| [ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 1003816100381610038161003816

10038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C]

4 [ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 100381610038161003816100381610038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C] minus cos (ln |P|)

=

1

4

[ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 100381610038161003816100381610038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C] [ln |P| sin (ln |P|) + cos (ln |P|)]

[ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 100381610038161003816100381610038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C]

2

(ln |P|)2

+

cos2 (ln |P|)

[ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 100381610038161003816100381610038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C]

2

(ln |P|)2

P = ℓ43

+ 2

lowast

119862 gt 0 2

lowast

119862 + 11989743

gt 0 2

lowast

119862 lt 1

(45)

where1198730and 119866

0are the values of119873(120585) and 119866(120585) at 120585 = 120585

0=

ℓ43 respectively On the other hand the formula for 1198731015840

120585in

(23a) at 120585 = 1205850results in

119873

1015840

1205850

=

1

(1198730+ 43)

4 [1198660+ (12) (ℓ

minus23minus ℓ43)]

(ℓ43

+ 4

lowast

119862)

2 (46)

while by the first of (23a) and (44) we have

1199111015840

1205850

=

1

2

119873

1015840

1205850

(1205850+ 4

lowast

119862) +

1

2

(1198730+

1

3

) (47)

or equivalently

1199111015840

1205850

=

2 [1198660+ (12) (ℓ

minus23minus ℓ43)]

(1198730+ 43) (ℓ

43+ 4

lowast

119862)

+

1

2

(1198730+

1

3

) (48)

The last relation (48) in combination together with (44)permits us to write

2 [1198660+ (12) (ℓ

minus23minus ℓ43)]

(1198730+ (43)) (ℓ

43+ 4

lowast

119862)

+

1

2

(1198730+

1

3

)

= 1 minus

1

4

ℓminus1minus ℓ

119898

(49)

The above three equations (46) to (49) constitute a nonlinear(transcendental) system for119873

01198660 and 120582 by means of which

one calculates these unknowns in terms of the parameters ℓand 119898 Thus the discriminant 119876

0(119876 at 120585 = 120585

0= ℓ43) given

in (27) is also known the fact that permits us to define thetype of the function119873(119894)(120585) for 120585 gt 120585

0= ℓ43 (24) to (26)This

completes the solution of the problem under consideration

which is the construction of the solution of the LB equationin the phase-plane (and thus in the physical plane) in themain interval [120585

0= ℓ43 120585] If this solution is not unique

inside [ℓ43 120585] then we follow step by step the methodologydeveloped in [15]

6 Conclusions

By a series of admissible functional transformations wereduce the nonlinear TF and LB equations to Abelrsquos equa-tions of the second kind of the normal form of (16) and(18) respectively These equations do not admit closed-formsolutions in terms of known (tabulated) functions Thisunsolvability is due to the fact that only very special formsof this kind of (16) and (18) can be solved in parametric form[4] Our goal is the development of the construction of theexact analytic solutions of the above equations based on amathematical technique leading to the derivation of closeform solutions for the Abel equation of the second kind of thenormal form [3 4] The proposed methodology constitutesthe intermediate integral of the TF and LB equations inthe phase and physical planes The reduction procedure inthe paper and the constructed solutions are very generaland can be applied to a large number of nonlinear ODEsin mathematical physics and nonlinear mechanics Also themethod employed is exact in accordance with the boundaryconditions of each problem under consideration and guidesus to the a priori approximate construction of the solutions inpower series some coefficients of which must be evaluated

With the proposed procedure the exact value of first orderderivative of TF equation is derived at 119909 = 0 In particularthe solution for the TF equation is finally constructed bycalculating 4 constants from the specific boundary conditions(4) (see (34a) (34b) (41) (42a) and (42b)) Equations(40) constitute the parametric solutions of the TF problemWith our method essentially a different way of calculating

8 Mathematical Problems in Engineering

1199101015840

119909(0) =

lowast

119861 (Section 4) is proposed with reference to themethods proposed in [17ndash21] where the value of

lowast

119861 has beencalculated having kept the solution from approximate seriesin which their coefficients had been calculated from theboundary conditions In conclusion in this paper an analyticmethod is developed that does not contradict previous resultsbut it is a different way of calculating them

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] I Langmuir and K B Blodgett ldquoCurrents limited by spacecharge between coaxial cylindersrdquo Physical Review vol 22 no4 pp 347ndash356 1923

[2] H T Davis Introduction to Nonlinear Differential and IntegralEquations Dover New York NY USA 1962

[3] E Kamke Differentialgleichungen I Gewohnliche Differential-gleichungen and II Partielle Differentialgleichungen Akademis-che Verlagsgesellschaft Leipzig Germany 1962

[4] A D Polyanin and V F Zaitsev Handbook of Exact Solutionsfor Ordinary Differential Equations CRCPress Boca Raton FlaUSA 1999

[5] K Horvath J N Fairchild K Kaczmarski and G Guio-chon ldquoMartin-Synge algorithm for the solution of equilibrium-dispersive model of liquid chromatographyrdquo Journal of Chro-matography A vol 1217 no 52 pp 8127ndash8135 2010

[6] Y Shang and X Zheng ldquoThe first-integral method and abun-dant explicit exact solutions to the zakharov equationsrdquo Journalof Applied Mathematics vol 2012 Article ID 818345 16 pages2012

[7] D B Graham and I H Cairns ldquoConstraints on the formationand structure of langmuir eigenmodes in the solar windrdquoPhysical Review Letters vol 111 no 12 Article ID 121101 2013

[8] L H Thomas ldquoThe calculation of atomic fieldsrdquoMathematicalProceedings of the Cambridge Philosophical Society vol 23 no5 pp 542ndash548 1927

[9] E Fermi ldquoStatistical method of investigating electrons inatomsrdquo Zeitschrift fur Physik vol 48 pp 73ndash79 1927

[10] T Tsurumi and MWadati ldquoDynamics of magnetically trappedboson-fermion mixturesrdquo Journal of the Physical Society ofJapan vol 69 no 1 pp 97ndash103 2000

[11] S Liao ldquoAn explicit analytic solution to the Thomas-Fermiequationrdquo Applied Mathematics and Computation vol 144 no2-3 pp 495ndash506 2003

[12] S Abbasbandy and C Bervillier ldquoAnalytic continuation ofTaylor series and the boundary value problems of some nonlin-ear ordinary differential equationsrdquo Applied Mathematics andComputation vol 218 no 5 pp 2178ndash2199 2011

[13] K Ourabah and M Tribeche ldquoRelativistic formulation of thegeneralized nonextensiveThomas-FermimodelrdquoPhysicaA vol393 pp 470ndash474 2014

[14] D E Panayotounakos ldquoExact analytic solutions of unsolvableclasses of first and second order nonlinear ODEs (part I Abelrsquosequations)rdquo Applied Mathematics Letters vol 18 no 2 pp 155ndash162 2005

[15] D E Panayotounakos N B Sotiropoulos A B Sotiropoulouand N D Panayotounakou ldquoExact analytic solutions of non-linear boundary value problems in fluid mechanics (Blasiusequations)rdquo Journal of Mathematical Physics vol 46 no 3Article ID 033101 2005

[16] D E Panayotounakos E ETheotokoglou and M P MarkakisldquoExact analytic solutions for the unforded damped duffingnonlinear oscillatorrdquo Comptes Rendus Mechanics Journal vol334 pp 311ndash316 2006

[17] V Bush and S H Caldwell ldquoThomas-Fermi equation solutionby the differential analyzerrdquo Physical Review vol 38 no 10 pp1898ndash1901 1931

[18] R P Feynman N Metropolis and E Teller ldquoEquations of stateof elements based on the generalized fermi-thomas theoryrdquoPhysical Review vol 75 no 10 pp 1561ndash1573 1949

[19] C A Coulson and N H March ldquoMomenta in atoms usingthe Thomas-Fermi methodrdquo Proceedings of the Physical SocietySection A vol 63 no 4 pp 367ndash374 1950

[20] S Kobayashi TMatsukuma S Nagai and K Umeda ldquoAccuratevalue of the initial slope of the ordinary TF functionrdquo Journal ofthe Physical Society of Japan vol 10 no 9 pp 759ndash762 1955

[21] S Kobayashi ldquoSome coefficients of the series expansion of theTFD functionrdquo Journal of the Physical Society of Japan vol 10no 9 pp 824ndash825 1955

[22] S A Sommerfeld ldquoIntegrazione Asintotica dellrsquoEquazione Dif-ferenziali diThomas-FermirdquoAccademia dei Lincei Atti-Rendic-onte vol 4 pp 85ndash110 1932

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

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OptimizationJournal of

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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

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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

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

that is the Bernoulli equation

11991013

=

exp ((1120585) int (119906 (120585) 120585) 119889120585)

119862 + (17) int exp [((1120585) int (119906 (120585) 120585) 119889120585)] 119889120585

119909 =

1

713

(

120585

119910

)

13

119909 ge 0 119910 ge 0 120585 = parameter (120585 ge 0)

119862 = second constant of integration

(40)

The last step of the above analysis concerns the calculationof the second constant of integration 119862 Thus from the firstof the boundary conditions (4) and the parametric solution(40) one writes

1199091997888rarr 0

1199101997888rarr 1

1205851997888rarr 0

lim1205851rarr0

(11991013)

=

lim1205851rarr0

exp ((1120585) int (119906 (120585) 120585) 119889120585)

lim1205851rarr0

119862 + (17) int exp [((1120585) int (119906 (120585) 120585) 119889120585)] 119889120585

(41)

and estimates the constant of integration 119862 By the second ofthe boundary conditions (4) one writes 119909

2rarr +infin 119910

2rarr 0

and 120585 rarr 1205852 Again through the parametric solution (40) one

writes

lim120585rarr1205852

[exp [1120585

int

119906 (120585)

120585

119889120585]] (42a)

and evaluates the value of 1205852 The evaluation of the second

constant of integration can be obtained again through theboundary condition 1119909

1rarr 0 rArr 119910

1rarr infin that is

lim120585rarr1205851

119862 +

1

7

int exp [(minus1120585

int

119906 (120585)

120585

119889120585)

1

12058513119889120585]

= 0

(42b)

Equations (41) and (42a) (42b) complete the solutions of theproblem under consideration that is the construction of aclosed-form parametric solution of the TF equation (3)

It is worthwhile to remark that since the solution isgiven in a closed form the mathematical methodology beingdeveloped results in the ad hoc definition of the TF function119910in approximate power series with 1199101015840

119909(0) =

lowast

119861 (lowast

119861 = a factor thatmust be determined) and with boundary conditions given by1199100(0) = 119909

0= 1 119910

1rarr 0 as 119909 rarr infin The value 1199101015840

119909(0) =

lowast

119861

can be evaluated by way of the already constructed closed-form solution This is a very interesting result in accordancewith previous results presented by Bush and Caldwell [17]Feynman et al [18] Coulson and March [19] Kobayashi etal [20] and Kobayashi [21]

5 Closed-Form Solutions of the LBEquation in the Physical Plane

It was already proved (Section 3) that the LB nonlinear ODE(1) has been reduced to an Abel ODE of the second kindof the normal form (15) It was also proved that this Abelequation admits exact analytic solutions given as in case ofthe TF nonlinear ODE by (24) to (26) Thus the solutionof the LB equation (1) in the phase plane is given by theformulae (24) to (26) if instead of the parenthesis (119860120585 +11986112058532) the quantity (120585minus124minus1205854) is introducedThe solution

of the original equation (1) in the physical plane is giventhrough the combination of the above prescribed formulaein combination with (17) As in case of the TF equationthe whole problem includes two constants of integration

lowast

119862

and 119862 which are to be determined by using the convenientboundary data of the problem under consideration In orderto define simultaneously the type of the function 119873

(119894)

(120585)

(119894 = 1 2 3) which must be selected in accordance with themodified previously developed formulae (24) to (26) andthus define the modified function 119866(120585) and the constant ofintegration

lowast

119862 wemust combine the aboveAbel solutionswiththe prescribed convenient boundary data

Based on what was previously developed in Section 3for the original (LB) equation (1) the following boundaryconditions hold true

at 119909 = 1199090

997904rArr 119910 = 1199100= ℓ

1199101015840

119909= 1199101015840

1199090

= 119898

(43)

where ℓ and119898 are arbitrary given values provided119898 = 0Based on these boundary conditions the substitution

(8) furnishes 119901(1199100) = 119901

0= 119898 while transformation (10)

furnishes 119908(1199100) = 119908

0= 11991013

01199010= ℓ13119898 too Thus the sub-

stitutions (13) (15) and the Abel nonlinear ODE (16) performthe following relations

1199080= 119899 (120585

0) = 1198990= ℓ13119898

1205850= 11991043

0= ℓ43

1198990= minus119911 (120585

0) = minus119911

0

997904rArr 1199110= minusℓ13119898

1199111015840

1205850

= minus

(14) (ℓminus23

minus ℓ43)

ℓ13119898

+ 1 = 1 minus

1

4

ℓminus1minus ℓ

119898

(44)

We are now able to evaluate all the before mentionedquantities Indeed (23a) and (23b) for 120585

0= ℓ43 1199110= minusℓ13119898

Mathematical Problems in Engineering 7

1199111015840

1205850

= 1 minus (ℓminus1minus ℓ)4119898 furnish the following two equations

1198730= minus(

2ℓ13119898

ℓ43

+ 4

lowast

119862

+

1

3

)

81198660+ 2 (ℓ

43minus ℓ13)P ln |P| [ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 1003816100381610038161003816

10038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C]

4 [ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 100381610038161003816100381610038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C] minus cos (ln |P|)

=

1

4

[ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 100381610038161003816100381610038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C] [ln |P| sin (ln |P|) + cos (ln |P|)]

[ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 100381610038161003816100381610038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C]

2

(ln |P|)2

+

cos2 (ln |P|)

[ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 100381610038161003816100381610038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C]

2

(ln |P|)2

P = ℓ43

+ 2

lowast

119862 gt 0 2

lowast

119862 + 11989743

gt 0 2

lowast

119862 lt 1

(45)

where1198730and 119866

0are the values of119873(120585) and 119866(120585) at 120585 = 120585

0=

ℓ43 respectively On the other hand the formula for 1198731015840

120585in

(23a) at 120585 = 1205850results in

119873

1015840

1205850

=

1

(1198730+ 43)

4 [1198660+ (12) (ℓ

minus23minus ℓ43)]

(ℓ43

+ 4

lowast

119862)

2 (46)

while by the first of (23a) and (44) we have

1199111015840

1205850

=

1

2

119873

1015840

1205850

(1205850+ 4

lowast

119862) +

1

2

(1198730+

1

3

) (47)

or equivalently

1199111015840

1205850

=

2 [1198660+ (12) (ℓ

minus23minus ℓ43)]

(1198730+ 43) (ℓ

43+ 4

lowast

119862)

+

1

2

(1198730+

1

3

) (48)

The last relation (48) in combination together with (44)permits us to write

2 [1198660+ (12) (ℓ

minus23minus ℓ43)]

(1198730+ (43)) (ℓ

43+ 4

lowast

119862)

+

1

2

(1198730+

1

3

)

= 1 minus

1

4

ℓminus1minus ℓ

119898

(49)

The above three equations (46) to (49) constitute a nonlinear(transcendental) system for119873

01198660 and 120582 by means of which

one calculates these unknowns in terms of the parameters ℓand 119898 Thus the discriminant 119876

0(119876 at 120585 = 120585

0= ℓ43) given

in (27) is also known the fact that permits us to define thetype of the function119873(119894)(120585) for 120585 gt 120585

0= ℓ43 (24) to (26)This

completes the solution of the problem under consideration

which is the construction of the solution of the LB equationin the phase-plane (and thus in the physical plane) in themain interval [120585

0= ℓ43 120585] If this solution is not unique

inside [ℓ43 120585] then we follow step by step the methodologydeveloped in [15]

6 Conclusions

By a series of admissible functional transformations wereduce the nonlinear TF and LB equations to Abelrsquos equa-tions of the second kind of the normal form of (16) and(18) respectively These equations do not admit closed-formsolutions in terms of known (tabulated) functions Thisunsolvability is due to the fact that only very special formsof this kind of (16) and (18) can be solved in parametric form[4] Our goal is the development of the construction of theexact analytic solutions of the above equations based on amathematical technique leading to the derivation of closeform solutions for the Abel equation of the second kind of thenormal form [3 4] The proposed methodology constitutesthe intermediate integral of the TF and LB equations inthe phase and physical planes The reduction procedure inthe paper and the constructed solutions are very generaland can be applied to a large number of nonlinear ODEsin mathematical physics and nonlinear mechanics Also themethod employed is exact in accordance with the boundaryconditions of each problem under consideration and guidesus to the a priori approximate construction of the solutions inpower series some coefficients of which must be evaluated

With the proposed procedure the exact value of first orderderivative of TF equation is derived at 119909 = 0 In particularthe solution for the TF equation is finally constructed bycalculating 4 constants from the specific boundary conditions(4) (see (34a) (34b) (41) (42a) and (42b)) Equations(40) constitute the parametric solutions of the TF problemWith our method essentially a different way of calculating

8 Mathematical Problems in Engineering

1199101015840

119909(0) =

lowast

119861 (Section 4) is proposed with reference to themethods proposed in [17ndash21] where the value of

lowast

119861 has beencalculated having kept the solution from approximate seriesin which their coefficients had been calculated from theboundary conditions In conclusion in this paper an analyticmethod is developed that does not contradict previous resultsbut it is a different way of calculating them

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] I Langmuir and K B Blodgett ldquoCurrents limited by spacecharge between coaxial cylindersrdquo Physical Review vol 22 no4 pp 347ndash356 1923

[2] H T Davis Introduction to Nonlinear Differential and IntegralEquations Dover New York NY USA 1962

[3] E Kamke Differentialgleichungen I Gewohnliche Differential-gleichungen and II Partielle Differentialgleichungen Akademis-che Verlagsgesellschaft Leipzig Germany 1962

[4] A D Polyanin and V F Zaitsev Handbook of Exact Solutionsfor Ordinary Differential Equations CRCPress Boca Raton FlaUSA 1999

[5] K Horvath J N Fairchild K Kaczmarski and G Guio-chon ldquoMartin-Synge algorithm for the solution of equilibrium-dispersive model of liquid chromatographyrdquo Journal of Chro-matography A vol 1217 no 52 pp 8127ndash8135 2010

[6] Y Shang and X Zheng ldquoThe first-integral method and abun-dant explicit exact solutions to the zakharov equationsrdquo Journalof Applied Mathematics vol 2012 Article ID 818345 16 pages2012

[7] D B Graham and I H Cairns ldquoConstraints on the formationand structure of langmuir eigenmodes in the solar windrdquoPhysical Review Letters vol 111 no 12 Article ID 121101 2013

[8] L H Thomas ldquoThe calculation of atomic fieldsrdquoMathematicalProceedings of the Cambridge Philosophical Society vol 23 no5 pp 542ndash548 1927

[9] E Fermi ldquoStatistical method of investigating electrons inatomsrdquo Zeitschrift fur Physik vol 48 pp 73ndash79 1927

[10] T Tsurumi and MWadati ldquoDynamics of magnetically trappedboson-fermion mixturesrdquo Journal of the Physical Society ofJapan vol 69 no 1 pp 97ndash103 2000

[11] S Liao ldquoAn explicit analytic solution to the Thomas-Fermiequationrdquo Applied Mathematics and Computation vol 144 no2-3 pp 495ndash506 2003

[12] S Abbasbandy and C Bervillier ldquoAnalytic continuation ofTaylor series and the boundary value problems of some nonlin-ear ordinary differential equationsrdquo Applied Mathematics andComputation vol 218 no 5 pp 2178ndash2199 2011

[13] K Ourabah and M Tribeche ldquoRelativistic formulation of thegeneralized nonextensiveThomas-FermimodelrdquoPhysicaA vol393 pp 470ndash474 2014

[14] D E Panayotounakos ldquoExact analytic solutions of unsolvableclasses of first and second order nonlinear ODEs (part I Abelrsquosequations)rdquo Applied Mathematics Letters vol 18 no 2 pp 155ndash162 2005

[15] D E Panayotounakos N B Sotiropoulos A B Sotiropoulouand N D Panayotounakou ldquoExact analytic solutions of non-linear boundary value problems in fluid mechanics (Blasiusequations)rdquo Journal of Mathematical Physics vol 46 no 3Article ID 033101 2005

[16] D E Panayotounakos E ETheotokoglou and M P MarkakisldquoExact analytic solutions for the unforded damped duffingnonlinear oscillatorrdquo Comptes Rendus Mechanics Journal vol334 pp 311ndash316 2006

[17] V Bush and S H Caldwell ldquoThomas-Fermi equation solutionby the differential analyzerrdquo Physical Review vol 38 no 10 pp1898ndash1901 1931

[18] R P Feynman N Metropolis and E Teller ldquoEquations of stateof elements based on the generalized fermi-thomas theoryrdquoPhysical Review vol 75 no 10 pp 1561ndash1573 1949

[19] C A Coulson and N H March ldquoMomenta in atoms usingthe Thomas-Fermi methodrdquo Proceedings of the Physical SocietySection A vol 63 no 4 pp 367ndash374 1950

[20] S Kobayashi TMatsukuma S Nagai and K Umeda ldquoAccuratevalue of the initial slope of the ordinary TF functionrdquo Journal ofthe Physical Society of Japan vol 10 no 9 pp 759ndash762 1955

[21] S Kobayashi ldquoSome coefficients of the series expansion of theTFD functionrdquo Journal of the Physical Society of Japan vol 10no 9 pp 824ndash825 1955

[22] S A Sommerfeld ldquoIntegrazione Asintotica dellrsquoEquazione Dif-ferenziali diThomas-FermirdquoAccademia dei Lincei Atti-Rendic-onte vol 4 pp 85ndash110 1932

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

Mathematical Problems in Engineering 7

1199111015840

1205850

= 1 minus (ℓminus1minus ℓ)4119898 furnish the following two equations

1198730= minus(

2ℓ13119898

ℓ43

+ 4

lowast

119862

+

1

3

)

81198660+ 2 (ℓ

43minus ℓ13)P ln |P| [ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 1003816100381610038161003816

10038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C]

4 [ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 100381610038161003816100381610038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C] minus cos (ln |P|)

=

1

4

[ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 100381610038161003816100381610038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C] [ln |P| sin (ln |P|) + cos (ln |P|)]

[ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 100381610038161003816100381610038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C]

2

(ln |P|)2

+

cos2 (ln |P|)

[ci (ln |P|) minus ln |P| + ln (ln |P|) minus ln (ln 100381610038161003816100381610038161 minus 2

lowast

119862

10038161003816100381610038161003816) minus C]

2

(ln |P|)2

P = ℓ43

+ 2

lowast

119862 gt 0 2

lowast

119862 + 11989743

gt 0 2

lowast

119862 lt 1

(45)

where1198730and 119866

0are the values of119873(120585) and 119866(120585) at 120585 = 120585

0=

ℓ43 respectively On the other hand the formula for 1198731015840

120585in

(23a) at 120585 = 1205850results in

119873

1015840

1205850

=

1

(1198730+ 43)

4 [1198660+ (12) (ℓ

minus23minus ℓ43)]

(ℓ43

+ 4

lowast

119862)

2 (46)

while by the first of (23a) and (44) we have

1199111015840

1205850

=

1

2

119873

1015840

1205850

(1205850+ 4

lowast

119862) +

1

2

(1198730+

1

3

) (47)

or equivalently

1199111015840

1205850

=

2 [1198660+ (12) (ℓ

minus23minus ℓ43)]

(1198730+ 43) (ℓ

43+ 4

lowast

119862)

+

1

2

(1198730+

1

3

) (48)

The last relation (48) in combination together with (44)permits us to write

2 [1198660+ (12) (ℓ

minus23minus ℓ43)]

(1198730+ (43)) (ℓ

43+ 4

lowast

119862)

+

1

2

(1198730+

1

3

)

= 1 minus

1

4

ℓminus1minus ℓ

119898

(49)

The above three equations (46) to (49) constitute a nonlinear(transcendental) system for119873

01198660 and 120582 by means of which

one calculates these unknowns in terms of the parameters ℓand 119898 Thus the discriminant 119876

0(119876 at 120585 = 120585

0= ℓ43) given

in (27) is also known the fact that permits us to define thetype of the function119873(119894)(120585) for 120585 gt 120585

0= ℓ43 (24) to (26)This

completes the solution of the problem under consideration

which is the construction of the solution of the LB equationin the phase-plane (and thus in the physical plane) in themain interval [120585

0= ℓ43 120585] If this solution is not unique

inside [ℓ43 120585] then we follow step by step the methodologydeveloped in [15]

6 Conclusions

By a series of admissible functional transformations wereduce the nonlinear TF and LB equations to Abelrsquos equa-tions of the second kind of the normal form of (16) and(18) respectively These equations do not admit closed-formsolutions in terms of known (tabulated) functions Thisunsolvability is due to the fact that only very special formsof this kind of (16) and (18) can be solved in parametric form[4] Our goal is the development of the construction of theexact analytic solutions of the above equations based on amathematical technique leading to the derivation of closeform solutions for the Abel equation of the second kind of thenormal form [3 4] The proposed methodology constitutesthe intermediate integral of the TF and LB equations inthe phase and physical planes The reduction procedure inthe paper and the constructed solutions are very generaland can be applied to a large number of nonlinear ODEsin mathematical physics and nonlinear mechanics Also themethod employed is exact in accordance with the boundaryconditions of each problem under consideration and guidesus to the a priori approximate construction of the solutions inpower series some coefficients of which must be evaluated

With the proposed procedure the exact value of first orderderivative of TF equation is derived at 119909 = 0 In particularthe solution for the TF equation is finally constructed bycalculating 4 constants from the specific boundary conditions(4) (see (34a) (34b) (41) (42a) and (42b)) Equations(40) constitute the parametric solutions of the TF problemWith our method essentially a different way of calculating

8 Mathematical Problems in Engineering

1199101015840

119909(0) =

lowast

119861 (Section 4) is proposed with reference to themethods proposed in [17ndash21] where the value of

lowast

119861 has beencalculated having kept the solution from approximate seriesin which their coefficients had been calculated from theboundary conditions In conclusion in this paper an analyticmethod is developed that does not contradict previous resultsbut it is a different way of calculating them

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] I Langmuir and K B Blodgett ldquoCurrents limited by spacecharge between coaxial cylindersrdquo Physical Review vol 22 no4 pp 347ndash356 1923

[2] H T Davis Introduction to Nonlinear Differential and IntegralEquations Dover New York NY USA 1962

[3] E Kamke Differentialgleichungen I Gewohnliche Differential-gleichungen and II Partielle Differentialgleichungen Akademis-che Verlagsgesellschaft Leipzig Germany 1962

[4] A D Polyanin and V F Zaitsev Handbook of Exact Solutionsfor Ordinary Differential Equations CRCPress Boca Raton FlaUSA 1999

[5] K Horvath J N Fairchild K Kaczmarski and G Guio-chon ldquoMartin-Synge algorithm for the solution of equilibrium-dispersive model of liquid chromatographyrdquo Journal of Chro-matography A vol 1217 no 52 pp 8127ndash8135 2010

[6] Y Shang and X Zheng ldquoThe first-integral method and abun-dant explicit exact solutions to the zakharov equationsrdquo Journalof Applied Mathematics vol 2012 Article ID 818345 16 pages2012

[7] D B Graham and I H Cairns ldquoConstraints on the formationand structure of langmuir eigenmodes in the solar windrdquoPhysical Review Letters vol 111 no 12 Article ID 121101 2013

[8] L H Thomas ldquoThe calculation of atomic fieldsrdquoMathematicalProceedings of the Cambridge Philosophical Society vol 23 no5 pp 542ndash548 1927

[9] E Fermi ldquoStatistical method of investigating electrons inatomsrdquo Zeitschrift fur Physik vol 48 pp 73ndash79 1927

[10] T Tsurumi and MWadati ldquoDynamics of magnetically trappedboson-fermion mixturesrdquo Journal of the Physical Society ofJapan vol 69 no 1 pp 97ndash103 2000

[11] S Liao ldquoAn explicit analytic solution to the Thomas-Fermiequationrdquo Applied Mathematics and Computation vol 144 no2-3 pp 495ndash506 2003

[12] S Abbasbandy and C Bervillier ldquoAnalytic continuation ofTaylor series and the boundary value problems of some nonlin-ear ordinary differential equationsrdquo Applied Mathematics andComputation vol 218 no 5 pp 2178ndash2199 2011

[13] K Ourabah and M Tribeche ldquoRelativistic formulation of thegeneralized nonextensiveThomas-FermimodelrdquoPhysicaA vol393 pp 470ndash474 2014

[14] D E Panayotounakos ldquoExact analytic solutions of unsolvableclasses of first and second order nonlinear ODEs (part I Abelrsquosequations)rdquo Applied Mathematics Letters vol 18 no 2 pp 155ndash162 2005

[15] D E Panayotounakos N B Sotiropoulos A B Sotiropoulouand N D Panayotounakou ldquoExact analytic solutions of non-linear boundary value problems in fluid mechanics (Blasiusequations)rdquo Journal of Mathematical Physics vol 46 no 3Article ID 033101 2005

[16] D E Panayotounakos E ETheotokoglou and M P MarkakisldquoExact analytic solutions for the unforded damped duffingnonlinear oscillatorrdquo Comptes Rendus Mechanics Journal vol334 pp 311ndash316 2006

[17] V Bush and S H Caldwell ldquoThomas-Fermi equation solutionby the differential analyzerrdquo Physical Review vol 38 no 10 pp1898ndash1901 1931

[18] R P Feynman N Metropolis and E Teller ldquoEquations of stateof elements based on the generalized fermi-thomas theoryrdquoPhysical Review vol 75 no 10 pp 1561ndash1573 1949

[19] C A Coulson and N H March ldquoMomenta in atoms usingthe Thomas-Fermi methodrdquo Proceedings of the Physical SocietySection A vol 63 no 4 pp 367ndash374 1950

[20] S Kobayashi TMatsukuma S Nagai and K Umeda ldquoAccuratevalue of the initial slope of the ordinary TF functionrdquo Journal ofthe Physical Society of Japan vol 10 no 9 pp 759ndash762 1955

[21] S Kobayashi ldquoSome coefficients of the series expansion of theTFD functionrdquo Journal of the Physical Society of Japan vol 10no 9 pp 824ndash825 1955

[22] S A Sommerfeld ldquoIntegrazione Asintotica dellrsquoEquazione Dif-ferenziali diThomas-FermirdquoAccademia dei Lincei Atti-Rendic-onte vol 4 pp 85ndash110 1932

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 Mathematical Problems in Engineering

1199101015840

119909(0) =

lowast

119861 (Section 4) is proposed with reference to themethods proposed in [17ndash21] where the value of

lowast

119861 has beencalculated having kept the solution from approximate seriesin which their coefficients had been calculated from theboundary conditions In conclusion in this paper an analyticmethod is developed that does not contradict previous resultsbut it is a different way of calculating them

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] I Langmuir and K B Blodgett ldquoCurrents limited by spacecharge between coaxial cylindersrdquo Physical Review vol 22 no4 pp 347ndash356 1923

[2] H T Davis Introduction to Nonlinear Differential and IntegralEquations Dover New York NY USA 1962

[3] E Kamke Differentialgleichungen I Gewohnliche Differential-gleichungen and II Partielle Differentialgleichungen Akademis-che Verlagsgesellschaft Leipzig Germany 1962

[4] A D Polyanin and V F Zaitsev Handbook of Exact Solutionsfor Ordinary Differential Equations CRCPress Boca Raton FlaUSA 1999

[5] K Horvath J N Fairchild K Kaczmarski and G Guio-chon ldquoMartin-Synge algorithm for the solution of equilibrium-dispersive model of liquid chromatographyrdquo Journal of Chro-matography A vol 1217 no 52 pp 8127ndash8135 2010

[6] Y Shang and X Zheng ldquoThe first-integral method and abun-dant explicit exact solutions to the zakharov equationsrdquo Journalof Applied Mathematics vol 2012 Article ID 818345 16 pages2012

[7] D B Graham and I H Cairns ldquoConstraints on the formationand structure of langmuir eigenmodes in the solar windrdquoPhysical Review Letters vol 111 no 12 Article ID 121101 2013

[8] L H Thomas ldquoThe calculation of atomic fieldsrdquoMathematicalProceedings of the Cambridge Philosophical Society vol 23 no5 pp 542ndash548 1927

[9] E Fermi ldquoStatistical method of investigating electrons inatomsrdquo Zeitschrift fur Physik vol 48 pp 73ndash79 1927

[10] T Tsurumi and MWadati ldquoDynamics of magnetically trappedboson-fermion mixturesrdquo Journal of the Physical Society ofJapan vol 69 no 1 pp 97ndash103 2000

[11] S Liao ldquoAn explicit analytic solution to the Thomas-Fermiequationrdquo Applied Mathematics and Computation vol 144 no2-3 pp 495ndash506 2003

[12] S Abbasbandy and C Bervillier ldquoAnalytic continuation ofTaylor series and the boundary value problems of some nonlin-ear ordinary differential equationsrdquo Applied Mathematics andComputation vol 218 no 5 pp 2178ndash2199 2011

[13] K Ourabah and M Tribeche ldquoRelativistic formulation of thegeneralized nonextensiveThomas-FermimodelrdquoPhysicaA vol393 pp 470ndash474 2014

[14] D E Panayotounakos ldquoExact analytic solutions of unsolvableclasses of first and second order nonlinear ODEs (part I Abelrsquosequations)rdquo Applied Mathematics Letters vol 18 no 2 pp 155ndash162 2005

[15] D E Panayotounakos N B Sotiropoulos A B Sotiropoulouand N D Panayotounakou ldquoExact analytic solutions of non-linear boundary value problems in fluid mechanics (Blasiusequations)rdquo Journal of Mathematical Physics vol 46 no 3Article ID 033101 2005

[16] D E Panayotounakos E ETheotokoglou and M P MarkakisldquoExact analytic solutions for the unforded damped duffingnonlinear oscillatorrdquo Comptes Rendus Mechanics Journal vol334 pp 311ndash316 2006

[17] V Bush and S H Caldwell ldquoThomas-Fermi equation solutionby the differential analyzerrdquo Physical Review vol 38 no 10 pp1898ndash1901 1931

[18] R P Feynman N Metropolis and E Teller ldquoEquations of stateof elements based on the generalized fermi-thomas theoryrdquoPhysical Review vol 75 no 10 pp 1561ndash1573 1949

[19] C A Coulson and N H March ldquoMomenta in atoms usingthe Thomas-Fermi methodrdquo Proceedings of the Physical SocietySection A vol 63 no 4 pp 367ndash374 1950

[20] S Kobayashi TMatsukuma S Nagai and K Umeda ldquoAccuratevalue of the initial slope of the ordinary TF functionrdquo Journal ofthe Physical Society of Japan vol 10 no 9 pp 759ndash762 1955

[21] S Kobayashi ldquoSome coefficients of the series expansion of theTFD functionrdquo Journal of the Physical Society of Japan vol 10no 9 pp 824ndash825 1955

[22] S A Sommerfeld ldquoIntegrazione Asintotica dellrsquoEquazione Dif-ferenziali diThomas-FermirdquoAccademia dei Lincei Atti-Rendic-onte vol 4 pp 85ndash110 1932

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