Research Article Gravity s Rainbow and Its Einstein

Research Article119865(119877) Gravityrsquos Rainbow and Its Einstein Counterpart

S H Hendi12 B Eslam Panah1 S Panahiyan13 and M Momennia1

1Physics Department and Biruni Observatory College of Sciences Shiraz University Shiraz 71454 Iran2Research Institute for Astronomy and Astrophysics of Maragha (RIAAM) PO Box 55134-441 Maragha Iran3Physics Department Shahid Beheshti University Tehran 19839 Iran

Correspondence should be addressed to S H Hendi hendishirazuacir

Received 20 June 2016 Revised 21 July 2016 Accepted 21 July 2016

Academic Editor Rong-Gen Cai

Copyright copy 2016 S H Hendi et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

Motivated by UV completion of general relativity with a modification of a geometry at high energy scale it is expected to havean energy dependent geometry In this paper we introduce charged black hole solutions with power Maxwell invariant sourcein the context of gravityrsquos rainbow In addition we investigate two classes of 119865(119877) gravityrsquos rainbow solutions At first we studyenergy dependent 119865(119877) gravity without energy-momentum tensor and then we obtain 119865(119877) gravityrsquos rainbow in the presence ofconformally invariant Maxwell source We study geometrical properties of the mentioned solutions and compare their results Wealso give some related comments regarding thermodynamical behavior of the obtained solutions and discuss thermal stability ofthe solutions

1 Introduction

An accelerated expansion of the universe was confirmed byvarious observational evidences The luminosity distance ofSupernovae type Ia [1 2] the anisotropy of cosmicmicrowavebackground radiation [3] and also wide surveys on galaxies[4] confirm such accelerated expansion On the other handbaryon oscillations [5] large scale structure formation [6]and weak lensing [7] also propose such an acceleratedexpansion of the universe

After discovery of such an expansion in 1998 under-standing its theoretical reasons presents one of the funda-mental open questions in physics Identifying the cause ofthis late time acceleration is a challenging problem in cosmol-ogy Physicists are interested in considering this acceleratedexpansion in a gravitational background and they proposedsome candidates to explain it For example a positive cosmo-logical constant leads to an accelerated expansion but it isplagued by the fine tuning problem [8ndash12] In other wordsthe left hand side of Einstein equations can modify by thecosmological constant as a geometrical modification or it canbe interpreted as a kinematic term on the right hand sidewith the equation of state parameter 119908 = minus1 By considering119908 lt minus13 for a source term it is possible to further

modify this approach This consideration has interpretationof dark energy which has been investigated in literature [13ndash19] In dark energy models the acceleration expansion ofthe universe is due to an unknown ingredient added to thecosmic pieThe effect of this unknown ingredient is extractedbymodifying the stress energy tensor of the Einstein equationwith a matter which is different from the usual matter andradiation components

On the other hand it is proposed that the presenceof accelerated expansion of the universe indicates that thestandard general relativity requires modification To do soone can generalize the Einstein field equations to obtain amodified version of gravity There are different branches ofmodified gravity with various motivations such as braneworld cosmology [20ndash22] Lovelock gravity [23ndash27] scalar-tensor theories [28ndash35] and 119865(119877) gravity [36ndash51]

Modifying general relativity opens a new way to alarge class of alternative theories of gravity ranging fromhigher dimensional physics [52ndash54] to non-minimally cou-pled (scalar) fields [55ndash58] Regarding various models ofmodified gravity we will be interested in 119865(119877) gravity[59ndash65] based on replacing the scalar curvature 119877 witha generic analytic function 119865(119877) in the Hilbert-Einsteinaction Some viable functional forms of 119865(119877) gravity may

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2016 Article ID 9813582 21 pageshttpdxdoiorg10115520169813582

2 Advances in High Energy Physics

be reconstructed starting from data and physically motivatedissues

However the field equations of 119865(119877) gravity are compli-cated fourth-order differential equations and it is not easy tofind exact solutions In addition adding amatter field to119865(119877)

gravity makes the field equations much more complicatedOn the other hand regarding constant curvature scalarmodelas a subclass of general 119865(119877) gravity can simplify the fieldequations Also one can extract exact solutions of 119865(119877)

gravity coupled to a traceless energy-momentum tensor withconstant curvature scalar [66] For example consideringexact solutions of 119865(119877) gravity with conformally invariantMaxwell (CIM) field as a matter source has been investigated[67 68]

General relativity coupled to a nonlinear electrodynamicsattracts significant attentions because of its specific prop-erties in gaugegravity coupling Interesting properties ofvarious models of the nonlinear electrodynamics have beeninvestigated before [69ndash81] Power Maxwell invariant (PMI)theory is one of the interesting branches of the nonlinearelectrodynamics and its Lagrangian is an arbitrary powerof Maxwell Lagrangian [82ndash84] The PMI theory is moreinteresting with regard to Maxwell field and for unit powerit reduces to linear Maxwell theory These nonlinear elec-trodynamics enjoy conformal invariancy when the power ofMaxwell invariant is a quarter of space-time dimensions andthis is one of the attractive properties of this theory In otherwords one can obtain traceless energy-momentum tensorfor a special case ldquopower = dimensions4rdquo which leadsto conformal invariancy It is notable to mention that thisidea has been considered to take advantage of the conformalsymmetry to construct the analogues of the four-dimensionalReissner-Nordstrom solutions with an inverse square electricfield in arbitrary dimensions [85]

From the gravitational point of view it is possible to showthat the electric charge and cosmological constant can beextracted simultaneously from pure 119865(119877) gravity (withoutmatter field 119879120583] = 0) [51] In this paper we are goingto obtain 119889-dimensional charged black hole solutions fromgravityrsquos rainbow and pure 119865(119877) gravityrsquos rainbow as well as119865(119877) gravityrsquos rainbow with CIM source and compare themto obtain their direct relation

In order to build up special relativity from Galilean the-ory one has to take into account an upper limit for velocity ofparticlesThe samemethod could be used to restrict particlesfrom obtaining energies no more than specific energy theso-called Planck energy scale This upper bound of energymay modify dispersion relation which is known as doublespecial relativity [86] Generalization of this doubly specialrelativity to incorporate curvature leads to gravityrsquos rainbow[87] In gravityrsquos rainbow space-time is a combination of thetemporal and spatial coordinates as well as energy functionsThe existence of such energy functions indicates that theparticle probing the space-time can acquire specific range ofenergies which in essence leads to formation of a rainbow ofenergy

There are several features for gravityrsquos rainbow whichhighlight the importance of such generalization Amongthem one can point modification in energy-momentum

dispersion relationwhich is supported by studies that are con-ducted in string theory [88] loop quantum gravity [89] andexperimental observation [90] Also existence of remnant forblack holes [91] is proposed to be a candidate for solving theinformation paradox [92] In addition this theory admits theusual uncertainty principle [93 94]

Recently there has been a growing interest in energydependent space-time [95ndash100] Different classes of blackholes have been investigated in the context of gravityrsquosrainbow [101ndash104] The hydrostatic equilibrium equation ofstars in the presence of gravityrsquos rainbow has been obtained[105] Furthermore a study regarding the gravityrsquos rainbowand compact stars has been done [106] In [107] the effectsof gravityrsquos rainbow for wormholes have been investigatedMoreover the influences of gravityrsquos rainbow on gravitationalforce have been investigated [108] Also Starobinskymodel of119865(119877) theory in gravityrsquos rainbow has been studied in [109] Inaddition gravityrsquos rainbow has interesting effects on the earlyuniverse [110ndash112]

Themainmotivations for studying black holes in the pres-ence of gravityrsquos rainbow are given as follows First of all dueto the high energy properties of the black holes it is necessaryto consider quantum corrections of classical perspectivesOne of the methods to include quantum corrections ofgravitational fields is by considering energy dependent space-time In fact it was shown that the quantum correction ofgravitational systems could be observed in dependency ofspace-time on the energy of particles probing it which isgravityrsquos rainbow point of view [93 113 114] Since we aremodifying our point of view to energy dependent space-timeit is expected to find its effects on the properties of blackholes especially in the context of their thermodynamicsThis is another motivation for considering gravityrsquos rainbowgeneralization Also there are specific achievements for grav-ityrsquos rainbow in the context of black holes and among themone can name the following modified uncertainty principle[93 94] existence of remnants for the black holes [91 102]furnishing a bridge toward Horava-Lifshitz gravity [115]providing possible solution toward information paradox [91]and finally being UV completion of Einstein gravity [116]In addition as it was pointed out before in the contextof cosmology it presents a possible solution toward BigBang singularity problem [110ndash112] On the other hand 119865(119877)gravity provides correction toward gravitational sector of theEinstein theory of gravity The importance of this correctionis highlighted in the context of black holes In order tohave better picture regarding the physical nature of the blackholes one may consider high energy regime effects as well(considering that the concepts of Hawking radiation werederived by studying black holes in semiclassicalquantumregime) Here we apply 119865(119877) gravityrsquos rainbow to find theeffects of 119865(119877) generalization as well as energy dependencyof space-time on the black hole solutions

The outline of our paper is as follows In Section 2we are going to investigate black hole solutions in Einsteingravityrsquos rainbow with PMI and CIM fields Then we wantto investigate conserved and thermodynamic quantities ofthe solutions and check the first law of thermodynamicsIn Section 3 we will obtain black hole solutions for 119865(119877)

Advances in High Energy Physics 3

gravityrsquos rainbow with CIM source and check the first lawof thermodynamics We also discuss thermal stability ofthese solutions and criteria governing stabilityinstability inSection 4Then we consider pure 119865(119877) gravityrsquos rainbow andcompare these solutions with 119865(119877) gravityrsquos rainbow withCIM source and give some related comments regarding itsthermodynamical behavior Finally we finish our paper withsome conclusions

2 Einstein Gravityrsquos Rainbow inthe Presence of PMI Field

Here we are going to introduce 119889-dimensional solutions ofthe Einstein gravityrsquos rainbow in the presence of PMI fieldwith the following Lagrangian

L = 119877 minus 2Λ + (120581F)119904 (1)

where 119877 and Λ are respectively the Ricci scalar and thecosmological constant In (1) the Maxwell invariant is F =

119865120583]119865120583] where 119865120583] = 120597120583119860] minus 120597]119860120583 is the electromagnetic

tensor field and 119860120583 is the gauge potential As we mentionedbefore in the limit 119904 = 1 with 120581 = minus1 Lagrangian (1)reduces to the Lagrangian of Einstein-Maxwell gravity Sincethe Maxwell invariant is negative henceforth we set 120581 = minus1

without loss of generality Using the variation principle wecan find the field equations the same as those obtained in[117]

21 Black Hole Solutions Here we will obtain charged rain-bow black hole solutions with negative cosmological constantin 119889-dimensions It is notable that the charged rainbowblack hole solutions in Einstein gravity coupled to nonlinearelectromagnetic fields have been studied in [118] In thispaper we want to extend the space-time to 119889-dimensions andobtain black hole solutions in the presence of PMI field as amatter source The rainbow metric for spherical symmetricspace-time in 119889-dimensions can be written as

1198891199042= minus

120595 (119903)

119891 (119864)21198891199052+

1

119892 (119864)2[

1198891199032

120595 (119903)+ 1199032119889Ω2] (2)

where

119889Ω2= 1198891205791 +

119889minus2

sum

119894=2

119894minus1

prod

119895=1

sin21205791198951198891205792

119894 (3)

Using metric (2) with the field equations one can obtainthe following solutions for the metric function as well asgauge potential

120595 (119903) = 1 minus119898

119903119889minus3minus

2Λ1199032

(119889 minus 1) (119889 minus 2) 119892 (119864)2

+

minus2((119889minus1)2)119892(119864)

119889minus3(119902119891(119864))

119889minus1

119903119889minus3ln(

119903

119897) 119904 =

119889 minus 1

2

1199032(2119904 minus 1)

2(2 [119902119891 (119864) 119892 (119864) (2119904 minus 119889 + 1)]

2119903(minus2119889+4)(2119904minus1)

(2119904 minus 1)2)119904

[(119889 minus 1) (119889 minus 2) minus 2 (119889 minus 2) 119904] 119892 (119864)2

otherwise

(4)

119860120583 = ℎ (119903) 120575119905

120583 (5)

where the consistent ℎ(119903) function is minus119902 ln(119903l) orminus119902119903(2119904minus119889+1)(2119904minus1) for 119904 = (119889 minus 1)2 and 119904 = (119889 minus 1)2

respectively In addition 119898 and 119902 are integration constantswhich are respectively related to themass and electric chargeof the black hole We should also mention that we consider119904 gt 12 for obtaining well-behaved electromagnetic fieldIt is notable that by replacing 119904 = 1 and 119892(119864) = 119891(119864) = 1

solutions (4) reduce to the following higher dimensionalReissner-Nordstrom black hole solutions

120595 (119903) = 1 minus119898


2Λ1199032

(119889 minus 1) (119889 minus 2)+

2 (119889 minus 3) 1199022

(119889 minus 2) 1199032(119889minus3)

(6)

In order to investigate the geometrical structure of thesesolutions we first look for the essential singularity(ies) TheRicci scalar can be written as

119877 =2119889

119889 minus 2Λ +

2((119889minus1)2)(119902119891(119864)119892(119864))

119889minus1

119903119889minus1 119904 =

119889 minus 1

2

4119904 minus 119889

119889 minus 2(2 [119902119891 (119864) 119892 (119864) (2119904 minus 119889 + 1)]

2119903(minus2119889+4)(2119904minus1)

(2119904 minus 1)2

)

119904

otherwise

(7)


and the behavior of Kretschmann scalar is

lim119903rarr0

119877120572120573120574120575119877120572120573120574120575

997888rarr infin

lim119903rarrinfin

119877120572120573120574120575119877120572120573120574120575

=8119889

(119889 minus 1) (119889 minus 2)2Λ2

(8)

therefore confirming that there is a curvature singularity at119903 = 0 In addition the Kretschmann scalar is (8119889(119889 minus 1)(119889 minus

2)2)Λ2 for 119903 rarr infin which confirms that the asymptotical

behavior of the charged rainbow black hole is adS It isworthwhile to mention that the asymptotical behavior ofthese solutions is independent of the rainbow functions andthe power of PMI source This independency comes fromthe fact that rainbow functions are sensible in high energyregime such as near-horizon and one expects to ignore itseffects far from the black hole (we only consider high energyregime) In addition at large distances the electric field ofMaxwell and PMI theories vanishes and therefore one mayexpect to ignore the effects of electric charge far from theorigin In order to investigate the possibility of the horizonwe plot the metric function versus 119903 in Figures 1 and 2 It isevident that depending on the choices of values for differentparameters wemay encounter two horizons (inner and outerhorizons) one extreme horizon and without horizon (nakedsingularity)

Special Case (119904 = 1198894) Here we are going to investigatethe special case 119904 = 1198894 the so-called conformally invariantMaxwell field (for more details see [68 85]) It was shownthat for 119904 = 1198894 the energy-momentum tensor willbe traceless and the corresponding electric field will beproportional to 119903

minus2 in arbitrary dimensions as it takes place

for the Maxwell field in 4 dimensionsTherefore we consider119904 = 1198894 into (4) and (5) to obtain

120595 (119903) = 1 minus119898


2Λ1199032

(119889 minus 1) (119889 minus 2) 119892 (119864)2

+2(119889minus4)4

(119902119891 (119864) 119892 (119864))1198892

119892 (119864)2119903119889minus2

(9)

ℎ (119903) = minus119902

119903 (10)

Considering 119892(119864) = 119891(119864) = 1 in the above equationwe obtain the higher dimensional Reissner-Nordstrom blackhole solution In order tomakemore investigations regardingthe properties of these solutions we plot the metric function(9) in Figure 3 It is evident that the behavior of Einstein-CIM-rainbowblack holes (see (9)) is similar to Einstein-PMI-rainbow black holes (see Figures 1 and 2)

22 Thermodynamics Here we are going to calculate theconserved and thermodynamic quantities and check the firstlaw of thermodynamics for such black hole In order to cal-culate thermodynamic quantities we start with temperatureUsing the definition of surface gravity one can calculate theHawking temperature of the black hole as

119879+ =1

2120587

radicminus1

2(nabla120583120594]) (nabla

120583120594]) (11)

where 120594 = 120597120597119905 is the Killing vector So the Hawkingtemperature for the rainbow black holes in the presence ofPMI source can be written as

119879+ = minus1

2120587119891 (119864)

Λ119903+

(119889 minus 2) 119892 (119864)minus

(119889 minus 3) 119892 (119864)

2119903+

minusT (12)

whereT is

T =

2(119889minus5)2

[119902119891 (119864) 119892 (119864)]119889

120587119902119891 (119864)2119892 (119864)2119903119889minus2+

119904 =119889 minus 1

2

2119904minus2

(2119904 minus 1) 119903+

(119889 minus 2) 120587119891 (119864) 119892 (119864)[119902119891 (119864) 119892 (119864) (119889 minus 2119904 minus 1)

(2119904 minus 1) 119903(119889minus2)(2119904minus1)

+

]

2119904

otherwise

(13)

in which 119903+ satisfies 119891(119903 = 119903+) = 0 Moreover the electricpotential 119880 is defined by [119 120]

119880 = 11986012058312059412058310038161003816100381610038161003816119903rarrreference minus 119860120583120594

12058310038161003816100381610038161003816119903=119903+ (14)

So for these black holes we obtain

119880 =

119902 ln(119903+

119897) 119904 =

119889 minus 1

2

119902119903(2119904minus3)(2119904minus1)

+otherwise

(15)

The entropy of the black holes satisfies the so-called arealaw of entropy in Einstein gravity It means that the blackholersquos entropy equals one-quarter of horizon area [121ndash123]Therefore the entropy of the black holes in 119889-dimensions is

119878 =1

4(

119903+

119892 (119864))

119889minus2

(16)

In order to obtain the electric charge of the black holesone can calculate the flux of the electromagnetic field atinfinity so we have


10 11 12 13 1409

r

minus01

0

01

02

03

04

120595(r)

(a)

minus01

0

01

02

03

04

120595(r)

10 11 12 13 1409

r

(b)

minus01

0

01

02

03

04

120595(r)

06 07 08 09 10 1105

r

(c)

Figure 1 120595(119903) versus 119903 for 119892(119864) = 11 119891(119864) = 11 Λ = minus1 and 119889 = 4 (a) for 119904 = 07 119902 = 1 119898 = 195 (doted line) 119898 = 187 (continuousline) and119898 = 180 (dashed line) (b) for 119904 = 07119898 = 2 119902 = 05 (doted line) 119902 = 07 (continuous line) and 119902 = 11 (dashed line) and (c) for119898 = 2 119902 = 1 119904 = 1125 (doted line) 119904 = 1108 (continuous line) and 119904 = 1095 (dashed line)

119876 =

minus1

1205872(119889minus3)2

(119889 minus 1) [119902119891 (119864)]119889minus2

119904 =119889 minus 1

2

119904 (2119904 minus 1)

8 (2119904 minus 119889 + 1) 120587119902119891 (119864) 119892 (119864)119889minus1

[radic2 (2119904 minus 119889 + 1) 119902119891 (119864) g (119864)

(2119904 minus 1)]

2119904

otherwise

(17)

The space-time introduced in (2) has boundaries withtime-like (120585 = 120597120597119905) Killing vector field It is straight-

forward to show that the total finite mass can be writtenas


minus005

0

005

010

015

120595(r)

06 07 08 0905

r

(a)

06 08 10 1204

r

minus02

minus01

0

01

02

03

04

05

06

120595(r)

(b)

Figure 2 120595(119903) versus 119903 for 119898 = 2 119902 = 08 Λ = minus1 and 119889 = 4 (a) for 119891(119864) = 100 119904 = 106 119892(119864) = 140 (dotted line) 119892(119864) = 086

(continuous line) and 119892(119864) = 068 (dashed line) and (b) for 119892(119864) = 10 119904 = 11 119891(119864) = 095 (dotted line) 119891(119864) = 102 (continuous line)and 119891(119864) = 110 (dashed line)

119872 =(119889 minus 2)119898

16120587119891 (119864) 119892 (119864)119889minus3

(18)

Now we are in a position to check the validity of thefirst law of thermodynamics To do so one can employ thefollowing relation

119889119872 (119878 119876) = (120597119872 (119878 119876)

120597119878)

119876

119889119878 + (120597119872(119878 119876)

120597119876)

119878

119889119876 (19)

It is a matter of calculation to show that the followingequalities hold

119879 = (120597119872

120597119878)

119876

119880 = (120597119872

120597119876)

119878

(20)

which confirm that the first law of thermodynamics is validfor the obtained thermodynamic and conserved quantitiesWe will address thermodynamic stability in Section 4

3 119865(119877) Gravityrsquos Rainbow in the Presence ofCIM Field

Here we consider 119865(119877) = 119877 + 119891(119877) gravityrsquos rainbow withthe CIM field as a matter source which leads to a traceless

energy-momentum tensor For 119889-dimensions the equationsof motion for 119865(119877) gravityrsquos rainbow with CIM source are

119877120583] (1 + 119891119877) minus

119892120583]

2119865 (119877) + (119892120583]nabla

2minus nabla120583nabla]) 119891119877

= 8120587T120583](21)

120597120583 (radicminus119892119865120583]F(1198894minus1)

) = 0 (22)

in which 119891119877 = 119889119891(119877)119889119877 In order to extract black holesolutions in 119865(119877) gravityrsquos rainbow coupled to the matterfield it is essential that we consider a traceless energy-momentum tensor

31 Black Hole Solutions We want to obtain the black holesolutions for constant scalar curvature (119877 = 1198770 = const)Using (5) and (22) with metric (2) one can find

ℎ (119903) =minusB

119903

119865tr =B

1199032

(23)

whereB is an integration constantRegarding the trace of (21) one finds 119877 = 1198770 =

119889119891(1198770)(2(1 + 119891119877) minus 119889) equiv 4Λ Substituting the mentioned1198770 into (21) we obtain the following equation

119877120583] (1 + 119891119877) minus1

119889119892120583]1198770 (1 + 119891119877) = 8120587T120583] (24)


1 2 30

r

minus1

0

1

2

3120595(r)

Figure 3 120595(119903) versus 119903 for 119902 = 1 119889 = 4 119891(119864) = 11 Λ = minus1119892(119864) = 10119898 = 300 (dotted line)119898 = 246 (continuous line) and119898 = 180 (dashed line)

Now considering metric (2) with (24) one can write thefield equations in the following forms

119892 (119864)2

119889 minus 211990312059510158401015840(119903) + 119892 (119864)

21205951015840(119903) +

2

119889 (119889 minus 2)1199031198770

=21198894

[B119892 (119864) 119891 (119864)]1198892

4 (1 + 119891119877) 119903119889minus1

119903119892 (119864)21205951015840(119903) + (119889 minus 3) 119892 (119864)

2[120595 (119903) minus 1] +

1199031198770

119889

= minus2(119889minus4)4

[B119892 (119864) 119891 (119864)]1198892

(1 + 119891119877) 119903119889minus2

(25)

which are corresponding to 119905119905 (or 119903119903) and 120593120593 (or 120579120579)components respectively After some calculations we canobtain the metric function in the following form

120595 (119903) = 1 minus119898

119903119889minus3+

2(119889minus4)4

(B119891 (119864) 119892 (119864))1198892

119892 (119864)2(1 + 119891119877) 119903

119889minus2

minus11987701199032

119889 (119889 minus 1) 119892 (119864)2

(26)

In order to have well-behaved solutions hereafter werestrict ourselves to 119891

1015840(1198770) = minus1 We are going to study

the general structure of the solutions For this purpose wemust investigate the behavior of the Kretschmann scalarThe Kretschmann scalar goes to infinity (infin) at the origin(lim119903rarr0119877120572120573120574120575119877

120572120573120574120575rarr infin) Also the Kretschmann scalar is

finite at infinity (lim119903rarrinfin119877120572120573120574120575119877120572120573120574120575

rarr (2119889(119889 minus 1))1198772

0) So

it shows that the Kretschmann scalar diverges at 119903 = 0 andis finite for 119903 = 0 Therefore there is a curvature singularitylocated at 119903 = 0 When we define 1198770 = (2119889(119889 minus 2))Λthe space-time will be asymptotically adS In order to havephysical solutions of this gravity we should restrict ourselvesto 119891119877 = minus1 Also for 119891119877 gt minus1 when we substitute 1198770 =

(2119889(119889 minus 2))Λ and B = 119902[1 + 119891119877]2119889 in (26) this solution

turns into the black hole solution of Einstein-CIM-gravityrsquosrainbow (9) with the same behavior In other words thesolutions obtained for 119865(119877) gravityrsquos rainbow in the presenceof CIM source (see (26)) are similar to the Einstein-CIM-rainbow solutions Therefore these black hole solutions haveat least one horizon

32 Thermodynamics Here considering 119865(119877) gravityrsquos rain-bow in the presence of CIM source we want to obtain theHawking temperature for the black hole solutions Using thedefinition of surface gravity and (26) one can calculate theHawking temperature as

119879+ =

(1 + 119891119877) [119889 (119889 minus 3) 119892 (119864)2minus 1198770119903

2

+] 119903119889minus2

+minus 119889 2

119889minus4[B119891 (119864) 119892 (119864)]

2119889

14

4119889120587119891 (119864) 119892 (119864) 119903119889minus1+

(1 + 119891119877)

(27)

For this theory the electric charge and the electricpotential of the conformally invariant 119865(119877) rainbow blackhole are respectively

119876 =11988921198894minus1

16120587(B119891 (119864)

119892 (119864))

1198892minus1

(28)

119880 =B

119903+

(29)

In the context of modified gravity theories the arealaw may be generalized and one can use the Wald entropy

associated with the Noether charge [124] In order to obtainthe entropy of black holes in 119865(119877) = 119877+119891(119877) theory one canuse a modification of the area law [125]

119878 =1

4(

119903+

119892 (119864))

119889minus2

(1 + 119891119877) (30)

which reveals that the area law does not hold for the obtainedblack hole solutions in 119865(119877) gravityrsquos rainbow


In addition using the Noether approach we find that thefinite mass can be obtained as

119872 =(119889 minus 2) (1 + 119891119877)119898

16120587119891 (119864) 119892 (119864)119889minus3

(31)

After calculating the conserved and thermodynamicquantities we can check the validity of the first law of blackhole thermodynamics Although 119865(119877) gravity and rainbowfunctionsmaymodify various quantities it is straightforwardto show that the modified conserved and thermodynamicquantities satisfy the first law of thermodynamics as 119889119872 =

119879119889119878 + 119880119889119876 The modified conserved and thermodynamicquantities suggest a deep connection between the horizonthermodynamics and geometrical properties in modifiedgravity

4 Thermodynamic Stability of CIM PMI and119865(119877) Models

In this section we employ the canonical ensemble approachtoward thermal stability and phase transition of the solutions

The canonical ensemble is based on studying the behaviorof heat capacity In this approach roots of the heat capacitywhich are exactly the same as the roots of temperatureare denoted as bound points which separate nonphysicalsolutions (having negative temperature) from physical ones(positive temperature) On the other hand the divergenciesof the heat capacity are denoted as second-order phasetransition points In other words in divergence point ofthe heat capacity system goes under a second-order phasetransition

The system is in thermal stable state if the signature ofheat capacity is positive In case of the negative heat capacitysystem may acquire stability by going under phase transitionor it may always be unstable which is known as nonphysicalcase The heat capacity is obtained as

119862119876 =119879

(12059721198721205971198782)119876

=119879

(120597119879120597119878)119876

(32)

Now considering (12) (16) (27) and (30) one can findthe following heat capacities

119862119876

=

minus(119889 minus 2) 119903

119889minus2

+

4119892119889minus2 (119864)(

[(119889 minus 3) 1198922(119864) minus 2Λ119903

2

+ (119889 minus 2)] 119903

119889minus3

+minus [radic2119902119891 (119864) 119892 (119864)]

119889minus1

[(119889 minus 3) 1198922(119864) + 2Λ1199032

+ (119889 minus 2)] 119903

119889minus3+

minus (119889 minus 2) [radic2119902119891 (119864) 119892 (119864)]119889minus1

) CIM

minus(119889 minus 2) 119903

119889minus2

+

4119892119889minus2 (119864)(

(119889 minus 2) (119889 minus 3) 1198922(119864) minus Θ (2119904 minus 1) 119903

2

+minus 2Λ119903

2

+

(119889 minus 2) (119889 minus 3) 1198922(119864) minus Θ (2119904 [119889 minus 3] + 1) 119903

2++ 2Λ1199032

+

) PMI

minus(119889 minus 2) (1 + 119891119877) 119903

119889minus2

+

4119892119889minus2 (119864)(

(1 + 119891119877) [(119889 minus 3) 1198922(119864) minus 2Λ119903

2

+ (119889 minus 2)] 119903

119889minus2

+minus4radic2119889minus4 (119902119891 (119864) 119892 (119864))

2119889

(1 + 119891119877) [(119889 minus 3) 1198922(119864) + 2Λ1199032

+ (119889 minus 2)] 119903

119889minus2+

minus ((119889 minus 1) (119889 minus 2))4radic2119889minus4 (119902119891 (119864) 119892 (119864))

2119889

) 119865 (119877)

(33)

in which

Θ = (2 [119902119891 (119864) 119892 (119864) (2119904 minus 119889 + 1)]

2119903(4minus2119889)(2119904minus1)

+

(2119904 minus 1)2

)

119904

(34)

Due to complexity of the obtained solutions it is notpossible to find the divergence and bound points analyticallyand therefore we employ numerical approach We presentthe results of our study through different diagrams (seeFigures 4ndash10)

Evidently for CIM and PMI the thermodynamical sta-bility and phase transitions are highly sensitive to variationsof the energy functions As one can see for these two casesby increasing energy functions the stability conditions willbe modified completely For small values of energy functionstwo divergencies and one bound point are observed (Figures4(a) 4(b) and 8(a)) The stable solutions exist betweenbound point and smaller divergency and also after largerdivergency Physical but unstable solutions are between twophase transition points and otherwise they are nonphysicalThedivergencies aremarking second-order phase transitionsBy increasing energy functions these behaviors will be

modified into two single states of nonphysical unstable andphysical stable solutions (Figures 4(c) and 8(a)) Interestinglysuch effects are not observed for 119865(119877) gravity In this case thevariations of energy functions act as a translation factor Inother words they shift the places of bound and divergencepoints (Figure 9(c))

As for the PMI solutions increasing PMI parameter 119904leads to modification in thermal structure of the solutionsFor small values of this parameter two regions of nonphysicalunstable and physical stable states exist (Figure 6(a)) Byincreasing 119904 a bound point and two singularities for the heatcapacity are formedThe bound point and smaller divergencyare decreasing functions of 119904 while larger divergency isan increasing function of it (see Figures 6 and 7) Similarbehavior is observed for 119891119877 in 119865(119877) gravity (Figures 9(a)and 9(b)) As for the effects of dimensionality in these threetheories the bound and divergence points are increasingfunctions of this parameter (see Figures 5 8(b) and 10)The only exception is an interesting behavior for smallerdivergence point for 119865(119877) where an abnormal behavior isobserved (see Figure 10(b))


001 002 003

minus010

minus005

0

005

010

(a)

01 02 03 04

minus1

minus05

0

05

1

(b)

05 1 15 2

minus1

minus05

0

05

1

(c)

Figure 4 For different scales (CIM case) 119862119876and 119879 (bold lines) versus 119903

+for 119902 = 1 Λ = minus1 and 119889 = 5 ((a) and (b)) 119892(119864) = 119891(119864) = 01

(continuous line) and 119892(119864) = 119891(119864) = 02 (dotted line) (c) 119892(119864) = 119891(119864) = 09 (continuous line) and 119892(119864) = 119891(119864) = 1 (dotted line)

5 Pure 119865(119877) Gravityrsquos Rainbow

In this section we are going to obtain the charged solutionsof pure 119865(119877) gravityrsquos rainbowTherefore we consider 119865(119877) =119877 + 119891(119877) gravity without matter field (Tmatt

120583] = 0) for thespherical symmetric space-time in 119889-dimensions

Using the field equation (21) and metric (2) one canobtain the following independent sourceless equations

11986510158401015840

119877+

J12119903120595 (119903)

1198651015840

119877minus

J22119903120595 (119903)

119865119877 =119865 (119877)

2120595 (119903) 119892 (119864)2

J11198651015840

119877minusJ2119865119877 =

119903119865 (119877)

119892 (119864)2

11986510158401015840

119877+

J3

119903120595 (119903)1198651015840

119877minus

(J3 minus 119889 + 3)

1199032120595 (119903)119865119877 =

119865 (119877)

2120595 (119903) 119892 (119864)2

(35)

where these equations are respectively corresponding to 119905119905119903119903 and 120593120593 components of gravitational field equation It isnotable that 119865119877 = 119889119865(119877)119889119877 prime and double prime denotethe first and second derivative with respect to 119903 and

J1 = 1199031205951015840(119903) + 2 (119889 minus 2) 120595 (119903)

J2 = 11990312059510158401015840(119903) + (119889 minus 2) 120595

1015840(119903)

J3 = 1199031205951015840(119903) + (119889 minus 3) 120595 (119903)

(36)

51 Black Hole Solutions Here we study black hole solutionsin pure 119865(119877) gravityrsquos rainbow with constant Ricci scalar(11986510158401015840119877

= 1198651015840

119877= 0) so it is easy to show that (35) reduce to the

following forms


1

05

0

minus05

minus1

01 02 03 040001 0002 0003 0004 0005 0006 0007 0008

minus02

minus01

0

01

02


+for 119902 = 1 Λ = minus1 and 119892(119864) = 119891(119864) = 01 119889 = 5 (continuous line)

and 119889 = 6 (dotted line)

1

05

02 04 06 08 10

minus05

minus1

(a)

05 1 15

1

05

0

minus05

minus1

(b)

Figure 6 For different scales (PMI case) 119862119876and 119879 (bold lines) versus 119903

+for 119902 = 1 Λ = minus1 119892(119864) = 119891(119864) = 07 and 119889 = 5 (a) 119904 = 09

(continuous line) 119904 = 1 (dotted line) and 119904 = 11 (dashed line) (b) 119904 = 12 (continuous line) 119904 = 13 (dotted line) and 119904 = 14 (dashed line)

119892 (119864)2[11990312059510158401015840(119903) + (119889 minus 2) 120595

1015840(119903)] 119865119877 = minus119903119865 (119877)

2119892 (119864)2[1199031205951015840(119903) + (119889 minus 3) (120595 (119903) minus 1)] 119865119877 = minus119903

2119865 (119877)

(37)

It is notable that there are differentmodels of119865(119877) gravitywhich may explain some of local (or global) properties ofthe universe One of these models which is supposed toexplain the positive acceleration of expanding universe is119865(119877) = 119877 minus 120583

4119877 model [63] Another model of 119865(119877)

gravity which was introduced by Kobayashi and Maeda [126]is 119865(119877) = 119877 + 120581119877

119899 (where 119899 gt 1) This model can resolvethe singularity problem arising in the strong gravity regime

On the other hand by adding an exponential correctionterm to the Einstein Lagrangian [40 127 128] one can provethat this model can satisfy both Solar system tests and highcurvature condition [129] Also one of the interestingmodelsthat passes all the theoretical and observational constraintsis known as the Starobinsky model (its functional form is119865(119877) = 119877 + 1205821198770((1 + 119877

21198772

0)minus119899

minus 1) [130 131]) It isnotable that this model could produce viable cosmologydifferent from the ΛCDM one at recent times and alsosatisfy Solar system and cosmological tests simultaneouslyMoreover its is worthwhile to mention that there are variousmodels of119865(119877) gravity inwhich thesemodels have interesting


01 02 03

minus4

minus2

0

2

4

115 120 125 130

minus4

minus2

0

2

4

Figure 7 For different scales (PMI case)119862119876and119879 (bold lines) versus 119903

+for 119902 = 1Λ = minus1 119892(119864) = 119891(119864) = 07 and 119889 = 5 119904 = 16 (continuous

line) 119904 = 17 (dotted line) and 119904 = 18 (dashed line)

minus02

minus04

04

02

0

05 1 15 2

(a)

minus01

minus03

minus02

minus04

04

05

05 1 15 2 25 3

02

03

0

01

(b)


+for 119902 = 1 Λ = minus1 and 119904 = 14 (a) 119889 = 5 119892(119864) = 119891(119864) = 07

(continuous line) 119892(119864) = 119891(119864) = 1 (dotted line) and 119892(119864) = 119891(119864) = 17 (dashed line) (b) 119892(119864) = 119891(119864) = 07 119889 = 5 (continuous line) and119889 = 7 (dotted line)

properties for describing our universe and also Solar system(see [132ndash148] for an incomplete list of references in thisdirection)

Besides another interesting model of 119865(119877) gravity wasintroduced byBamba et al [149]They constructed119865(119877) grav-ity theory corresponding to the Weyl invariant two-scalarfield theory (119865(119877) = (119890

120578(120593(119877))12)[1minus120593(119877)]119877minus119890

2120578(120593(119877))119869(120593(119877))

see [149] for more details) Their model can have theantigravity regions in which the Weyl curvature invariantdoes not diverge at the Big Crunch and Big Bang singularitiesAlso Nojiri and Odintsov investigated the antievaporation ofSchwarzschild-de Sitter and Reissner-Nordstrom black holes

by several interesting 119865(119877) models such as 119865(119877) = 11987721205812+

11989101198724minus2119899

119877119899+ 1198912119877

2 [150] and 119865(119877) = (11987721205812)(1 minus 1198771198770) [151]

In order to obtain an exact solution we should choose aproper model of 119865(119877) In this regard we use the followinginteresting models of 119865(119877) gravity with appropriate proper-ties

Type 1

119865 (119877) = 119877 minus 120582119890minus120585119877

+ 120578119877119899 (38)

Type 2

119865 (119877) = 119877 + 120572119877119899minus 1205731198772minus119899 (39)


8

6

4

2

0

minus2

minus4

04 06 08 10 12 14 16

(a)

4

2

0

minus2

minus4

04 06 08 10 12 14

(b)

04 06 08 10 12 14 16

3

2

1

0

minus1

minus2

minus3

(c)

Figure 9 For different scales (119865(119877) case) 119862119876and 119879 (bold lines) versus 119903

+for 119902 = 1 Λ = minus1 and 119889 = 5 (a) 119892(119864) = 119891(119864) = 07 119891

119877= 0

(continuous line) 119891119877= 09 (dotted line) and 119891

119877= 1 (dashed line) (b) 119892(119864) = 119891(119864) = 07 119891

119877= 14 (continuous line) 119891

119877= 15 (dotted line)

and 119891119877= 16 (dashed line) (c) 119891

119877= 15 119892(119864) = 119891(119864) = 07 (continuous line) 119892(119864) = 119891(119864) = 08 (dotted line) and 119892(119864) = 119891(119864) = 09

(dashed line)

Type 3

119865 (119877) = 119877 minus 1198982

1198881 (1198771198982)119899

1198882 (1198771198982)119899+ 1

(40)

Type 4

119865 (119877) = 119877 minus 119886 [119890minus119887119877

minus 1] + 119888119877119873 119890119887119877

minus 1

119890119887119877 + 1198901198871198770(41)

Type 1 This model includes an additional exponential term(120582119890minus120585119877) It was shown that such model enjoys validity of theSolar system tests and high curvature condition [129] whereas

it suffers instability for its solutions In order to remove suchinstability it is possible to add a correction term (120578119877119899 with119899 gt 1) such as the one introduced by Kobayashi and Maeda[126] in which the singularity problem in strong gravityregime is removed It is worthwhile to mention that for 119899 =

2 this model can be used to explain inflation mechanism[127 128 130] Type 1 model and some of its properties havebeen investigated in [51 68]

Type 2 Considering a scalar potential with nonzero valueof residual vacuum energy another model of 119865(119877) gravitywas proposed by Artymowski and Lalak [152] The reasonfor such consideration was to have a source for dark energyThis model is a stable minimum of the Einstein frame scalar


minus1

minus05

05

045 050 055

1

0

(a)

070 072 076074 078

minus1

minus05

05

1

0

(b)

minus1

minus05

05

15 25

1

1 2 3

0

(c)


+for 119902 = 1 Λ = minus1 119892(119864) = 119891(119864) = 07 and 119891

119877= 15 (a) 119889 = 6

(continuous line) 119889 = 7 (dotted line) and 119889 = 8 (dashed line)

potential of the auxiliary field The results of this model areconsistent with Planck data and also lack eternal inflationIt should be pointed out that here 120572 and 120573 are positiveconstants and 119899 is within the range [(12)(1 + radic3) 2] (see[43 152] for more details)

Type 3 The third model was introduced by Hu and Sawicki[134] This model provides a description regarding acceler-ated expansion without cosmological constant In additionthis model enjoys the validity of both cosmological and Solarsystem tests in the small-field limit It should be pointed outthat 119899 in this model is positive valued (119899 gt 0) where forthe case 119899 = 1 it covers the mCDTT with a cosmologicalconstant at high curvature regimes Furthermore for 119899 =

2 the inverse curvature squared model is included [153]

with a cosmological constant In this model 1198881 and 1198882 aredimensionless parameters and119898

2 is the mass scale (for moredetails see [134])

Type 4 The last model is able to describe both inflation inthe early universe and the recent accelerated expansion [40]In addition this model passes all local tests such as stabilityof spherical body solutions and generation of a very heavypositive mass for the additional scalar degree of freedom andnonviolation of Newtonrsquos law In this model119873 gt 2 and 119888 is apositive quantity (see [40] for more details)

To find the metric function 120595(119903) we use the componentsof (37) with the models under consideration Using (37) withmetric (2) one can obtain a general solution for these puregravity models in the following form


120595 (119903) = 1 minus2Λ1199032

(119889 minus 1) (119889 minus 2) 119892 (119864)2minus

119898

119903119889minus3

+2(119889minus4)4

[119902119891 (119864) 119892 (119864)]1198892

119892 (119864)2119903119889minus2

(42)

In order to satisfy all components of the field equations(see (37)) we should set the parameters of 119865(119877) model suchthat the following equations are satisfied

Type 1

119877[(119899 minus119889

2) 120578119877119899minus1

minus119889 minus 2

2] 119890120585119877

+ 120582(119889

2+ 120585119877) = 0

(1 + 119899120578119877119899minus1

) 119890120585119877

+ 120582120585 = 0

(43)

Type 2

[120572(119899 minus119889

2)119877119899minus

119889 minus 2

2119877]119877119899+ (119899 +

119889 minus 4

2)1205731198772

= 0

(120572119899119877119899+ 119877)119877

119899+ (119899 minus 2) 120573119877

2= 0

(44)

Type 3

1198882 [119889

119889 minus 211989821198881 minus 1198882119877](

119877

1198982)

2119899

minus [211989821198881

119889 minus 2(119899 minus

119889

2) + 21198882119877](

119877

1198982)

119899

minus 119877 = 0

[1198882

2(

119877

1198982)

2119899

+ 1]119877 + [21198882119877 minus 11989821198991198881] (

119877

1198982)

119899

= 0

(45)

Type 4

[119888 (119873 minus119889

2)119877119873minus

119889 minus 2

2119877 minus

119889

2119886] 1198903119887119877

+ 119886(119887119877 +119889

2)

sdot 11989021198871198770

+ 119888 [(119887119877 + 119873 minus119889

2) 1198901198871198770 + 119887119877 minus 119873 +

119889

2]119877119873

minus [(119889 minus 2) 119877 + 119886119889] 1198901198871198770 + 119886(119887119877 +

119889

2) 1198902119887119877

+ [119886 (2119887119877 + 119889) minus 119888 (119873 minus119889

2)119877119873

minus (119886119889

2minus

(119889 minus 2) 119877

2) 1198901198871198770] 119890119887(119877+1198770) = 0

(1 + 119888119873119877119873minus1

) 1198903119887119877

+ [1198901198871198770 (2119886119887 + 119890

1198871198770)

minus 119888119873119877119873minus1

1198901198871198770] 119890119887119877

+ [119888119877119873minus1

(1198901198871198770

(119873 + 119887119877) minus 119873 + 119887119877) + 119886119887 + 21198901198871198770]

sdot 1198902119887119877

= 0

(46)

Solving (43) (44) (45) and (46) we obtain the parame-ters of 119865(119877)models as follows

Type 1

120582 =119877 (119899 minus 1) 119890

120585119877

119899 + 120585119877

120578 = minus(1 + 120585119877)

119877119899minus1 (119899 + 120585119877)

(47)

Type 2

120572 = minus1

2119877119899minus1

120573 =119877119899minus1

2

(48)

Type 3

1198982= 119877(

119899 minus 1

1198882

)

minus1119899

1198881 = 119899(119899 minus 1

1198882

)

1119899minus1

(49)

Type 4

119888 =

[119890119887119877

+ 1198901198871198770]2

(119887119877 + 1 minus 119890119887119877)

119877119873minus1 [119873 (119890119887119877 + 1198901198871198770) + 119887119877] (119890119887119877 minus 1)

119886 = minus

119877 (119873 minus 1) 1198902119887119877

+ [119887119877 (1 + 1198901198871198770) + (119873 minus 1) (119890

1198871198770 minus 1)] 119890119887119877 119890119887119877

(119890119887119877 minus 1)2[119887119877 + 119873 (119890119887119877 + 1198901198871198770)]

(50)

Equation (42) is similar to the solutions obtained forEinstein-CIM-gravityrsquos rainbow (see (9)) Calculations show

that theRicci scalar is119877 = (2119889(119889minus2))Λ and theKretschmannscalar has the following behaviors


1 2 30

r

minus2

minus1

0

1

2

3

4120595(r)

Figure 11 120595(119903) versus 119903 for 119898 = 2 119889 = 4 119891(119864) = 11 Λ = minus1119892(119864) = 11 119902 = 050 (dotted line) 119902 = 085 (continuous line) and119902 = 110 (dashed line)

lim119903rarr0

119877120572120573120574120575119877120572120573120574120575

997888rarr infin

lim119903rarrinfin

119877120572120573120574120575119877120572120573120574120575

997888rarr8119889

(119889 minus 1) (119889 minus 2)2Λ2

(51)

which confirm that there is a curvature singularity at 119903 =

0 In addition we find that the asymptotical behavior ofthe mentioned space-time is similar to (a)dS Einstein-CIM-rainbow black holes In other words one can extract chargedsolutions from the pure gravity if the 119865(119877) model and itsparameters are chosen suitably

For more investigation of these solutions we plot 120595(119903)versus 119903 Figure 11 shows that the mentioned singularity canbe covered with two horizons (Cauchy horizon and eventhorizon) or one horizon (extreme black hole) and otherwisewe encounter a naked singularity

52 Thermodynamics In order to check the first law ofthermodynamics for the obtained charged black hole firstwe investigate the Hawking temperature of the pure 119865(119877)

gravityrsquos rainbow black holes on the outer horizon 119903+ as

119879+ =

119903119889minus2

+[(119889 minus 3) 119892

2(119864) minus 2Λ119903

2

+ (119889 minus 2)] minus 2

(119889minus4)4[119892119891 (119864) 119892 (119864)]

1198892

4120587119891 (119864) 119892 (119864) 119903119889minus1+

(52)

Using the method introduced in [125] for the obtainedsolutions of pure 119865(119877) gravityrsquos rainbow with 119865(119877) = 119865119877 = 0one may get zero entropy (119878 = (1198604)119865119877 = 0 119860 being thehorizon area)Therefore in order to resolve this problem andto obtain correct nonzero entropy one may consider the first

law of thermodynamics as a fundamental principle In otherwords we can use 119889119878 = (1119879)119889119872 Using 120597120597119905 as a Killingvector one can extract the following relation for the finitemass as

119872 =(119889 minus 2)119898

16120587119891 (119864) 119892 (119864)119889minus3

=

[(119889 minus 2) 1198922(119864) minus 2Λ119903

2

+ (119889 minus 1)] 119903

119889minus2

++ 2(119889minus4)4

(119889 minus 2) [119892119891 (119864) 119892 (119864)]1198892

16120587119891 (119864) 119892 (119864)119889minus1

119903+

(53)

Hence we can write

119889119872 =

119903119889minus3

+[((119889 minus 2) (119889 minus 3) 119892

2(119864) minus 2Λ119903

2

+) 119903119889minus2

+minus 2(119889minus4)4

(119889 minus 2) (119892119891 (119864) 119892 (119864))1198892

]

16120587119891 (119864) 119892119889minus1

(119864) 119903119889minus1+

119889119903+(54)

so we can obtain the entropy in the following form

119878 = int1

119879119889119872 =

1

4(

119903+

119892 (119864))

119889minus2

(55)

where it is the area law for the entropy In other wordsconsidering 119865(119877) gravityrsquos rainbow with 119865119877 = 0 one may usethe so-called area law instead of modified area law (see (30))[125]

53 DK Stability Here we are going to discuss the Dolgov-Kawasaki (DK) stability of these solutions DK stability in119865(119877) gravity has been investigated in literature [61 125 135154ndash156] It has been shown that there is no stable groundstate for models of 119865(119877) gravity when 119865(119877) = 0 and also119865119877 = 119889119865(119877)119889119877 = 0 [157] It is notable that we find that119865(119877) = 119865119877 = 0 for the obtained solutions (in order toobtain the charged black hole solution in 119865(119877) gravity withconstant Ricci scalarNojiri andOdintsov in [151] showed that


minus04

minus02

02

04

FRR

0 1 2minus1

120585

(a)

b

minus4

minus2

0

2

4

FRR

2515 2

(b)

Figure 12 Type 1 (a) 119865119877119877

versus 120585 forΛ = minus1 and 119899 = 2 (dashed line) and 119899 = 3 (continuous line) Type 4 (b) 119865119877119877

versus 119887 for119873 = 4 1198770= 4

Λ = minus06 (dashed line) and Λ = minus05 (continuous line)

119865(119877) and 119865119877 must be zero) On the other hand it has beenshown that 119865119877119877 = 119889

2119865(119877)119889119877

2 is related to the effective massof the dynamical field of the Ricci scalar (see [158 159] formore details) So the positive effective mass is a requirement

usually referred to DK stability criterion which leads to stabledynamical field [160 161] In order to check this stability wecalculate the second derivative of 119865(119877) functions with respectto the Ricci scalar for specific models in the following forms

119865119877119877

=

minus(119899 minus 1) [120585119877 (119899 + 120585119877) + 119899]

119877 (119899 + 120585119877) Type 1

minus (119899 minus 1)2

119877 Type 2

119899 minus 1

119877 Type 3

1198872119877 [(119899 minus 1) 119890

2119887119877+ (1198901198871198770 (119887R + 119899 minus 1) + 119887119877 minus 119899 + 1) 119890

119887119877minus (119899 minus 1) 119890

1198871198770]

(119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770]

minus

(119887119877 + 1 minus 119890119887119877) 119899 (1 minus 119899) 119887

3119887119877+ 12059111198902119887119877

minus 1205912119890119887(119877+1198770) + 119899 (119899 minus 1) 119890

21198871198770

(119890119887119877 + 1198901198871198770) (119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770] 119877

Type 4

(56)

where

1205911 = [2119899 (1 minus 119899) + 119887119877 (119887119877 minus 2119899)] 1198901198871198770 + 119899 (1 minus 119899)

+ 119887119877 (119887119877 minus 2119899)

1205912 = [119899 (119899 minus 1) + 119887119877 (119887119877 + 2119899)] 1198901198871198770 + 2119899 (1 minus 119899)

+ 119887119877 (119887119877 + 2119899)

(57)

In order to have DK stability 119865119877119877must be positive (119865119877119877 gt0) As one can see for Types 2 and 3 models the sign of119865119877119877 depends on the sign of Ricci scalar In other words byconsideringΛ lt 0 andΛ gt 0 the obtained solutions have DKstability for Types 2 and 3 respectively On the other hand forTypes 1 and 4 the sign of119865119877119877 is not clear for arbitrary values ofparameters So we plot119865119877119877 in Figure 12 to analyze the sign of119865119877119877 These figures show that one may obtain stable solutions

for special values of parameters of models In other words wecan set free parameters to obtain stable models

6 Conclusions

In this paper we have considered higher dimensionalEinstein-PMI and Einstein-CIM theories in the presence ofenergy dependent space-time We obtained metric functionsand discussed geometrical properties as well as thermody-namical quantitiesWe showed that despite the contributionsof the gravityrsquos rainbow in thermodynamical quantities thefirst law of black holes thermodynamics was valid Next westudied pure 119865(119877) gravity as well as 119865(119877) gravity with CIMfield in the presence of gravityrsquos rainbow We pointed outthat in case of pure 119865(119877) gravityrsquos rainbow for the suitablechoices of different parameters one can obtain the electrical


charge and cosmological constant simultaneously In otherwords we showed that the pure 119865(119877) gravityrsquos rainbowwas equivalent to the Einstein-CIM-gravityrsquos rainbow Alsoin case of calculations of thermodynamical quantities forthe pure 119865(119877) gravityrsquos rainbow we pointed out that forthese classes of black holes one should employ area law forobtaining entropy instead of the usual modified approachesin case of 119865119877 = 0

In addition we conducted a study regarding the thermalstability of the obtained solutions in this paper We pointedout that variations of energy functions in PMI class of thesolutions lead to modifications in stability conditions phasetransition points and thermal structure of the solutionswhile the effects of such variations in 119865(119877) gravity weretranslation-like We also observed that PMI parameter 119904and 119865(119877) gravity parameter 119891119877 were modifying factors instability conditions and phase transition points Regardingthe effect of dimensionality we found that it has a translation-like behavior too An abnormal behavior was observed forsmaller divergency in 119865(119877)model

Regarding the results of this paper one may regardextended phase space and 119875-119881 criticality of the solutionsIn addition we can investigate different viable models of119865(119877) gravity to study the possible abnormal behavior inthermal stability Moreover it is interesting to use a suitablelocal transformation with appropriate boundary conditionsto obtain the so-called Nariai space-time [162 163] anddiscuss the antievaporation process [164ndash167]We leave theseissues for future work

Competing Interests

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

Acknowledgments

The authors thank Shiraz University Research Council Thiswork has been supported financially by the Research Institutefor Astronomy and Astrophysics of Maragha Iran

References

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[2] S Perlmutter G Aldering G Goldhaber et al ldquoMeasurementsofΩ andΛ from42high-redshift supernovaerdquoTheAstrophysicalJournal vol 517 pp 565ndash586 1999

[3] D N Spergel R Bean O Dore et al ldquoThree-year WilkinsonMicrowave Anisotropy Probe (WMAP) observations implica-tions for cosmologyrdquoTheAstrophysical Journal Supplement vol170 no 2 p 377 2007

[4] S Cole W J Percival J A Peacock et al ldquoThe 2dF GalaxyRedshift Survey power-spectrum analysis of the final data setand cosmological implicationsrdquo Monthly Notices of the RoyalAstronomical Society vol 362 no 2 pp 505ndash534 2005

[5] D J Eisenstein I Zehavi D W Hogg et al ldquoDetection of thebaryon acoustic peak in the large-scale correlation function ofSDSS luminous red galaxiesrdquoTheAstrophysical Journal vol 633no 2 p 560 2005

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[7] B Jain and A Taylor ldquoCross-correlation tomography measur-ing dark energy evolution with weak lensingrdquo Physical ReviewLetters vol 91 no 14 p 141302 2003

[8] V Sahni and A Starobinsky ldquoThe case for a postive cosmolog-ical 120582-termrdquo International Journal of Modern Physics D vol 9no 4 pp 373ndash443 2000

[9] E J Copeland M Sami and S Tsujikawa ldquoDynamics of darkenergyrdquo International Journal ofModern Physics D GravitationAstrophysics Cosmology vol 15 no 11 pp 1753ndash1935 2006

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[14] J Garriga and V F Mukhanov ldquoPerturbations in k-inflationrdquoPhysics Letters B vol 458 no 2-3 pp 219ndash225 1999

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[17] A Dev J S Alcaniz and D Jain ldquoCosmological consequencesof a Chaplygin gas dark energyrdquo Physical Review D vol 67Article ID 023515 2003

[18] R J Scherrer ldquoPurely kinetic k essence as unified dark matterrdquoPhysical Review Letters vol 93 Article ID 011301 2004

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[22] M Demetrian ldquoFalse vacuum decay in a brane world cosmo-logical modelrdquoGeneral Relativity and Gravitation vol 38 no 5pp 953ndash962 2006

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[26] D C Zou S J Zhang and BWang ldquoHolographic charged fluiddual to third order Lovelock gravityrdquo Physical Review D vol 87no 8 Article ID 084032 2013

[27] J XMo andWB Liu ldquoPndashV criticality of topological black holesin Lovelock-Born-Infeld gravityrdquoTheEuropean Physical JournalC vol 74 article 2836 2014

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[38] M Akbar and R-G Cai ldquoThermodynamic behavior of fieldequations for f (R) gravityrdquo Physics Letters B vol 648 no 2-3pp 243ndash248 2007

[39] K Bamba and S D Odintsov ldquoInflation and late-time cosmicacceleration in non-minimal Maxwell-F(R) gravity and thegeneration of large-scale magnetic fieldsrdquo Journal of Cosmologyand Astroparticle Physics vol 2008 no 4 article 024 2008

[40] G Cognola E Elizalde S Nojiri S D Odintsov L Sebastianiand S Zerbini ldquoClass of viable modified f(R) gravities describ-ing inflation and the onset of accelerated expansionrdquo PhysicalReview DmdashParticles Fields Gravitation and Cosmology vol 77no 4 Article ID 046009 2008

[41] S Nojiri and S D Odintsov ldquoFuture evolution and finite-time singularities in F(R) gravity unifying inflation and cosmicaccelerationrdquo Physical Review D vol 78 no 4 Article ID046006 2008

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[43] A De Felice and S Tsujikawa ldquof (R) theoriesrdquo Living Reviews inRelativity vol 13 article 3 2010

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[45] T P Sotiriou and V Faraoni ldquo119891(119877) theories of gravityrdquo Reviewsof Modern Physics vol 82 no 1 pp 451ndash497 2010

[46] S Nojiri and S D Odintsov ldquoUnified cosmic history inmodified gravity from F(R) theory to Lorentz non-invariantmodelsrdquo Physics Reports vol 505 no 2ndash4 pp 59ndash144 2011

[47] S Capozziello and M De Laurentis ldquoExtended theories ofgravityrdquo Physics Reports vol 509 no 4-5 pp 167ndash321 2011

[48] M Chaichian A Ghalee and J Kluson ldquoRestricted f(R) gravityand its cosmological implicationsrdquo Physical Review D vol 93no 10 Article ID 104020 2016

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[50] S Nojiri S D Odintsov and V K Oikonomou ldquoUnimodularf(R) gravityrdquo Journal of Cosmology and Astroparticle Physics vol2016 no 5 p 46 2016

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[66] T Moon Y S Myung and E J Son ldquof (R) black holesrdquo GeneralRelativity and Gravitation vol 43 no 11 pp 3079ndash3098 2011

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[72] I Z Stefanov S S Yazadjiev and M D Todorov ldquoScalar-tensor black holes coupled to Euler-Heisenberg nonlinearelectrodynamicsrdquo Modern Physics Letters A vol 22 no 17 pp1217ndash1231 2007

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[75] S H Hendi ldquoAsymptotic charged BTZ black hole solutionsrdquoJournal of High Energy Physics vol 2012 no 3 article 065 2012

[76] J Diaz-Alonso and D Rubiera-Garcia ldquoThermodynamic anal-ysis of black hole solutions in gravitating nonlinear electrody-namicsrdquo General Relativity and Gravitation vol 45 no 10 pp1901ndash1950 2013

[77] S H Mazharimousavi M Halilsoy and O Gurtug ldquoA newEinstein-nonlinear electrodynamics solution in 2 + 1 dimen-sionsrdquo European Physical Journal C vol 74 article 2735 2014

[78] S I Kruglov ldquoAmodel of nonlinear electrodynamicsrdquoAnnals ofPhysics vol 353 pp 299ndash306 2015

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[81] S H Hendi B Eslam Panah and S Panahiyan ldquoEinstein-Born-Infeld-massive gravity adS-black hole solutions and theirthermodynamical propertiesrdquo Journal of High Energy Physicsvol 2015 no 11 article 157 pp 1ndash29 2015

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[87] J-J Peng and S-Q Wu ldquoCovariant anomaly and Hawkingradiation from the modified black hole in the rainbow gravitytheoryrdquo General Relativity and Gravitation vol 40 no 12 pp2619ndash2626 2008

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[91] A F Ali M Faizal and M M Khalil ldquoRemnant for all blackobjects due to gravityrsquos rainbowrdquoNuclear Physics B vol 894 pp341ndash360 2015

[92] A F Ali M Faizal and B Majumder ldquoAbsence of an effectiveHorizon for black holes in Gravityrsquos rainbowrdquo EurophysicsLetters vol 109 no 2 Article ID 20001 2015

[93] Y Ling X Li and H Zhang ldquoThermodynamics of modifiedblack holes from gravityrsquos rainbowrdquo Modern Physics Letters Avol 22 no 36 pp 2749ndash2756 2007

[94] H Li Y Ling andXHan ldquoModified (A)dS Schwarzschild blackholes in rainbow spacetimerdquo Classical and Quantum Gravityvol 26 no 6 Article ID 065004 2009

[95] R Garattini and B Majumder ldquoNaked singularities are notsingular in distorted gravityrdquo Nuclear Physics B vol 884 pp125ndash141 2014

[96] Z Chang and SWang ldquoNearly scale-invariant power spectrumand quantum cosmological perturbations in the gravityrsquos rain-bow scenariordquo The European Physical Journal C vol 75 article259 2015

[97] G Santos G Gubitosi and G Amelino-Camelia ldquoOn theinitial singularity problem in rainbow cosmologyrdquo Journal ofCosmology and Astroparticle Physics vol 2015 no 8 article 0052015

[98] A F Ali and M M Khalil ldquoA proposal for testing gravityrsquosrainbowrdquo EPL (Europhysics Letters) vol 110 no 2 Article ID20009 2015

[99] G Yadav B Komal and B R Majhi ldquoRainbow Rindler metricand Unruh effectrdquo httpsarxivorgabs160501499

[100] S H Hendi S Panahiyan B Eslam Panah M Faizal and MMomennia ldquoCritical behavior of charged black holes in Gauss-Bonnet gravityrsquos rainbowrdquo Physical Review D vol 94 no 2Article ID 024028 2016

[101] P Galan and G A Mena Marugan ldquoEntropy and temperatureof black holes in a gravityrsquos rainbowrdquo Physical Review D vol 74no 4 Article ID 044035 2006


[102] A F Ali ldquoBlack hole remnant from gravityrsquos rainbowrdquo PhysicalReview D vol 89 no 10 Article ID 104040 2014

[103] Y Gim and W Kim ldquoBlack hole complementarity in gravityrsquosrainbowrdquo Journal of Cosmology and Astroparticle Physics vol 5article 002 2015

[104] S H Hendi S Panahiyan and B Eslam Panah ldquoChargedblack hole solutions in Gauss-Bonnet-massive gravityrdquo Journalof High Energy Physics vol 2016 article 129 2016

[105] S H Hendi G H Bordbar B Eslam Panah and S PanahiyanldquoModified TOV in gravityrsquos rainbow properties of neutronstars and dynamical stability conditionsrdquo httpsarxivorgabs150905145

[106] R Garattini and GMandanici ldquoGravityrsquos rainbow and compactstarsrdquo httparxivorgabs160100879

[107] R Garattini and F S N Lobo ldquoGravityrsquos rainbow andtraversable wormholesrdquo httparxivorgabs151204470

[108] A S Sefiedgar ldquoHow can rainbow gravity affect on gravitationalforcerdquo httparxivorgabs151208372v1

[109] A Chatrabhuti V Yingcharoenrat and P Channuie ldquoStarobin-sky model in rainbow gravityrdquo Physical Review D vol 93 no 4Article ID 043515 2016

[110] A Awad A F Ali and B Majumder ldquoNonsingular rainbowuniversesrdquo Journal of Cosmology and Astroparticle Physics vol2013 no 10 article 052 2013

[111] Y Ling ldquoRainbowuniverserdquo Journal of Cosmology andAstropar-ticle Physics vol 2007 no 8 article 017 2007

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[115] R Garattini and E N Saridakis ldquoGravityrsquos Rainbow a bridgetowardsHoravandashLifshitz gravityrdquoTheEuropeanPhysical JournalC vol 75 no 7 p 343 2015


[117] S H Hendi ldquoRotating black branes in the presence of nonlinearelectromagnetic fieldrdquoThe European Physical Journal C vol 69no 1-2 pp 281ndash288 2010

[118] S H Hendi S Panahiyan B Eslam Panah and M MomennialdquoThermodynamic instability of nonlinearly charged black holesin gravityrsquos rainbowrdquo The European Physical Journal C vol 76article 150 2016

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[125] G Cognola E Elizalde S Nojiri S D Odintsov and SZerbini ldquoOne-loop 119891(119877) gravity in de Sitter universerdquo Journalof Cosmology and Astroparticle Physics vol 2005 no 2 p 102005

[126] T Kobayashi and K Maeda ldquoCan higher curvature correctionscure the singularity problem in f (R) gravityrdquo Physical ReviewD vol 79 Article ID 024009 2009

[127] S Nojiri and S D Odintsov ldquoNon-singular modified gravityunifying inflation with late-time acceleration and universalityof viscous ratio bound in F(R) theoryrdquo Progress of TheoreticalPhysics Supplements vol 190 pp 155ndash178 2011

[128] E Elizalde S Nojiri S D Odintsov L Sebastiani and SZerbini ldquoNonsingular exponential gravity a simple theory forearly- and late-time accelerated expansionrdquo Physical Review Dvol 83 no 8 Article ID 086006 14 pages 2011

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[130] A A Starobinsky and H-J Schmidt ldquoOn a general vacuumsolution of fourth-order gravityrdquo Classical and Quantum Grav-ity vol 4 no 3 pp 695ndash702 1987

[131] A A Starobinsky ldquoDisappearing cosmological constant in f(R)gravityrdquo Journal of Experimental andTheoretical Physics vol 86no 3 pp 157ndash163 2007

[132] P Zhang ldquoTesting gravity against the early time integratedSachs-Wolfe effectrdquo Physical Review D vol 73 no 12 ArticleID 123504 5 pages 2006

[133] S A Appleby and R A Battye ldquoDo consistent F(R) modelsmimic general relativity plus Λrdquo Physics Letters B vol 654 no1-2 pp 7ndash12 2007

[134] W Hu and I Sawicki ldquoParametrized post-Friedmann frame-work for modified gravityrdquo Physical Review D vol 76 no 10Article ID 104043 12 pages 2007

[135] L Amendola R Gannouji D Polarski and S TsujikawaldquoConditions for the cosmological viability of 119891(119877) dark energymodelsrdquo Physical Review DmdashParticles Fields Gravitation andCosmology vol 75 no 8 Article ID 083504 2007

[136] S Tsujikawa ldquoObservational signatures of f (R) dark energymodels that satisfy cosmological and local gravity constraintsrdquoPhysical Review D vol 77 no 2 Article ID 023507 13 pages2008

[137] E V Linder ldquoExponential gravityrdquo Physical Review D vol 80no 12 Article ID 123528 6 pages 2009

[138] S Nojiri S D Odintsov and D Saez-Gomez ldquoCosmologicalreconstruction of realistic modified F(R) gravitiesrdquo PhysicsLetters B vol 681 no 1 pp 74ndash80 2009

[139] K Bamba C Q Geng and C C Lee ldquoGeneric feature of futurecrossing of phantom divide in viable f (R) gravity modelsrdquoJournal of Cosmology and Astroparticle Physics vol 2010 no 11article 001 2010

[140] Z Girones A Marchetti O Mena C Pena-Garay and N RiusldquoCosmological data analysis of f (R) gravity modelsrdquo Journal ofCosmology andAstroparticle Physics vol 2010 no 11 article 0042010

[141] J H He and B Wang ldquoRevisiting f (R) gravity models thatreproduceΛCDMexpansionrdquoPhysical ReviewD vol 87 ArticleID 023508 2013

[142] Q G Huang ldquoA polynomial f (R) inflation modelrdquo Journal ofCosmology and Astroparticle Physics vol 2014 no 2 article 0352014


[143] S D Odintsov and V K Oikonomou ldquoSingular inflationaryuniverse from F(R) gravityrdquo Physical Review D vol 92 no 12Article ID 124024 22 pages 2015

[144] K Bamba and S D Odintsov ldquoInflationary cosmology inmodified gravity theoriesrdquo Symmetry vol 7 no 1 pp 220ndash2402015

[145] S I Kruglov ldquoAnewF(R) gravitymodelrdquoAstrophysics and SpaceScience vol 358 article 48 2015

[146] M Kusakabe S Koh K S Kim andM K Cheoun ldquoCorrectedconstraints on big bang nucleosynthesis in a modified gravitymodel of 119891(119877) prop 119877

119899rdquo Physical Review D vol 91 no 10 ArticleID 104023 7 pages 2015

[147] C Q Geng C C Lee and J L Shen ldquoMatter power spectrain viable f (R) gravity models with massive neutrinosrdquo PhysicsLetters B vol 740 pp 285ndash290 2015

[148] K Dutta S Panda and A Patel ldquoViability of an arctan modelof f (R) gravity for late-time cosmologyrdquo Physical Review D vol94 no 2 Article ID 024016 8 pages 2016

[149] K Bamba S Nojiri S D Odintsov and D Saez-Gomez ldquoPos-sible antigravity regions in F(R) theoryrdquo Physics Letters B vol730 pp 136ndash140 2014

[150] S Nojiri and S D Odintsov ldquoAnti-evaporation ofSchwarzschildndashde Sitter black holes in F(R) gravitrdquo Classicaland Quantum Gravity vol 30 no 12 Article ID 125003 2013

[151] S Nojiri and S D Odintsov ldquoInstabilities and anti-evaporationof Reissner-Nordstrom black holes in modified 119891(119877) gravityrdquoPhysics Letters B vol 735 pp 376ndash382 2014

[152] M Artymowski and Z Lalak ldquoInflation and dark energy fromf (R) gravityrdquo Journal of Cosmology andAstroparticle Physics vol2014 no 9 article 036 2014

[153] O Mena J Santiago and J Weller ldquoConstraining inverse-curvature gravity with supernovaerdquo Physical Review Letters vol96 no 4 Article ID 041103 4 pages 2006

[154] D Bazeia B C da Cunha R Menezes and A Yu PetrovldquoPerturbative aspects and conformal solutions of F(R) gravityrdquoPhysics Letters B vol 649 no 5-6 pp 445ndash453 2007

[155] T Rador ldquoAcceleration of the Universe via f (R) gravities andthe stability of extra dimensionsrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 75 no 6 Article ID064033 2007

[156] V Faraoni ldquoPhase space geometry in scalar-tensor cosmologyrdquoAnnals of Physics vol 317 no 2 pp 366ndash382 2005

[157] H-J Schmidt ldquoComparing self-interacting scalar fields and R +R3 cosmological modelsrdquo Astronomische Nachrichten vol 308no 3 pp 183ndash188 1987

[158] A D Dolgov and M Kawasaki ldquoCan modified gravity explainaccelerated cosmic expansionrdquo Physics Letters B vol 573 pp1ndash4 2003

[159] I Sawicki andWHu ldquoStability of cosmological solutions in f (R)models of gravityrdquo Physical Review D vol 75 no 12 Article ID127502 2007

[160] V Faraoni ldquoMatter instability in modified gravityrdquo PhysicalReview D vol 74 no 10 Article ID 104017 4 pages 2006

[161] O Bertolami and M Carvalho Sequeira ldquoEnergy conditionsand stability in 119891(119877) theories of gravity with nonminimalcoupling to matterrdquo Physical Review D vol 79 no 10 ArticleID 104010 2009

[162] H Nariai ldquoOn some static solutions of Einsteinrsquos gravitationalfield equations in a spherically symmetric caserdquo Science Reportsof the Tohoku Imperial University vol 34 p 160 1950

[163] H Nariai ldquoOn a new cosmological solution of einsteinrsquos fieldequations of gravitationrdquo Science Reports of the Tohoku ImperialUniversity vol 35 p 62 1951

[164] S Nojiri and S D Odintsov ldquoEffective action for conformalscalars and antievaporation of black holesrdquo International Jour-nal of Modern Physics A vol 14 no 8 pp 1293ndash1304 1999

[165] S Nojiri and S D Odintsov ldquoQuantum evolution ofSchwarzschild-de Sitter (Nariai) black holesrdquo Physical ReviewD vol 59 no 4 Article ID 044026 1999

[166] R Casadio and C Germani ldquoGravitational collapse and blackhole evolution do holographic black holes eventually lsquoanti-evaporatersquordquo Progress of Theoretical Physics vol 114 no 1 pp23ndash56 2005

[167] L Sebastiani D Momeni R Myrzakulov and S D OdintsovldquoInstabilities and (anti)-evaporation of Schwarzschild-de Sitterblack holes in modified gravityrdquo Physical Review DmdashParticlesFields Gravitation and Cosmology vol 88 no 10 Article ID104022 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



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AerodynamicsJournal of

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be reconstructed starting from data and physically motivatedissues

However the field equations of 119865(119877) gravity are compli-cated fourth-order differential equations and it is not easy tofind exact solutions In addition adding amatter field to119865(119877)

gravity makes the field equations much more complicatedOn the other hand regarding constant curvature scalarmodelas a subclass of general 119865(119877) gravity can simplify the fieldequations Also one can extract exact solutions of 119865(119877)

gravity coupled to a traceless energy-momentum tensor withconstant curvature scalar [66] For example consideringexact solutions of 119865(119877) gravity with conformally invariantMaxwell (CIM) field as a matter source has been investigated[67 68]

General relativity coupled to a nonlinear electrodynamicsattracts significant attentions because of its specific prop-erties in gaugegravity coupling Interesting properties ofvarious models of the nonlinear electrodynamics have beeninvestigated before [69ndash81] Power Maxwell invariant (PMI)theory is one of the interesting branches of the nonlinearelectrodynamics and its Lagrangian is an arbitrary powerof Maxwell Lagrangian [82ndash84] The PMI theory is moreinteresting with regard to Maxwell field and for unit powerit reduces to linear Maxwell theory These nonlinear elec-trodynamics enjoy conformal invariancy when the power ofMaxwell invariant is a quarter of space-time dimensions andthis is one of the attractive properties of this theory In otherwords one can obtain traceless energy-momentum tensorfor a special case ldquopower = dimensions4rdquo which leadsto conformal invariancy It is notable to mention that thisidea has been considered to take advantage of the conformalsymmetry to construct the analogues of the four-dimensionalReissner-Nordstrom solutions with an inverse square electricfield in arbitrary dimensions [85]

From the gravitational point of view it is possible to showthat the electric charge and cosmological constant can beextracted simultaneously from pure 119865(119877) gravity (withoutmatter field 119879120583] = 0) [51] In this paper we are goingto obtain 119889-dimensional charged black hole solutions fromgravityrsquos rainbow and pure 119865(119877) gravityrsquos rainbow as well as119865(119877) gravityrsquos rainbow with CIM source and compare themto obtain their direct relation

In order to build up special relativity from Galilean the-ory one has to take into account an upper limit for velocity ofparticlesThe samemethod could be used to restrict particlesfrom obtaining energies no more than specific energy theso-called Planck energy scale This upper bound of energymay modify dispersion relation which is known as doublespecial relativity [86] Generalization of this doubly specialrelativity to incorporate curvature leads to gravityrsquos rainbow[87] In gravityrsquos rainbow space-time is a combination of thetemporal and spatial coordinates as well as energy functionsThe existence of such energy functions indicates that theparticle probing the space-time can acquire specific range ofenergies which in essence leads to formation of a rainbow ofenergy

There are several features for gravityrsquos rainbow whichhighlight the importance of such generalization Amongthem one can point modification in energy-momentum

dispersion relationwhich is supported by studies that are con-ducted in string theory [88] loop quantum gravity [89] andexperimental observation [90] Also existence of remnant forblack holes [91] is proposed to be a candidate for solving theinformation paradox [92] In addition this theory admits theusual uncertainty principle [93 94]

Recently there has been a growing interest in energydependent space-time [95ndash100] Different classes of blackholes have been investigated in the context of gravityrsquosrainbow [101ndash104] The hydrostatic equilibrium equation ofstars in the presence of gravityrsquos rainbow has been obtained[105] Furthermore a study regarding the gravityrsquos rainbowand compact stars has been done [106] In [107] the effectsof gravityrsquos rainbow for wormholes have been investigatedMoreover the influences of gravityrsquos rainbow on gravitationalforce have been investigated [108] Also Starobinskymodel of119865(119877) theory in gravityrsquos rainbow has been studied in [109] Inaddition gravityrsquos rainbow has interesting effects on the earlyuniverse [110ndash112]

Themainmotivations for studying black holes in the pres-ence of gravityrsquos rainbow are given as follows First of all dueto the high energy properties of the black holes it is necessaryto consider quantum corrections of classical perspectivesOne of the methods to include quantum corrections ofgravitational fields is by considering energy dependent space-time In fact it was shown that the quantum correction ofgravitational systems could be observed in dependency ofspace-time on the energy of particles probing it which isgravityrsquos rainbow point of view [93 113 114] Since we aremodifying our point of view to energy dependent space-timeit is expected to find its effects on the properties of blackholes especially in the context of their thermodynamicsThis is another motivation for considering gravityrsquos rainbowgeneralization Also there are specific achievements for grav-ityrsquos rainbow in the context of black holes and among themone can name the following modified uncertainty principle[93 94] existence of remnants for the black holes [91 102]furnishing a bridge toward Horava-Lifshitz gravity [115]providing possible solution toward information paradox [91]and finally being UV completion of Einstein gravity [116]In addition as it was pointed out before in the contextof cosmology it presents a possible solution toward BigBang singularity problem [110ndash112] On the other hand 119865(119877)gravity provides correction toward gravitational sector of theEinstein theory of gravity The importance of this correctionis highlighted in the context of black holes In order tohave better picture regarding the physical nature of the blackholes one may consider high energy regime effects as well(considering that the concepts of Hawking radiation werederived by studying black holes in semiclassicalquantumregime) Here we apply 119865(119877) gravityrsquos rainbow to find theeffects of 119865(119877) generalization as well as energy dependencyof space-time on the black hole solutions

The outline of our paper is as follows In Section 2we are going to investigate black hole solutions in Einsteingravityrsquos rainbow with PMI and CIM fields Then we wantto investigate conserved and thermodynamic quantities ofthe solutions and check the first law of thermodynamicsIn Section 3 we will obtain black hole solutions for 119865(119877)





L = 119877 minus 2Λ + (120581F)119904 (1)






1198891199042= minus

120595 (119903)

119891 (119864)21198891199052+

1

119892 (119864)2[

1198891199032

120595 (119903)+ 1199032119889Ω2] (2)

where

119889Ω2= 1198891205791 +

119889minus2

sum

119894=2

119894minus1

prod

119895=1

sin21205791198951198891205792

119894 (3)


120595 (119903) = 1 minus119898


2Λ1199032

(119889 minus 1) (119889 minus 2) 119892 (119864)2

+

minus2((119889minus1)2)119892(119864)

119889minus3(119902119891(119864))

119889minus1

119903119889minus3ln(

119903

119897) 119904 =

119889 minus 1

2

1199032(2119904 minus 1)

2(2 [119902119891 (119864) 119892 (119864) (2119904 minus 119889 + 1)]

2119903(minus2119889+4)(2119904minus1)

(2119904 minus 1)2)119904


otherwise

(4)

119860120583 = ℎ (119903) 120575119905

120583 (5)




120595 (119903) = 1 minus119898


2Λ1199032

(119889 minus 1) (119889 minus 2)+

2 (119889 minus 3) 1199022

(119889 minus 2) 1199032(119889minus3)

(6)


119877 =2119889

119889 minus 2Λ +

2((119889minus1)2)(119902119891(119864)119892(119864))

119889minus1

119903119889minus1 119904 =

119889 minus 1

2

4119904 minus 119889

119889 minus 2(2 [119902119891 (119864) 119892 (119864) (2119904 minus 119889 + 1)]

2119903(minus2119889+4)(2119904minus1)

(2119904 minus 1)2

)

119904

otherwise

(7)



lim119903rarr0

119877120572120573120574120575119877120572120573120574120575

997888rarr infin

lim119903rarrinfin

119877120572120573120574120575119877120572120573120574120575

=8119889

(119889 minus 1) (119889 minus 2)2Λ2

(8)







120595 (119903) = 1 minus119898


2Λ1199032

(119889 minus 1) (119889 minus 2) 119892 (119864)2

+2(119889minus4)4

(119902119891 (119864) 119892 (119864))1198892

119892 (119864)2119903119889minus2

(9)

ℎ (119903) = minus119902

119903 (10)



119879+ =1

2120587

radicminus1

2(nabla120583120594]) (nabla

120583120594]) (11)


119879+ = minus1

2120587119891 (119864)

Λ119903+

(119889 minus 2) 119892 (119864)minus

(119889 minus 3) 119892 (119864)

2119903+

minusT (12)

whereT is

T =

2(119889minus5)2

[119902119891 (119864) 119892 (119864)]119889

120587119902119891 (119864)2119892 (119864)2119903119889minus2+

119904 =119889 minus 1

2

2119904minus2

(2119904 minus 1) 119903+



+

]

2119904

otherwise

(13)



12058310038161003816100381610038161003816119903=119903+ (14)


119880 =

119902 ln(119903+

119897) 119904 =

119889 minus 1

2

119902119903(2119904minus3)(2119904minus1)

+otherwise

(15)


119878 =1

4(

119903+

119892 (119864))

119889minus2

(16)



10 11 12 13 1409

r

minus01

0

01

02

03

04

120595(r)

(a)

minus01

0

01

02

03

04

120595(r)

10 11 12 13 1409

r

(b)

minus01

0

01

02

03

04

120595(r)

06 07 08 09 10 1105

r

(c)


119876 =

minus1

1205872(119889minus3)2

(119889 minus 1) [119902119891 (119864)]119889minus2

119904 =119889 minus 1

2

119904 (2119904 minus 1)

8 (2119904 minus 119889 + 1) 120587119902119891 (119864) 119892 (119864)119889minus1

[radic2 (2119904 minus 119889 + 1) 119902119891 (119864) g (119864)

(2119904 minus 1)]

2119904

otherwise

(17)




minus005

0

005

010

015

120595(r)

06 07 08 0905

r

(a)

06 08 10 1204

r

minus02

minus01

0

01

02

03

04

05

06

120595(r)

(b)



119872 =(119889 minus 2)119898

16120587119891 (119864) 119892 (119864)119889minus3

(18)


119889119872 (119878 119876) = (120597119872 (119878 119876)

120597119878)

119876

119889119878 + (120597119872(119878 119876)

120597119876)

119878

119889119876 (19)


119879 = (120597119872

120597119878)

119876

119880 = (120597119872

120597119876)

119878

(20)





119877120583] (1 + 119891119877) minus

119892120583]

2119865 (119877) + (119892120583]nabla


= 8120587T120583](21)


) = 0 (22)



ℎ (119903) =minusB

119903

119865tr =B

1199032

(23)



119877120583] (1 + 119891119877) minus1

119889119892120583]1198770 (1 + 119891119877) = 8120587T120583] (24)


1 2 30

r

minus1

0

1

2

3120595(r)



119892 (119864)2

119889 minus 211990312059510158401015840(119903) + 119892 (119864)

21205951015840(119903) +

2

119889 (119889 minus 2)1199031198770

=21198894

[B119892 (119864) 119891 (119864)]1198892

4 (1 + 119891119877) 119903119889minus1

119903119892 (119864)21205951015840(119903) + (119889 minus 3) 119892 (119864)

2[120595 (119903) minus 1] +

1199031198770

119889


[B119892 (119864) 119891 (119864)]1198892

(1 + 119891119877) 119903119889minus2

(25)


120595 (119903) = 1 minus119898

119903119889minus3+

2(119889minus4)4

(B119891 (119864) 119892 (119864))1198892

119892 (119864)2(1 + 119891119877) 119903

119889minus2

minus11987701199032

119889 (119889 minus 1) 119892 (119864)2

(26)






rarr (2119889(119889 minus 1))1198772

0) So





119879+ =

(1 + 119891119877) [119889 (119889 minus 3) 119892 (119864)2minus 1198770119903

2

+] 119903119889minus2

+minus 119889 2

119889minus4[B119891 (119864) 119892 (119864)]

2119889

14

4119889120587119891 (119864) 119892 (119864) 119903119889minus1+

(1 + 119891119877)

(27)


119876 =11988921198894minus1

16120587(B119891 (119864)

119892 (119864))

1198892minus1

(28)

119880 =B

119903+

(29)



119878 =1

4(

119903+

119892 (119864))

119889minus2

(1 + 119891119877) (30)




119872 =(119889 minus 2) (1 + 119891119877)119898

16120587119891 (119864) 119892 (119864)119889minus3

(31)







119862119876 =119879

(12059721198721205971198782)119876

=119879

(120597119879120597119878)119876

(32)


119862119876

=

minus(119889 minus 2) 119903

119889minus2

+

4119892119889minus2 (119864)(

[(119889 minus 3) 1198922(119864) minus 2Λ119903

2

+ (119889 minus 2)] 119903

119889minus3

+minus [radic2119902119891 (119864) 119892 (119864)]

119889minus1

[(119889 minus 3) 1198922(119864) + 2Λ1199032

+ (119889 minus 2)] 119903

119889minus3+


) CIM

minus(119889 minus 2) 119903

119889minus2

+

4119892119889minus2 (119864)(


2

+minus 2Λ119903

2

+


2++ 2Λ1199032

+

) PMI

minus(119889 minus 2) (1 + 119891119877) 119903

119889minus2

+

4119892119889minus2 (119864)(

(1 + 119891119877) [(119889 minus 3) 1198922(119864) minus 2Λ119903

2

+ (119889 minus 2)] 119903

119889minus2


2119889

(1 + 119891119877) [(119889 minus 3) 1198922(119864) + 2Λ1199032

+ (119889 minus 2)] 119903

119889minus2+


2119889

) 119865 (119877)

(33)

in which

Θ = (2 [119902119891 (119864) 119892 (119864) (2119904 minus 119889 + 1)]

2119903(4minus2119889)(2119904minus1)

+

(2119904 minus 1)2

)

119904

(34)






001 002 003

minus010

minus005

0

005

010

(a)

01 02 03 04

minus1

minus05

0

05

1

(b)

05 1 15 2

minus1

minus05

0

05

1

(c)








11986510158401015840

119877+

J12119903120595 (119903)

1198651015840

119877minus

J22119903120595 (119903)

119865119877 =119865 (119877)

2120595 (119903) 119892 (119864)2

J11198651015840

119877minusJ2119865119877 =

119903119865 (119877)

119892 (119864)2

11986510158401015840

119877+

J3

119903120595 (119903)1198651015840

119877minus

(J3 minus 119889 + 3)

1199032120595 (119903)119865119877 =

119865 (119877)

2120595 (119903) 119892 (119864)2

(35)


J1 = 1199031205951015840(119903) + 2 (119889 minus 2) 120595 (119903)

J2 = 11990312059510158401015840(119903) + (119889 minus 2) 120595

1015840(119903)

J3 = 1199031205951015840(119903) + (119889 minus 3) 120595 (119903)

(36)


= 1198651015840


following forms


1

05

0

minus05

minus1

01 02 03 040001 0002 0003 0004 0005 0006 0007 0008

minus02

minus01

0

01

02




1

05

02 04 06 08 10

minus05

minus1

(a)

05 1 15

1

05

0

minus05

minus1

(b)


+for 119902 = 1 Λ = minus1 119892(119864) = 119891(119864) = 07 and 119889 = 5 (a) 119904 = 09


119892 (119864)2[11990312059510158401015840(119903) + (119889 minus 2) 120595

1015840(119903)] 119865119877 = minus119903119865 (119877)


2119865 (119877)

(37)






21198772

0)minus119899



01 02 03

minus4

minus2

0

2

4

115 120 125 130

minus4

minus2

0

2

4




minus02

minus04

04

02

0

05 1 15 2

(a)

minus01

minus03

minus02

minus04

04

05

05 1 15 2 25 3

02

03

0

01

(b)


+for 119902 = 1 Λ = minus1 and 119904 = 14 (a) 119889 = 5 119892(119864) = 119891(119864) = 07




120578(120593(119877))12)[1minus120593(119877)]119877minus119890

2120578(120593(119877))119869(120593(119877))



11989101198724minus2119899

119877119899+ 1198912119877

2 [150] and 119865(119877) = (11987721205812)(1 minus 1198771198770) [151]


Type 1

119865 (119877) = 119877 minus 120582119890minus120585119877

+ 120578119877119899 (38)

Type 2

119865 (119877) = 119877 + 120572119877119899minus 1205731198772minus119899 (39)


8

6

4

2

0

minus2

minus4

04 06 08 10 12 14 16

(a)

4

2

0

minus2

minus4

04 06 08 10 12 14

(b)

04 06 08 10 12 14 16

3

2

1

0

minus1

minus2

minus3

(c)


+for 119902 = 1 Λ = minus1 and 119889 = 5 (a) 119892(119864) = 119891(119864) = 07 119891

119877= 0


119877= 1 (dashed line) (b) 119892(119864) = 119891(119864) = 07 119891



and 119891119877= 16 (dashed line) (c) 119891


(dashed line)

Type 3

119865 (119877) = 119877 minus 1198982

1198881 (1198771198982)119899

1198882 (1198771198982)119899+ 1

(40)

Type 4

119865 (119877) = 119877 minus 119886 [119890minus119887119877

minus 1] + 119888119877119873 119890119887119877

minus 1

119890119887119877 + 1198901198871198770(41)






minus1

minus05

05

045 050 055

1

0

(a)

070 072 076074 078

minus1

minus05

05

1

0

(b)

minus1

minus05

05

15 25

1

1 2 3

0

(c)


+for 119902 = 1 Λ = minus1 119892(119864) = 119891(119864) = 07 and 119891

119877= 15 (a) 119889 = 6










120595 (119903) = 1 minus2Λ1199032


119898

119903119889minus3

+2(119889minus4)4

[119902119891 (119864) 119892 (119864)]1198892

119892 (119864)2119903119889minus2

(42)


Type 1

119877[(119899 minus119889

2) 120578119877119899minus1

minus119889 minus 2

2] 119890120585119877

+ 120582(119889

2+ 120585119877) = 0

(1 + 119899120578119877119899minus1

) 119890120585119877

+ 120582120585 = 0

(43)

Type 2

[120572(119899 minus119889

2)119877119899minus

119889 minus 2

2119877]119877119899+ (119899 +

119889 minus 4

2)1205731198772

= 0

(120572119899119877119899+ 119877)119877

119899+ (119899 minus 2) 120573119877

2= 0

(44)

Type 3

1198882 [119889

119889 minus 211989821198881 minus 1198882119877](

119877

1198982)

2119899

minus [211989821198881

119889 minus 2(119899 minus

119889

2) + 21198882119877](

119877

1198982)

119899

minus 119877 = 0

[1198882

2(

119877

1198982)

2119899

+ 1]119877 + [21198882119877 minus 11989821198991198881] (

119877

1198982)

119899

= 0

(45)

Type 4

[119888 (119873 minus119889

2)119877119873minus

119889 minus 2

2119877 minus

119889

2119886] 1198903119887119877

+ 119886(119887119877 +119889

2)

sdot 11989021198871198770

+ 119888 [(119887119877 + 119873 minus119889

2) 1198901198871198770 + 119887119877 minus 119873 +

119889

2]119877119873

minus [(119889 minus 2) 119877 + 119886119889] 1198901198871198770 + 119886(119887119877 +

119889

2) 1198902119887119877

+ [119886 (2119887119877 + 119889) minus 119888 (119873 minus119889

2)119877119873

minus (119886119889

2minus

(119889 minus 2) 119877

2) 1198901198871198770] 119890119887(119877+1198770) = 0

(1 + 119888119873119877119873minus1

) 1198903119887119877

+ [1198901198871198770 (2119886119887 + 119890

1198871198770)

minus 119888119873119877119873minus1

1198901198871198770] 119890119887119877

+ [119888119877119873minus1

(1198901198871198770

(119873 + 119887119877) minus 119873 + 119887119877) + 119886119887 + 21198901198871198770]

sdot 1198902119887119877

= 0

(46)


Type 1

120582 =119877 (119899 minus 1) 119890

120585119877

119899 + 120585119877

120578 = minus(1 + 120585119877)

119877119899minus1 (119899 + 120585119877)

(47)

Type 2

120572 = minus1

2119877119899minus1

120573 =119877119899minus1

2

(48)

Type 3

1198982= 119877(

119899 minus 1

1198882

)

minus1119899

1198881 = 119899(119899 minus 1

1198882

)

1119899minus1

(49)

Type 4

119888 =

[119890119887119877

+ 1198901198871198770]2

(119887119877 + 1 minus 119890119887119877)

119877119873minus1 [119873 (119890119887119877 + 1198901198871198770) + 119887119877] (119890119887119877 minus 1)

119886 = minus

119877 (119873 minus 1) 1198902119887119877

+ [119887119877 (1 + 1198901198871198770) + (119873 minus 1) (119890

1198871198770 minus 1)] 119890119887119877 119890119887119877

(119890119887119877 minus 1)2[119887119877 + 119873 (119890119887119877 + 1198901198871198770)]

(50)




1 2 30

r

minus2

minus1

0

1

2

3

4120595(r)


lim119903rarr0

119877120572120573120574120575119877120572120573120574120575

997888rarr infin

lim119903rarrinfin

119877120572120573120574120575119877120572120573120574120575

997888rarr8119889

(119889 minus 1) (119889 minus 2)2Λ2

(51)






119879+ =

119903119889minus2

+[(119889 minus 3) 119892

2(119864) minus 2Λ119903

2

+ (119889 minus 2)] minus 2

(119889minus4)4[119892119891 (119864) 119892 (119864)]

1198892

4120587119891 (119864) 119892 (119864) 119903119889minus1+

(52)



119872 =(119889 minus 2)119898

16120587119891 (119864) 119892 (119864)119889minus3

=

[(119889 minus 2) 1198922(119864) minus 2Λ119903

2

+ (119889 minus 1)] 119903

119889minus2

++ 2(119889minus4)4

(119889 minus 2) [119892119891 (119864) 119892 (119864)]1198892

16120587119891 (119864) 119892 (119864)119889minus1

119903+

(53)

Hence we can write

119889119872 =

119903119889minus3

+[((119889 minus 2) (119889 minus 3) 119892

2(119864) minus 2Λ119903

2

+) 119903119889minus2


(119889 minus 2) (119892119891 (119864) 119892 (119864))1198892

]

16120587119891 (119864) 119892119889minus1

(119864) 119903119889minus1+

119889119903+(54)


119878 = int1

119879119889119872 =

1

4(

119903+

119892 (119864))

119889minus2

(55)




minus04

minus02

02

04

FRR

0 1 2minus1

120585

(a)

b

minus4

minus2

0

2

4

FRR

2515 2

(b)

Figure 12 Type 1 (a) 119865119877119877


versus 119887 for119873 = 4 1198770= 4



2119865(119877)119889119877



119865119877119877

=

minus(119899 minus 1) [120585119877 (119899 + 120585119877) + 119899]

119877 (119899 + 120585119877) Type 1


119877 Type 2

119899 minus 1

119877 Type 3

1198872119877 [(119899 minus 1) 119890

2119887119877+ (1198901198871198770 (119887R + 119899 minus 1) + 119887119877 minus 119899 + 1) 119890

119887119877minus (119899 minus 1) 119890

1198871198770]

(119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770]

minus

(119887119877 + 1 minus 119890119887119877) 119899 (1 minus 119899) 119887

3119887119877+ 12059111198902119887119877

minus 1205912119890119887(119877+1198770) + 119899 (119899 minus 1) 119890

21198871198770

(119890119887119877 + 1198901198871198770) (119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770] 119877

Type 4

(56)

where


+ 119887119877 (119887119877 minus 2119899)

1205912 = [119899 (119899 minus 1) + 119887119877 (119887119877 + 2119899)] 1198901198871198770 + 2119899 (1 minus 119899)

+ 119887119877 (119887119877 + 2119899)

(57)



6 Conclusions






Competing Interests


Acknowledgments


References


















































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







L = 119877 minus 2Λ + (120581F)119904 (1)






1198891199042= minus

120595 (119903)

119891 (119864)21198891199052+

1

119892 (119864)2[

1198891199032

120595 (119903)+ 1199032119889Ω2] (2)

where

119889Ω2= 1198891205791 +

119889minus2

sum

119894=2

119894minus1

prod

119895=1

sin21205791198951198891205792

119894 (3)


120595 (119903) = 1 minus119898


2Λ1199032

(119889 minus 1) (119889 minus 2) 119892 (119864)2

+

minus2((119889minus1)2)119892(119864)

119889minus3(119902119891(119864))

119889minus1

119903119889minus3ln(

119903

119897) 119904 =

119889 minus 1

2

1199032(2119904 minus 1)

2(2 [119902119891 (119864) 119892 (119864) (2119904 minus 119889 + 1)]

2119903(minus2119889+4)(2119904minus1)

(2119904 minus 1)2)119904


otherwise

(4)

119860120583 = ℎ (119903) 120575119905

120583 (5)




120595 (119903) = 1 minus119898


2Λ1199032

(119889 minus 1) (119889 minus 2)+

2 (119889 minus 3) 1199022

(119889 minus 2) 1199032(119889minus3)

(6)


119877 =2119889

119889 minus 2Λ +

2((119889minus1)2)(119902119891(119864)119892(119864))

119889minus1

119903119889minus1 119904 =

119889 minus 1

2

4119904 minus 119889

119889 minus 2(2 [119902119891 (119864) 119892 (119864) (2119904 minus 119889 + 1)]

2119903(minus2119889+4)(2119904minus1)

(2119904 minus 1)2

)

119904

otherwise

(7)



lim119903rarr0

119877120572120573120574120575119877120572120573120574120575

997888rarr infin

lim119903rarrinfin

119877120572120573120574120575119877120572120573120574120575

=8119889

(119889 minus 1) (119889 minus 2)2Λ2

(8)







120595 (119903) = 1 minus119898


2Λ1199032

(119889 minus 1) (119889 minus 2) 119892 (119864)2

+2(119889minus4)4

(119902119891 (119864) 119892 (119864))1198892

119892 (119864)2119903119889minus2

(9)

ℎ (119903) = minus119902

119903 (10)



119879+ =1

2120587

radicminus1

2(nabla120583120594]) (nabla

120583120594]) (11)


119879+ = minus1

2120587119891 (119864)

Λ119903+

(119889 minus 2) 119892 (119864)minus

(119889 minus 3) 119892 (119864)

2119903+

minusT (12)

whereT is

T =

2(119889minus5)2

[119902119891 (119864) 119892 (119864)]119889

120587119902119891 (119864)2119892 (119864)2119903119889minus2+

119904 =119889 minus 1

2

2119904minus2

(2119904 minus 1) 119903+



+

]

2119904

otherwise

(13)



12058310038161003816100381610038161003816119903=119903+ (14)


119880 =

119902 ln(119903+

119897) 119904 =

119889 minus 1

2

119902119903(2119904minus3)(2119904minus1)

+otherwise

(15)


119878 =1

4(

119903+

119892 (119864))

119889minus2

(16)



10 11 12 13 1409

r

minus01

0

01

02

03

04

120595(r)

(a)

minus01

0

01

02

03

04

120595(r)

10 11 12 13 1409

r

(b)

minus01

0

01

02

03

04

120595(r)

06 07 08 09 10 1105

r

(c)


119876 =

minus1

1205872(119889minus3)2

(119889 minus 1) [119902119891 (119864)]119889minus2

119904 =119889 minus 1

2

119904 (2119904 minus 1)

8 (2119904 minus 119889 + 1) 120587119902119891 (119864) 119892 (119864)119889minus1

[radic2 (2119904 minus 119889 + 1) 119902119891 (119864) g (119864)

(2119904 minus 1)]

2119904

otherwise

(17)




minus005

0

005

010

015

120595(r)

06 07 08 0905

r

(a)

06 08 10 1204

r

minus02

minus01

0

01

02

03

04

05

06

120595(r)

(b)



119872 =(119889 minus 2)119898

16120587119891 (119864) 119892 (119864)119889minus3

(18)


119889119872 (119878 119876) = (120597119872 (119878 119876)

120597119878)

119876

119889119878 + (120597119872(119878 119876)

120597119876)

119878

119889119876 (19)


119879 = (120597119872

120597119878)

119876

119880 = (120597119872

120597119876)

119878

(20)





119877120583] (1 + 119891119877) minus

119892120583]

2119865 (119877) + (119892120583]nabla


= 8120587T120583](21)


) = 0 (22)



ℎ (119903) =minusB

119903

119865tr =B

1199032

(23)



119877120583] (1 + 119891119877) minus1

119889119892120583]1198770 (1 + 119891119877) = 8120587T120583] (24)


1 2 30

r

minus1

0

1

2

3120595(r)



119892 (119864)2

119889 minus 211990312059510158401015840(119903) + 119892 (119864)

21205951015840(119903) +

2

119889 (119889 minus 2)1199031198770

=21198894

[B119892 (119864) 119891 (119864)]1198892

4 (1 + 119891119877) 119903119889minus1

119903119892 (119864)21205951015840(119903) + (119889 minus 3) 119892 (119864)

2[120595 (119903) minus 1] +

1199031198770

119889


[B119892 (119864) 119891 (119864)]1198892

(1 + 119891119877) 119903119889minus2

(25)


120595 (119903) = 1 minus119898

119903119889minus3+

2(119889minus4)4

(B119891 (119864) 119892 (119864))1198892

119892 (119864)2(1 + 119891119877) 119903

119889minus2

minus11987701199032

119889 (119889 minus 1) 119892 (119864)2

(26)






rarr (2119889(119889 minus 1))1198772

0) So





119879+ =

(1 + 119891119877) [119889 (119889 minus 3) 119892 (119864)2minus 1198770119903

2

+] 119903119889minus2

+minus 119889 2

119889minus4[B119891 (119864) 119892 (119864)]

2119889

14

4119889120587119891 (119864) 119892 (119864) 119903119889minus1+

(1 + 119891119877)

(27)


119876 =11988921198894minus1

16120587(B119891 (119864)

119892 (119864))

1198892minus1

(28)

119880 =B

119903+

(29)



119878 =1

4(

119903+

119892 (119864))

119889minus2

(1 + 119891119877) (30)




119872 =(119889 minus 2) (1 + 119891119877)119898

16120587119891 (119864) 119892 (119864)119889minus3

(31)







119862119876 =119879

(12059721198721205971198782)119876

=119879

(120597119879120597119878)119876

(32)


119862119876

=

minus(119889 minus 2) 119903

119889minus2

+

4119892119889minus2 (119864)(

[(119889 minus 3) 1198922(119864) minus 2Λ119903

2

+ (119889 minus 2)] 119903

119889minus3

+minus [radic2119902119891 (119864) 119892 (119864)]

119889minus1

[(119889 minus 3) 1198922(119864) + 2Λ1199032

+ (119889 minus 2)] 119903

119889minus3+


) CIM

minus(119889 minus 2) 119903

119889minus2

+

4119892119889minus2 (119864)(


2

+minus 2Λ119903

2

+


2++ 2Λ1199032

+

) PMI

minus(119889 minus 2) (1 + 119891119877) 119903

119889minus2

+

4119892119889minus2 (119864)(

(1 + 119891119877) [(119889 minus 3) 1198922(119864) minus 2Λ119903

2

+ (119889 minus 2)] 119903

119889minus2


2119889

(1 + 119891119877) [(119889 minus 3) 1198922(119864) + 2Λ1199032

+ (119889 minus 2)] 119903

119889minus2+


2119889

) 119865 (119877)

(33)

in which

Θ = (2 [119902119891 (119864) 119892 (119864) (2119904 minus 119889 + 1)]

2119903(4minus2119889)(2119904minus1)

+

(2119904 minus 1)2

)

119904

(34)






001 002 003

minus010

minus005

0

005

010

(a)

01 02 03 04

minus1

minus05

0

05

1

(b)

05 1 15 2

minus1

minus05

0

05

1

(c)








11986510158401015840

119877+

J12119903120595 (119903)

1198651015840

119877minus

J22119903120595 (119903)

119865119877 =119865 (119877)

2120595 (119903) 119892 (119864)2

J11198651015840

119877minusJ2119865119877 =

119903119865 (119877)

119892 (119864)2

11986510158401015840

119877+

J3

119903120595 (119903)1198651015840

119877minus

(J3 minus 119889 + 3)

1199032120595 (119903)119865119877 =

119865 (119877)

2120595 (119903) 119892 (119864)2

(35)


J1 = 1199031205951015840(119903) + 2 (119889 minus 2) 120595 (119903)

J2 = 11990312059510158401015840(119903) + (119889 minus 2) 120595

1015840(119903)

J3 = 1199031205951015840(119903) + (119889 minus 3) 120595 (119903)

(36)


= 1198651015840


following forms


1

05

0

minus05

minus1

01 02 03 040001 0002 0003 0004 0005 0006 0007 0008

minus02

minus01

0

01

02




1

05

02 04 06 08 10

minus05

minus1

(a)

05 1 15

1

05

0

minus05

minus1

(b)


+for 119902 = 1 Λ = minus1 119892(119864) = 119891(119864) = 07 and 119889 = 5 (a) 119904 = 09


119892 (119864)2[11990312059510158401015840(119903) + (119889 minus 2) 120595

1015840(119903)] 119865119877 = minus119903119865 (119877)


2119865 (119877)

(37)






21198772

0)minus119899



01 02 03

minus4

minus2

0

2

4

115 120 125 130

minus4

minus2

0

2

4




minus02

minus04

04

02

0

05 1 15 2

(a)

minus01

minus03

minus02

minus04

04

05

05 1 15 2 25 3

02

03

0

01

(b)


+for 119902 = 1 Λ = minus1 and 119904 = 14 (a) 119889 = 5 119892(119864) = 119891(119864) = 07




120578(120593(119877))12)[1minus120593(119877)]119877minus119890

2120578(120593(119877))119869(120593(119877))



11989101198724minus2119899

119877119899+ 1198912119877

2 [150] and 119865(119877) = (11987721205812)(1 minus 1198771198770) [151]


Type 1

119865 (119877) = 119877 minus 120582119890minus120585119877

+ 120578119877119899 (38)

Type 2

119865 (119877) = 119877 + 120572119877119899minus 1205731198772minus119899 (39)


8

6

4

2

0

minus2

minus4

04 06 08 10 12 14 16

(a)

4

2

0

minus2

minus4

04 06 08 10 12 14

(b)

04 06 08 10 12 14 16

3

2

1

0

minus1

minus2

minus3

(c)


+for 119902 = 1 Λ = minus1 and 119889 = 5 (a) 119892(119864) = 119891(119864) = 07 119891

119877= 0


119877= 1 (dashed line) (b) 119892(119864) = 119891(119864) = 07 119891



and 119891119877= 16 (dashed line) (c) 119891


(dashed line)

Type 3

119865 (119877) = 119877 minus 1198982

1198881 (1198771198982)119899

1198882 (1198771198982)119899+ 1

(40)

Type 4

119865 (119877) = 119877 minus 119886 [119890minus119887119877

minus 1] + 119888119877119873 119890119887119877

minus 1

119890119887119877 + 1198901198871198770(41)






minus1

minus05

05

045 050 055

1

0

(a)

070 072 076074 078

minus1

minus05

05

1

0

(b)

minus1

minus05

05

15 25

1

1 2 3

0

(c)


+for 119902 = 1 Λ = minus1 119892(119864) = 119891(119864) = 07 and 119891

119877= 15 (a) 119889 = 6










120595 (119903) = 1 minus2Λ1199032


119898

119903119889minus3

+2(119889minus4)4

[119902119891 (119864) 119892 (119864)]1198892

119892 (119864)2119903119889minus2

(42)


Type 1

119877[(119899 minus119889

2) 120578119877119899minus1

minus119889 minus 2

2] 119890120585119877

+ 120582(119889

2+ 120585119877) = 0

(1 + 119899120578119877119899minus1

) 119890120585119877

+ 120582120585 = 0

(43)

Type 2

[120572(119899 minus119889

2)119877119899minus

119889 minus 2

2119877]119877119899+ (119899 +

119889 minus 4

2)1205731198772

= 0

(120572119899119877119899+ 119877)119877

119899+ (119899 minus 2) 120573119877

2= 0

(44)

Type 3

1198882 [119889

119889 minus 211989821198881 minus 1198882119877](

119877

1198982)

2119899

minus [211989821198881

119889 minus 2(119899 minus

119889

2) + 21198882119877](

119877

1198982)

119899

minus 119877 = 0

[1198882

2(

119877

1198982)

2119899

+ 1]119877 + [21198882119877 minus 11989821198991198881] (

119877

1198982)

119899

= 0

(45)

Type 4

[119888 (119873 minus119889

2)119877119873minus

119889 minus 2

2119877 minus

119889

2119886] 1198903119887119877

+ 119886(119887119877 +119889

2)

sdot 11989021198871198770

+ 119888 [(119887119877 + 119873 minus119889

2) 1198901198871198770 + 119887119877 minus 119873 +

119889

2]119877119873

minus [(119889 minus 2) 119877 + 119886119889] 1198901198871198770 + 119886(119887119877 +

119889

2) 1198902119887119877

+ [119886 (2119887119877 + 119889) minus 119888 (119873 minus119889

2)119877119873

minus (119886119889

2minus

(119889 minus 2) 119877

2) 1198901198871198770] 119890119887(119877+1198770) = 0

(1 + 119888119873119877119873minus1

) 1198903119887119877

+ [1198901198871198770 (2119886119887 + 119890

1198871198770)

minus 119888119873119877119873minus1

1198901198871198770] 119890119887119877

+ [119888119877119873minus1

(1198901198871198770

(119873 + 119887119877) minus 119873 + 119887119877) + 119886119887 + 21198901198871198770]

sdot 1198902119887119877

= 0

(46)


Type 1

120582 =119877 (119899 minus 1) 119890

120585119877

119899 + 120585119877

120578 = minus(1 + 120585119877)

119877119899minus1 (119899 + 120585119877)

(47)

Type 2

120572 = minus1

2119877119899minus1

120573 =119877119899minus1

2

(48)

Type 3

1198982= 119877(

119899 minus 1

1198882

)

minus1119899

1198881 = 119899(119899 minus 1

1198882

)

1119899minus1

(49)

Type 4

119888 =

[119890119887119877

+ 1198901198871198770]2

(119887119877 + 1 minus 119890119887119877)

119877119873minus1 [119873 (119890119887119877 + 1198901198871198770) + 119887119877] (119890119887119877 minus 1)

119886 = minus

119877 (119873 minus 1) 1198902119887119877

+ [119887119877 (1 + 1198901198871198770) + (119873 minus 1) (119890

1198871198770 minus 1)] 119890119887119877 119890119887119877

(119890119887119877 minus 1)2[119887119877 + 119873 (119890119887119877 + 1198901198871198770)]

(50)




1 2 30

r

minus2

minus1

0

1

2

3

4120595(r)


lim119903rarr0

119877120572120573120574120575119877120572120573120574120575

997888rarr infin

lim119903rarrinfin

119877120572120573120574120575119877120572120573120574120575

997888rarr8119889

(119889 minus 1) (119889 minus 2)2Λ2

(51)






119879+ =

119903119889minus2

+[(119889 minus 3) 119892

2(119864) minus 2Λ119903

2

+ (119889 minus 2)] minus 2

(119889minus4)4[119892119891 (119864) 119892 (119864)]

1198892

4120587119891 (119864) 119892 (119864) 119903119889minus1+

(52)



119872 =(119889 minus 2)119898

16120587119891 (119864) 119892 (119864)119889minus3

=

[(119889 minus 2) 1198922(119864) minus 2Λ119903

2

+ (119889 minus 1)] 119903

119889minus2

++ 2(119889minus4)4

(119889 minus 2) [119892119891 (119864) 119892 (119864)]1198892

16120587119891 (119864) 119892 (119864)119889minus1

119903+

(53)

Hence we can write

119889119872 =

119903119889minus3

+[((119889 minus 2) (119889 minus 3) 119892

2(119864) minus 2Λ119903

2

+) 119903119889minus2


(119889 minus 2) (119892119891 (119864) 119892 (119864))1198892

]

16120587119891 (119864) 119892119889minus1

(119864) 119903119889minus1+

119889119903+(54)


119878 = int1

119879119889119872 =

1

4(

119903+

119892 (119864))

119889minus2

(55)




minus04

minus02

02

04

FRR

0 1 2minus1

120585

(a)

b

minus4

minus2

0

2

4

FRR

2515 2

(b)

Figure 12 Type 1 (a) 119865119877119877


versus 119887 for119873 = 4 1198770= 4



2119865(119877)119889119877



119865119877119877

=

minus(119899 minus 1) [120585119877 (119899 + 120585119877) + 119899]

119877 (119899 + 120585119877) Type 1


119877 Type 2

119899 minus 1

119877 Type 3

1198872119877 [(119899 minus 1) 119890

2119887119877+ (1198901198871198770 (119887R + 119899 minus 1) + 119887119877 minus 119899 + 1) 119890

119887119877minus (119899 minus 1) 119890

1198871198770]

(119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770]

minus

(119887119877 + 1 minus 119890119887119877) 119899 (1 minus 119899) 119887

3119887119877+ 12059111198902119887119877

minus 1205912119890119887(119877+1198770) + 119899 (119899 minus 1) 119890

21198871198770

(119890119887119877 + 1198901198871198770) (119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770] 119877

Type 4

(56)

where


+ 119887119877 (119887119877 minus 2119899)

1205912 = [119899 (119899 minus 1) + 119887119877 (119887119877 + 2119899)] 1198901198871198770 + 2119899 (1 minus 119899)

+ 119887119877 (119887119877 + 2119899)

(57)



6 Conclusions






Competing Interests


Acknowledgments


References


















































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





lim119903rarr0

119877120572120573120574120575119877120572120573120574120575

997888rarr infin

lim119903rarrinfin

119877120572120573120574120575119877120572120573120574120575

=8119889

(119889 minus 1) (119889 minus 2)2Λ2

(8)







120595 (119903) = 1 minus119898


2Λ1199032

(119889 minus 1) (119889 minus 2) 119892 (119864)2

+2(119889minus4)4

(119902119891 (119864) 119892 (119864))1198892

119892 (119864)2119903119889minus2

(9)

ℎ (119903) = minus119902

119903 (10)



119879+ =1

2120587

radicminus1

2(nabla120583120594]) (nabla

120583120594]) (11)


119879+ = minus1

2120587119891 (119864)

Λ119903+

(119889 minus 2) 119892 (119864)minus

(119889 minus 3) 119892 (119864)

2119903+

minusT (12)

whereT is

T =

2(119889minus5)2

[119902119891 (119864) 119892 (119864)]119889

120587119902119891 (119864)2119892 (119864)2119903119889minus2+

119904 =119889 minus 1

2

2119904minus2

(2119904 minus 1) 119903+



+

]

2119904

otherwise

(13)



12058310038161003816100381610038161003816119903=119903+ (14)


119880 =

119902 ln(119903+

119897) 119904 =

119889 minus 1

2

119902119903(2119904minus3)(2119904minus1)

+otherwise

(15)


119878 =1

4(

119903+

119892 (119864))

119889minus2

(16)



10 11 12 13 1409

r

minus01

0

01

02

03

04

120595(r)

(a)

minus01

0

01

02

03

04

120595(r)

10 11 12 13 1409

r

(b)

minus01

0

01

02

03

04

120595(r)

06 07 08 09 10 1105

r

(c)


119876 =

minus1

1205872(119889minus3)2

(119889 minus 1) [119902119891 (119864)]119889minus2

119904 =119889 minus 1

2

119904 (2119904 minus 1)

8 (2119904 minus 119889 + 1) 120587119902119891 (119864) 119892 (119864)119889minus1

[radic2 (2119904 minus 119889 + 1) 119902119891 (119864) g (119864)

(2119904 minus 1)]

2119904

otherwise

(17)




minus005

0

005

010

015

120595(r)

06 07 08 0905

r

(a)

06 08 10 1204

r

minus02

minus01

0

01

02

03

04

05

06

120595(r)

(b)



119872 =(119889 minus 2)119898

16120587119891 (119864) 119892 (119864)119889minus3

(18)


119889119872 (119878 119876) = (120597119872 (119878 119876)

120597119878)

119876

119889119878 + (120597119872(119878 119876)

120597119876)

119878

119889119876 (19)


119879 = (120597119872

120597119878)

119876

119880 = (120597119872

120597119876)

119878

(20)





119877120583] (1 + 119891119877) minus

119892120583]

2119865 (119877) + (119892120583]nabla


= 8120587T120583](21)


) = 0 (22)



ℎ (119903) =minusB

119903

119865tr =B

1199032

(23)



119877120583] (1 + 119891119877) minus1

119889119892120583]1198770 (1 + 119891119877) = 8120587T120583] (24)


1 2 30

r

minus1

0

1

2

3120595(r)



119892 (119864)2

119889 minus 211990312059510158401015840(119903) + 119892 (119864)

21205951015840(119903) +

2

119889 (119889 minus 2)1199031198770

=21198894

[B119892 (119864) 119891 (119864)]1198892

4 (1 + 119891119877) 119903119889minus1

119903119892 (119864)21205951015840(119903) + (119889 minus 3) 119892 (119864)

2[120595 (119903) minus 1] +

1199031198770

119889


[B119892 (119864) 119891 (119864)]1198892

(1 + 119891119877) 119903119889minus2

(25)


120595 (119903) = 1 minus119898

119903119889minus3+

2(119889minus4)4

(B119891 (119864) 119892 (119864))1198892

119892 (119864)2(1 + 119891119877) 119903

119889minus2

minus11987701199032

119889 (119889 minus 1) 119892 (119864)2

(26)






rarr (2119889(119889 minus 1))1198772

0) So





119879+ =

(1 + 119891119877) [119889 (119889 minus 3) 119892 (119864)2minus 1198770119903

2

+] 119903119889minus2

+minus 119889 2

119889minus4[B119891 (119864) 119892 (119864)]

2119889

14

4119889120587119891 (119864) 119892 (119864) 119903119889minus1+

(1 + 119891119877)

(27)


119876 =11988921198894minus1

16120587(B119891 (119864)

119892 (119864))

1198892minus1

(28)

119880 =B

119903+

(29)



119878 =1

4(

119903+

119892 (119864))

119889minus2

(1 + 119891119877) (30)




119872 =(119889 minus 2) (1 + 119891119877)119898

16120587119891 (119864) 119892 (119864)119889minus3

(31)







119862119876 =119879

(12059721198721205971198782)119876

=119879

(120597119879120597119878)119876

(32)


119862119876

=

minus(119889 minus 2) 119903

119889minus2

+

4119892119889minus2 (119864)(

[(119889 minus 3) 1198922(119864) minus 2Λ119903

2

+ (119889 minus 2)] 119903

119889minus3

+minus [radic2119902119891 (119864) 119892 (119864)]

119889minus1

[(119889 minus 3) 1198922(119864) + 2Λ1199032

+ (119889 minus 2)] 119903

119889minus3+


) CIM

minus(119889 minus 2) 119903

119889minus2

+

4119892119889minus2 (119864)(


2

+minus 2Λ119903

2

+


2++ 2Λ1199032

+

) PMI

minus(119889 minus 2) (1 + 119891119877) 119903

119889minus2

+

4119892119889minus2 (119864)(

(1 + 119891119877) [(119889 minus 3) 1198922(119864) minus 2Λ119903

2

+ (119889 minus 2)] 119903

119889minus2


2119889

(1 + 119891119877) [(119889 minus 3) 1198922(119864) + 2Λ1199032

+ (119889 minus 2)] 119903

119889minus2+


2119889

) 119865 (119877)

(33)

in which

Θ = (2 [119902119891 (119864) 119892 (119864) (2119904 minus 119889 + 1)]

2119903(4minus2119889)(2119904minus1)

+

(2119904 minus 1)2

)

119904

(34)






001 002 003

minus010

minus005

0

005

010

(a)

01 02 03 04

minus1

minus05

0

05

1

(b)

05 1 15 2

minus1

minus05

0

05

1

(c)








11986510158401015840

119877+

J12119903120595 (119903)

1198651015840

119877minus

J22119903120595 (119903)

119865119877 =119865 (119877)

2120595 (119903) 119892 (119864)2

J11198651015840

119877minusJ2119865119877 =

119903119865 (119877)

119892 (119864)2

11986510158401015840

119877+

J3

119903120595 (119903)1198651015840

119877minus

(J3 minus 119889 + 3)

1199032120595 (119903)119865119877 =

119865 (119877)

2120595 (119903) 119892 (119864)2

(35)


J1 = 1199031205951015840(119903) + 2 (119889 minus 2) 120595 (119903)

J2 = 11990312059510158401015840(119903) + (119889 minus 2) 120595

1015840(119903)

J3 = 1199031205951015840(119903) + (119889 minus 3) 120595 (119903)

(36)


= 1198651015840


following forms


1

05

0

minus05

minus1

01 02 03 040001 0002 0003 0004 0005 0006 0007 0008

minus02

minus01

0

01

02




1

05

02 04 06 08 10

minus05

minus1

(a)

05 1 15

1

05

0

minus05

minus1

(b)


+for 119902 = 1 Λ = minus1 119892(119864) = 119891(119864) = 07 and 119889 = 5 (a) 119904 = 09


119892 (119864)2[11990312059510158401015840(119903) + (119889 minus 2) 120595

1015840(119903)] 119865119877 = minus119903119865 (119877)


2119865 (119877)

(37)






21198772

0)minus119899



01 02 03

minus4

minus2

0

2

4

115 120 125 130

minus4

minus2

0

2

4




minus02

minus04

04

02

0

05 1 15 2

(a)

minus01

minus03

minus02

minus04

04

05

05 1 15 2 25 3

02

03

0

01

(b)


+for 119902 = 1 Λ = minus1 and 119904 = 14 (a) 119889 = 5 119892(119864) = 119891(119864) = 07




120578(120593(119877))12)[1minus120593(119877)]119877minus119890

2120578(120593(119877))119869(120593(119877))



11989101198724minus2119899

119877119899+ 1198912119877

2 [150] and 119865(119877) = (11987721205812)(1 minus 1198771198770) [151]


Type 1

119865 (119877) = 119877 minus 120582119890minus120585119877

+ 120578119877119899 (38)

Type 2

119865 (119877) = 119877 + 120572119877119899minus 1205731198772minus119899 (39)


8

6

4

2

0

minus2

minus4

04 06 08 10 12 14 16

(a)

4

2

0

minus2

minus4

04 06 08 10 12 14

(b)

04 06 08 10 12 14 16

3

2

1

0

minus1

minus2

minus3

(c)


+for 119902 = 1 Λ = minus1 and 119889 = 5 (a) 119892(119864) = 119891(119864) = 07 119891

119877= 0


119877= 1 (dashed line) (b) 119892(119864) = 119891(119864) = 07 119891



and 119891119877= 16 (dashed line) (c) 119891


(dashed line)

Type 3

119865 (119877) = 119877 minus 1198982

1198881 (1198771198982)119899

1198882 (1198771198982)119899+ 1

(40)

Type 4

119865 (119877) = 119877 minus 119886 [119890minus119887119877

minus 1] + 119888119877119873 119890119887119877

minus 1

119890119887119877 + 1198901198871198770(41)






minus1

minus05

05

045 050 055

1

0

(a)

070 072 076074 078

minus1

minus05

05

1

0

(b)

minus1

minus05

05

15 25

1

1 2 3

0

(c)


+for 119902 = 1 Λ = minus1 119892(119864) = 119891(119864) = 07 and 119891

119877= 15 (a) 119889 = 6










120595 (119903) = 1 minus2Λ1199032


119898

119903119889minus3

+2(119889minus4)4

[119902119891 (119864) 119892 (119864)]1198892

119892 (119864)2119903119889minus2

(42)


Type 1

119877[(119899 minus119889

2) 120578119877119899minus1

minus119889 minus 2

2] 119890120585119877

+ 120582(119889

2+ 120585119877) = 0

(1 + 119899120578119877119899minus1

) 119890120585119877

+ 120582120585 = 0

(43)

Type 2

[120572(119899 minus119889

2)119877119899minus

119889 minus 2

2119877]119877119899+ (119899 +

119889 minus 4

2)1205731198772

= 0

(120572119899119877119899+ 119877)119877

119899+ (119899 minus 2) 120573119877

2= 0

(44)

Type 3

1198882 [119889

119889 minus 211989821198881 minus 1198882119877](

119877

1198982)

2119899

minus [211989821198881

119889 minus 2(119899 minus

119889

2) + 21198882119877](

119877

1198982)

119899

minus 119877 = 0

[1198882

2(

119877

1198982)

2119899

+ 1]119877 + [21198882119877 minus 11989821198991198881] (

119877

1198982)

119899

= 0

(45)

Type 4

[119888 (119873 minus119889

2)119877119873minus

119889 minus 2

2119877 minus

119889

2119886] 1198903119887119877

+ 119886(119887119877 +119889

2)

sdot 11989021198871198770

+ 119888 [(119887119877 + 119873 minus119889

2) 1198901198871198770 + 119887119877 minus 119873 +

119889

2]119877119873

minus [(119889 minus 2) 119877 + 119886119889] 1198901198871198770 + 119886(119887119877 +

119889

2) 1198902119887119877

+ [119886 (2119887119877 + 119889) minus 119888 (119873 minus119889

2)119877119873

minus (119886119889

2minus

(119889 minus 2) 119877

2) 1198901198871198770] 119890119887(119877+1198770) = 0

(1 + 119888119873119877119873minus1

) 1198903119887119877

+ [1198901198871198770 (2119886119887 + 119890

1198871198770)

minus 119888119873119877119873minus1

1198901198871198770] 119890119887119877

+ [119888119877119873minus1

(1198901198871198770

(119873 + 119887119877) minus 119873 + 119887119877) + 119886119887 + 21198901198871198770]

sdot 1198902119887119877

= 0

(46)


Type 1

120582 =119877 (119899 minus 1) 119890

120585119877

119899 + 120585119877

120578 = minus(1 + 120585119877)

119877119899minus1 (119899 + 120585119877)

(47)

Type 2

120572 = minus1

2119877119899minus1

120573 =119877119899minus1

2

(48)

Type 3

1198982= 119877(

119899 minus 1

1198882

)

minus1119899

1198881 = 119899(119899 minus 1

1198882

)

1119899minus1

(49)

Type 4

119888 =

[119890119887119877

+ 1198901198871198770]2

(119887119877 + 1 minus 119890119887119877)

119877119873minus1 [119873 (119890119887119877 + 1198901198871198770) + 119887119877] (119890119887119877 minus 1)

119886 = minus

119877 (119873 minus 1) 1198902119887119877

+ [119887119877 (1 + 1198901198871198770) + (119873 minus 1) (119890

1198871198770 minus 1)] 119890119887119877 119890119887119877

(119890119887119877 minus 1)2[119887119877 + 119873 (119890119887119877 + 1198901198871198770)]

(50)




1 2 30

r

minus2

minus1

0

1

2

3

4120595(r)


lim119903rarr0

119877120572120573120574120575119877120572120573120574120575

997888rarr infin

lim119903rarrinfin

119877120572120573120574120575119877120572120573120574120575

997888rarr8119889

(119889 minus 1) (119889 minus 2)2Λ2

(51)






119879+ =

119903119889minus2

+[(119889 minus 3) 119892

2(119864) minus 2Λ119903

2

+ (119889 minus 2)] minus 2

(119889minus4)4[119892119891 (119864) 119892 (119864)]

1198892

4120587119891 (119864) 119892 (119864) 119903119889minus1+

(52)



119872 =(119889 minus 2)119898

16120587119891 (119864) 119892 (119864)119889minus3

=

[(119889 minus 2) 1198922(119864) minus 2Λ119903

2

+ (119889 minus 1)] 119903

119889minus2

++ 2(119889minus4)4

(119889 minus 2) [119892119891 (119864) 119892 (119864)]1198892

16120587119891 (119864) 119892 (119864)119889minus1

119903+

(53)

Hence we can write

119889119872 =

119903119889minus3

+[((119889 minus 2) (119889 minus 3) 119892

2(119864) minus 2Λ119903

2

+) 119903119889minus2


(119889 minus 2) (119892119891 (119864) 119892 (119864))1198892

]

16120587119891 (119864) 119892119889minus1

(119864) 119903119889minus1+

119889119903+(54)


119878 = int1

119879119889119872 =

1

4(

119903+

119892 (119864))

119889minus2

(55)




minus04

minus02

02

04

FRR

0 1 2minus1

120585

(a)

b

minus4

minus2

0

2

4

FRR

2515 2

(b)

Figure 12 Type 1 (a) 119865119877119877


versus 119887 for119873 = 4 1198770= 4



2119865(119877)119889119877



119865119877119877

=

minus(119899 minus 1) [120585119877 (119899 + 120585119877) + 119899]

119877 (119899 + 120585119877) Type 1


119877 Type 2

119899 minus 1

119877 Type 3

1198872119877 [(119899 minus 1) 119890

2119887119877+ (1198901198871198770 (119887R + 119899 minus 1) + 119887119877 minus 119899 + 1) 119890

119887119877minus (119899 minus 1) 119890

1198871198770]

(119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770]

minus

(119887119877 + 1 minus 119890119887119877) 119899 (1 minus 119899) 119887

3119887119877+ 12059111198902119887119877

minus 1205912119890119887(119877+1198770) + 119899 (119899 minus 1) 119890

21198871198770

(119890119887119877 + 1198901198871198770) (119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770] 119877

Type 4

(56)

where


+ 119887119877 (119887119877 minus 2119899)

1205912 = [119899 (119899 minus 1) + 119887119877 (119887119877 + 2119899)] 1198901198871198770 + 2119899 (1 minus 119899)

+ 119887119877 (119887119877 + 2119899)

(57)



6 Conclusions






Competing Interests


Acknowledgments


References


















































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




10 11 12 13 1409

r

minus01

0

01

02

03

04

120595(r)

(a)

minus01

0

01

02

03

04

120595(r)

10 11 12 13 1409

r

(b)

minus01

0

01

02

03

04

120595(r)

06 07 08 09 10 1105

r

(c)


119876 =

minus1

1205872(119889minus3)2

(119889 minus 1) [119902119891 (119864)]119889minus2

119904 =119889 minus 1

2

119904 (2119904 minus 1)

8 (2119904 minus 119889 + 1) 120587119902119891 (119864) 119892 (119864)119889minus1

[radic2 (2119904 minus 119889 + 1) 119902119891 (119864) g (119864)

(2119904 minus 1)]

2119904

otherwise

(17)




minus005

0

005

010

015

120595(r)

06 07 08 0905

r

(a)

06 08 10 1204

r

minus02

minus01

0

01

02

03

04

05

06

120595(r)

(b)



119872 =(119889 minus 2)119898

16120587119891 (119864) 119892 (119864)119889minus3

(18)


119889119872 (119878 119876) = (120597119872 (119878 119876)

120597119878)

119876

119889119878 + (120597119872(119878 119876)

120597119876)

119878

119889119876 (19)


119879 = (120597119872

120597119878)

119876

119880 = (120597119872

120597119876)

119878

(20)





119877120583] (1 + 119891119877) minus

119892120583]

2119865 (119877) + (119892120583]nabla


= 8120587T120583](21)


) = 0 (22)



ℎ (119903) =minusB

119903

119865tr =B

1199032

(23)



119877120583] (1 + 119891119877) minus1

119889119892120583]1198770 (1 + 119891119877) = 8120587T120583] (24)


1 2 30

r

minus1

0

1

2

3120595(r)



119892 (119864)2

119889 minus 211990312059510158401015840(119903) + 119892 (119864)

21205951015840(119903) +

2

119889 (119889 minus 2)1199031198770

=21198894

[B119892 (119864) 119891 (119864)]1198892

4 (1 + 119891119877) 119903119889minus1

119903119892 (119864)21205951015840(119903) + (119889 minus 3) 119892 (119864)

2[120595 (119903) minus 1] +

1199031198770

119889


[B119892 (119864) 119891 (119864)]1198892

(1 + 119891119877) 119903119889minus2

(25)


120595 (119903) = 1 minus119898

119903119889minus3+

2(119889minus4)4

(B119891 (119864) 119892 (119864))1198892

119892 (119864)2(1 + 119891119877) 119903

119889minus2

minus11987701199032

119889 (119889 minus 1) 119892 (119864)2

(26)






rarr (2119889(119889 minus 1))1198772

0) So





119879+ =

(1 + 119891119877) [119889 (119889 minus 3) 119892 (119864)2minus 1198770119903

2

+] 119903119889minus2

+minus 119889 2

119889minus4[B119891 (119864) 119892 (119864)]

2119889

14

4119889120587119891 (119864) 119892 (119864) 119903119889minus1+

(1 + 119891119877)

(27)


119876 =11988921198894minus1

16120587(B119891 (119864)

119892 (119864))

1198892minus1

(28)

119880 =B

119903+

(29)



119878 =1

4(

119903+

119892 (119864))

119889minus2

(1 + 119891119877) (30)




119872 =(119889 minus 2) (1 + 119891119877)119898

16120587119891 (119864) 119892 (119864)119889minus3

(31)







119862119876 =119879

(12059721198721205971198782)119876

=119879

(120597119879120597119878)119876

(32)


119862119876

=

minus(119889 minus 2) 119903

119889minus2

+

4119892119889minus2 (119864)(

[(119889 minus 3) 1198922(119864) minus 2Λ119903

2

+ (119889 minus 2)] 119903

119889minus3

+minus [radic2119902119891 (119864) 119892 (119864)]

119889minus1

[(119889 minus 3) 1198922(119864) + 2Λ1199032

+ (119889 minus 2)] 119903

119889minus3+


) CIM

minus(119889 minus 2) 119903

119889minus2

+

4119892119889minus2 (119864)(


2

+minus 2Λ119903

2

+


2++ 2Λ1199032

+

) PMI

minus(119889 minus 2) (1 + 119891119877) 119903

119889minus2

+

4119892119889minus2 (119864)(

(1 + 119891119877) [(119889 minus 3) 1198922(119864) minus 2Λ119903

2

+ (119889 minus 2)] 119903

119889minus2


2119889

(1 + 119891119877) [(119889 minus 3) 1198922(119864) + 2Λ1199032

+ (119889 minus 2)] 119903

119889minus2+


2119889

) 119865 (119877)

(33)

in which

Θ = (2 [119902119891 (119864) 119892 (119864) (2119904 minus 119889 + 1)]

2119903(4minus2119889)(2119904minus1)

+

(2119904 minus 1)2

)

119904

(34)






001 002 003

minus010

minus005

0

005

010

(a)

01 02 03 04

minus1

minus05

0

05

1

(b)

05 1 15 2

minus1

minus05

0

05

1

(c)








11986510158401015840

119877+

J12119903120595 (119903)

1198651015840

119877minus

J22119903120595 (119903)

119865119877 =119865 (119877)

2120595 (119903) 119892 (119864)2

J11198651015840

119877minusJ2119865119877 =

119903119865 (119877)

119892 (119864)2

11986510158401015840

119877+

J3

119903120595 (119903)1198651015840

119877minus

(J3 minus 119889 + 3)

1199032120595 (119903)119865119877 =

119865 (119877)

2120595 (119903) 119892 (119864)2

(35)


J1 = 1199031205951015840(119903) + 2 (119889 minus 2) 120595 (119903)

J2 = 11990312059510158401015840(119903) + (119889 minus 2) 120595

1015840(119903)

J3 = 1199031205951015840(119903) + (119889 minus 3) 120595 (119903)

(36)


= 1198651015840


following forms


1

05

0

minus05

minus1

01 02 03 040001 0002 0003 0004 0005 0006 0007 0008

minus02

minus01

0

01

02




1

05

02 04 06 08 10

minus05

minus1

(a)

05 1 15

1

05

0

minus05

minus1

(b)


+for 119902 = 1 Λ = minus1 119892(119864) = 119891(119864) = 07 and 119889 = 5 (a) 119904 = 09


119892 (119864)2[11990312059510158401015840(119903) + (119889 minus 2) 120595

1015840(119903)] 119865119877 = minus119903119865 (119877)


2119865 (119877)

(37)






21198772

0)minus119899



01 02 03

minus4

minus2

0

2

4

115 120 125 130

minus4

minus2

0

2

4




minus02

minus04

04

02

0

05 1 15 2

(a)

minus01

minus03

minus02

minus04

04

05

05 1 15 2 25 3

02

03

0

01

(b)


+for 119902 = 1 Λ = minus1 and 119904 = 14 (a) 119889 = 5 119892(119864) = 119891(119864) = 07




120578(120593(119877))12)[1minus120593(119877)]119877minus119890

2120578(120593(119877))119869(120593(119877))



11989101198724minus2119899

119877119899+ 1198912119877

2 [150] and 119865(119877) = (11987721205812)(1 minus 1198771198770) [151]


Type 1

119865 (119877) = 119877 minus 120582119890minus120585119877

+ 120578119877119899 (38)

Type 2

119865 (119877) = 119877 + 120572119877119899minus 1205731198772minus119899 (39)


8

6

4

2

0

minus2

minus4

04 06 08 10 12 14 16

(a)

4

2

0

minus2

minus4

04 06 08 10 12 14

(b)

04 06 08 10 12 14 16

3

2

1

0

minus1

minus2

minus3

(c)


+for 119902 = 1 Λ = minus1 and 119889 = 5 (a) 119892(119864) = 119891(119864) = 07 119891

119877= 0


119877= 1 (dashed line) (b) 119892(119864) = 119891(119864) = 07 119891



and 119891119877= 16 (dashed line) (c) 119891


(dashed line)

Type 3

119865 (119877) = 119877 minus 1198982

1198881 (1198771198982)119899

1198882 (1198771198982)119899+ 1

(40)

Type 4

119865 (119877) = 119877 minus 119886 [119890minus119887119877

minus 1] + 119888119877119873 119890119887119877

minus 1

119890119887119877 + 1198901198871198770(41)






minus1

minus05

05

045 050 055

1

0

(a)

070 072 076074 078

minus1

minus05

05

1

0

(b)

minus1

minus05

05

15 25

1

1 2 3

0

(c)


+for 119902 = 1 Λ = minus1 119892(119864) = 119891(119864) = 07 and 119891

119877= 15 (a) 119889 = 6










120595 (119903) = 1 minus2Λ1199032


119898

119903119889minus3

+2(119889minus4)4

[119902119891 (119864) 119892 (119864)]1198892

119892 (119864)2119903119889minus2

(42)


Type 1

119877[(119899 minus119889

2) 120578119877119899minus1

minus119889 minus 2

2] 119890120585119877

+ 120582(119889

2+ 120585119877) = 0

(1 + 119899120578119877119899minus1

) 119890120585119877

+ 120582120585 = 0

(43)

Type 2

[120572(119899 minus119889

2)119877119899minus

119889 minus 2

2119877]119877119899+ (119899 +

119889 minus 4

2)1205731198772

= 0

(120572119899119877119899+ 119877)119877

119899+ (119899 minus 2) 120573119877

2= 0

(44)

Type 3

1198882 [119889

119889 minus 211989821198881 minus 1198882119877](

119877

1198982)

2119899

minus [211989821198881

119889 minus 2(119899 minus

119889

2) + 21198882119877](

119877

1198982)

119899

minus 119877 = 0

[1198882

2(

119877

1198982)

2119899

+ 1]119877 + [21198882119877 minus 11989821198991198881] (

119877

1198982)

119899

= 0

(45)

Type 4

[119888 (119873 minus119889

2)119877119873minus

119889 minus 2

2119877 minus

119889

2119886] 1198903119887119877

+ 119886(119887119877 +119889

2)

sdot 11989021198871198770

+ 119888 [(119887119877 + 119873 minus119889

2) 1198901198871198770 + 119887119877 minus 119873 +

119889

2]119877119873

minus [(119889 minus 2) 119877 + 119886119889] 1198901198871198770 + 119886(119887119877 +

119889

2) 1198902119887119877

+ [119886 (2119887119877 + 119889) minus 119888 (119873 minus119889

2)119877119873

minus (119886119889

2minus

(119889 minus 2) 119877

2) 1198901198871198770] 119890119887(119877+1198770) = 0

(1 + 119888119873119877119873minus1

) 1198903119887119877

+ [1198901198871198770 (2119886119887 + 119890

1198871198770)

minus 119888119873119877119873minus1

1198901198871198770] 119890119887119877

+ [119888119877119873minus1

(1198901198871198770

(119873 + 119887119877) minus 119873 + 119887119877) + 119886119887 + 21198901198871198770]

sdot 1198902119887119877

= 0

(46)


Type 1

120582 =119877 (119899 minus 1) 119890

120585119877

119899 + 120585119877

120578 = minus(1 + 120585119877)

119877119899minus1 (119899 + 120585119877)

(47)

Type 2

120572 = minus1

2119877119899minus1

120573 =119877119899minus1

2

(48)

Type 3

1198982= 119877(

119899 minus 1

1198882

)

minus1119899

1198881 = 119899(119899 minus 1

1198882

)

1119899minus1

(49)

Type 4

119888 =

[119890119887119877

+ 1198901198871198770]2

(119887119877 + 1 minus 119890119887119877)

119877119873minus1 [119873 (119890119887119877 + 1198901198871198770) + 119887119877] (119890119887119877 minus 1)

119886 = minus

119877 (119873 minus 1) 1198902119887119877

+ [119887119877 (1 + 1198901198871198770) + (119873 minus 1) (119890

1198871198770 minus 1)] 119890119887119877 119890119887119877

(119890119887119877 minus 1)2[119887119877 + 119873 (119890119887119877 + 1198901198871198770)]

(50)




1 2 30

r

minus2

minus1

0

1

2

3

4120595(r)


lim119903rarr0

119877120572120573120574120575119877120572120573120574120575

997888rarr infin

lim119903rarrinfin

119877120572120573120574120575119877120572120573120574120575

997888rarr8119889

(119889 minus 1) (119889 minus 2)2Λ2

(51)






119879+ =

119903119889minus2

+[(119889 minus 3) 119892

2(119864) minus 2Λ119903

2

+ (119889 minus 2)] minus 2

(119889minus4)4[119892119891 (119864) 119892 (119864)]

1198892

4120587119891 (119864) 119892 (119864) 119903119889minus1+

(52)



119872 =(119889 minus 2)119898

16120587119891 (119864) 119892 (119864)119889minus3

=

[(119889 minus 2) 1198922(119864) minus 2Λ119903

2

+ (119889 minus 1)] 119903

119889minus2

++ 2(119889minus4)4

(119889 minus 2) [119892119891 (119864) 119892 (119864)]1198892

16120587119891 (119864) 119892 (119864)119889minus1

119903+

(53)

Hence we can write

119889119872 =

119903119889minus3

+[((119889 minus 2) (119889 minus 3) 119892

2(119864) minus 2Λ119903

2

+) 119903119889minus2


(119889 minus 2) (119892119891 (119864) 119892 (119864))1198892

]

16120587119891 (119864) 119892119889minus1

(119864) 119903119889minus1+

119889119903+(54)


119878 = int1

119879119889119872 =

1

4(

119903+

119892 (119864))

119889minus2

(55)




minus04

minus02

02

04

FRR

0 1 2minus1

120585

(a)

b

minus4

minus2

0

2

4

FRR

2515 2

(b)

Figure 12 Type 1 (a) 119865119877119877


versus 119887 for119873 = 4 1198770= 4



2119865(119877)119889119877



119865119877119877

=

minus(119899 minus 1) [120585119877 (119899 + 120585119877) + 119899]

119877 (119899 + 120585119877) Type 1


119877 Type 2

119899 minus 1

119877 Type 3

1198872119877 [(119899 minus 1) 119890

2119887119877+ (1198901198871198770 (119887R + 119899 minus 1) + 119887119877 minus 119899 + 1) 119890

119887119877minus (119899 minus 1) 119890

1198871198770]

(119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770]

minus

(119887119877 + 1 minus 119890119887119877) 119899 (1 minus 119899) 119887

3119887119877+ 12059111198902119887119877

minus 1205912119890119887(119877+1198770) + 119899 (119899 minus 1) 119890

21198871198770

(119890119887119877 + 1198901198871198770) (119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770] 119877

Type 4

(56)

where


+ 119887119877 (119887119877 minus 2119899)

1205912 = [119899 (119899 minus 1) + 119887119877 (119887119877 + 2119899)] 1198901198871198770 + 2119899 (1 minus 119899)

+ 119887119877 (119887119877 + 2119899)

(57)



6 Conclusions






Competing Interests


Acknowledgments


References


















































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




minus005

0

005

010

015

120595(r)

06 07 08 0905

r

(a)

06 08 10 1204

r

minus02

minus01

0

01

02

03

04

05

06

120595(r)

(b)



119872 =(119889 minus 2)119898

16120587119891 (119864) 119892 (119864)119889minus3

(18)


119889119872 (119878 119876) = (120597119872 (119878 119876)

120597119878)

119876

119889119878 + (120597119872(119878 119876)

120597119876)

119878

119889119876 (19)


119879 = (120597119872

120597119878)

119876

119880 = (120597119872

120597119876)

119878

(20)





119877120583] (1 + 119891119877) minus

119892120583]

2119865 (119877) + (119892120583]nabla


= 8120587T120583](21)


) = 0 (22)



ℎ (119903) =minusB

119903

119865tr =B

1199032

(23)



119877120583] (1 + 119891119877) minus1

119889119892120583]1198770 (1 + 119891119877) = 8120587T120583] (24)


1 2 30

r

minus1

0

1

2

3120595(r)



119892 (119864)2

119889 minus 211990312059510158401015840(119903) + 119892 (119864)

21205951015840(119903) +

2

119889 (119889 minus 2)1199031198770

=21198894

[B119892 (119864) 119891 (119864)]1198892

4 (1 + 119891119877) 119903119889minus1

119903119892 (119864)21205951015840(119903) + (119889 minus 3) 119892 (119864)

2[120595 (119903) minus 1] +

1199031198770

119889


[B119892 (119864) 119891 (119864)]1198892

(1 + 119891119877) 119903119889minus2

(25)


120595 (119903) = 1 minus119898

119903119889minus3+

2(119889minus4)4

(B119891 (119864) 119892 (119864))1198892

119892 (119864)2(1 + 119891119877) 119903

119889minus2

minus11987701199032

119889 (119889 minus 1) 119892 (119864)2

(26)






rarr (2119889(119889 minus 1))1198772

0) So





119879+ =

(1 + 119891119877) [119889 (119889 minus 3) 119892 (119864)2minus 1198770119903

2

+] 119903119889minus2

+minus 119889 2

119889minus4[B119891 (119864) 119892 (119864)]

2119889

14

4119889120587119891 (119864) 119892 (119864) 119903119889minus1+

(1 + 119891119877)

(27)


119876 =11988921198894minus1

16120587(B119891 (119864)

119892 (119864))

1198892minus1

(28)

119880 =B

119903+

(29)



119878 =1

4(

119903+

119892 (119864))

119889minus2

(1 + 119891119877) (30)




119872 =(119889 minus 2) (1 + 119891119877)119898

16120587119891 (119864) 119892 (119864)119889minus3

(31)







119862119876 =119879

(12059721198721205971198782)119876

=119879

(120597119879120597119878)119876

(32)


119862119876

=

minus(119889 minus 2) 119903

119889minus2

+

4119892119889minus2 (119864)(

[(119889 minus 3) 1198922(119864) minus 2Λ119903

2

+ (119889 minus 2)] 119903

119889minus3

+minus [radic2119902119891 (119864) 119892 (119864)]

119889minus1

[(119889 minus 3) 1198922(119864) + 2Λ1199032

+ (119889 minus 2)] 119903

119889minus3+


) CIM

minus(119889 minus 2) 119903

119889minus2

+

4119892119889minus2 (119864)(


2

+minus 2Λ119903

2

+


2++ 2Λ1199032

+

) PMI

minus(119889 minus 2) (1 + 119891119877) 119903

119889minus2

+

4119892119889minus2 (119864)(

(1 + 119891119877) [(119889 minus 3) 1198922(119864) minus 2Λ119903

2

+ (119889 minus 2)] 119903

119889minus2


2119889

(1 + 119891119877) [(119889 minus 3) 1198922(119864) + 2Λ1199032

+ (119889 minus 2)] 119903

119889minus2+


2119889

) 119865 (119877)

(33)

in which

Θ = (2 [119902119891 (119864) 119892 (119864) (2119904 minus 119889 + 1)]

2119903(4minus2119889)(2119904minus1)

+

(2119904 minus 1)2

)

119904

(34)






001 002 003

minus010

minus005

0

005

010

(a)

01 02 03 04

minus1

minus05

0

05

1

(b)

05 1 15 2

minus1

minus05

0

05

1

(c)








11986510158401015840

119877+

J12119903120595 (119903)

1198651015840

119877minus

J22119903120595 (119903)

119865119877 =119865 (119877)

2120595 (119903) 119892 (119864)2

J11198651015840

119877minusJ2119865119877 =

119903119865 (119877)

119892 (119864)2

11986510158401015840

119877+

J3

119903120595 (119903)1198651015840

119877minus

(J3 minus 119889 + 3)

1199032120595 (119903)119865119877 =

119865 (119877)

2120595 (119903) 119892 (119864)2

(35)


J1 = 1199031205951015840(119903) + 2 (119889 minus 2) 120595 (119903)

J2 = 11990312059510158401015840(119903) + (119889 minus 2) 120595

1015840(119903)

J3 = 1199031205951015840(119903) + (119889 minus 3) 120595 (119903)

(36)


= 1198651015840


following forms


1

05

0

minus05

minus1

01 02 03 040001 0002 0003 0004 0005 0006 0007 0008

minus02

minus01

0

01

02




1

05

02 04 06 08 10

minus05

minus1

(a)

05 1 15

1

05

0

minus05

minus1

(b)


+for 119902 = 1 Λ = minus1 119892(119864) = 119891(119864) = 07 and 119889 = 5 (a) 119904 = 09


119892 (119864)2[11990312059510158401015840(119903) + (119889 minus 2) 120595

1015840(119903)] 119865119877 = minus119903119865 (119877)


2119865 (119877)

(37)






21198772

0)minus119899



01 02 03

minus4

minus2

0

2

4

115 120 125 130

minus4

minus2

0

2

4




minus02

minus04

04

02

0

05 1 15 2

(a)

minus01

minus03

minus02

minus04

04

05

05 1 15 2 25 3

02

03

0

01

(b)


+for 119902 = 1 Λ = minus1 and 119904 = 14 (a) 119889 = 5 119892(119864) = 119891(119864) = 07




120578(120593(119877))12)[1minus120593(119877)]119877minus119890

2120578(120593(119877))119869(120593(119877))



11989101198724minus2119899

119877119899+ 1198912119877

2 [150] and 119865(119877) = (11987721205812)(1 minus 1198771198770) [151]


Type 1

119865 (119877) = 119877 minus 120582119890minus120585119877

+ 120578119877119899 (38)

Type 2

119865 (119877) = 119877 + 120572119877119899minus 1205731198772minus119899 (39)


8

6

4

2

0

minus2

minus4

04 06 08 10 12 14 16

(a)

4

2

0

minus2

minus4

04 06 08 10 12 14

(b)

04 06 08 10 12 14 16

3

2

1

0

minus1

minus2

minus3

(c)


+for 119902 = 1 Λ = minus1 and 119889 = 5 (a) 119892(119864) = 119891(119864) = 07 119891

119877= 0


119877= 1 (dashed line) (b) 119892(119864) = 119891(119864) = 07 119891



and 119891119877= 16 (dashed line) (c) 119891


(dashed line)

Type 3

119865 (119877) = 119877 minus 1198982

1198881 (1198771198982)119899

1198882 (1198771198982)119899+ 1

(40)

Type 4

119865 (119877) = 119877 minus 119886 [119890minus119887119877

minus 1] + 119888119877119873 119890119887119877

minus 1

119890119887119877 + 1198901198871198770(41)






minus1

minus05

05

045 050 055

1

0

(a)

070 072 076074 078

minus1

minus05

05

1

0

(b)

minus1

minus05

05

15 25

1

1 2 3

0

(c)


+for 119902 = 1 Λ = minus1 119892(119864) = 119891(119864) = 07 and 119891

119877= 15 (a) 119889 = 6










120595 (119903) = 1 minus2Λ1199032


119898

119903119889minus3

+2(119889minus4)4

[119902119891 (119864) 119892 (119864)]1198892

119892 (119864)2119903119889minus2

(42)


Type 1

119877[(119899 minus119889

2) 120578119877119899minus1

minus119889 minus 2

2] 119890120585119877

+ 120582(119889

2+ 120585119877) = 0

(1 + 119899120578119877119899minus1

) 119890120585119877

+ 120582120585 = 0

(43)

Type 2

[120572(119899 minus119889

2)119877119899minus

119889 minus 2

2119877]119877119899+ (119899 +

119889 minus 4

2)1205731198772

= 0

(120572119899119877119899+ 119877)119877

119899+ (119899 minus 2) 120573119877

2= 0

(44)

Type 3

1198882 [119889

119889 minus 211989821198881 minus 1198882119877](

119877

1198982)

2119899

minus [211989821198881

119889 minus 2(119899 minus

119889

2) + 21198882119877](

119877

1198982)

119899

minus 119877 = 0

[1198882

2(

119877

1198982)

2119899

+ 1]119877 + [21198882119877 minus 11989821198991198881] (

119877

1198982)

119899

= 0

(45)

Type 4

[119888 (119873 minus119889

2)119877119873minus

119889 minus 2

2119877 minus

119889

2119886] 1198903119887119877

+ 119886(119887119877 +119889

2)

sdot 11989021198871198770

+ 119888 [(119887119877 + 119873 minus119889

2) 1198901198871198770 + 119887119877 minus 119873 +

119889

2]119877119873

minus [(119889 minus 2) 119877 + 119886119889] 1198901198871198770 + 119886(119887119877 +

119889

2) 1198902119887119877

+ [119886 (2119887119877 + 119889) minus 119888 (119873 minus119889

2)119877119873

minus (119886119889

2minus

(119889 minus 2) 119877

2) 1198901198871198770] 119890119887(119877+1198770) = 0

(1 + 119888119873119877119873minus1

) 1198903119887119877

+ [1198901198871198770 (2119886119887 + 119890

1198871198770)

minus 119888119873119877119873minus1

1198901198871198770] 119890119887119877

+ [119888119877119873minus1

(1198901198871198770

(119873 + 119887119877) minus 119873 + 119887119877) + 119886119887 + 21198901198871198770]

sdot 1198902119887119877

= 0

(46)


Type 1

120582 =119877 (119899 minus 1) 119890

120585119877

119899 + 120585119877

120578 = minus(1 + 120585119877)

119877119899minus1 (119899 + 120585119877)

(47)

Type 2

120572 = minus1

2119877119899minus1

120573 =119877119899minus1

2

(48)

Type 3

1198982= 119877(

119899 minus 1

1198882

)

minus1119899

1198881 = 119899(119899 minus 1

1198882

)

1119899minus1

(49)

Type 4

119888 =

[119890119887119877

+ 1198901198871198770]2

(119887119877 + 1 minus 119890119887119877)

119877119873minus1 [119873 (119890119887119877 + 1198901198871198770) + 119887119877] (119890119887119877 minus 1)

119886 = minus

119877 (119873 minus 1) 1198902119887119877

+ [119887119877 (1 + 1198901198871198770) + (119873 minus 1) (119890

1198871198770 minus 1)] 119890119887119877 119890119887119877

(119890119887119877 minus 1)2[119887119877 + 119873 (119890119887119877 + 1198901198871198770)]

(50)




1 2 30

r

minus2

minus1

0

1

2

3

4120595(r)


lim119903rarr0

119877120572120573120574120575119877120572120573120574120575

997888rarr infin

lim119903rarrinfin

119877120572120573120574120575119877120572120573120574120575

997888rarr8119889

(119889 minus 1) (119889 minus 2)2Λ2

(51)






119879+ =

119903119889minus2

+[(119889 minus 3) 119892

2(119864) minus 2Λ119903

2

+ (119889 minus 2)] minus 2

(119889minus4)4[119892119891 (119864) 119892 (119864)]

1198892

4120587119891 (119864) 119892 (119864) 119903119889minus1+

(52)



119872 =(119889 minus 2)119898

16120587119891 (119864) 119892 (119864)119889minus3

=

[(119889 minus 2) 1198922(119864) minus 2Λ119903

2

+ (119889 minus 1)] 119903

119889minus2

++ 2(119889minus4)4

(119889 minus 2) [119892119891 (119864) 119892 (119864)]1198892

16120587119891 (119864) 119892 (119864)119889minus1

119903+

(53)

Hence we can write

119889119872 =

119903119889minus3

+[((119889 minus 2) (119889 minus 3) 119892

2(119864) minus 2Λ119903

2

+) 119903119889minus2


(119889 minus 2) (119892119891 (119864) 119892 (119864))1198892

]

16120587119891 (119864) 119892119889minus1

(119864) 119903119889minus1+

119889119903+(54)


119878 = int1

119879119889119872 =

1

4(

119903+

119892 (119864))

119889minus2

(55)




minus04

minus02

02

04

FRR

0 1 2minus1

120585

(a)

b

minus4

minus2

0

2

4

FRR

2515 2

(b)

Figure 12 Type 1 (a) 119865119877119877


versus 119887 for119873 = 4 1198770= 4



2119865(119877)119889119877



119865119877119877

=

minus(119899 minus 1) [120585119877 (119899 + 120585119877) + 119899]

119877 (119899 + 120585119877) Type 1


119877 Type 2

119899 minus 1

119877 Type 3

1198872119877 [(119899 minus 1) 119890

2119887119877+ (1198901198871198770 (119887R + 119899 minus 1) + 119887119877 minus 119899 + 1) 119890

119887119877minus (119899 minus 1) 119890

1198871198770]

(119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770]

minus

(119887119877 + 1 minus 119890119887119877) 119899 (1 minus 119899) 119887

3119887119877+ 12059111198902119887119877

minus 1205912119890119887(119877+1198770) + 119899 (119899 minus 1) 119890

21198871198770

(119890119887119877 + 1198901198871198770) (119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770] 119877

Type 4

(56)

where


+ 119887119877 (119887119877 minus 2119899)

1205912 = [119899 (119899 minus 1) + 119887119877 (119887119877 + 2119899)] 1198901198871198770 + 2119899 (1 minus 119899)

+ 119887119877 (119887119877 + 2119899)

(57)



6 Conclusions






Competing Interests


Acknowledgments


References


















































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




1 2 30

r

minus1

0

1

2

3120595(r)



119892 (119864)2

119889 minus 211990312059510158401015840(119903) + 119892 (119864)

21205951015840(119903) +

2

119889 (119889 minus 2)1199031198770

=21198894

[B119892 (119864) 119891 (119864)]1198892

4 (1 + 119891119877) 119903119889minus1

119903119892 (119864)21205951015840(119903) + (119889 minus 3) 119892 (119864)

2[120595 (119903) minus 1] +

1199031198770

119889


[B119892 (119864) 119891 (119864)]1198892

(1 + 119891119877) 119903119889minus2

(25)


120595 (119903) = 1 minus119898

119903119889minus3+

2(119889minus4)4

(B119891 (119864) 119892 (119864))1198892

119892 (119864)2(1 + 119891119877) 119903

119889minus2

minus11987701199032

119889 (119889 minus 1) 119892 (119864)2

(26)






rarr (2119889(119889 minus 1))1198772

0) So





119879+ =

(1 + 119891119877) [119889 (119889 minus 3) 119892 (119864)2minus 1198770119903

2

+] 119903119889minus2

+minus 119889 2

119889minus4[B119891 (119864) 119892 (119864)]

2119889

14

4119889120587119891 (119864) 119892 (119864) 119903119889minus1+

(1 + 119891119877)

(27)


119876 =11988921198894minus1

16120587(B119891 (119864)

119892 (119864))

1198892minus1

(28)

119880 =B

119903+

(29)



119878 =1

4(

119903+

119892 (119864))

119889minus2

(1 + 119891119877) (30)




119872 =(119889 minus 2) (1 + 119891119877)119898

16120587119891 (119864) 119892 (119864)119889minus3

(31)







119862119876 =119879

(12059721198721205971198782)119876

=119879

(120597119879120597119878)119876

(32)


119862119876

=

minus(119889 minus 2) 119903

119889minus2

+

4119892119889minus2 (119864)(

[(119889 minus 3) 1198922(119864) minus 2Λ119903

2

+ (119889 minus 2)] 119903

119889minus3

+minus [radic2119902119891 (119864) 119892 (119864)]

119889minus1

[(119889 minus 3) 1198922(119864) + 2Λ1199032

+ (119889 minus 2)] 119903

119889minus3+


) CIM

minus(119889 minus 2) 119903

119889minus2

+

4119892119889minus2 (119864)(


2

+minus 2Λ119903

2

+


2++ 2Λ1199032

+

) PMI

minus(119889 minus 2) (1 + 119891119877) 119903

119889minus2

+

4119892119889minus2 (119864)(

(1 + 119891119877) [(119889 minus 3) 1198922(119864) minus 2Λ119903

2

+ (119889 minus 2)] 119903

119889minus2


2119889

(1 + 119891119877) [(119889 minus 3) 1198922(119864) + 2Λ1199032

+ (119889 minus 2)] 119903

119889minus2+


2119889

) 119865 (119877)

(33)

in which

Θ = (2 [119902119891 (119864) 119892 (119864) (2119904 minus 119889 + 1)]

2119903(4minus2119889)(2119904minus1)

+

(2119904 minus 1)2

)

119904

(34)






001 002 003

minus010

minus005

0

005

010

(a)

01 02 03 04

minus1

minus05

0

05

1

(b)

05 1 15 2

minus1

minus05

0

05

1

(c)








11986510158401015840

119877+

J12119903120595 (119903)

1198651015840

119877minus

J22119903120595 (119903)

119865119877 =119865 (119877)

2120595 (119903) 119892 (119864)2

J11198651015840

119877minusJ2119865119877 =

119903119865 (119877)

119892 (119864)2

11986510158401015840

119877+

J3

119903120595 (119903)1198651015840

119877minus

(J3 minus 119889 + 3)

1199032120595 (119903)119865119877 =

119865 (119877)

2120595 (119903) 119892 (119864)2

(35)


J1 = 1199031205951015840(119903) + 2 (119889 minus 2) 120595 (119903)

J2 = 11990312059510158401015840(119903) + (119889 minus 2) 120595

1015840(119903)

J3 = 1199031205951015840(119903) + (119889 minus 3) 120595 (119903)

(36)


= 1198651015840


following forms


1

05

0

minus05

minus1

01 02 03 040001 0002 0003 0004 0005 0006 0007 0008

minus02

minus01

0

01

02




1

05

02 04 06 08 10

minus05

minus1

(a)

05 1 15

1

05

0

minus05

minus1

(b)


+for 119902 = 1 Λ = minus1 119892(119864) = 119891(119864) = 07 and 119889 = 5 (a) 119904 = 09


119892 (119864)2[11990312059510158401015840(119903) + (119889 minus 2) 120595

1015840(119903)] 119865119877 = minus119903119865 (119877)


2119865 (119877)

(37)






21198772

0)minus119899



01 02 03

minus4

minus2

0

2

4

115 120 125 130

minus4

minus2

0

2

4




minus02

minus04

04

02

0

05 1 15 2

(a)

minus01

minus03

minus02

minus04

04

05

05 1 15 2 25 3

02

03

0

01

(b)


+for 119902 = 1 Λ = minus1 and 119904 = 14 (a) 119889 = 5 119892(119864) = 119891(119864) = 07




120578(120593(119877))12)[1minus120593(119877)]119877minus119890

2120578(120593(119877))119869(120593(119877))



11989101198724minus2119899

119877119899+ 1198912119877

2 [150] and 119865(119877) = (11987721205812)(1 minus 1198771198770) [151]


Type 1

119865 (119877) = 119877 minus 120582119890minus120585119877

+ 120578119877119899 (38)

Type 2

119865 (119877) = 119877 + 120572119877119899minus 1205731198772minus119899 (39)


8

6

4

2

0

minus2

minus4

04 06 08 10 12 14 16

(a)

4

2

0

minus2

minus4

04 06 08 10 12 14

(b)

04 06 08 10 12 14 16

3

2

1

0

minus1

minus2

minus3

(c)


+for 119902 = 1 Λ = minus1 and 119889 = 5 (a) 119892(119864) = 119891(119864) = 07 119891

119877= 0


119877= 1 (dashed line) (b) 119892(119864) = 119891(119864) = 07 119891



and 119891119877= 16 (dashed line) (c) 119891


(dashed line)

Type 3

119865 (119877) = 119877 minus 1198982

1198881 (1198771198982)119899

1198882 (1198771198982)119899+ 1

(40)

Type 4

119865 (119877) = 119877 minus 119886 [119890minus119887119877

minus 1] + 119888119877119873 119890119887119877

minus 1

119890119887119877 + 1198901198871198770(41)






minus1

minus05

05

045 050 055

1

0

(a)

070 072 076074 078

minus1

minus05

05

1

0

(b)

minus1

minus05

05

15 25

1

1 2 3

0

(c)


+for 119902 = 1 Λ = minus1 119892(119864) = 119891(119864) = 07 and 119891

119877= 15 (a) 119889 = 6










120595 (119903) = 1 minus2Λ1199032


119898

119903119889minus3

+2(119889minus4)4

[119902119891 (119864) 119892 (119864)]1198892

119892 (119864)2119903119889minus2

(42)


Type 1

119877[(119899 minus119889

2) 120578119877119899minus1

minus119889 minus 2

2] 119890120585119877

+ 120582(119889

2+ 120585119877) = 0

(1 + 119899120578119877119899minus1

) 119890120585119877

+ 120582120585 = 0

(43)

Type 2

[120572(119899 minus119889

2)119877119899minus

119889 minus 2

2119877]119877119899+ (119899 +

119889 minus 4

2)1205731198772

= 0

(120572119899119877119899+ 119877)119877

119899+ (119899 minus 2) 120573119877

2= 0

(44)

Type 3

1198882 [119889

119889 minus 211989821198881 minus 1198882119877](

119877

1198982)

2119899

minus [211989821198881

119889 minus 2(119899 minus

119889

2) + 21198882119877](

119877

1198982)

119899

minus 119877 = 0

[1198882

2(

119877

1198982)

2119899

+ 1]119877 + [21198882119877 minus 11989821198991198881] (

119877

1198982)

119899

= 0

(45)

Type 4

[119888 (119873 minus119889

2)119877119873minus

119889 minus 2

2119877 minus

119889

2119886] 1198903119887119877

+ 119886(119887119877 +119889

2)

sdot 11989021198871198770

+ 119888 [(119887119877 + 119873 minus119889

2) 1198901198871198770 + 119887119877 minus 119873 +

119889

2]119877119873

minus [(119889 minus 2) 119877 + 119886119889] 1198901198871198770 + 119886(119887119877 +

119889

2) 1198902119887119877

+ [119886 (2119887119877 + 119889) minus 119888 (119873 minus119889

2)119877119873

minus (119886119889

2minus

(119889 minus 2) 119877

2) 1198901198871198770] 119890119887(119877+1198770) = 0

(1 + 119888119873119877119873minus1

) 1198903119887119877

+ [1198901198871198770 (2119886119887 + 119890

1198871198770)

minus 119888119873119877119873minus1

1198901198871198770] 119890119887119877

+ [119888119877119873minus1

(1198901198871198770

(119873 + 119887119877) minus 119873 + 119887119877) + 119886119887 + 21198901198871198770]

sdot 1198902119887119877

= 0

(46)


Type 1

120582 =119877 (119899 minus 1) 119890

120585119877

119899 + 120585119877

120578 = minus(1 + 120585119877)

119877119899minus1 (119899 + 120585119877)

(47)

Type 2

120572 = minus1

2119877119899minus1

120573 =119877119899minus1

2

(48)

Type 3

1198982= 119877(

119899 minus 1

1198882

)

minus1119899

1198881 = 119899(119899 minus 1

1198882

)

1119899minus1

(49)

Type 4

119888 =

[119890119887119877

+ 1198901198871198770]2

(119887119877 + 1 minus 119890119887119877)

119877119873minus1 [119873 (119890119887119877 + 1198901198871198770) + 119887119877] (119890119887119877 minus 1)

119886 = minus

119877 (119873 minus 1) 1198902119887119877

+ [119887119877 (1 + 1198901198871198770) + (119873 minus 1) (119890

1198871198770 minus 1)] 119890119887119877 119890119887119877

(119890119887119877 minus 1)2[119887119877 + 119873 (119890119887119877 + 1198901198871198770)]

(50)




1 2 30

r

minus2

minus1

0

1

2

3

4120595(r)


lim119903rarr0

119877120572120573120574120575119877120572120573120574120575

997888rarr infin

lim119903rarrinfin

119877120572120573120574120575119877120572120573120574120575

997888rarr8119889

(119889 minus 1) (119889 minus 2)2Λ2

(51)






119879+ =

119903119889minus2

+[(119889 minus 3) 119892

2(119864) minus 2Λ119903

2

+ (119889 minus 2)] minus 2

(119889minus4)4[119892119891 (119864) 119892 (119864)]

1198892

4120587119891 (119864) 119892 (119864) 119903119889minus1+

(52)



119872 =(119889 minus 2)119898

16120587119891 (119864) 119892 (119864)119889minus3

=

[(119889 minus 2) 1198922(119864) minus 2Λ119903

2

+ (119889 minus 1)] 119903

119889minus2

++ 2(119889minus4)4

(119889 minus 2) [119892119891 (119864) 119892 (119864)]1198892

16120587119891 (119864) 119892 (119864)119889minus1

119903+

(53)

Hence we can write

119889119872 =

119903119889minus3

+[((119889 minus 2) (119889 minus 3) 119892

2(119864) minus 2Λ119903

2

+) 119903119889minus2


(119889 minus 2) (119892119891 (119864) 119892 (119864))1198892

]

16120587119891 (119864) 119892119889minus1

(119864) 119903119889minus1+

119889119903+(54)


119878 = int1

119879119889119872 =

1

4(

119903+

119892 (119864))

119889minus2

(55)




minus04

minus02

02

04

FRR

0 1 2minus1

120585

(a)

b

minus4

minus2

0

2

4

FRR

2515 2

(b)

Figure 12 Type 1 (a) 119865119877119877


versus 119887 for119873 = 4 1198770= 4



2119865(119877)119889119877



119865119877119877

=

minus(119899 minus 1) [120585119877 (119899 + 120585119877) + 119899]

119877 (119899 + 120585119877) Type 1


119877 Type 2

119899 minus 1

119877 Type 3

1198872119877 [(119899 minus 1) 119890

2119887119877+ (1198901198871198770 (119887R + 119899 minus 1) + 119887119877 minus 119899 + 1) 119890

119887119877minus (119899 minus 1) 119890

1198871198770]

(119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770]

minus

(119887119877 + 1 minus 119890119887119877) 119899 (1 minus 119899) 119887

3119887119877+ 12059111198902119887119877

minus 1205912119890119887(119877+1198770) + 119899 (119899 minus 1) 119890

21198871198770

(119890119887119877 + 1198901198871198770) (119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770] 119877

Type 4

(56)

where


+ 119887119877 (119887119877 minus 2119899)

1205912 = [119899 (119899 minus 1) + 119887119877 (119887119877 + 2119899)] 1198901198871198770 + 2119899 (1 minus 119899)

+ 119887119877 (119887119877 + 2119899)

(57)



6 Conclusions






Competing Interests


Acknowledgments


References


















































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119872 =(119889 minus 2) (1 + 119891119877)119898

16120587119891 (119864) 119892 (119864)119889minus3

(31)







119862119876 =119879

(12059721198721205971198782)119876

=119879

(120597119879120597119878)119876

(32)


119862119876

=

minus(119889 minus 2) 119903

119889minus2

+

4119892119889minus2 (119864)(

[(119889 minus 3) 1198922(119864) minus 2Λ119903

2

+ (119889 minus 2)] 119903

119889minus3

+minus [radic2119902119891 (119864) 119892 (119864)]

119889minus1

[(119889 minus 3) 1198922(119864) + 2Λ1199032

+ (119889 minus 2)] 119903

119889minus3+


) CIM

minus(119889 minus 2) 119903

119889minus2

+

4119892119889minus2 (119864)(


2

+minus 2Λ119903

2

+


2++ 2Λ1199032

+

) PMI

minus(119889 minus 2) (1 + 119891119877) 119903

119889minus2

+

4119892119889minus2 (119864)(

(1 + 119891119877) [(119889 minus 3) 1198922(119864) minus 2Λ119903

2

+ (119889 minus 2)] 119903

119889minus2


2119889

(1 + 119891119877) [(119889 minus 3) 1198922(119864) + 2Λ1199032

+ (119889 minus 2)] 119903

119889minus2+


2119889

) 119865 (119877)

(33)

in which

Θ = (2 [119902119891 (119864) 119892 (119864) (2119904 minus 119889 + 1)]

2119903(4minus2119889)(2119904minus1)

+

(2119904 minus 1)2

)

119904

(34)






001 002 003

minus010

minus005

0

005

010

(a)

01 02 03 04

minus1

minus05

0

05

1

(b)

05 1 15 2

minus1

minus05

0

05

1

(c)








11986510158401015840

119877+

J12119903120595 (119903)

1198651015840

119877minus

J22119903120595 (119903)

119865119877 =119865 (119877)

2120595 (119903) 119892 (119864)2

J11198651015840

119877minusJ2119865119877 =

119903119865 (119877)

119892 (119864)2

11986510158401015840

119877+

J3

119903120595 (119903)1198651015840

119877minus

(J3 minus 119889 + 3)

1199032120595 (119903)119865119877 =

119865 (119877)

2120595 (119903) 119892 (119864)2

(35)


J1 = 1199031205951015840(119903) + 2 (119889 minus 2) 120595 (119903)

J2 = 11990312059510158401015840(119903) + (119889 minus 2) 120595

1015840(119903)

J3 = 1199031205951015840(119903) + (119889 minus 3) 120595 (119903)

(36)


= 1198651015840


following forms


1

05

0

minus05

minus1

01 02 03 040001 0002 0003 0004 0005 0006 0007 0008

minus02

minus01

0

01

02




1

05

02 04 06 08 10

minus05

minus1

(a)

05 1 15

1

05

0

minus05

minus1

(b)


+for 119902 = 1 Λ = minus1 119892(119864) = 119891(119864) = 07 and 119889 = 5 (a) 119904 = 09


119892 (119864)2[11990312059510158401015840(119903) + (119889 minus 2) 120595

1015840(119903)] 119865119877 = minus119903119865 (119877)


2119865 (119877)

(37)






21198772

0)minus119899



01 02 03

minus4

minus2

0

2

4

115 120 125 130

minus4

minus2

0

2

4




minus02

minus04

04

02

0

05 1 15 2

(a)

minus01

minus03

minus02

minus04

04

05

05 1 15 2 25 3

02

03

0

01

(b)


+for 119902 = 1 Λ = minus1 and 119904 = 14 (a) 119889 = 5 119892(119864) = 119891(119864) = 07




120578(120593(119877))12)[1minus120593(119877)]119877minus119890

2120578(120593(119877))119869(120593(119877))



11989101198724minus2119899

119877119899+ 1198912119877

2 [150] and 119865(119877) = (11987721205812)(1 minus 1198771198770) [151]


Type 1

119865 (119877) = 119877 minus 120582119890minus120585119877

+ 120578119877119899 (38)

Type 2

119865 (119877) = 119877 + 120572119877119899minus 1205731198772minus119899 (39)


8

6

4

2

0

minus2

minus4

04 06 08 10 12 14 16

(a)

4

2

0

minus2

minus4

04 06 08 10 12 14

(b)

04 06 08 10 12 14 16

3

2

1

0

minus1

minus2

minus3

(c)


+for 119902 = 1 Λ = minus1 and 119889 = 5 (a) 119892(119864) = 119891(119864) = 07 119891

119877= 0


119877= 1 (dashed line) (b) 119892(119864) = 119891(119864) = 07 119891



and 119891119877= 16 (dashed line) (c) 119891


(dashed line)

Type 3

119865 (119877) = 119877 minus 1198982

1198881 (1198771198982)119899

1198882 (1198771198982)119899+ 1

(40)

Type 4

119865 (119877) = 119877 minus 119886 [119890minus119887119877

minus 1] + 119888119877119873 119890119887119877

minus 1

119890119887119877 + 1198901198871198770(41)






minus1

minus05

05

045 050 055

1

0

(a)

070 072 076074 078

minus1

minus05

05

1

0

(b)

minus1

minus05

05

15 25

1

1 2 3

0

(c)


+for 119902 = 1 Λ = minus1 119892(119864) = 119891(119864) = 07 and 119891

119877= 15 (a) 119889 = 6










120595 (119903) = 1 minus2Λ1199032


119898

119903119889minus3

+2(119889minus4)4

[119902119891 (119864) 119892 (119864)]1198892

119892 (119864)2119903119889minus2

(42)


Type 1

119877[(119899 minus119889

2) 120578119877119899minus1

minus119889 minus 2

2] 119890120585119877

+ 120582(119889

2+ 120585119877) = 0

(1 + 119899120578119877119899minus1

) 119890120585119877

+ 120582120585 = 0

(43)

Type 2

[120572(119899 minus119889

2)119877119899minus

119889 minus 2

2119877]119877119899+ (119899 +

119889 minus 4

2)1205731198772

= 0

(120572119899119877119899+ 119877)119877

119899+ (119899 minus 2) 120573119877

2= 0

(44)

Type 3

1198882 [119889

119889 minus 211989821198881 minus 1198882119877](

119877

1198982)

2119899

minus [211989821198881

119889 minus 2(119899 minus

119889

2) + 21198882119877](

119877

1198982)

119899

minus 119877 = 0

[1198882

2(

119877

1198982)

2119899

+ 1]119877 + [21198882119877 minus 11989821198991198881] (

119877

1198982)

119899

= 0

(45)

Type 4

[119888 (119873 minus119889

2)119877119873minus

119889 minus 2

2119877 minus

119889

2119886] 1198903119887119877

+ 119886(119887119877 +119889

2)

sdot 11989021198871198770

+ 119888 [(119887119877 + 119873 minus119889

2) 1198901198871198770 + 119887119877 minus 119873 +

119889

2]119877119873

minus [(119889 minus 2) 119877 + 119886119889] 1198901198871198770 + 119886(119887119877 +

119889

2) 1198902119887119877

+ [119886 (2119887119877 + 119889) minus 119888 (119873 minus119889

2)119877119873

minus (119886119889

2minus

(119889 minus 2) 119877

2) 1198901198871198770] 119890119887(119877+1198770) = 0

(1 + 119888119873119877119873minus1

) 1198903119887119877

+ [1198901198871198770 (2119886119887 + 119890

1198871198770)

minus 119888119873119877119873minus1

1198901198871198770] 119890119887119877

+ [119888119877119873minus1

(1198901198871198770

(119873 + 119887119877) minus 119873 + 119887119877) + 119886119887 + 21198901198871198770]

sdot 1198902119887119877

= 0

(46)


Type 1

120582 =119877 (119899 minus 1) 119890

120585119877

119899 + 120585119877

120578 = minus(1 + 120585119877)

119877119899minus1 (119899 + 120585119877)

(47)

Type 2

120572 = minus1

2119877119899minus1

120573 =119877119899minus1

2

(48)

Type 3

1198982= 119877(

119899 minus 1

1198882

)

minus1119899

1198881 = 119899(119899 minus 1

1198882

)

1119899minus1

(49)

Type 4

119888 =

[119890119887119877

+ 1198901198871198770]2

(119887119877 + 1 minus 119890119887119877)

119877119873minus1 [119873 (119890119887119877 + 1198901198871198770) + 119887119877] (119890119887119877 minus 1)

119886 = minus

119877 (119873 minus 1) 1198902119887119877

+ [119887119877 (1 + 1198901198871198770) + (119873 minus 1) (119890

1198871198770 minus 1)] 119890119887119877 119890119887119877

(119890119887119877 minus 1)2[119887119877 + 119873 (119890119887119877 + 1198901198871198770)]

(50)




1 2 30

r

minus2

minus1

0

1

2

3

4120595(r)


lim119903rarr0

119877120572120573120574120575119877120572120573120574120575

997888rarr infin

lim119903rarrinfin

119877120572120573120574120575119877120572120573120574120575

997888rarr8119889

(119889 minus 1) (119889 minus 2)2Λ2

(51)






119879+ =

119903119889minus2

+[(119889 minus 3) 119892

2(119864) minus 2Λ119903

2

+ (119889 minus 2)] minus 2

(119889minus4)4[119892119891 (119864) 119892 (119864)]

1198892

4120587119891 (119864) 119892 (119864) 119903119889minus1+

(52)



119872 =(119889 minus 2)119898

16120587119891 (119864) 119892 (119864)119889minus3

=

[(119889 minus 2) 1198922(119864) minus 2Λ119903

2

+ (119889 minus 1)] 119903

119889minus2

++ 2(119889minus4)4

(119889 minus 2) [119892119891 (119864) 119892 (119864)]1198892

16120587119891 (119864) 119892 (119864)119889minus1

119903+

(53)

Hence we can write

119889119872 =

119903119889minus3

+[((119889 minus 2) (119889 minus 3) 119892

2(119864) minus 2Λ119903

2

+) 119903119889minus2


(119889 minus 2) (119892119891 (119864) 119892 (119864))1198892

]

16120587119891 (119864) 119892119889minus1

(119864) 119903119889minus1+

119889119903+(54)


119878 = int1

119879119889119872 =

1

4(

119903+

119892 (119864))

119889minus2

(55)




minus04

minus02

02

04

FRR

0 1 2minus1

120585

(a)

b

minus4

minus2

0

2

4

FRR

2515 2

(b)

Figure 12 Type 1 (a) 119865119877119877


versus 119887 for119873 = 4 1198770= 4



2119865(119877)119889119877



119865119877119877

=

minus(119899 minus 1) [120585119877 (119899 + 120585119877) + 119899]

119877 (119899 + 120585119877) Type 1


119877 Type 2

119899 minus 1

119877 Type 3

1198872119877 [(119899 minus 1) 119890

2119887119877+ (1198901198871198770 (119887R + 119899 minus 1) + 119887119877 minus 119899 + 1) 119890

119887119877minus (119899 minus 1) 119890

1198871198770]

(119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770]

minus

(119887119877 + 1 minus 119890119887119877) 119899 (1 minus 119899) 119887

3119887119877+ 12059111198902119887119877

minus 1205912119890119887(119877+1198770) + 119899 (119899 minus 1) 119890

21198871198770

(119890119887119877 + 1198901198871198770) (119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770] 119877

Type 4

(56)

where


+ 119887119877 (119887119877 minus 2119899)

1205912 = [119899 (119899 minus 1) + 119887119877 (119887119877 + 2119899)] 1198901198871198770 + 2119899 (1 minus 119899)

+ 119887119877 (119887119877 + 2119899)

(57)



6 Conclusions






Competing Interests


Acknowledgments


References


















































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




001 002 003

minus010

minus005

0

005

010

(a)

01 02 03 04

minus1

minus05

0

05

1

(b)

05 1 15 2

minus1

minus05

0

05

1

(c)








11986510158401015840

119877+

J12119903120595 (119903)

1198651015840

119877minus

J22119903120595 (119903)

119865119877 =119865 (119877)

2120595 (119903) 119892 (119864)2

J11198651015840

119877minusJ2119865119877 =

119903119865 (119877)

119892 (119864)2

11986510158401015840

119877+

J3

119903120595 (119903)1198651015840

119877minus

(J3 minus 119889 + 3)

1199032120595 (119903)119865119877 =

119865 (119877)

2120595 (119903) 119892 (119864)2

(35)


J1 = 1199031205951015840(119903) + 2 (119889 minus 2) 120595 (119903)

J2 = 11990312059510158401015840(119903) + (119889 minus 2) 120595

1015840(119903)

J3 = 1199031205951015840(119903) + (119889 minus 3) 120595 (119903)

(36)


= 1198651015840


following forms


1

05

0

minus05

minus1

01 02 03 040001 0002 0003 0004 0005 0006 0007 0008

minus02

minus01

0

01

02




1

05

02 04 06 08 10

minus05

minus1

(a)

05 1 15

1

05

0

minus05

minus1

(b)


+for 119902 = 1 Λ = minus1 119892(119864) = 119891(119864) = 07 and 119889 = 5 (a) 119904 = 09


119892 (119864)2[11990312059510158401015840(119903) + (119889 minus 2) 120595

1015840(119903)] 119865119877 = minus119903119865 (119877)


2119865 (119877)

(37)






21198772

0)minus119899



01 02 03

minus4

minus2

0

2

4

115 120 125 130

minus4

minus2

0

2

4




minus02

minus04

04

02

0

05 1 15 2

(a)

minus01

minus03

minus02

minus04

04

05

05 1 15 2 25 3

02

03

0

01

(b)


+for 119902 = 1 Λ = minus1 and 119904 = 14 (a) 119889 = 5 119892(119864) = 119891(119864) = 07




120578(120593(119877))12)[1minus120593(119877)]119877minus119890

2120578(120593(119877))119869(120593(119877))



11989101198724minus2119899

119877119899+ 1198912119877

2 [150] and 119865(119877) = (11987721205812)(1 minus 1198771198770) [151]


Type 1

119865 (119877) = 119877 minus 120582119890minus120585119877

+ 120578119877119899 (38)

Type 2

119865 (119877) = 119877 + 120572119877119899minus 1205731198772minus119899 (39)


8

6

4

2

0

minus2

minus4

04 06 08 10 12 14 16

(a)

4

2

0

minus2

minus4

04 06 08 10 12 14

(b)

04 06 08 10 12 14 16

3

2

1

0

minus1

minus2

minus3

(c)


+for 119902 = 1 Λ = minus1 and 119889 = 5 (a) 119892(119864) = 119891(119864) = 07 119891

119877= 0


119877= 1 (dashed line) (b) 119892(119864) = 119891(119864) = 07 119891



and 119891119877= 16 (dashed line) (c) 119891


(dashed line)

Type 3

119865 (119877) = 119877 minus 1198982

1198881 (1198771198982)119899

1198882 (1198771198982)119899+ 1

(40)

Type 4

119865 (119877) = 119877 minus 119886 [119890minus119887119877

minus 1] + 119888119877119873 119890119887119877

minus 1

119890119887119877 + 1198901198871198770(41)






minus1

minus05

05

045 050 055

1

0

(a)

070 072 076074 078

minus1

minus05

05

1

0

(b)

minus1

minus05

05

15 25

1

1 2 3

0

(c)


+for 119902 = 1 Λ = minus1 119892(119864) = 119891(119864) = 07 and 119891

119877= 15 (a) 119889 = 6










120595 (119903) = 1 minus2Λ1199032


119898

119903119889minus3

+2(119889minus4)4

[119902119891 (119864) 119892 (119864)]1198892

119892 (119864)2119903119889minus2

(42)


Type 1

119877[(119899 minus119889

2) 120578119877119899minus1

minus119889 minus 2

2] 119890120585119877

+ 120582(119889

2+ 120585119877) = 0

(1 + 119899120578119877119899minus1

) 119890120585119877

+ 120582120585 = 0

(43)

Type 2

[120572(119899 minus119889

2)119877119899minus

119889 minus 2

2119877]119877119899+ (119899 +

119889 minus 4

2)1205731198772

= 0

(120572119899119877119899+ 119877)119877

119899+ (119899 minus 2) 120573119877

2= 0

(44)

Type 3

1198882 [119889

119889 minus 211989821198881 minus 1198882119877](

119877

1198982)

2119899

minus [211989821198881

119889 minus 2(119899 minus

119889

2) + 21198882119877](

119877

1198982)

119899

minus 119877 = 0

[1198882

2(

119877

1198982)

2119899

+ 1]119877 + [21198882119877 minus 11989821198991198881] (

119877

1198982)

119899

= 0

(45)

Type 4

[119888 (119873 minus119889

2)119877119873minus

119889 minus 2

2119877 minus

119889

2119886] 1198903119887119877

+ 119886(119887119877 +119889

2)

sdot 11989021198871198770

+ 119888 [(119887119877 + 119873 minus119889

2) 1198901198871198770 + 119887119877 minus 119873 +

119889

2]119877119873

minus [(119889 minus 2) 119877 + 119886119889] 1198901198871198770 + 119886(119887119877 +

119889

2) 1198902119887119877

+ [119886 (2119887119877 + 119889) minus 119888 (119873 minus119889

2)119877119873

minus (119886119889

2minus

(119889 minus 2) 119877

2) 1198901198871198770] 119890119887(119877+1198770) = 0

(1 + 119888119873119877119873minus1

) 1198903119887119877

+ [1198901198871198770 (2119886119887 + 119890

1198871198770)

minus 119888119873119877119873minus1

1198901198871198770] 119890119887119877

+ [119888119877119873minus1

(1198901198871198770

(119873 + 119887119877) minus 119873 + 119887119877) + 119886119887 + 21198901198871198770]

sdot 1198902119887119877

= 0

(46)


Type 1

120582 =119877 (119899 minus 1) 119890

120585119877

119899 + 120585119877

120578 = minus(1 + 120585119877)

119877119899minus1 (119899 + 120585119877)

(47)

Type 2

120572 = minus1

2119877119899minus1

120573 =119877119899minus1

2

(48)

Type 3

1198982= 119877(

119899 minus 1

1198882

)

minus1119899

1198881 = 119899(119899 minus 1

1198882

)

1119899minus1

(49)

Type 4

119888 =

[119890119887119877

+ 1198901198871198770]2

(119887119877 + 1 minus 119890119887119877)

119877119873minus1 [119873 (119890119887119877 + 1198901198871198770) + 119887119877] (119890119887119877 minus 1)

119886 = minus

119877 (119873 minus 1) 1198902119887119877

+ [119887119877 (1 + 1198901198871198770) + (119873 minus 1) (119890

1198871198770 minus 1)] 119890119887119877 119890119887119877

(119890119887119877 minus 1)2[119887119877 + 119873 (119890119887119877 + 1198901198871198770)]

(50)




1 2 30

r

minus2

minus1

0

1

2

3

4120595(r)


lim119903rarr0

119877120572120573120574120575119877120572120573120574120575

997888rarr infin

lim119903rarrinfin

119877120572120573120574120575119877120572120573120574120575

997888rarr8119889

(119889 minus 1) (119889 minus 2)2Λ2

(51)






119879+ =

119903119889minus2

+[(119889 minus 3) 119892

2(119864) minus 2Λ119903

2

+ (119889 minus 2)] minus 2

(119889minus4)4[119892119891 (119864) 119892 (119864)]

1198892

4120587119891 (119864) 119892 (119864) 119903119889minus1+

(52)



119872 =(119889 minus 2)119898

16120587119891 (119864) 119892 (119864)119889minus3

=

[(119889 minus 2) 1198922(119864) minus 2Λ119903

2

+ (119889 minus 1)] 119903

119889minus2

++ 2(119889minus4)4

(119889 minus 2) [119892119891 (119864) 119892 (119864)]1198892

16120587119891 (119864) 119892 (119864)119889minus1

119903+

(53)

Hence we can write

119889119872 =

119903119889minus3

+[((119889 minus 2) (119889 minus 3) 119892

2(119864) minus 2Λ119903

2

+) 119903119889minus2


(119889 minus 2) (119892119891 (119864) 119892 (119864))1198892

]

16120587119891 (119864) 119892119889minus1

(119864) 119903119889minus1+

119889119903+(54)


119878 = int1

119879119889119872 =

1

4(

119903+

119892 (119864))

119889minus2

(55)




minus04

minus02

02

04

FRR

0 1 2minus1

120585

(a)

b

minus4

minus2

0

2

4

FRR

2515 2

(b)

Figure 12 Type 1 (a) 119865119877119877


versus 119887 for119873 = 4 1198770= 4



2119865(119877)119889119877



119865119877119877

=

minus(119899 minus 1) [120585119877 (119899 + 120585119877) + 119899]

119877 (119899 + 120585119877) Type 1


119877 Type 2

119899 minus 1

119877 Type 3

1198872119877 [(119899 minus 1) 119890

2119887119877+ (1198901198871198770 (119887R + 119899 minus 1) + 119887119877 minus 119899 + 1) 119890

119887119877minus (119899 minus 1) 119890

1198871198770]

(119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770]

minus

(119887119877 + 1 minus 119890119887119877) 119899 (1 minus 119899) 119887

3119887119877+ 12059111198902119887119877

minus 1205912119890119887(119877+1198770) + 119899 (119899 minus 1) 119890

21198871198770

(119890119887119877 + 1198901198871198770) (119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770] 119877

Type 4

(56)

where


+ 119887119877 (119887119877 minus 2119899)

1205912 = [119899 (119899 minus 1) + 119887119877 (119887119877 + 2119899)] 1198901198871198770 + 2119899 (1 minus 119899)

+ 119887119877 (119887119877 + 2119899)

(57)



6 Conclusions






Competing Interests


Acknowledgments


References


















































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




1

05

0

minus05

minus1

01 02 03 040001 0002 0003 0004 0005 0006 0007 0008

minus02

minus01

0

01

02




1

05

02 04 06 08 10

minus05

minus1

(a)

05 1 15

1

05

0

minus05

minus1

(b)


+for 119902 = 1 Λ = minus1 119892(119864) = 119891(119864) = 07 and 119889 = 5 (a) 119904 = 09


119892 (119864)2[11990312059510158401015840(119903) + (119889 minus 2) 120595

1015840(119903)] 119865119877 = minus119903119865 (119877)


2119865 (119877)

(37)






21198772

0)minus119899



01 02 03

minus4

minus2

0

2

4

115 120 125 130

minus4

minus2

0

2

4




minus02

minus04

04

02

0

05 1 15 2

(a)

minus01

minus03

minus02

minus04

04

05

05 1 15 2 25 3

02

03

0

01

(b)


+for 119902 = 1 Λ = minus1 and 119904 = 14 (a) 119889 = 5 119892(119864) = 119891(119864) = 07




120578(120593(119877))12)[1minus120593(119877)]119877minus119890

2120578(120593(119877))119869(120593(119877))



11989101198724minus2119899

119877119899+ 1198912119877

2 [150] and 119865(119877) = (11987721205812)(1 minus 1198771198770) [151]


Type 1

119865 (119877) = 119877 minus 120582119890minus120585119877

+ 120578119877119899 (38)

Type 2

119865 (119877) = 119877 + 120572119877119899minus 1205731198772minus119899 (39)


8

6

4

2

0

minus2

minus4

04 06 08 10 12 14 16

(a)

4

2

0

minus2

minus4

04 06 08 10 12 14

(b)

04 06 08 10 12 14 16

3

2

1

0

minus1

minus2

minus3

(c)


+for 119902 = 1 Λ = minus1 and 119889 = 5 (a) 119892(119864) = 119891(119864) = 07 119891

119877= 0


119877= 1 (dashed line) (b) 119892(119864) = 119891(119864) = 07 119891



and 119891119877= 16 (dashed line) (c) 119891


(dashed line)

Type 3

119865 (119877) = 119877 minus 1198982

1198881 (1198771198982)119899

1198882 (1198771198982)119899+ 1

(40)

Type 4

119865 (119877) = 119877 minus 119886 [119890minus119887119877

minus 1] + 119888119877119873 119890119887119877

minus 1

119890119887119877 + 1198901198871198770(41)






minus1

minus05

05

045 050 055

1

0

(a)

070 072 076074 078

minus1

minus05

05

1

0

(b)

minus1

minus05

05

15 25

1

1 2 3

0

(c)


+for 119902 = 1 Λ = minus1 119892(119864) = 119891(119864) = 07 and 119891

119877= 15 (a) 119889 = 6










120595 (119903) = 1 minus2Λ1199032


119898

119903119889minus3

+2(119889minus4)4

[119902119891 (119864) 119892 (119864)]1198892

119892 (119864)2119903119889minus2

(42)


Type 1

119877[(119899 minus119889

2) 120578119877119899minus1

minus119889 minus 2

2] 119890120585119877

+ 120582(119889

2+ 120585119877) = 0

(1 + 119899120578119877119899minus1

) 119890120585119877

+ 120582120585 = 0

(43)

Type 2

[120572(119899 minus119889

2)119877119899minus

119889 minus 2

2119877]119877119899+ (119899 +

119889 minus 4

2)1205731198772

= 0

(120572119899119877119899+ 119877)119877

119899+ (119899 minus 2) 120573119877

2= 0

(44)

Type 3

1198882 [119889

119889 minus 211989821198881 minus 1198882119877](

119877

1198982)

2119899

minus [211989821198881

119889 minus 2(119899 minus

119889

2) + 21198882119877](

119877

1198982)

119899

minus 119877 = 0

[1198882

2(

119877

1198982)

2119899

+ 1]119877 + [21198882119877 minus 11989821198991198881] (

119877

1198982)

119899

= 0

(45)

Type 4

[119888 (119873 minus119889

2)119877119873minus

119889 minus 2

2119877 minus

119889

2119886] 1198903119887119877

+ 119886(119887119877 +119889

2)

sdot 11989021198871198770

+ 119888 [(119887119877 + 119873 minus119889

2) 1198901198871198770 + 119887119877 minus 119873 +

119889

2]119877119873

minus [(119889 minus 2) 119877 + 119886119889] 1198901198871198770 + 119886(119887119877 +

119889

2) 1198902119887119877

+ [119886 (2119887119877 + 119889) minus 119888 (119873 minus119889

2)119877119873

minus (119886119889

2minus

(119889 minus 2) 119877

2) 1198901198871198770] 119890119887(119877+1198770) = 0

(1 + 119888119873119877119873minus1

) 1198903119887119877

+ [1198901198871198770 (2119886119887 + 119890

1198871198770)

minus 119888119873119877119873minus1

1198901198871198770] 119890119887119877

+ [119888119877119873minus1

(1198901198871198770

(119873 + 119887119877) minus 119873 + 119887119877) + 119886119887 + 21198901198871198770]

sdot 1198902119887119877

= 0

(46)


Type 1

120582 =119877 (119899 minus 1) 119890

120585119877

119899 + 120585119877

120578 = minus(1 + 120585119877)

119877119899minus1 (119899 + 120585119877)

(47)

Type 2

120572 = minus1

2119877119899minus1

120573 =119877119899minus1

2

(48)

Type 3

1198982= 119877(

119899 minus 1

1198882

)

minus1119899

1198881 = 119899(119899 minus 1

1198882

)

1119899minus1

(49)

Type 4

119888 =

[119890119887119877

+ 1198901198871198770]2

(119887119877 + 1 minus 119890119887119877)

119877119873minus1 [119873 (119890119887119877 + 1198901198871198770) + 119887119877] (119890119887119877 minus 1)

119886 = minus

119877 (119873 minus 1) 1198902119887119877

+ [119887119877 (1 + 1198901198871198770) + (119873 minus 1) (119890

1198871198770 minus 1)] 119890119887119877 119890119887119877

(119890119887119877 minus 1)2[119887119877 + 119873 (119890119887119877 + 1198901198871198770)]

(50)




1 2 30

r

minus2

minus1

0

1

2

3

4120595(r)


lim119903rarr0

119877120572120573120574120575119877120572120573120574120575

997888rarr infin

lim119903rarrinfin

119877120572120573120574120575119877120572120573120574120575

997888rarr8119889

(119889 minus 1) (119889 minus 2)2Λ2

(51)






119879+ =

119903119889minus2

+[(119889 minus 3) 119892

2(119864) minus 2Λ119903

2

+ (119889 minus 2)] minus 2

(119889minus4)4[119892119891 (119864) 119892 (119864)]

1198892

4120587119891 (119864) 119892 (119864) 119903119889minus1+

(52)



119872 =(119889 minus 2)119898

16120587119891 (119864) 119892 (119864)119889minus3

=

[(119889 minus 2) 1198922(119864) minus 2Λ119903

2

+ (119889 minus 1)] 119903

119889minus2

++ 2(119889minus4)4

(119889 minus 2) [119892119891 (119864) 119892 (119864)]1198892

16120587119891 (119864) 119892 (119864)119889minus1

119903+

(53)

Hence we can write

119889119872 =

119903119889minus3

+[((119889 minus 2) (119889 minus 3) 119892

2(119864) minus 2Λ119903

2

+) 119903119889minus2


(119889 minus 2) (119892119891 (119864) 119892 (119864))1198892

]

16120587119891 (119864) 119892119889minus1

(119864) 119903119889minus1+

119889119903+(54)


119878 = int1

119879119889119872 =

1

4(

119903+

119892 (119864))

119889minus2

(55)




minus04

minus02

02

04

FRR

0 1 2minus1

120585

(a)

b

minus4

minus2

0

2

4

FRR

2515 2

(b)

Figure 12 Type 1 (a) 119865119877119877


versus 119887 for119873 = 4 1198770= 4



2119865(119877)119889119877



119865119877119877

=

minus(119899 minus 1) [120585119877 (119899 + 120585119877) + 119899]

119877 (119899 + 120585119877) Type 1


119877 Type 2

119899 minus 1

119877 Type 3

1198872119877 [(119899 minus 1) 119890

2119887119877+ (1198901198871198770 (119887R + 119899 minus 1) + 119887119877 minus 119899 + 1) 119890

119887119877minus (119899 minus 1) 119890

1198871198770]

(119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770]

minus

(119887119877 + 1 minus 119890119887119877) 119899 (1 minus 119899) 119887

3119887119877+ 12059111198902119887119877

minus 1205912119890119887(119877+1198770) + 119899 (119899 minus 1) 119890

21198871198770

(119890119887119877 + 1198901198871198770) (119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770] 119877

Type 4

(56)

where


+ 119887119877 (119887119877 minus 2119899)

1205912 = [119899 (119899 minus 1) + 119887119877 (119887119877 + 2119899)] 1198901198871198770 + 2119899 (1 minus 119899)

+ 119887119877 (119887119877 + 2119899)

(57)



6 Conclusions






Competing Interests


Acknowledgments


References


















































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




01 02 03

minus4

minus2

0

2

4

115 120 125 130

minus4

minus2

0

2

4




minus02

minus04

04

02

0

05 1 15 2

(a)

minus01

minus03

minus02

minus04

04

05

05 1 15 2 25 3

02

03

0

01

(b)


+for 119902 = 1 Λ = minus1 and 119904 = 14 (a) 119889 = 5 119892(119864) = 119891(119864) = 07




120578(120593(119877))12)[1minus120593(119877)]119877minus119890

2120578(120593(119877))119869(120593(119877))



11989101198724minus2119899

119877119899+ 1198912119877

2 [150] and 119865(119877) = (11987721205812)(1 minus 1198771198770) [151]


Type 1

119865 (119877) = 119877 minus 120582119890minus120585119877

+ 120578119877119899 (38)

Type 2

119865 (119877) = 119877 + 120572119877119899minus 1205731198772minus119899 (39)


8

6

4

2

0

minus2

minus4

04 06 08 10 12 14 16

(a)

4

2

0

minus2

minus4

04 06 08 10 12 14

(b)

04 06 08 10 12 14 16

3

2

1

0

minus1

minus2

minus3

(c)


+for 119902 = 1 Λ = minus1 and 119889 = 5 (a) 119892(119864) = 119891(119864) = 07 119891

119877= 0


119877= 1 (dashed line) (b) 119892(119864) = 119891(119864) = 07 119891



and 119891119877= 16 (dashed line) (c) 119891


(dashed line)

Type 3

119865 (119877) = 119877 minus 1198982

1198881 (1198771198982)119899

1198882 (1198771198982)119899+ 1

(40)

Type 4

119865 (119877) = 119877 minus 119886 [119890minus119887119877

minus 1] + 119888119877119873 119890119887119877

minus 1

119890119887119877 + 1198901198871198770(41)






minus1

minus05

05

045 050 055

1

0

(a)

070 072 076074 078

minus1

minus05

05

1

0

(b)

minus1

minus05

05

15 25

1

1 2 3

0

(c)


+for 119902 = 1 Λ = minus1 119892(119864) = 119891(119864) = 07 and 119891

119877= 15 (a) 119889 = 6










120595 (119903) = 1 minus2Λ1199032


119898

119903119889minus3

+2(119889minus4)4

[119902119891 (119864) 119892 (119864)]1198892

119892 (119864)2119903119889minus2

(42)


Type 1

119877[(119899 minus119889

2) 120578119877119899minus1

minus119889 minus 2

2] 119890120585119877

+ 120582(119889

2+ 120585119877) = 0

(1 + 119899120578119877119899minus1

) 119890120585119877

+ 120582120585 = 0

(43)

Type 2

[120572(119899 minus119889

2)119877119899minus

119889 minus 2

2119877]119877119899+ (119899 +

119889 minus 4

2)1205731198772

= 0

(120572119899119877119899+ 119877)119877

119899+ (119899 minus 2) 120573119877

2= 0

(44)

Type 3

1198882 [119889

119889 minus 211989821198881 minus 1198882119877](

119877

1198982)

2119899

minus [211989821198881

119889 minus 2(119899 minus

119889

2) + 21198882119877](

119877

1198982)

119899

minus 119877 = 0

[1198882

2(

119877

1198982)

2119899

+ 1]119877 + [21198882119877 minus 11989821198991198881] (

119877

1198982)

119899

= 0

(45)

Type 4

[119888 (119873 minus119889

2)119877119873minus

119889 minus 2

2119877 minus

119889

2119886] 1198903119887119877

+ 119886(119887119877 +119889

2)

sdot 11989021198871198770

+ 119888 [(119887119877 + 119873 minus119889

2) 1198901198871198770 + 119887119877 minus 119873 +

119889

2]119877119873

minus [(119889 minus 2) 119877 + 119886119889] 1198901198871198770 + 119886(119887119877 +

119889

2) 1198902119887119877

+ [119886 (2119887119877 + 119889) minus 119888 (119873 minus119889

2)119877119873

minus (119886119889

2minus

(119889 minus 2) 119877

2) 1198901198871198770] 119890119887(119877+1198770) = 0

(1 + 119888119873119877119873minus1

) 1198903119887119877

+ [1198901198871198770 (2119886119887 + 119890

1198871198770)

minus 119888119873119877119873minus1

1198901198871198770] 119890119887119877

+ [119888119877119873minus1

(1198901198871198770

(119873 + 119887119877) minus 119873 + 119887119877) + 119886119887 + 21198901198871198770]

sdot 1198902119887119877

= 0

(46)


Type 1

120582 =119877 (119899 minus 1) 119890

120585119877

119899 + 120585119877

120578 = minus(1 + 120585119877)

119877119899minus1 (119899 + 120585119877)

(47)

Type 2

120572 = minus1

2119877119899minus1

120573 =119877119899minus1

2

(48)

Type 3

1198982= 119877(

119899 minus 1

1198882

)

minus1119899

1198881 = 119899(119899 minus 1

1198882

)

1119899minus1

(49)

Type 4

119888 =

[119890119887119877

+ 1198901198871198770]2

(119887119877 + 1 minus 119890119887119877)

119877119873minus1 [119873 (119890119887119877 + 1198901198871198770) + 119887119877] (119890119887119877 minus 1)

119886 = minus

119877 (119873 minus 1) 1198902119887119877

+ [119887119877 (1 + 1198901198871198770) + (119873 minus 1) (119890

1198871198770 minus 1)] 119890119887119877 119890119887119877

(119890119887119877 minus 1)2[119887119877 + 119873 (119890119887119877 + 1198901198871198770)]

(50)




1 2 30

r

minus2

minus1

0

1

2

3

4120595(r)


lim119903rarr0

119877120572120573120574120575119877120572120573120574120575

997888rarr infin

lim119903rarrinfin

119877120572120573120574120575119877120572120573120574120575

997888rarr8119889

(119889 minus 1) (119889 minus 2)2Λ2

(51)






119879+ =

119903119889minus2

+[(119889 minus 3) 119892

2(119864) minus 2Λ119903

2

+ (119889 minus 2)] minus 2

(119889minus4)4[119892119891 (119864) 119892 (119864)]

1198892

4120587119891 (119864) 119892 (119864) 119903119889minus1+

(52)



119872 =(119889 minus 2)119898

16120587119891 (119864) 119892 (119864)119889minus3

=

[(119889 minus 2) 1198922(119864) minus 2Λ119903

2

+ (119889 minus 1)] 119903

119889minus2

++ 2(119889minus4)4

(119889 minus 2) [119892119891 (119864) 119892 (119864)]1198892

16120587119891 (119864) 119892 (119864)119889minus1

119903+

(53)

Hence we can write

119889119872 =

119903119889minus3

+[((119889 minus 2) (119889 minus 3) 119892

2(119864) minus 2Λ119903

2

+) 119903119889minus2


(119889 minus 2) (119892119891 (119864) 119892 (119864))1198892

]

16120587119891 (119864) 119892119889minus1

(119864) 119903119889minus1+

119889119903+(54)


119878 = int1

119879119889119872 =

1

4(

119903+

119892 (119864))

119889minus2

(55)




minus04

minus02

02

04

FRR

0 1 2minus1

120585

(a)

b

minus4

minus2

0

2

4

FRR

2515 2

(b)

Figure 12 Type 1 (a) 119865119877119877


versus 119887 for119873 = 4 1198770= 4



2119865(119877)119889119877



119865119877119877

=

minus(119899 minus 1) [120585119877 (119899 + 120585119877) + 119899]

119877 (119899 + 120585119877) Type 1


119877 Type 2

119899 minus 1

119877 Type 3

1198872119877 [(119899 minus 1) 119890

2119887119877+ (1198901198871198770 (119887R + 119899 minus 1) + 119887119877 minus 119899 + 1) 119890

119887119877minus (119899 minus 1) 119890

1198871198770]

(119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770]

minus

(119887119877 + 1 minus 119890119887119877) 119899 (1 minus 119899) 119887

3119887119877+ 12059111198902119887119877

minus 1205912119890119887(119877+1198770) + 119899 (119899 minus 1) 119890

21198871198770

(119890119887119877 + 1198901198871198770) (119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770] 119877

Type 4

(56)

where


+ 119887119877 (119887119877 minus 2119899)

1205912 = [119899 (119899 minus 1) + 119887119877 (119887119877 + 2119899)] 1198901198871198770 + 2119899 (1 minus 119899)

+ 119887119877 (119887119877 + 2119899)

(57)



6 Conclusions






Competing Interests


Acknowledgments


References


















































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




8

6

4

2

0

minus2

minus4

04 06 08 10 12 14 16

(a)

4

2

0

minus2

minus4

04 06 08 10 12 14

(b)

04 06 08 10 12 14 16

3

2

1

0

minus1

minus2

minus3

(c)


+for 119902 = 1 Λ = minus1 and 119889 = 5 (a) 119892(119864) = 119891(119864) = 07 119891

119877= 0


119877= 1 (dashed line) (b) 119892(119864) = 119891(119864) = 07 119891



and 119891119877= 16 (dashed line) (c) 119891


(dashed line)

Type 3

119865 (119877) = 119877 minus 1198982

1198881 (1198771198982)119899

1198882 (1198771198982)119899+ 1

(40)

Type 4

119865 (119877) = 119877 minus 119886 [119890minus119887119877

minus 1] + 119888119877119873 119890119887119877

minus 1

119890119887119877 + 1198901198871198770(41)






minus1

minus05

05

045 050 055

1

0

(a)

070 072 076074 078

minus1

minus05

05

1

0

(b)

minus1

minus05

05

15 25

1

1 2 3

0

(c)


+for 119902 = 1 Λ = minus1 119892(119864) = 119891(119864) = 07 and 119891

119877= 15 (a) 119889 = 6










120595 (119903) = 1 minus2Λ1199032


119898

119903119889minus3

+2(119889minus4)4

[119902119891 (119864) 119892 (119864)]1198892

119892 (119864)2119903119889minus2

(42)


Type 1

119877[(119899 minus119889

2) 120578119877119899minus1

minus119889 minus 2

2] 119890120585119877

+ 120582(119889

2+ 120585119877) = 0

(1 + 119899120578119877119899minus1

) 119890120585119877

+ 120582120585 = 0

(43)

Type 2

[120572(119899 minus119889

2)119877119899minus

119889 minus 2

2119877]119877119899+ (119899 +

119889 minus 4

2)1205731198772

= 0

(120572119899119877119899+ 119877)119877

119899+ (119899 minus 2) 120573119877

2= 0

(44)

Type 3

1198882 [119889

119889 minus 211989821198881 minus 1198882119877](

119877

1198982)

2119899

minus [211989821198881

119889 minus 2(119899 minus

119889

2) + 21198882119877](

119877

1198982)

119899

minus 119877 = 0

[1198882

2(

119877

1198982)

2119899

+ 1]119877 + [21198882119877 minus 11989821198991198881] (

119877

1198982)

119899

= 0

(45)

Type 4

[119888 (119873 minus119889

2)119877119873minus

119889 minus 2

2119877 minus

119889

2119886] 1198903119887119877

+ 119886(119887119877 +119889

2)

sdot 11989021198871198770

+ 119888 [(119887119877 + 119873 minus119889

2) 1198901198871198770 + 119887119877 minus 119873 +

119889

2]119877119873

minus [(119889 minus 2) 119877 + 119886119889] 1198901198871198770 + 119886(119887119877 +

119889

2) 1198902119887119877

+ [119886 (2119887119877 + 119889) minus 119888 (119873 minus119889

2)119877119873

minus (119886119889

2minus

(119889 minus 2) 119877

2) 1198901198871198770] 119890119887(119877+1198770) = 0

(1 + 119888119873119877119873minus1

) 1198903119887119877

+ [1198901198871198770 (2119886119887 + 119890

1198871198770)

minus 119888119873119877119873minus1

1198901198871198770] 119890119887119877

+ [119888119877119873minus1

(1198901198871198770

(119873 + 119887119877) minus 119873 + 119887119877) + 119886119887 + 21198901198871198770]

sdot 1198902119887119877

= 0

(46)


Type 1

120582 =119877 (119899 minus 1) 119890

120585119877

119899 + 120585119877

120578 = minus(1 + 120585119877)

119877119899minus1 (119899 + 120585119877)

(47)

Type 2

120572 = minus1

2119877119899minus1

120573 =119877119899minus1

2

(48)

Type 3

1198982= 119877(

119899 minus 1

1198882

)

minus1119899

1198881 = 119899(119899 minus 1

1198882

)

1119899minus1

(49)

Type 4

119888 =

[119890119887119877

+ 1198901198871198770]2

(119887119877 + 1 minus 119890119887119877)

119877119873minus1 [119873 (119890119887119877 + 1198901198871198770) + 119887119877] (119890119887119877 minus 1)

119886 = minus

119877 (119873 minus 1) 1198902119887119877

+ [119887119877 (1 + 1198901198871198770) + (119873 minus 1) (119890

1198871198770 minus 1)] 119890119887119877 119890119887119877

(119890119887119877 minus 1)2[119887119877 + 119873 (119890119887119877 + 1198901198871198770)]

(50)




1 2 30

r

minus2

minus1

0

1

2

3

4120595(r)


lim119903rarr0

119877120572120573120574120575119877120572120573120574120575

997888rarr infin

lim119903rarrinfin

119877120572120573120574120575119877120572120573120574120575

997888rarr8119889

(119889 minus 1) (119889 minus 2)2Λ2

(51)






119879+ =

119903119889minus2

+[(119889 minus 3) 119892

2(119864) minus 2Λ119903

2

+ (119889 minus 2)] minus 2

(119889minus4)4[119892119891 (119864) 119892 (119864)]

1198892

4120587119891 (119864) 119892 (119864) 119903119889minus1+

(52)



119872 =(119889 minus 2)119898

16120587119891 (119864) 119892 (119864)119889minus3

=

[(119889 minus 2) 1198922(119864) minus 2Λ119903

2

+ (119889 minus 1)] 119903

119889minus2

++ 2(119889minus4)4

(119889 minus 2) [119892119891 (119864) 119892 (119864)]1198892

16120587119891 (119864) 119892 (119864)119889minus1

119903+

(53)

Hence we can write

119889119872 =

119903119889minus3

+[((119889 minus 2) (119889 minus 3) 119892

2(119864) minus 2Λ119903

2

+) 119903119889minus2


(119889 minus 2) (119892119891 (119864) 119892 (119864))1198892

]

16120587119891 (119864) 119892119889minus1

(119864) 119903119889minus1+

119889119903+(54)


119878 = int1

119879119889119872 =

1

4(

119903+

119892 (119864))

119889minus2

(55)




minus04

minus02

02

04

FRR

0 1 2minus1

120585

(a)

b

minus4

minus2

0

2

4

FRR

2515 2

(b)

Figure 12 Type 1 (a) 119865119877119877


versus 119887 for119873 = 4 1198770= 4



2119865(119877)119889119877



119865119877119877

=

minus(119899 minus 1) [120585119877 (119899 + 120585119877) + 119899]

119877 (119899 + 120585119877) Type 1


119877 Type 2

119899 minus 1

119877 Type 3

1198872119877 [(119899 minus 1) 119890

2119887119877+ (1198901198871198770 (119887R + 119899 minus 1) + 119887119877 minus 119899 + 1) 119890

119887119877minus (119899 minus 1) 119890

1198871198770]

(119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770]

minus

(119887119877 + 1 minus 119890119887119877) 119899 (1 minus 119899) 119887

3119887119877+ 12059111198902119887119877

minus 1205912119890119887(119877+1198770) + 119899 (119899 minus 1) 119890

21198871198770

(119890119887119877 + 1198901198871198770) (119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770] 119877

Type 4

(56)

where


+ 119887119877 (119887119877 minus 2119899)

1205912 = [119899 (119899 minus 1) + 119887119877 (119887119877 + 2119899)] 1198901198871198770 + 2119899 (1 minus 119899)

+ 119887119877 (119887119877 + 2119899)

(57)



6 Conclusions






Competing Interests


Acknowledgments


References


















































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




minus1

minus05

05

045 050 055

1

0

(a)

070 072 076074 078

minus1

minus05

05

1

0

(b)

minus1

minus05

05

15 25

1

1 2 3

0

(c)


+for 119902 = 1 Λ = minus1 119892(119864) = 119891(119864) = 07 and 119891

119877= 15 (a) 119889 = 6










120595 (119903) = 1 minus2Λ1199032


119898

119903119889minus3

+2(119889minus4)4

[119902119891 (119864) 119892 (119864)]1198892

119892 (119864)2119903119889minus2

(42)


Type 1

119877[(119899 minus119889

2) 120578119877119899minus1

minus119889 minus 2

2] 119890120585119877

+ 120582(119889

2+ 120585119877) = 0

(1 + 119899120578119877119899minus1

) 119890120585119877

+ 120582120585 = 0

(43)

Type 2

[120572(119899 minus119889

2)119877119899minus

119889 minus 2

2119877]119877119899+ (119899 +

119889 minus 4

2)1205731198772

= 0

(120572119899119877119899+ 119877)119877

119899+ (119899 minus 2) 120573119877

2= 0

(44)

Type 3

1198882 [119889

119889 minus 211989821198881 minus 1198882119877](

119877

1198982)

2119899

minus [211989821198881

119889 minus 2(119899 minus

119889

2) + 21198882119877](

119877

1198982)

119899

minus 119877 = 0

[1198882

2(

119877

1198982)

2119899

+ 1]119877 + [21198882119877 minus 11989821198991198881] (

119877

1198982)

119899

= 0

(45)

Type 4

[119888 (119873 minus119889

2)119877119873minus

119889 minus 2

2119877 minus

119889

2119886] 1198903119887119877

+ 119886(119887119877 +119889

2)

sdot 11989021198871198770

+ 119888 [(119887119877 + 119873 minus119889

2) 1198901198871198770 + 119887119877 minus 119873 +

119889

2]119877119873

minus [(119889 minus 2) 119877 + 119886119889] 1198901198871198770 + 119886(119887119877 +

119889

2) 1198902119887119877

+ [119886 (2119887119877 + 119889) minus 119888 (119873 minus119889

2)119877119873

minus (119886119889

2minus

(119889 minus 2) 119877

2) 1198901198871198770] 119890119887(119877+1198770) = 0

(1 + 119888119873119877119873minus1

) 1198903119887119877

+ [1198901198871198770 (2119886119887 + 119890

1198871198770)

minus 119888119873119877119873minus1

1198901198871198770] 119890119887119877

+ [119888119877119873minus1

(1198901198871198770

(119873 + 119887119877) minus 119873 + 119887119877) + 119886119887 + 21198901198871198770]

sdot 1198902119887119877

= 0

(46)


Type 1

120582 =119877 (119899 minus 1) 119890

120585119877

119899 + 120585119877

120578 = minus(1 + 120585119877)

119877119899minus1 (119899 + 120585119877)

(47)

Type 2

120572 = minus1

2119877119899minus1

120573 =119877119899minus1

2

(48)

Type 3

1198982= 119877(

119899 minus 1

1198882

)

minus1119899

1198881 = 119899(119899 minus 1

1198882

)

1119899minus1

(49)

Type 4

119888 =

[119890119887119877

+ 1198901198871198770]2

(119887119877 + 1 minus 119890119887119877)

119877119873minus1 [119873 (119890119887119877 + 1198901198871198770) + 119887119877] (119890119887119877 minus 1)

119886 = minus

119877 (119873 minus 1) 1198902119887119877

+ [119887119877 (1 + 1198901198871198770) + (119873 minus 1) (119890

1198871198770 minus 1)] 119890119887119877 119890119887119877

(119890119887119877 minus 1)2[119887119877 + 119873 (119890119887119877 + 1198901198871198770)]

(50)




1 2 30

r

minus2

minus1

0

1

2

3

4120595(r)


lim119903rarr0

119877120572120573120574120575119877120572120573120574120575

997888rarr infin

lim119903rarrinfin

119877120572120573120574120575119877120572120573120574120575

997888rarr8119889

(119889 minus 1) (119889 minus 2)2Λ2

(51)






119879+ =

119903119889minus2

+[(119889 minus 3) 119892

2(119864) minus 2Λ119903

2

+ (119889 minus 2)] minus 2

(119889minus4)4[119892119891 (119864) 119892 (119864)]

1198892

4120587119891 (119864) 119892 (119864) 119903119889minus1+

(52)



119872 =(119889 minus 2)119898

16120587119891 (119864) 119892 (119864)119889minus3

=

[(119889 minus 2) 1198922(119864) minus 2Λ119903

2

+ (119889 minus 1)] 119903

119889minus2

++ 2(119889minus4)4

(119889 minus 2) [119892119891 (119864) 119892 (119864)]1198892

16120587119891 (119864) 119892 (119864)119889minus1

119903+

(53)

Hence we can write

119889119872 =

119903119889minus3

+[((119889 minus 2) (119889 minus 3) 119892

2(119864) minus 2Λ119903

2

+) 119903119889minus2


(119889 minus 2) (119892119891 (119864) 119892 (119864))1198892

]

16120587119891 (119864) 119892119889minus1

(119864) 119903119889minus1+

119889119903+(54)


119878 = int1

119879119889119872 =

1

4(

119903+

119892 (119864))

119889minus2

(55)




minus04

minus02

02

04

FRR

0 1 2minus1

120585

(a)

b

minus4

minus2

0

2

4

FRR

2515 2

(b)

Figure 12 Type 1 (a) 119865119877119877


versus 119887 for119873 = 4 1198770= 4



2119865(119877)119889119877



119865119877119877

=

minus(119899 minus 1) [120585119877 (119899 + 120585119877) + 119899]

119877 (119899 + 120585119877) Type 1


119877 Type 2

119899 minus 1

119877 Type 3

1198872119877 [(119899 minus 1) 119890

2119887119877+ (1198901198871198770 (119887R + 119899 minus 1) + 119887119877 minus 119899 + 1) 119890

119887119877minus (119899 minus 1) 119890

1198871198770]

(119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770]

minus

(119887119877 + 1 minus 119890119887119877) 119899 (1 minus 119899) 119887

3119887119877+ 12059111198902119887119877

minus 1205912119890119887(119877+1198770) + 119899 (119899 minus 1) 119890

21198871198770

(119890119887119877 + 1198901198871198770) (119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770] 119877

Type 4

(56)

where


+ 119887119877 (119887119877 minus 2119899)

1205912 = [119899 (119899 minus 1) + 119887119877 (119887119877 + 2119899)] 1198901198871198770 + 2119899 (1 minus 119899)

+ 119887119877 (119887119877 + 2119899)

(57)



6 Conclusions






Competing Interests


Acknowledgments


References


















































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




120595 (119903) = 1 minus2Λ1199032


119898

119903119889minus3

+2(119889minus4)4

[119902119891 (119864) 119892 (119864)]1198892

119892 (119864)2119903119889minus2

(42)


Type 1

119877[(119899 minus119889

2) 120578119877119899minus1

minus119889 minus 2

2] 119890120585119877

+ 120582(119889

2+ 120585119877) = 0

(1 + 119899120578119877119899minus1

) 119890120585119877

+ 120582120585 = 0

(43)

Type 2

[120572(119899 minus119889

2)119877119899minus

119889 minus 2

2119877]119877119899+ (119899 +

119889 minus 4

2)1205731198772

= 0

(120572119899119877119899+ 119877)119877

119899+ (119899 minus 2) 120573119877

2= 0

(44)

Type 3

1198882 [119889

119889 minus 211989821198881 minus 1198882119877](

119877

1198982)

2119899

minus [211989821198881

119889 minus 2(119899 minus

119889

2) + 21198882119877](

119877

1198982)

119899

minus 119877 = 0

[1198882

2(

119877

1198982)

2119899

+ 1]119877 + [21198882119877 minus 11989821198991198881] (

119877

1198982)

119899

= 0

(45)

Type 4

[119888 (119873 minus119889

2)119877119873minus

119889 minus 2

2119877 minus

119889

2119886] 1198903119887119877

+ 119886(119887119877 +119889

2)

sdot 11989021198871198770

+ 119888 [(119887119877 + 119873 minus119889

2) 1198901198871198770 + 119887119877 minus 119873 +

119889

2]119877119873

minus [(119889 minus 2) 119877 + 119886119889] 1198901198871198770 + 119886(119887119877 +

119889

2) 1198902119887119877

+ [119886 (2119887119877 + 119889) minus 119888 (119873 minus119889

2)119877119873

minus (119886119889

2minus

(119889 minus 2) 119877

2) 1198901198871198770] 119890119887(119877+1198770) = 0

(1 + 119888119873119877119873minus1

) 1198903119887119877

+ [1198901198871198770 (2119886119887 + 119890

1198871198770)

minus 119888119873119877119873minus1

1198901198871198770] 119890119887119877

+ [119888119877119873minus1

(1198901198871198770

(119873 + 119887119877) minus 119873 + 119887119877) + 119886119887 + 21198901198871198770]

sdot 1198902119887119877

= 0

(46)


Type 1

120582 =119877 (119899 minus 1) 119890

120585119877

119899 + 120585119877

120578 = minus(1 + 120585119877)

119877119899minus1 (119899 + 120585119877)

(47)

Type 2

120572 = minus1

2119877119899minus1

120573 =119877119899minus1

2

(48)

Type 3

1198982= 119877(

119899 minus 1

1198882

)

minus1119899

1198881 = 119899(119899 minus 1

1198882

)

1119899minus1

(49)

Type 4

119888 =

[119890119887119877

+ 1198901198871198770]2

(119887119877 + 1 minus 119890119887119877)

119877119873minus1 [119873 (119890119887119877 + 1198901198871198770) + 119887119877] (119890119887119877 minus 1)

119886 = minus

119877 (119873 minus 1) 1198902119887119877

+ [119887119877 (1 + 1198901198871198770) + (119873 minus 1) (119890

1198871198770 minus 1)] 119890119887119877 119890119887119877

(119890119887119877 minus 1)2[119887119877 + 119873 (119890119887119877 + 1198901198871198770)]

(50)




1 2 30

r

minus2

minus1

0

1

2

3

4120595(r)


lim119903rarr0

119877120572120573120574120575119877120572120573120574120575

997888rarr infin

lim119903rarrinfin

119877120572120573120574120575119877120572120573120574120575

997888rarr8119889

(119889 minus 1) (119889 minus 2)2Λ2

(51)






119879+ =

119903119889minus2

+[(119889 minus 3) 119892

2(119864) minus 2Λ119903

2

+ (119889 minus 2)] minus 2

(119889minus4)4[119892119891 (119864) 119892 (119864)]

1198892

4120587119891 (119864) 119892 (119864) 119903119889minus1+

(52)



119872 =(119889 minus 2)119898

16120587119891 (119864) 119892 (119864)119889minus3

=

[(119889 minus 2) 1198922(119864) minus 2Λ119903

2

+ (119889 minus 1)] 119903

119889minus2

++ 2(119889minus4)4

(119889 minus 2) [119892119891 (119864) 119892 (119864)]1198892

16120587119891 (119864) 119892 (119864)119889minus1

119903+

(53)

Hence we can write

119889119872 =

119903119889minus3

+[((119889 minus 2) (119889 minus 3) 119892

2(119864) minus 2Λ119903

2

+) 119903119889minus2


(119889 minus 2) (119892119891 (119864) 119892 (119864))1198892

]

16120587119891 (119864) 119892119889minus1

(119864) 119903119889minus1+

119889119903+(54)


119878 = int1

119879119889119872 =

1

4(

119903+

119892 (119864))

119889minus2

(55)




minus04

minus02

02

04

FRR

0 1 2minus1

120585

(a)

b

minus4

minus2

0

2

4

FRR

2515 2

(b)

Figure 12 Type 1 (a) 119865119877119877


versus 119887 for119873 = 4 1198770= 4



2119865(119877)119889119877



119865119877119877

=

minus(119899 minus 1) [120585119877 (119899 + 120585119877) + 119899]

119877 (119899 + 120585119877) Type 1


119877 Type 2

119899 minus 1

119877 Type 3

1198872119877 [(119899 minus 1) 119890

2119887119877+ (1198901198871198770 (119887R + 119899 minus 1) + 119887119877 minus 119899 + 1) 119890

119887119877minus (119899 minus 1) 119890

1198871198770]

(119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770]

minus

(119887119877 + 1 minus 119890119887119877) 119899 (1 minus 119899) 119887

3119887119877+ 12059111198902119887119877

minus 1205912119890119887(119877+1198770) + 119899 (119899 minus 1) 119890

21198871198770

(119890119887119877 + 1198901198871198770) (119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770] 119877

Type 4

(56)

where


+ 119887119877 (119887119877 minus 2119899)

1205912 = [119899 (119899 minus 1) + 119887119877 (119887119877 + 2119899)] 1198901198871198770 + 2119899 (1 minus 119899)

+ 119887119877 (119887119877 + 2119899)

(57)



6 Conclusions






Competing Interests


Acknowledgments


References


















































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




1 2 30

r

minus2

minus1

0

1

2

3

4120595(r)


lim119903rarr0

119877120572120573120574120575119877120572120573120574120575

997888rarr infin

lim119903rarrinfin

119877120572120573120574120575119877120572120573120574120575

997888rarr8119889

(119889 minus 1) (119889 minus 2)2Λ2

(51)






119879+ =

119903119889minus2

+[(119889 minus 3) 119892

2(119864) minus 2Λ119903

2

+ (119889 minus 2)] minus 2

(119889minus4)4[119892119891 (119864) 119892 (119864)]

1198892

4120587119891 (119864) 119892 (119864) 119903119889minus1+

(52)



119872 =(119889 minus 2)119898

16120587119891 (119864) 119892 (119864)119889minus3

=

[(119889 minus 2) 1198922(119864) minus 2Λ119903

2

+ (119889 minus 1)] 119903

119889minus2

++ 2(119889minus4)4

(119889 minus 2) [119892119891 (119864) 119892 (119864)]1198892

16120587119891 (119864) 119892 (119864)119889minus1

119903+

(53)

Hence we can write

119889119872 =

119903119889minus3

+[((119889 minus 2) (119889 minus 3) 119892

2(119864) minus 2Λ119903

2

+) 119903119889minus2


(119889 minus 2) (119892119891 (119864) 119892 (119864))1198892

]

16120587119891 (119864) 119892119889minus1

(119864) 119903119889minus1+

119889119903+(54)


119878 = int1

119879119889119872 =

1

4(

119903+

119892 (119864))

119889minus2

(55)




minus04

minus02

02

04

FRR

0 1 2minus1

120585

(a)

b

minus4

minus2

0

2

4

FRR

2515 2

(b)

Figure 12 Type 1 (a) 119865119877119877


versus 119887 for119873 = 4 1198770= 4



2119865(119877)119889119877



119865119877119877

=

minus(119899 minus 1) [120585119877 (119899 + 120585119877) + 119899]

119877 (119899 + 120585119877) Type 1


119877 Type 2

119899 minus 1

119877 Type 3

1198872119877 [(119899 minus 1) 119890

2119887119877+ (1198901198871198770 (119887R + 119899 minus 1) + 119887119877 minus 119899 + 1) 119890

119887119877minus (119899 minus 1) 119890

1198871198770]

(119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770]

minus

(119887119877 + 1 minus 119890119887119877) 119899 (1 minus 119899) 119887

3119887119877+ 12059111198902119887119877

minus 1205912119890119887(119877+1198770) + 119899 (119899 minus 1) 119890

21198871198770

(119890119887119877 + 1198901198871198770) (119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770] 119877

Type 4

(56)

where


+ 119887119877 (119887119877 minus 2119899)

1205912 = [119899 (119899 minus 1) + 119887119877 (119887119877 + 2119899)] 1198901198871198770 + 2119899 (1 minus 119899)

+ 119887119877 (119887119877 + 2119899)

(57)



6 Conclusions






Competing Interests


Acknowledgments


References


















































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




minus04

minus02

02

04

FRR

0 1 2minus1

120585

(a)

b

minus4

minus2

0

2

4

FRR

2515 2

(b)

Figure 12 Type 1 (a) 119865119877119877


versus 119887 for119873 = 4 1198770= 4



2119865(119877)119889119877



119865119877119877

=

minus(119899 minus 1) [120585119877 (119899 + 120585119877) + 119899]

119877 (119899 + 120585119877) Type 1


119877 Type 2

119899 minus 1

119877 Type 3

1198872119877 [(119899 minus 1) 119890

2119887119877+ (1198901198871198770 (119887R + 119899 minus 1) + 119887119877 minus 119899 + 1) 119890

119887119877minus (119899 minus 1) 119890

1198871198770]

(119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770]

minus

(119887119877 + 1 minus 119890119887119877) 119899 (1 minus 119899) 119887

3119887119877+ 12059111198902119887119877

minus 1205912119890119887(119877+1198770) + 119899 (119899 minus 1) 119890

21198871198770

(119890119887119877 + 1198901198871198770) (119890119887119877 minus 1)2[119899119890119887119877 + (119899 + 119887119877) 119890

1198871198770] 119877

Type 4

(56)

where


+ 119887119877 (119887119877 minus 2119899)

1205912 = [119899 (119899 minus 1) + 119887119877 (119887119877 + 2119899)] 1198901198871198770 + 2119899 (1 minus 119899)

+ 119887119877 (119887119877 + 2119899)

(57)



6 Conclusions






Competing Interests


Acknowledgments


References


















































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







Competing Interests


Acknowledgments


References


















































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




























































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




















































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics













































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



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