Research ArticleInvariant Inhomogeneous Bianchi Type-ICosmological Models with Electromagnetic FieldsUsing Lie Group Analysis in Lyra Geometry

Ahmad T Ali12

1 Department of Mathematics Faculty of Science King Abdulaziz University PO Box 80203 Jeddah 21589 Saudi Arabia2Mathematics Department Faculty of Science Al-Azhar University Nasr City Cairo 11884 Egypt

Correspondence should be addressed to Ahmad T Ali atali71yahoocom

Received 28 December 2013 Accepted 3 June 2014 Published 19 June 2014

Academic Editor Maria Bruzon

Copyright copy 2014 Ahmad T AliThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Wefind a new class of invariant inhomogeneous Bianchi type-I cosmologicalmodels in electromagnetic fieldwith variablemagneticpermeability For this Lie group analysis method is used to identify the generators that leave the given system of nonlinear partialdifferential equations (NLPDEs) (Einstein field equations) invariant With the help of canonical variables associated with thesegenerators the assigned system of PDEs is reduced to ordinary differential equations (ODEs) whose simple solutions providenontrivial solutions of the original system A new class of exact (invariant-similarity) solutions have been obtained by consideringthe potentials of metric and displacement field as functions of coordinates 119909 and 119905 We have assumed that 119865

12is only nonvanishing

component of electromagnetic field tensor119865119894119895TheMaxwell equations show that 119865

12is the function of 119909 alone whereas themagnetic

permeability 120583 is the function of 119909 and 119905 both The physical behavior of the obtained model is discussed

1 Introduction

The inhomogeneous cosmological models play a significantrole in understanding some essential features of the universesuch as the formation of galaxies during the early stages ofevolution and process of homogenization Therefore it willbe interesting to study inhomogeneous cosmological modelsThe best-known inhomogeneous cosmological model is theLemaitre-Tolman model (or LT model) which deals with thestudy of structure in the universe by means of exact solu-tions of Einsteinrsquos field equations Some other known exactsolutions of inhomogeneous cosmological models are theSzekeresmetric Szafronmetric Stephani metric Kantowski-Sachs metric Barnes metric Kustaanheimo-Qvist metricand Senovilla metric [1]

Einsteinrsquos general theory relativity is based on Rieman-nian geometry If one modifies the Riemannian geometrythen Einsteinrsquos field equations will be changed automati-cally from its original form Modifications of Riemanniangeometry have developed to solve the problems such asunification of gravitation with electromagnetism problems

arising when the gravitational field is coupled to matterfields and singularities of standard cosmology In recentyears there has been considerable interest in alternativetheory of gravitation to explain the above-unsolved problemsLong ago since 1951 Lyra [2] proposed a modification ofRiemannian geometry by introducing a gauge function intothe structureless manifold that bears a close resemblance toWeylrsquos geometry

Using the above modification of Riemannian geometrySen [3 4] and Sen and Dunn [5] proposed a new scalartensor theory of gravitation and constructed it very similar toEinstein field equations Based on Lyrarsquos geometry the fieldequations can be written as [3 4]

119877119894119895minus

1

2

119892119894119895119877 +

3

2

120601119894120601119895minus

3

4

119892119894119895120601119896120601119896= minus120594119879

119894119895 (1)

where 120601119894is the displacement vector and other symbols have

their usual meaning as in Riemannian geometryHalford [6] has argued that the nature of constant

displacement field 120601119894in Lyrarsquos geometry is very similar to

cosmological constant Λ in the normal general relativistic

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 918927 8 pageshttpdxdoiorg1011552014918927

2 Abstract and Applied Analysis

theory Halford also predicted that the present theory willprovide the same effects within observational limits as far asthe classical solar system tests are concerned as well as testsbased on the linearized form of field equations For a reviewon Lyra geometry one can see [7]

Recently Pradhan et al [8ndash14] Casana et al [15]Rahaman et al [16] Bali and Chandnani [17 18] Kumar andSingh [19] Yadav et al [20] Rao et al [21] Zia and Singh [22]and Ali and Rahaman [23] have studied cosmological modelsbased on Lyrarsquos geometry in various contexts

To study the nonlinear physical phenomena [24ndash27] it isimportant to search the exact solutions of nonlinear PDEsOvsiannikov [28] is the pioneer who had observed that theusual Lie infinitesimal invariance approach could as wellbe employed in order to construct symmetry groups [29ndash31] The symmetry groups of a differential equation couldbe defined as the groups of continuous transformations thatlead a given family of equations invariant [32ndash35] and areproved to be important to solve the nonlinear equations of themodels to describe complex physical phenomena in variousfields of science especially in fluid mechanics solid statephysics plasma physics plasma wave and general relativity

In this paper we have obtained exact solutions of Ein-steinrsquosmodified field equations in inhomogeneous space-timeBianchi type-I cosmological model within the frame work ofLyrarsquos geometry in the presence ofmagnetic fieldwith variablemagnetic permeability and time varying displacement vector120573(119909 119905) using the so-called symmetry analysis method Sincethe field equations are highly nonlinear differential equationstherefore symmetry analysis method can be successfullyapplied to nonlinear differential equations The similarity(invariant) solutions help to reduce the independent variablesof the problem and therefore we employ this method inthe investigation of exact solution of the field equationsIn general invariant solutions will transform the system ofnonlinear PDEs into a system of ODEs We attempted tofind a new class of exact (invariant) solutions for the fieldequations based on Lyra geometry

The scheme of the paper is as follows Magnetizedinhomogeneous Bianchi type-I cosmological model withvariable magnetic permeability based on Lyra geometry isintroduced in Section 2 In Section 3 we have performedsymmetry analysis and have obtained isovector fields forEinstein field equations under consideration In Section 4 wefound new class of exact (invariant) solutions for Einsteinfield equations Section 5 is devoted to study of some physicaland geometrical properties of the model

2 The Metric and Field Equations

We consider Bianchi type-Imetric with the convention (1199090=

119905 1199091= 119909 119909

2= 119910 119909

3= 119911) in the form

1198891199042= 1198891199052minus 11986021198891199092minus 11986121198891199102minus 11986221198891199112 (2)

where119860 is a function of 119905 only while 119861 and 119862 are functions of119909 and 119905 Without loss of generality we can put the followingtransformation

119861 = 119860119891 119862 = 119860119892 (3)

where 119891 and 119892 are functions of 119909 and 119905 The volume elementof model (2) is given by

119881 = radicminus119892 = 1198603119891119892 (4)

The four-acceleration vector the rotation the expansionscalar and the shear scalar characterizing the four-velocityvector field 119906119894 satisfying the relation in comoving coordinatesystem

119892119894119895119906119894119906119895= 1 119906

119894= 119906119894= (1 0 0 0) (5)

respectively have the usual definitions as given by Raychaud-huri [36]

119894= 119906119894119895119906119895

120596119894119895

= 119906[119894119895]

+ [119894119906119895]

Θ = 119906119894

119894

1205902=

1

2

120590119894119895120590119894119895

(6)

where

120590119894119895

= 119906(119894119895)

+ (119894119906119895)

minus

1

3

Θ (119892119894119895+ 119906119894119906119895) (7)

In view of metric (2) the four-acceleration vector therotation the expansion scalar and the shear scalar given by(6) can be written in a comoving coordinates system as

119894= 0

120596119894119895

= 0

Θ =

3

119860

+

119891119905

119891

+

119892119905

119892

1205902=

119860

(

2

119860

+

4119891119905

3119891

+

4119892119905

3119892

) +

51198912

119905

91198912

+

119891119905119892119905

9119891119892

+

51198922

119905

91198922

(8)

where the nonvanishing components of the shear tensor 120590119895

119894

are

1205901

1= minus

119891119905

3119891

minus

119892119905

3119891

1205902

2=

2119891119905

3119891

minus

119892119905

3119892

1205903

3=

2119892119905

3119892

minus

119891119905

3119891

1205904

4= minus

2

119860

minus

2119891119905

3119891

minus

2119892119905

3119892

(9)

To study the cosmological model we use the field equationsin Lyra geometry given in (1) in which the displacement fieldvector 120601

119894is given by

120601119894= (120573 (119909 119905) 0 0 0) (10)

Abstract and Applied Analysis 3

119879119894119895is the energy momentum tensor given by

119879119894119895

= (120588 + 119901) 119906119894119906119895minus 119901119892119894119895+ 119864119894119895 (11)

where 119864119894119895is the electromagnetic field given by Lichnerowicz

[37]

119864119894119895

= 120583 [ℎ119897ℎ119897(119906119894119906119895minus

1

2

119892119894119895) + ℎ119894ℎ119895] (12)

Here 120588 and 119901 are the energy density and isotropic pressurerespectively while 120583 is the magnetic permeability and ℎ

119894is

the magnetic flux vector defined by

ℎ119894=

radicminus119892

2120583

120598119894119895119896119897

119865119896119897119906119895 (13)

119865119894119895is the electromagnetic field tensor and 120598

119894119895119896119897is a Levi-Civita

tensor density If we consider the current flow along 119911-axisthen 119865

12is only nonvanishing component of 119865

119894119895 Then the

Maxwell equations

119865119894119895119896

+ 119865119895119896119894

+ 119865119896119894119895

= 0

[

1

120583

119865119894119895]

119895

= 119869119894

(14)

require 11986512

to be function of 119909 alone [38] We assume themagnetic permeability as a function of both 119909 and 119905 Here thesemicolon represents a covariant differentiation

For the line element (2) field equation (1) can be reducedto the following system of NLPDEs

1198641=

119891119909119905

119891

+

119892119909119905

119892

= 0

1198642=

119891119905119905

119891

+

119891119905119892119905

119891119892

+

1

1198602(

119892119909119909

119892

minus

119891119909119892119909

119891119892

) +

3119891119905

119860119891

= 0

(15)

120594120588 +

3

4

1205732=

119892119905119905

2119892

+

3119891119905119892119905

2119891119892

minus

1

1198602(

119891119909119909

2119891

+

119892119909119909

119892

+

3119891119909119892119909

2119891119892

)

+

119860

(

2119891119905

119891

+

119892119905

2119892

+

3

119860

)

120594119901 +

3

4

1205732=

1

21198602(

119891119909119909

119891

+

119891119909119892119909

119891119892

) minus

119891119905119905

119891

minus

119892119905119905

2119892

minus

119891119905119892119905

2119891119892

minus

119860

(

3119891119905

119891

+

119860

) minus

2

119860

1205941198652

12

12058311986041198912

=

1

1198602(

119891119909119909

119891

minus

119891119909119892119909

119891119892

) +

119892119905119905

119892

+

119891119905119892119905

119891119892

+

3119892119905

119860119892

(16)

3 Symmetry Analysis Method

Equations (15)ndash(16) are highly nonlinear PDEs and hence itis so difficult to handle since there exist no standard methodsfor obtaining analytical solutionThe system (15) is nonlinear

PDEs of second order for the two unknowns 119891 and 119892 If wesolve this system then we can get the solution of the fieldequations In order to obtain an exact solution of the systemof nonlinear PDEs (15) we will use the symmetry analysismethod For this we write

119909lowast

119894= 119909119894+ 120598120585119894(119909119895 119906120573) + o (120598

2)

119906lowast

120572= 119906120572+ 120598120578120572(119909119895 119906120573) + o (120598

2)

119894 119895 120572 120573 = 1 2 (17)

as the infinitesimal Lie point transformations We haveassumed that the system (15) is invariant under the trans-formations given in (17) The corresponding infinitesimalgenerator of Lie groups (symmetries) is given by

119883 =

2

sum

119894=1

120585119894

120597

120597119909119894

+

2

sum

120572=1

120578120572

120597

120597119906120572

(18)

where 1199091

= 119909 1199092

= 119905 1199061

= 119891 and 1199062

= 119892 The coefficients1205851 1205852 1205781 and 120578

2are the functions of 119909 119905 119891 and 119892 These

coefficients are the components of infinitesimals symmetriescorresponding to 119909 119905119891 and119892 respectively to be determinedfrom the invariance conditions

Pr(2)119883(119864119898)

10038161003816100381610038161003816119864119898=0

= 0 (19)

where 119864119898

= 0119898 = 1 2 are the system (15) under study andPr(2) is the second prolongation of the symmetries 119883 Sinceour equations (15) are at most of order two therefore weneed second order prolongation of the infinitesimal generatorin (19) It is worth noting that the 2nd order prolongation isgiven by

Pr(2)119883 = 119883 +

2

sum

119894=1

2

sum

120572=1

120578120572119894

120597

120597119906120572119894

+

2

sum

119895=1

2

sum

119894=1

2

sum

120572=1

120578120572119894119895

120597

120597119906120572119894119895

(20)

where

120578120572119894

= 119863119894[

[

120578120572minus

2

sum

119895=1

120585119895119906120572119895

]

]

+

2

sum

119895=1

120585119895119906120572119894119895

120578120572119894119895

= 119863119894119895[120578120572minus

2

sum

119896=1

120585119896119906120572119896

] +

2

sum

119896=1

120585119896119906120572119894119895119896

(21)

The operator 119863119894(119863119894119895) is called the total derivative (Hach

operator) and takes the following form

119863119894=

120597

120597119909119894

+

2

sum

120572=1

119906120572119894

120597

120597119906120572

+

2

sum

119895=1

2

sum

120572=1

119906120572119894119895

120597

120597119906120572119895

(22)

where 119863119894119895

= 119863119894(119863119895) = 119863

119895(119863119894) = 119863

119895119894and 119906

120572119894= 120597119906120572120597119909119894

Expanding the system (19) with the aid of Mathematicaprogram along with the original system (15) to eliminate 119891

119909119905

and 119892119909119909

while we set the coefficients involving 119891119909 119891119905 119891119909119909 119891119905119905

119892119909119892119905119892119909119905 and119892

119905119905and various products equal zero these gives

rise to the essential set of overdetermined equations Solvingthe set of these determining equations the components ofsymmetries take the following form

1205851= 1198881119909 + 1198882 120585

2= 1198883119905 + 1198884

1205781= 0 120578

2= 1198885119892

(23)

4 Abstract and Applied Analysis

where 119888119894 119894 = 1 2 5 are arbitrary constants and the

function 119860(119905) must equal

119860 (119905) = 11988861205852(119905) exp [minus119888

1int

119889119905

1205852(119905)

] (24)

Therefore 119860(119905) becomes

119860 (119905) = 1198886(1198883119905 + 1198884)1minus(11988811198883)

if 1198883

= 0

119860 (119905) = 1198887exp [minus

1198881

1198884

119905] if 1198883= 0

(25)

where 1198886and 1198887= 11988841198886are arbitrary constants

4 Invariant Solutions

The characteristic equations corresponding to the symme-tries (23) are given by

119889119909

1198881119909 + 1198882

=

119889119905

1198883119905 + 1198884

=

119889119891

0

=

119889119892

1198885119892

(26)

By solving the above system we have the following four cases

Case 1 When 1198881

= 0 and 1198883

= 0 the similarity variable andsimilarity functions can be written as follows

120585 =

119909 + 119886

(119905 + 119887)119888 119891 (119909 119905) = Ψ (120585)

119892 (119909 119905) = (119909 + 119886)119889Φ (120585)

(27)

where 119886 = 11988821198881 119887 = 119888

41198883 119888 = 119888

11198883 and 119889 = 119888

51198881are

arbitrary constants In this case 119860(119905) = 119902(119905 + 119887)1minus119888 where

119902 = 11988861198881minus119888

3 Substituting the transformations (27) in the field

(15) leads to the following system of ODEs

120585Ψ10158401015840+ Ψ1015840

Ψ

+

120585Φ10158401015840+ (1 + 119889)Φ

1015840

Φ

= 0 (28)

(119888211990221205852minus 1) 120585Φ

1015840Ψ1015840

ΦΨ

+

2119889Φ1015840+ 120585Φ10158401015840

Φ

+

119888211990221205853Ψ10158401015840minus [119889 + 2119888 (1 minus 2119888) 119902

21205852]Ψ1015840

Ψ

=

119889 (1 minus 119889)

120585

(29)

If one solves the system of second order NLPDEs (28)-(29)one can obtain the exact solutions of the original Einstein fieldequations (15) corresponding to reduction (30)

Case 2 When 1198881

= 0 and 1198883

= 0 the similarity variable andsimilarity functions can be written as follows

120585 = (119909 + 119886) exp [119887119905] 119891 (119909 119905) = Ψ (120585)

119892 (119909 119905) = (119909 + 119886)119888Φ (120585)

(30)

where 119886 = 11988821198881 119887 = minus119888

11198884 and 119888 = 119888

51198881are arbitrary

constants In this case 119860(119905) = 119889 exp[119887119905] where 119889 = 1198887

Substituting transformations (30) in the field equations (15)leads to the following system of ODEs

120585Ψ10158401015840+ Ψ1015840

Ψ

+

120585Φ10158401015840+ (1 + 119888)Φ

1015840

Φ

= 0 (31)

(119887211988921205852minus 1) 120585Φ

1015840Ψ1015840

ΦΨ

+

2119888Φ1015840+ 120585Φ10158401015840

Φ

+

119887211988921205853Ψ10158401015840+ (4119887211988921205852minus 119888)Ψ

1015840

Ψ

=

119888 (1 minus 119888)

120585

(32)

If one solves the system of second order NLPDEs (31)-(32)one can obtain the exact solutions of the original Einstein fieldequations (15) corresponding to reduction (30)

Case 3 When 1198881

= 0 and 1198883

= 0 the similarity variable andsimilarity functions can be written as follows

120585 =

exp [119886119909]

119905 + 119887

119891 (119909 119905) = Ψ (120585)

119892 (119909 119905) = Φ (120585) exp [119886119909]

(33)

where 119886 = 11988831198882and 119887 = 119888

41198863are arbitrary constants In this

case we have 119860(119905) = 119888(119905 + 119887) where 119888 = 1198886 Substituting the

transformations (33) in the field equations (15) leads to thefollowing system of ODEs

120585Ψ10158401015840+ Ψ1015840

Ψ

+

120585Φ10158401015840+ 2Φ1015840

Φ

= 0

120585 [

(1198882minus 1198862) 120585Φ1015840Ψ1015840

ΦΨ

+

1198862(3Φ1015840+ 120585Φ10158401015840)

Φ

+

1198882120585Ψ10158401015840minus (1198882+ 1198862)Ψ1015840

Ψ

] + 1198862= 0

(34)

Without loss of generality we can take the following usefultransformation

120585 = exp [120579]

119889Ψ

119889120585

= exp [minus120579]

119889Ψ

119889120579

1198892Ψ

1198891205852

= exp [minus2120579] (

1198892Ψ

1198891205792

minus

119889Ψ

119889120579

)

(35)

Then the system of ODEs (34) transforms to

Ψ

Ψ

+

Φ + Φ

Φ

= 0 (36)

(1198882minus 1198862) ΦΨ

ΦΨ

+

1198862(2Φ + Φ)

Φ

+

1198882Ψ minus (2119888

2+ 1198862) Ψ

Ψ

+ 1198862= 0

(37)

Equation (36) can be written in the following form

Ψ = minus

Ψ

Φ

(Φ + Φ) (38)

Abstract and Applied Analysis 5

From the above equation if we substitute Ψ in (37) we canobtain the following form

(1198882minus 1198862) [

ΦΨ

ΦΨ

minus

Φ

Φ

] + (21198862minus 1198882)(

Φ

Φ

)

minus (21198882+ 1198862)(

Ψ

Ψ

) + 1198862= 0

(39)

Equation (39) is a nonlinear ODE which is very difficult tosolve However it is worth noting that this equation is easyto solve when 119888 = 119886 In this case we can integrate (39) andobtain the following

Φ (120579) = 1198861Ψ3(120579) exp [minus120579] (40)

where 1198861is an arbitrary constant of integration Substituting

(40) in (36) we have the following ODE of the function Ψ

only

Ψ(4Ψ minus 3Ψ) + 6Ψ2= 0 (41)

The general solution of the above equation is

Ψ (120579) = 1198863(1198862+ exp [

3120579

4

])

25

(42)

where 1198862and 1198863are arbitrary constants of integration Now

by using (40) and the inverse of transformations (35) and (33)we can find the solution as follows

119860 (119905) = 119886 (119905 + 119887)

119891 (119909 119905) = 1198863(1198862+ [

exp [119886119909]

119905 + 119887

]

34

)

25

119892 (119909 119905) = 11988611198863(119905 + 119887) (119886

2+ [

exp [119886119909]

119905 + 119887

]

34

)

65

(43)

It is observed from (43) and (3) that the line element (2) canbe written in the following form

1198891199042= 1198891199052minus 1198862(119905 + 119887)

21198891199092minus 1198892(119905 + 119887)

2

times (1198862+ [

exp [119886119909]

119905 + 119887

]

34

)

45

1198891199102

minus 1199022(119905 + 119887)

4(1198862+ [

exp [119886119909]

119905 + 119887

]

34

)

125

1198891199112

(44)

where 119886 119887 119889 = 1198861198863 119902 = 119886119886

11198863 and 119886

2are arbitrary constants

Case 4 When 1198881

= 1198883

= 0 the similarity variable andsimilarity functions can be written as follows

120585 = 119886119909 + 119887119905 119891 (119909 119905) = Ψ (120585)

119892 (119909 119905) = Φ (120585) exp [119888119909]

(45)

where 119886 = 1198884 119887 = minus119888

2 and 119888 = 119888

51198882are arbitrary constants In

this case we have119860(119905) = 1198887 Substituting transformations (45)

in field equations (15) leads to the following system of ODEs

119886Ψ10158401015840

Ψ

+

119886Φ10158401015840+ 119888Φ1015840

Φ

= 0 (46)

(11988721199032minus 1198862)Φ1015840Ψ1015840

ΦΨ

+

119886 (2119888Φ1015840+ 119886Φ10158401015840)

Φ

+

11988721199032Ψ10158401015840minus 119886119888Ψ

1015840

Ψ

+ 1198882= 0

(47)

Equation (46) can be written in the following form

Ψ10158401015840

= minus

Ψ

119886Φ

(119886Φ10158401015840+ 119888Φ1015840) (48)

From the above equation if we substitute Ψ in (47) we canobtain the following form

(11988721199032minus 1198862) [

Φ1015840Ψ1015840

ΦΨ

minus

Φ10158401015840

Φ

] + (21198862minus 11988721199032)(

119888Φ1015840

119886Φ

)

minus 119886119888(

Ψ1015840

Ψ

) + 1198882= 0

(49)

Equation (49) is a nonlinear ODE which is very difficult tosolve However it is worth noting that this equation is easyto solve when 119886 = 119887119903 In this case we can integrate (49) andobtain the following

Φ (120585) = 1198861Ψ (120585) exp [minus

119888120585

119887119903

] (50)

where 1198861is an arbitrary constant of integration Substituting

(50) in (46) we have the following ODE of function Ψ only

2119887119903Ψ10158401015840

= 119888Ψ (51)

The general solution of the above equation is

Ψ (120585) = 1198863+ 1198862exp [

119888120585

2119887119903

] (52)

where 1198862and 1198863are arbitrary constants of integration Now

by using (50) and (33) we can find the solution as follows

119860 (119905) = 119903

119891 (119909 119905) = 1198863+ 1198862exp [

119888 (119905 + 119903119909)

2119887119903

]

119892 (119909 119905) = 1198861exp [minus

119888119905

119903

] (1198863+ 1198862exp [

119888 (119905 + 119903119909)

2119887119903

])

(53)

It is observed from (53) and (3) that the line element (2) canbe written in the following form

1198891199042= 1198891199052minus 11990321198891199092minus 1199032(1198863+ 1198862exp [

119888 (119905 + 119903119909)

2119887119903

])

2

times (1198891199102+ 1198862

1exp [minus

2119888119905

119903

] 1198891199112)

(54)

where 119903 119888 119887 1198861 1198862 and 119886

3are arbitrary constants

6 Abstract and Applied Analysis

5 Physical Properties of the Model

The field equations (15)ndash(16) constitute a system of fivehighly nonlinear differential equations with seven unknownsvariables 119860 119891 119892 119901 120588 120583 and 120573 The symmetries giveone condition (24) for function 119860 Therefore one physicallyreasonable condition amongst these parameters is required toobtain explicit solutions of the field equations Let us assumethat the density 120588 and the pressure 119901 are related by barotropicequation of state

119901 = 120582120588 0 le 120582 le 1 (55)

51 ForModel (44) Using (43) in the Einstein field equations(16) with taking into account condition (55) the expressionsfor density 120588 pressure 119901 magnetic permeability 120583 anddisplacement field 120573 are given by

120588 (119909 119905) =

9

5120594 (1 minus 120582)

times[

[

51198862+ [exp [119886119909] (119905 + 119887)]

34

(119905 + 119887)2(1198862+ [exp [119886119909] (119905 + 119887)]

34

)

]

]

119901 (119909 119905) =

9120582

5120594 (1 minus 120582)

times[

[

51198862+ [exp [119886119909] (119905 + 119887)]

34

(119905 + 119887)2(1198862+ [exp [119886119909] (119905 + 119887)]

34

)

]

]

120583 (119909 119905) =

1205941198652

12(119909)

3119886211988621198892(119905 + 119887)

2(1198862+ [

exp [119886119909]

119905 + 119887

]

34

)

15

1205732(119909 119905) =

2

15

[

[

(51198862(5 + 13120582)

+ 2 (2 + 7120582) [

exp [119886119909]

119905 + 119887

]

34

)

times ((120582 minus 1) (119905 + 119887)2

times(1198862+ [

exp [119886119909]

119905 + 119887

]

34

))

minus1

]

(56)

For the line element (44) using (4) (8) and (9) we have thefollowing physical properties The volume element is

119881 = 119886119889119902(119905 + 119887)4(1198862+ [

exp [119886119909]

119905 + 119887

]

34

)

85

(57)

The expansion scalar which determines the volume behaviorof the fluid is given by

Θ =

2

5

[

[

101198862+ 7[exp [119886119909] (119905 + 119887)]

34

(119905 + 119887) (1198862+ [exp [119886119909] (119905 + 119887)]

34

)

]

]

(58)

The nonvanishing components of the shear tensor 120590119895119894 are

1205901

1=

[exp [119886119909] (119905 + 119887)]34

minus 51198862

15 (119905 + 119887) (1198862+ [exp [119886119909] (119905 + 119887)]

34

)

1205902

2= minus

1

30

[

[

101198862+ [exp [119886119909] (119905 + 119887)]

34

(119905 + 119887) (1198862+ [exp [119886119909] (119905 + 119887)]

34

)

]

]

1205903

3=

41198862+ 7[exp [119886119909] (119905 + 119887)]

34

6 (119905 + 119887) (1198862+ [exp [119886119909] (119905 + 119887)]

34

)

1205904

4= minus

4

15

[

[

101198862+ 7[exp [119886119909] (119905 + 119887)]

34

(119905 + 119887) (1198862+ [exp [119886119909] (119905 + 119887)]

34

)

]

]

(59)

Hence the shear scalar 120590 is given by

1205902= (3500119886

2

2+ 4630119886

2[exp [119886119909] (119905 + 119887)]

34

+1607[exp [119886119909] (119905 + 119887)]32

)

times (900(119905 + 119887)2(1198862+ [exp [119886119909] (119905 + 119887)]

34

)

2

)

minus1

(60)

The model does not admit acceleration and rotation since119894= 0 and 120596

119894119895= 0 We can see that

1205904

4

Θ

= minus

2

3

(61)

which is a constant of proportional We found also that

lim119905rarrinfin

1205902

Θ

=

5radic35

2

= 0 (62)

and this means that there is no possibility that the universemay got isotropized in some later time that is it remainsanisotropic for all times

52 ForModel (54) Using (53) in the Einstein field equations(16) with taking into account condition (55) the expressions

Abstract and Applied Analysis 7

for density 120588 pressure 119901 magnetic permeability 120583 anddisplacement field 120573 are given by

120588 (119909 119905) =

1198882

1205941199032(120582 minus 1)

(

1198882exp [119888 (119905 + 119903119909) 2119903] minus 119888

3

1198882exp [119888 (119905 + 119903119909) 2119903] + 119888

3

)

119901 (119909 119905) =

1205821198882

1205941199032(120582 minus 1)

(

1198882exp [119888 (119905 + 119903119909) 2119903] minus 119888

3

1198882exp [119888 (119905 + 119903119909) 2119903] + 119888

3

)

120583 (119909 119905) =

1205941198652

12(119909)

119888311988821199032

(1198882exp [

119888 (119905 + 119903119909)

2119903

] + 1198883)

minus1

1205732(119909 119905) =

21198882

31199032(120582 minus 1)

times (

1198883(1 + 120582) minus 2120582119888

2exp [119888 (119905 + 119903119909) 2119903]

1198883+ 1198882exp [119888 (119905 + 119903119909) 2119903]

)

(63)

For the line element (54) using (4) (8) and (9) we have thefollowing physical properties The volume element is

119881 = 11988811199033 exp [minus

119888119905

119903

] (1198883+ 1198882exp [

119888 (119905 + 119903119909)

2119903

])

2

(64)

The expansion scalar which determines the volume behaviorof the fluid is given by

Θ = minus

1198881198883

119903

(1198883+ 1198882exp [

119888 (119905 + 119903119909)

2119903

])

minus1

(65)

The nonvanishing components of the shear tensor 120590119895119894 satisfy

1205901

1

Θ

= minus

1

3

1205902

2

Θ

= minus(

1

3

+

1198882

21198883

exp [

119888 (119905 + 119903119909)

2119903

])

1205903

3

Θ

=

2

3

+

1198882

21198883

exp [

119888 (119905 + 119903119909)

2119903

]

1205904

4

Θ

= minus

2

3

(66)

Hence the shear scalar 120590 is given by

1205902=

1198882

41199032+

111198882

31198882

361199032

(1198883+ 1198882exp [

119888 (119905 + 119903119909)

2119903

])

minus2

(67)

The model does not admit acceleration and rotation since119894= 0 and 120596

119894119895= 0

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah under Grant no130-016-D1434 The author therefore acknowledges withthanks DSR technical and financial support

References

[1] A Krasinski Inhomogeneous Cosmological Models CambridgeUniversity Press Cambridge UK 1997

[2] G Lyra ldquoUber eine Modifikation der Riemannschen Geome-trierdquoMathematische Zeitschrift vol 54 pp 52ndash64 1951

[3] D K Sen ldquoA static cosmological modelrdquo Zeitschrift fur PhysikC Particles and Fields vol 149 pp 311ndash323 1957

[4] D K Sen ldquoOn geodesics of a modified Riemannian manifoldrdquoCanadian Mathematical Society vol 3 pp 255ndash261 1960

[5] D K Sen and K A Dunn ldquoA scalar-tensor theory of gravitationin a modified Riemannian manifoldrdquo Journal of MathematicalPhysics vol 12 pp 578ndash586 1971

[6] W D Halford ldquoCosmological theory based on Lyrarsquos geometryrdquoAustralian Journal of Physics vol 23 no 5 pp 863ndash870 1970

[7] S S De and F Rahaman Finsler Geometry of Hadrons and LyraGeometry Cosmological Aspects Lambert Academic Publish-ing Saarbrucken Germany 2012

[8] A Pradhan I Aotemshi and G P Singh ldquoPlane symmetricdomain wall in Lyra geometryrdquo Astrophysics and Space Sciencevol 288 no 3 pp 315ndash325 2003

[9] A Pradhan and A K Vishwakarma ldquoA new class of LRSBianchi type-I cosmological models in Lyra geometryrdquo Journalof Geometry and Physics vol 49 no 3-4 pp 332ndash342 2004

[10] A Pradhan and S S Kumhar ldquoPlane symmetric inhomoge-neous perfect fluid universe with electromagnetic field in Lyrageometryrdquo Astrophysics and Space Science vol 321 no 2 pp137ndash146 2009

[11] A Pradhan and P Ram ldquoA plane-symmetric magnetizedinhomogeneous cosmological models of perfect fluid distribu-tion with variable magnetic permeability in Lyra geometryrdquoInternational Journal of Theoretical Physics vol 48 no 11 pp3188ndash3201 2009

[12] A Pradhan H Amirhashchi and H Zainuddin ldquoA new classof inhomogeneous cosmological models with electromagneticfield in normal gauge for Lyras manifoldrdquo International Journalof Theoretical Physics vol 50 no 1 pp 56ndash69 2011

[13] P Anirudh A Singh and R S Singh ldquoA plane-symmetric mag-netized inhomogeneous cosmological model of perfect fluiddistribution with variable magnetic permeabilityrdquo RomanianReports of Physics vol 56 no 1-2 pp 297ndash307 2011

[14] A Pradhan and A K Singh ldquoAnisotropic Bianchi type-I stringcosmological models in normal Gauge for Lyras manifoldwith constant deceleration parameterrdquo International Journal ofTheoretical Physics vol 50 no 3 pp 916ndash933 2011

[15] R Casana C AM deMelo andBM Pimentel ldquoSpinorial fieldand Lyra geometryrdquoAstrophysics and Space Science vol 305 no2 pp 125ndash132 2006

[16] F Rahaman B C Bhui and G Bag ldquoCan Lyra geometryexplain the singularity free as well as accelerating UniverserdquoAstrophysics and Space Science vol 295 no 4 pp 507ndash513 2005

[17] R Bali and N K Chandnani ldquoBianchi type-I cosmologicalmodel for perfect fluid distribution in Lyra geometryrdquo Journal of

8 Abstract and Applied Analysis

Mathematical Physics vol 49 no 3 Article ID 032502 8 pages2008

[18] R Bali andN K Chandnani ldquoBianchi typeV barotropic perfectfluid cosmological model in Lyra geometryrdquo InternationalJournal ofTheoretical Physics vol 48 no 5 pp 1523ndash1533 2009

[19] S Kumar andC P Singh ldquoAn exact Bianchi type-I cosmologicalmodel in Lyrarsquos manifoldrdquo International Journal of ModernPhysics A vol 23 no 6 pp 813ndash822 2008

[20] A K Yadav A Pradhan and A Singh ldquoLRS Bianchi type-II massive string cosmological models in general relativityrdquoRomanianReports of Physics vol 56 no 7-8 pp 1019ndash1034 2011

[21] V U M Rao T Vinutha and M V Santhi ldquoBianchi type-Vcosmological model with perfect fluid using negative constantdeceleration parameter in a scalar tensor theory based on LyraManifoldrdquo Astrophysics and Space Science vol 314 no 1ndash3 pp213ndash216 2008

[22] R Zia and R P Singh ldquoBulk viscous inhomogeneous cosmo-logical models with electromagnetic field in Lyra geometryrdquoRomanian Journal of Physics vol 57 no 3-4 pp 761ndash778 2012

[23] A T Ali and F Rahaman ldquoNew class of magnetized inho-mogeneous Bianchi type-I cosmological model with variablemagnetic permeability in Lyra geometryrdquo International Journalof Theoretical Physics vol 52 no 11 pp 4055ndash4067 2013

[24] A T Ali ldquoNew generalized Jacobi elliptic function rationalexpansion methodrdquo Journal of Computational and AppliedMathematics vol 235 no 14 pp 4117ndash4127 2011

[25] A T Ali and E R Hassan ldquoGeneral Expa-function methodfor nonlinear evolution equationsrdquo Applied Mathematics andComputation vol 217 no 2 pp 451ndash459 2010

[26] M F El-Sabbagh and A T Ali ldquoNew exact solutions for (3+1)-dimensional Kadomtsev-Petviashvili equation and generalized(2+1)-dimensional Boussinesq equationrdquo International Journalof Nonlinear Sciences andNumerical Simulation vol 6 no 2 pp151ndash162 2005

[27] M F El-Sabbagh and A T Ali ldquoNew generalized Jacobi ellipticfunction expansion methodrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 9 pp 1758ndash17662008

[28] L V Ovsiannikov Group Analysis of Differential Equationstranslated by Y Chapovsky edited by W F Ames AcademicPress New York NY USA 1982

[29] G W Bluman and S Kumei Symmetries and DifferentialEquations in Applied Sciences Springer New York NY USA1989

[30] N H Ibragimov Transformation Groups Applied to Mathemat-ical Physics D Reidel Dordrecht The Netherlands 1985

[31] P J Olver Applications of Lie Groups to Differential Equationsvol 107 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 1993

[32] A T Ali A K Yadav and S R Mahmoud ldquoSome planesymmetric inhomogeneous cosmological models in the scalar-tensor theory of gravitationrdquoAstrophysics and Space Science vol349 no 1 pp 539ndash547 2014

[33] A T Ali ldquoNew exact solutions of the Einstein vacuumequationsfor rotating axially symmetric fieldsrdquoPhysica Scripta vol 79 no3 Article ID 035006 2009

[34] S K Attallah M F El-Sabbagh and A T Ali ldquoIsovectorfields and similarity solutions of Einstein vacuum equationsfor rotating fieldsrdquo Communications in Nonlinear Science andNumerical Simulation vol 12 no 7 pp 1153ndash1161 2007

[35] K SMekheimer S Z Husseny A T Ali andA E Abo-ElkhairldquoLie point symmetries and similarity solutions for an electricallyconducting Jeffrey fluidrdquo Physica Scripta vol 83 no 1 ArticleID 015017 2011

[36] A K Raychaudhuri Theoritical Cosmology Oxford SciencePublications Oxford UK 1979

[37] A Lichnerowicz Relativistic Hydrodynamics and Magneto-Hydro-Dynamics W A Benjamin New York NY USA 1967

[38] A Pradhan and P Mathur ldquoInhomogeneous perfect fluiduniverse with electromagnetic field in Lyra geometryrdquo Fizika Bvol 18 no 4 pp 243ndash264 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

theory Halford also predicted that the present theory willprovide the same effects within observational limits as far asthe classical solar system tests are concerned as well as testsbased on the linearized form of field equations For a reviewon Lyra geometry one can see [7]

Recently Pradhan et al [8ndash14] Casana et al [15]Rahaman et al [16] Bali and Chandnani [17 18] Kumar andSingh [19] Yadav et al [20] Rao et al [21] Zia and Singh [22]and Ali and Rahaman [23] have studied cosmological modelsbased on Lyrarsquos geometry in various contexts

To study the nonlinear physical phenomena [24ndash27] it isimportant to search the exact solutions of nonlinear PDEsOvsiannikov [28] is the pioneer who had observed that theusual Lie infinitesimal invariance approach could as wellbe employed in order to construct symmetry groups [29ndash31] The symmetry groups of a differential equation couldbe defined as the groups of continuous transformations thatlead a given family of equations invariant [32ndash35] and areproved to be important to solve the nonlinear equations of themodels to describe complex physical phenomena in variousfields of science especially in fluid mechanics solid statephysics plasma physics plasma wave and general relativity

In this paper we have obtained exact solutions of Ein-steinrsquosmodified field equations in inhomogeneous space-timeBianchi type-I cosmological model within the frame work ofLyrarsquos geometry in the presence ofmagnetic fieldwith variablemagnetic permeability and time varying displacement vector120573(119909 119905) using the so-called symmetry analysis method Sincethe field equations are highly nonlinear differential equationstherefore symmetry analysis method can be successfullyapplied to nonlinear differential equations The similarity(invariant) solutions help to reduce the independent variablesof the problem and therefore we employ this method inthe investigation of exact solution of the field equationsIn general invariant solutions will transform the system ofnonlinear PDEs into a system of ODEs We attempted tofind a new class of exact (invariant) solutions for the fieldequations based on Lyra geometry

The scheme of the paper is as follows Magnetizedinhomogeneous Bianchi type-I cosmological model withvariable magnetic permeability based on Lyra geometry isintroduced in Section 2 In Section 3 we have performedsymmetry analysis and have obtained isovector fields forEinstein field equations under consideration In Section 4 wefound new class of exact (invariant) solutions for Einsteinfield equations Section 5 is devoted to study of some physicaland geometrical properties of the model

2 The Metric and Field Equations

We consider Bianchi type-Imetric with the convention (1199090=

119905 1199091= 119909 119909

2= 119910 119909

3= 119911) in the form

1198891199042= 1198891199052minus 11986021198891199092minus 11986121198891199102minus 11986221198891199112 (2)

where119860 is a function of 119905 only while 119861 and 119862 are functions of119909 and 119905 Without loss of generality we can put the followingtransformation

119861 = 119860119891 119862 = 119860119892 (3)

where 119891 and 119892 are functions of 119909 and 119905 The volume elementof model (2) is given by

119881 = radicminus119892 = 1198603119891119892 (4)

The four-acceleration vector the rotation the expansionscalar and the shear scalar characterizing the four-velocityvector field 119906119894 satisfying the relation in comoving coordinatesystem

119892119894119895119906119894119906119895= 1 119906

119894= 119906119894= (1 0 0 0) (5)

respectively have the usual definitions as given by Raychaud-huri [36]

119894= 119906119894119895119906119895

120596119894119895

= 119906[119894119895]

+ [119894119906119895]

Θ = 119906119894

119894

1205902=

1

2

120590119894119895120590119894119895

(6)

where

120590119894119895

= 119906(119894119895)

+ (119894119906119895)

minus

1

3

Θ (119892119894119895+ 119906119894119906119895) (7)

In view of metric (2) the four-acceleration vector therotation the expansion scalar and the shear scalar given by(6) can be written in a comoving coordinates system as

119894= 0

120596119894119895

= 0

Θ =

3

119860

+

119891119905

119891

+

119892119905

119892

1205902=

119860

(

2

119860

+

4119891119905

3119891

+

4119892119905

3119892

) +

51198912

119905

91198912

+

119891119905119892119905

9119891119892

+

51198922

119905

91198922

(8)

where the nonvanishing components of the shear tensor 120590119895

119894

are

1205901

1= minus

119891119905

3119891

minus

119892119905

3119891

1205902

2=

2119891119905

3119891

minus

119892119905

3119892

1205903

3=

2119892119905

3119892

minus

119891119905

3119891

1205904

4= minus

2

119860

minus

2119891119905

3119891

minus

2119892119905

3119892

(9)

To study the cosmological model we use the field equationsin Lyra geometry given in (1) in which the displacement fieldvector 120601

119894is given by

120601119894= (120573 (119909 119905) 0 0 0) (10)

Abstract and Applied Analysis 3

119879119894119895is the energy momentum tensor given by

119879119894119895

= (120588 + 119901) 119906119894119906119895minus 119901119892119894119895+ 119864119894119895 (11)

where 119864119894119895is the electromagnetic field given by Lichnerowicz

[37]

119864119894119895

= 120583 [ℎ119897ℎ119897(119906119894119906119895minus

1

2

119892119894119895) + ℎ119894ℎ119895] (12)

Here 120588 and 119901 are the energy density and isotropic pressurerespectively while 120583 is the magnetic permeability and ℎ

119894is

the magnetic flux vector defined by

ℎ119894=

radicminus119892

2120583

120598119894119895119896119897

119865119896119897119906119895 (13)

119865119894119895is the electromagnetic field tensor and 120598

119894119895119896119897is a Levi-Civita

tensor density If we consider the current flow along 119911-axisthen 119865

12is only nonvanishing component of 119865

119894119895 Then the

Maxwell equations

119865119894119895119896

+ 119865119895119896119894

+ 119865119896119894119895

= 0

[

1

120583

119865119894119895]

119895

= 119869119894

(14)

require 11986512

to be function of 119909 alone [38] We assume themagnetic permeability as a function of both 119909 and 119905 Here thesemicolon represents a covariant differentiation

For the line element (2) field equation (1) can be reducedto the following system of NLPDEs

1198641=

119891119909119905

119891

+

119892119909119905

119892

= 0

1198642=

119891119905119905

119891

+

119891119905119892119905

119891119892

+

1

1198602(

119892119909119909

119892

minus

119891119909119892119909

119891119892

) +

3119891119905

119860119891

= 0

(15)

120594120588 +

3

4

1205732=

119892119905119905

2119892

+

3119891119905119892119905

2119891119892

minus

1

1198602(

119891119909119909

2119891

+

119892119909119909

119892

+

3119891119909119892119909

2119891119892

)

+

119860

(

2119891119905

119891

+

119892119905

2119892

+

3

119860

)

120594119901 +

3

4

1205732=

1

21198602(

119891119909119909

119891

+

119891119909119892119909

119891119892

) minus

119891119905119905

119891

minus

119892119905119905

2119892

minus

119891119905119892119905

2119891119892

minus

119860

(

3119891119905

119891

+

119860

) minus

2

119860

1205941198652

12

12058311986041198912

=

1

1198602(

119891119909119909

119891

minus

119891119909119892119909

119891119892

) +

119892119905119905

119892

+

119891119905119892119905

119891119892

+

3119892119905

119860119892

(16)

3 Symmetry Analysis Method

Equations (15)ndash(16) are highly nonlinear PDEs and hence itis so difficult to handle since there exist no standard methodsfor obtaining analytical solutionThe system (15) is nonlinear

PDEs of second order for the two unknowns 119891 and 119892 If wesolve this system then we can get the solution of the fieldequations In order to obtain an exact solution of the systemof nonlinear PDEs (15) we will use the symmetry analysismethod For this we write

119909lowast

119894= 119909119894+ 120598120585119894(119909119895 119906120573) + o (120598

2)

119906lowast

120572= 119906120572+ 120598120578120572(119909119895 119906120573) + o (120598

2)

119894 119895 120572 120573 = 1 2 (17)

as the infinitesimal Lie point transformations We haveassumed that the system (15) is invariant under the trans-formations given in (17) The corresponding infinitesimalgenerator of Lie groups (symmetries) is given by

119883 =

2

sum

119894=1

120585119894

120597

120597119909119894

+

2

sum

120572=1

120578120572

120597

120597119906120572

(18)

where 1199091

= 119909 1199092

= 119905 1199061

= 119891 and 1199062

= 119892 The coefficients1205851 1205852 1205781 and 120578

2are the functions of 119909 119905 119891 and 119892 These

coefficients are the components of infinitesimals symmetriescorresponding to 119909 119905119891 and119892 respectively to be determinedfrom the invariance conditions

Pr(2)119883(119864119898)

10038161003816100381610038161003816119864119898=0

= 0 (19)

where 119864119898

= 0119898 = 1 2 are the system (15) under study andPr(2) is the second prolongation of the symmetries 119883 Sinceour equations (15) are at most of order two therefore weneed second order prolongation of the infinitesimal generatorin (19) It is worth noting that the 2nd order prolongation isgiven by

Pr(2)119883 = 119883 +

2

sum

119894=1

2

sum

120572=1

120578120572119894

120597

120597119906120572119894

+

2

sum

119895=1

2

sum

119894=1

2

sum

120572=1

120578120572119894119895

120597

120597119906120572119894119895

(20)

where

120578120572119894

= 119863119894[

[

120578120572minus

2

sum

119895=1

120585119895119906120572119895

]

]

+

2

sum

119895=1

120585119895119906120572119894119895

120578120572119894119895

= 119863119894119895[120578120572minus

2

sum

119896=1

120585119896119906120572119896

] +

2

sum

119896=1

120585119896119906120572119894119895119896

(21)

The operator 119863119894(119863119894119895) is called the total derivative (Hach

operator) and takes the following form

119863119894=

120597

120597119909119894

+

2

sum

120572=1

119906120572119894

120597

120597119906120572

+

2

sum

119895=1

2

sum

120572=1

119906120572119894119895

120597

120597119906120572119895

(22)

where 119863119894119895

= 119863119894(119863119895) = 119863

119895(119863119894) = 119863

119895119894and 119906

120572119894= 120597119906120572120597119909119894

Expanding the system (19) with the aid of Mathematicaprogram along with the original system (15) to eliminate 119891

119909119905

and 119892119909119909

while we set the coefficients involving 119891119909 119891119905 119891119909119909 119891119905119905

119892119909119892119905119892119909119905 and119892

119905119905and various products equal zero these gives

rise to the essential set of overdetermined equations Solvingthe set of these determining equations the components ofsymmetries take the following form

1205851= 1198881119909 + 1198882 120585

2= 1198883119905 + 1198884

1205781= 0 120578

2= 1198885119892

(23)

4 Abstract and Applied Analysis

where 119888119894 119894 = 1 2 5 are arbitrary constants and the

function 119860(119905) must equal

119860 (119905) = 11988861205852(119905) exp [minus119888

1int

119889119905

1205852(119905)

] (24)

Therefore 119860(119905) becomes

119860 (119905) = 1198886(1198883119905 + 1198884)1minus(11988811198883)

if 1198883

= 0

119860 (119905) = 1198887exp [minus

1198881

1198884

119905] if 1198883= 0

(25)

where 1198886and 1198887= 11988841198886are arbitrary constants

4 Invariant Solutions

The characteristic equations corresponding to the symme-tries (23) are given by

119889119909

1198881119909 + 1198882

=

119889119905

1198883119905 + 1198884

=

119889119891

0

=

119889119892

1198885119892

(26)

By solving the above system we have the following four cases

Case 1 When 1198881

= 0 and 1198883

= 0 the similarity variable andsimilarity functions can be written as follows

120585 =

119909 + 119886

(119905 + 119887)119888 119891 (119909 119905) = Ψ (120585)

119892 (119909 119905) = (119909 + 119886)119889Φ (120585)

(27)

where 119886 = 11988821198881 119887 = 119888

41198883 119888 = 119888

11198883 and 119889 = 119888

51198881are

arbitrary constants In this case 119860(119905) = 119902(119905 + 119887)1minus119888 where

119902 = 11988861198881minus119888

3 Substituting the transformations (27) in the field

(15) leads to the following system of ODEs

120585Ψ10158401015840+ Ψ1015840

Ψ

+

120585Φ10158401015840+ (1 + 119889)Φ

1015840

Φ

= 0 (28)

(119888211990221205852minus 1) 120585Φ

1015840Ψ1015840

ΦΨ

+

2119889Φ1015840+ 120585Φ10158401015840

Φ

+

119888211990221205853Ψ10158401015840minus [119889 + 2119888 (1 minus 2119888) 119902

21205852]Ψ1015840

Ψ

=

119889 (1 minus 119889)

120585

(29)

If one solves the system of second order NLPDEs (28)-(29)one can obtain the exact solutions of the original Einstein fieldequations (15) corresponding to reduction (30)

Case 2 When 1198881

= 0 and 1198883

= 0 the similarity variable andsimilarity functions can be written as follows

120585 = (119909 + 119886) exp [119887119905] 119891 (119909 119905) = Ψ (120585)

119892 (119909 119905) = (119909 + 119886)119888Φ (120585)

(30)

where 119886 = 11988821198881 119887 = minus119888

11198884 and 119888 = 119888

51198881are arbitrary

constants In this case 119860(119905) = 119889 exp[119887119905] where 119889 = 1198887

Substituting transformations (30) in the field equations (15)leads to the following system of ODEs

120585Ψ10158401015840+ Ψ1015840

Ψ

+

120585Φ10158401015840+ (1 + 119888)Φ

1015840

Φ

= 0 (31)

(119887211988921205852minus 1) 120585Φ

1015840Ψ1015840

ΦΨ

+

2119888Φ1015840+ 120585Φ10158401015840

Φ

+

119887211988921205853Ψ10158401015840+ (4119887211988921205852minus 119888)Ψ

1015840

Ψ

=

119888 (1 minus 119888)

120585

(32)

If one solves the system of second order NLPDEs (31)-(32)one can obtain the exact solutions of the original Einstein fieldequations (15) corresponding to reduction (30)

Case 3 When 1198881

= 0 and 1198883

= 0 the similarity variable andsimilarity functions can be written as follows

120585 =

exp [119886119909]

119905 + 119887

119891 (119909 119905) = Ψ (120585)

119892 (119909 119905) = Φ (120585) exp [119886119909]

(33)

where 119886 = 11988831198882and 119887 = 119888

41198863are arbitrary constants In this

case we have 119860(119905) = 119888(119905 + 119887) where 119888 = 1198886 Substituting the

transformations (33) in the field equations (15) leads to thefollowing system of ODEs

120585Ψ10158401015840+ Ψ1015840

Ψ

+

120585Φ10158401015840+ 2Φ1015840

Φ

= 0

120585 [

(1198882minus 1198862) 120585Φ1015840Ψ1015840

ΦΨ

+

1198862(3Φ1015840+ 120585Φ10158401015840)

Φ

+

1198882120585Ψ10158401015840minus (1198882+ 1198862)Ψ1015840

Ψ

] + 1198862= 0

(34)

Without loss of generality we can take the following usefultransformation

120585 = exp [120579]

119889Ψ

119889120585

= exp [minus120579]

119889Ψ

119889120579

1198892Ψ

1198891205852

= exp [minus2120579] (

1198892Ψ

1198891205792

minus

119889Ψ

119889120579

)

(35)

Then the system of ODEs (34) transforms to

Ψ

Ψ

+

Φ + Φ

Φ

= 0 (36)

(1198882minus 1198862) ΦΨ

ΦΨ

+

1198862(2Φ + Φ)

Φ

+

1198882Ψ minus (2119888

2+ 1198862) Ψ

Ψ

+ 1198862= 0

(37)

Equation (36) can be written in the following form

Ψ = minus

Ψ

Φ

(Φ + Φ) (38)

Abstract and Applied Analysis 5

From the above equation if we substitute Ψ in (37) we canobtain the following form

(1198882minus 1198862) [

ΦΨ

ΦΨ

minus

Φ

Φ

] + (21198862minus 1198882)(

Φ

Φ

)

minus (21198882+ 1198862)(

Ψ

Ψ

) + 1198862= 0

(39)

Equation (39) is a nonlinear ODE which is very difficult tosolve However it is worth noting that this equation is easyto solve when 119888 = 119886 In this case we can integrate (39) andobtain the following

Φ (120579) = 1198861Ψ3(120579) exp [minus120579] (40)

where 1198861is an arbitrary constant of integration Substituting

(40) in (36) we have the following ODE of the function Ψ

only

Ψ(4Ψ minus 3Ψ) + 6Ψ2= 0 (41)

The general solution of the above equation is

Ψ (120579) = 1198863(1198862+ exp [

3120579

4

])

25

(42)

where 1198862and 1198863are arbitrary constants of integration Now

by using (40) and the inverse of transformations (35) and (33)we can find the solution as follows

119860 (119905) = 119886 (119905 + 119887)

119891 (119909 119905) = 1198863(1198862+ [

exp [119886119909]

119905 + 119887

]

34

)

25

119892 (119909 119905) = 11988611198863(119905 + 119887) (119886

2+ [

exp [119886119909]

119905 + 119887

]

34

)

65

(43)

It is observed from (43) and (3) that the line element (2) canbe written in the following form

1198891199042= 1198891199052minus 1198862(119905 + 119887)

21198891199092minus 1198892(119905 + 119887)

2

times (1198862+ [

exp [119886119909]

119905 + 119887

]

34

)

45

1198891199102

minus 1199022(119905 + 119887)

4(1198862+ [

exp [119886119909]

119905 + 119887

]

34

)

125

1198891199112

(44)

where 119886 119887 119889 = 1198861198863 119902 = 119886119886

11198863 and 119886

2are arbitrary constants

Case 4 When 1198881

= 1198883

= 0 the similarity variable andsimilarity functions can be written as follows

120585 = 119886119909 + 119887119905 119891 (119909 119905) = Ψ (120585)

119892 (119909 119905) = Φ (120585) exp [119888119909]

(45)

where 119886 = 1198884 119887 = minus119888

2 and 119888 = 119888

51198882are arbitrary constants In

this case we have119860(119905) = 1198887 Substituting transformations (45)

in field equations (15) leads to the following system of ODEs

119886Ψ10158401015840

Ψ

+

119886Φ10158401015840+ 119888Φ1015840

Φ

= 0 (46)

(11988721199032minus 1198862)Φ1015840Ψ1015840

ΦΨ

+

119886 (2119888Φ1015840+ 119886Φ10158401015840)

Φ

+

11988721199032Ψ10158401015840minus 119886119888Ψ

1015840

Ψ

+ 1198882= 0

(47)

Equation (46) can be written in the following form

Ψ10158401015840

= minus

Ψ

119886Φ

(119886Φ10158401015840+ 119888Φ1015840) (48)

From the above equation if we substitute Ψ in (47) we canobtain the following form

(11988721199032minus 1198862) [

Φ1015840Ψ1015840

ΦΨ

minus

Φ10158401015840

Φ

] + (21198862minus 11988721199032)(

119888Φ1015840

119886Φ

)

minus 119886119888(

Ψ1015840

Ψ

) + 1198882= 0

(49)

Equation (49) is a nonlinear ODE which is very difficult tosolve However it is worth noting that this equation is easyto solve when 119886 = 119887119903 In this case we can integrate (49) andobtain the following

Φ (120585) = 1198861Ψ (120585) exp [minus

119888120585

119887119903

] (50)

where 1198861is an arbitrary constant of integration Substituting

(50) in (46) we have the following ODE of function Ψ only

2119887119903Ψ10158401015840

= 119888Ψ (51)

The general solution of the above equation is

Ψ (120585) = 1198863+ 1198862exp [

119888120585

2119887119903

] (52)

where 1198862and 1198863are arbitrary constants of integration Now

by using (50) and (33) we can find the solution as follows

119860 (119905) = 119903

119891 (119909 119905) = 1198863+ 1198862exp [

119888 (119905 + 119903119909)

2119887119903

]

119892 (119909 119905) = 1198861exp [minus

119888119905

119903

] (1198863+ 1198862exp [

119888 (119905 + 119903119909)

2119887119903

])

(53)

It is observed from (53) and (3) that the line element (2) canbe written in the following form

1198891199042= 1198891199052minus 11990321198891199092minus 1199032(1198863+ 1198862exp [

119888 (119905 + 119903119909)

2119887119903

])

2

times (1198891199102+ 1198862

1exp [minus

2119888119905

119903

] 1198891199112)

(54)

where 119903 119888 119887 1198861 1198862 and 119886

3are arbitrary constants

6 Abstract and Applied Analysis

5 Physical Properties of the Model

The field equations (15)ndash(16) constitute a system of fivehighly nonlinear differential equations with seven unknownsvariables 119860 119891 119892 119901 120588 120583 and 120573 The symmetries giveone condition (24) for function 119860 Therefore one physicallyreasonable condition amongst these parameters is required toobtain explicit solutions of the field equations Let us assumethat the density 120588 and the pressure 119901 are related by barotropicequation of state

119901 = 120582120588 0 le 120582 le 1 (55)

51 ForModel (44) Using (43) in the Einstein field equations(16) with taking into account condition (55) the expressionsfor density 120588 pressure 119901 magnetic permeability 120583 anddisplacement field 120573 are given by

120588 (119909 119905) =

9

5120594 (1 minus 120582)

times[

[

51198862+ [exp [119886119909] (119905 + 119887)]

34

(119905 + 119887)2(1198862+ [exp [119886119909] (119905 + 119887)]

34

)

]

]

119901 (119909 119905) =

9120582

5120594 (1 minus 120582)

times[

[

51198862+ [exp [119886119909] (119905 + 119887)]

34

(119905 + 119887)2(1198862+ [exp [119886119909] (119905 + 119887)]

34

)

]

]

120583 (119909 119905) =

1205941198652

12(119909)

3119886211988621198892(119905 + 119887)

2(1198862+ [

exp [119886119909]

119905 + 119887

]

34

)

15

1205732(119909 119905) =

2

15

[

[

(51198862(5 + 13120582)

+ 2 (2 + 7120582) [

exp [119886119909]

119905 + 119887

]

34

)

times ((120582 minus 1) (119905 + 119887)2

times(1198862+ [

exp [119886119909]

119905 + 119887

]

34

))

minus1

]

(56)

For the line element (44) using (4) (8) and (9) we have thefollowing physical properties The volume element is

119881 = 119886119889119902(119905 + 119887)4(1198862+ [

exp [119886119909]

119905 + 119887

]

34

)

85

(57)

The expansion scalar which determines the volume behaviorof the fluid is given by

Θ =

2

5

[

[

101198862+ 7[exp [119886119909] (119905 + 119887)]

34

(119905 + 119887) (1198862+ [exp [119886119909] (119905 + 119887)]

34

)

]

]

(58)

The nonvanishing components of the shear tensor 120590119895119894 are

1205901

1=

[exp [119886119909] (119905 + 119887)]34

minus 51198862

15 (119905 + 119887) (1198862+ [exp [119886119909] (119905 + 119887)]

34

)

1205902

2= minus

1

30

[

[

101198862+ [exp [119886119909] (119905 + 119887)]

34

(119905 + 119887) (1198862+ [exp [119886119909] (119905 + 119887)]

34

)

]

]

1205903

3=

41198862+ 7[exp [119886119909] (119905 + 119887)]

34

6 (119905 + 119887) (1198862+ [exp [119886119909] (119905 + 119887)]

34

)

1205904

4= minus

4

15

[

[

101198862+ 7[exp [119886119909] (119905 + 119887)]

34

(119905 + 119887) (1198862+ [exp [119886119909] (119905 + 119887)]

34

)

]

]

(59)

Hence the shear scalar 120590 is given by

1205902= (3500119886

2

2+ 4630119886

2[exp [119886119909] (119905 + 119887)]

34

+1607[exp [119886119909] (119905 + 119887)]32

)

times (900(119905 + 119887)2(1198862+ [exp [119886119909] (119905 + 119887)]

34

)

2

)

minus1

(60)

The model does not admit acceleration and rotation since119894= 0 and 120596

119894119895= 0 We can see that

1205904

4

Θ

= minus

2

3

(61)

which is a constant of proportional We found also that

lim119905rarrinfin

1205902

Θ

=

5radic35

2

= 0 (62)

and this means that there is no possibility that the universemay got isotropized in some later time that is it remainsanisotropic for all times

52 ForModel (54) Using (53) in the Einstein field equations(16) with taking into account condition (55) the expressions

Abstract and Applied Analysis 7

for density 120588 pressure 119901 magnetic permeability 120583 anddisplacement field 120573 are given by

120588 (119909 119905) =

1198882

1205941199032(120582 minus 1)

(

1198882exp [119888 (119905 + 119903119909) 2119903] minus 119888

3

1198882exp [119888 (119905 + 119903119909) 2119903] + 119888

3

)

119901 (119909 119905) =

1205821198882

1205941199032(120582 minus 1)

(

1198882exp [119888 (119905 + 119903119909) 2119903] minus 119888

3

1198882exp [119888 (119905 + 119903119909) 2119903] + 119888

3

)

120583 (119909 119905) =

1205941198652

12(119909)

119888311988821199032

(1198882exp [

119888 (119905 + 119903119909)

2119903

] + 1198883)

minus1

1205732(119909 119905) =

21198882

31199032(120582 minus 1)

times (

1198883(1 + 120582) minus 2120582119888

2exp [119888 (119905 + 119903119909) 2119903]

1198883+ 1198882exp [119888 (119905 + 119903119909) 2119903]

)

(63)

For the line element (54) using (4) (8) and (9) we have thefollowing physical properties The volume element is

119881 = 11988811199033 exp [minus

119888119905

119903

] (1198883+ 1198882exp [

119888 (119905 + 119903119909)

2119903

])

2

(64)

The expansion scalar which determines the volume behaviorof the fluid is given by

Θ = minus

1198881198883

119903

(1198883+ 1198882exp [

119888 (119905 + 119903119909)

2119903

])

minus1

(65)

The nonvanishing components of the shear tensor 120590119895119894 satisfy

1205901

1

Θ

= minus

1

3

1205902

2

Θ

= minus(

1

3

+

1198882

21198883

exp [

119888 (119905 + 119903119909)

2119903

])

1205903

3

Θ

=

2

3

+

1198882

21198883

exp [

119888 (119905 + 119903119909)

2119903

]

1205904

4

Θ

= minus

2

3

(66)

Hence the shear scalar 120590 is given by

1205902=

1198882

41199032+

111198882

31198882

361199032

(1198883+ 1198882exp [

119888 (119905 + 119903119909)

2119903

])

minus2

(67)

The model does not admit acceleration and rotation since119894= 0 and 120596

119894119895= 0

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah under Grant no130-016-D1434 The author therefore acknowledges withthanks DSR technical and financial support

References

[1] A Krasinski Inhomogeneous Cosmological Models CambridgeUniversity Press Cambridge UK 1997

[2] G Lyra ldquoUber eine Modifikation der Riemannschen Geome-trierdquoMathematische Zeitschrift vol 54 pp 52ndash64 1951

[3] D K Sen ldquoA static cosmological modelrdquo Zeitschrift fur PhysikC Particles and Fields vol 149 pp 311ndash323 1957

[4] D K Sen ldquoOn geodesics of a modified Riemannian manifoldrdquoCanadian Mathematical Society vol 3 pp 255ndash261 1960

[5] D K Sen and K A Dunn ldquoA scalar-tensor theory of gravitationin a modified Riemannian manifoldrdquo Journal of MathematicalPhysics vol 12 pp 578ndash586 1971

[6] W D Halford ldquoCosmological theory based on Lyrarsquos geometryrdquoAustralian Journal of Physics vol 23 no 5 pp 863ndash870 1970

[7] S S De and F Rahaman Finsler Geometry of Hadrons and LyraGeometry Cosmological Aspects Lambert Academic Publish-ing Saarbrucken Germany 2012

[8] A Pradhan I Aotemshi and G P Singh ldquoPlane symmetricdomain wall in Lyra geometryrdquo Astrophysics and Space Sciencevol 288 no 3 pp 315ndash325 2003

[9] A Pradhan and A K Vishwakarma ldquoA new class of LRSBianchi type-I cosmological models in Lyra geometryrdquo Journalof Geometry and Physics vol 49 no 3-4 pp 332ndash342 2004

[10] A Pradhan and S S Kumhar ldquoPlane symmetric inhomoge-neous perfect fluid universe with electromagnetic field in Lyrageometryrdquo Astrophysics and Space Science vol 321 no 2 pp137ndash146 2009

[11] A Pradhan and P Ram ldquoA plane-symmetric magnetizedinhomogeneous cosmological models of perfect fluid distribu-tion with variable magnetic permeability in Lyra geometryrdquoInternational Journal of Theoretical Physics vol 48 no 11 pp3188ndash3201 2009

[12] A Pradhan H Amirhashchi and H Zainuddin ldquoA new classof inhomogeneous cosmological models with electromagneticfield in normal gauge for Lyras manifoldrdquo International Journalof Theoretical Physics vol 50 no 1 pp 56ndash69 2011

[13] P Anirudh A Singh and R S Singh ldquoA plane-symmetric mag-netized inhomogeneous cosmological model of perfect fluiddistribution with variable magnetic permeabilityrdquo RomanianReports of Physics vol 56 no 1-2 pp 297ndash307 2011

[14] A Pradhan and A K Singh ldquoAnisotropic Bianchi type-I stringcosmological models in normal Gauge for Lyras manifoldwith constant deceleration parameterrdquo International Journal ofTheoretical Physics vol 50 no 3 pp 916ndash933 2011

[15] R Casana C AM deMelo andBM Pimentel ldquoSpinorial fieldand Lyra geometryrdquoAstrophysics and Space Science vol 305 no2 pp 125ndash132 2006

[16] F Rahaman B C Bhui and G Bag ldquoCan Lyra geometryexplain the singularity free as well as accelerating UniverserdquoAstrophysics and Space Science vol 295 no 4 pp 507ndash513 2005

[17] R Bali and N K Chandnani ldquoBianchi type-I cosmologicalmodel for perfect fluid distribution in Lyra geometryrdquo Journal of

8 Abstract and Applied Analysis

Mathematical Physics vol 49 no 3 Article ID 032502 8 pages2008

[18] R Bali andN K Chandnani ldquoBianchi typeV barotropic perfectfluid cosmological model in Lyra geometryrdquo InternationalJournal ofTheoretical Physics vol 48 no 5 pp 1523ndash1533 2009

[19] S Kumar andC P Singh ldquoAn exact Bianchi type-I cosmologicalmodel in Lyrarsquos manifoldrdquo International Journal of ModernPhysics A vol 23 no 6 pp 813ndash822 2008

[20] A K Yadav A Pradhan and A Singh ldquoLRS Bianchi type-II massive string cosmological models in general relativityrdquoRomanianReports of Physics vol 56 no 7-8 pp 1019ndash1034 2011

[21] V U M Rao T Vinutha and M V Santhi ldquoBianchi type-Vcosmological model with perfect fluid using negative constantdeceleration parameter in a scalar tensor theory based on LyraManifoldrdquo Astrophysics and Space Science vol 314 no 1ndash3 pp213ndash216 2008

[22] R Zia and R P Singh ldquoBulk viscous inhomogeneous cosmo-logical models with electromagnetic field in Lyra geometryrdquoRomanian Journal of Physics vol 57 no 3-4 pp 761ndash778 2012

[23] A T Ali and F Rahaman ldquoNew class of magnetized inho-mogeneous Bianchi type-I cosmological model with variablemagnetic permeability in Lyra geometryrdquo International Journalof Theoretical Physics vol 52 no 11 pp 4055ndash4067 2013

[24] A T Ali ldquoNew generalized Jacobi elliptic function rationalexpansion methodrdquo Journal of Computational and AppliedMathematics vol 235 no 14 pp 4117ndash4127 2011

[25] A T Ali and E R Hassan ldquoGeneral Expa-function methodfor nonlinear evolution equationsrdquo Applied Mathematics andComputation vol 217 no 2 pp 451ndash459 2010

[26] M F El-Sabbagh and A T Ali ldquoNew exact solutions for (3+1)-dimensional Kadomtsev-Petviashvili equation and generalized(2+1)-dimensional Boussinesq equationrdquo International Journalof Nonlinear Sciences andNumerical Simulation vol 6 no 2 pp151ndash162 2005

[27] M F El-Sabbagh and A T Ali ldquoNew generalized Jacobi ellipticfunction expansion methodrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 9 pp 1758ndash17662008

[28] L V Ovsiannikov Group Analysis of Differential Equationstranslated by Y Chapovsky edited by W F Ames AcademicPress New York NY USA 1982

[29] G W Bluman and S Kumei Symmetries and DifferentialEquations in Applied Sciences Springer New York NY USA1989

[30] N H Ibragimov Transformation Groups Applied to Mathemat-ical Physics D Reidel Dordrecht The Netherlands 1985

[31] P J Olver Applications of Lie Groups to Differential Equationsvol 107 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 1993

[32] A T Ali A K Yadav and S R Mahmoud ldquoSome planesymmetric inhomogeneous cosmological models in the scalar-tensor theory of gravitationrdquoAstrophysics and Space Science vol349 no 1 pp 539ndash547 2014

[33] A T Ali ldquoNew exact solutions of the Einstein vacuumequationsfor rotating axially symmetric fieldsrdquoPhysica Scripta vol 79 no3 Article ID 035006 2009

[34] S K Attallah M F El-Sabbagh and A T Ali ldquoIsovectorfields and similarity solutions of Einstein vacuum equationsfor rotating fieldsrdquo Communications in Nonlinear Science andNumerical Simulation vol 12 no 7 pp 1153ndash1161 2007

[35] K SMekheimer S Z Husseny A T Ali andA E Abo-ElkhairldquoLie point symmetries and similarity solutions for an electricallyconducting Jeffrey fluidrdquo Physica Scripta vol 83 no 1 ArticleID 015017 2011

[36] A K Raychaudhuri Theoritical Cosmology Oxford SciencePublications Oxford UK 1979

[37] A Lichnerowicz Relativistic Hydrodynamics and Magneto-Hydro-Dynamics W A Benjamin New York NY USA 1967

[38] A Pradhan and P Mathur ldquoInhomogeneous perfect fluiduniverse with electromagnetic field in Lyra geometryrdquo Fizika Bvol 18 no 4 pp 243ndash264 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

119879119894119895is the energy momentum tensor given by

119879119894119895

= (120588 + 119901) 119906119894119906119895minus 119901119892119894119895+ 119864119894119895 (11)

where 119864119894119895is the electromagnetic field given by Lichnerowicz

[37]

119864119894119895

= 120583 [ℎ119897ℎ119897(119906119894119906119895minus

1

2

119892119894119895) + ℎ119894ℎ119895] (12)

Here 120588 and 119901 are the energy density and isotropic pressurerespectively while 120583 is the magnetic permeability and ℎ

119894is

the magnetic flux vector defined by

ℎ119894=

radicminus119892

2120583

120598119894119895119896119897

119865119896119897119906119895 (13)

119865119894119895is the electromagnetic field tensor and 120598

119894119895119896119897is a Levi-Civita

tensor density If we consider the current flow along 119911-axisthen 119865

12is only nonvanishing component of 119865

119894119895 Then the

Maxwell equations

119865119894119895119896

+ 119865119895119896119894

+ 119865119896119894119895

= 0

[

1

120583

119865119894119895]

119895

= 119869119894

(14)

require 11986512

to be function of 119909 alone [38] We assume themagnetic permeability as a function of both 119909 and 119905 Here thesemicolon represents a covariant differentiation

For the line element (2) field equation (1) can be reducedto the following system of NLPDEs

1198641=

119891119909119905

119891

+

119892119909119905

119892

= 0

1198642=

119891119905119905

119891

+

119891119905119892119905

119891119892

+

1

1198602(

119892119909119909

119892

minus

119891119909119892119909

119891119892

) +

3119891119905

119860119891

= 0

(15)

120594120588 +

3

4

1205732=

119892119905119905

2119892

+

3119891119905119892119905

2119891119892

minus

1

1198602(

119891119909119909

2119891

+

119892119909119909

119892

+

3119891119909119892119909

2119891119892

)

+

119860

(

2119891119905

119891

+

119892119905

2119892

+

3

119860

)

120594119901 +

3

4

1205732=

1

21198602(

119891119909119909

119891

+

119891119909119892119909

119891119892

) minus

119891119905119905

119891

minus

119892119905119905

2119892

minus

119891119905119892119905

2119891119892

minus

119860

(

3119891119905

119891

+

119860

) minus

2

119860

1205941198652

12

12058311986041198912

=

1

1198602(

119891119909119909

119891

minus

119891119909119892119909

119891119892

) +

119892119905119905

119892

+

119891119905119892119905

119891119892

+

3119892119905

119860119892

(16)

3 Symmetry Analysis Method

Equations (15)ndash(16) are highly nonlinear PDEs and hence itis so difficult to handle since there exist no standard methodsfor obtaining analytical solutionThe system (15) is nonlinear

PDEs of second order for the two unknowns 119891 and 119892 If wesolve this system then we can get the solution of the fieldequations In order to obtain an exact solution of the systemof nonlinear PDEs (15) we will use the symmetry analysismethod For this we write

119909lowast

119894= 119909119894+ 120598120585119894(119909119895 119906120573) + o (120598

2)

119906lowast

120572= 119906120572+ 120598120578120572(119909119895 119906120573) + o (120598

2)

119894 119895 120572 120573 = 1 2 (17)

as the infinitesimal Lie point transformations We haveassumed that the system (15) is invariant under the trans-formations given in (17) The corresponding infinitesimalgenerator of Lie groups (symmetries) is given by

119883 =

2

sum

119894=1

120585119894

120597

120597119909119894

+

2

sum

120572=1

120578120572

120597

120597119906120572

(18)

where 1199091

= 119909 1199092

= 119905 1199061

= 119891 and 1199062

= 119892 The coefficients1205851 1205852 1205781 and 120578

2are the functions of 119909 119905 119891 and 119892 These

coefficients are the components of infinitesimals symmetriescorresponding to 119909 119905119891 and119892 respectively to be determinedfrom the invariance conditions

Pr(2)119883(119864119898)

10038161003816100381610038161003816119864119898=0

= 0 (19)

where 119864119898

= 0119898 = 1 2 are the system (15) under study andPr(2) is the second prolongation of the symmetries 119883 Sinceour equations (15) are at most of order two therefore weneed second order prolongation of the infinitesimal generatorin (19) It is worth noting that the 2nd order prolongation isgiven by

Pr(2)119883 = 119883 +

2

sum

119894=1

2

sum

120572=1

120578120572119894

120597

120597119906120572119894

+

2

sum

119895=1

2

sum

119894=1

2

sum

120572=1

120578120572119894119895

120597

120597119906120572119894119895

(20)

where

120578120572119894

= 119863119894[

[

120578120572minus

2

sum

119895=1

120585119895119906120572119895

]

]

+

2

sum

119895=1

120585119895119906120572119894119895

120578120572119894119895

= 119863119894119895[120578120572minus

2

sum

119896=1

120585119896119906120572119896

] +

2

sum

119896=1

120585119896119906120572119894119895119896

(21)

The operator 119863119894(119863119894119895) is called the total derivative (Hach

operator) and takes the following form

119863119894=

120597

120597119909119894

+

2

sum

120572=1

119906120572119894

120597

120597119906120572

+

2

sum

119895=1

2

sum

120572=1

119906120572119894119895

120597

120597119906120572119895

(22)

where 119863119894119895

= 119863119894(119863119895) = 119863

119895(119863119894) = 119863

119895119894and 119906

120572119894= 120597119906120572120597119909119894

Expanding the system (19) with the aid of Mathematicaprogram along with the original system (15) to eliminate 119891

119909119905

and 119892119909119909

while we set the coefficients involving 119891119909 119891119905 119891119909119909 119891119905119905

119892119909119892119905119892119909119905 and119892

119905119905and various products equal zero these gives

rise to the essential set of overdetermined equations Solvingthe set of these determining equations the components ofsymmetries take the following form

1205851= 1198881119909 + 1198882 120585

2= 1198883119905 + 1198884

1205781= 0 120578

2= 1198885119892

(23)

4 Abstract and Applied Analysis

where 119888119894 119894 = 1 2 5 are arbitrary constants and the

function 119860(119905) must equal

119860 (119905) = 11988861205852(119905) exp [minus119888

1int

119889119905

1205852(119905)

] (24)

Therefore 119860(119905) becomes

119860 (119905) = 1198886(1198883119905 + 1198884)1minus(11988811198883)

if 1198883

= 0

119860 (119905) = 1198887exp [minus

1198881

1198884

119905] if 1198883= 0

(25)

where 1198886and 1198887= 11988841198886are arbitrary constants

4 Invariant Solutions

The characteristic equations corresponding to the symme-tries (23) are given by

119889119909

1198881119909 + 1198882

=

119889119905

1198883119905 + 1198884

=

119889119891

0

=

119889119892

1198885119892

(26)

By solving the above system we have the following four cases

Case 1 When 1198881

= 0 and 1198883

= 0 the similarity variable andsimilarity functions can be written as follows

120585 =

119909 + 119886

(119905 + 119887)119888 119891 (119909 119905) = Ψ (120585)

119892 (119909 119905) = (119909 + 119886)119889Φ (120585)

(27)

where 119886 = 11988821198881 119887 = 119888

41198883 119888 = 119888

11198883 and 119889 = 119888

51198881are

arbitrary constants In this case 119860(119905) = 119902(119905 + 119887)1minus119888 where

119902 = 11988861198881minus119888

3 Substituting the transformations (27) in the field

(15) leads to the following system of ODEs

120585Ψ10158401015840+ Ψ1015840

Ψ

+

120585Φ10158401015840+ (1 + 119889)Φ

1015840

Φ

= 0 (28)

(119888211990221205852minus 1) 120585Φ

1015840Ψ1015840

ΦΨ

+

2119889Φ1015840+ 120585Φ10158401015840

Φ

+

119888211990221205853Ψ10158401015840minus [119889 + 2119888 (1 minus 2119888) 119902

21205852]Ψ1015840

Ψ

=

119889 (1 minus 119889)

120585

(29)

If one solves the system of second order NLPDEs (28)-(29)one can obtain the exact solutions of the original Einstein fieldequations (15) corresponding to reduction (30)

Case 2 When 1198881

= 0 and 1198883

= 0 the similarity variable andsimilarity functions can be written as follows

120585 = (119909 + 119886) exp [119887119905] 119891 (119909 119905) = Ψ (120585)

119892 (119909 119905) = (119909 + 119886)119888Φ (120585)

(30)

where 119886 = 11988821198881 119887 = minus119888

11198884 and 119888 = 119888

51198881are arbitrary

constants In this case 119860(119905) = 119889 exp[119887119905] where 119889 = 1198887

Substituting transformations (30) in the field equations (15)leads to the following system of ODEs

120585Ψ10158401015840+ Ψ1015840

Ψ

+

120585Φ10158401015840+ (1 + 119888)Φ

1015840

Φ

= 0 (31)

(119887211988921205852minus 1) 120585Φ

1015840Ψ1015840

ΦΨ

+

2119888Φ1015840+ 120585Φ10158401015840

Φ

+

119887211988921205853Ψ10158401015840+ (4119887211988921205852minus 119888)Ψ

1015840

Ψ

=

119888 (1 minus 119888)

120585

(32)

If one solves the system of second order NLPDEs (31)-(32)one can obtain the exact solutions of the original Einstein fieldequations (15) corresponding to reduction (30)

Case 3 When 1198881

= 0 and 1198883

= 0 the similarity variable andsimilarity functions can be written as follows

120585 =

exp [119886119909]

119905 + 119887

119891 (119909 119905) = Ψ (120585)

119892 (119909 119905) = Φ (120585) exp [119886119909]

(33)

where 119886 = 11988831198882and 119887 = 119888

41198863are arbitrary constants In this

case we have 119860(119905) = 119888(119905 + 119887) where 119888 = 1198886 Substituting the

transformations (33) in the field equations (15) leads to thefollowing system of ODEs

120585Ψ10158401015840+ Ψ1015840

Ψ

+

120585Φ10158401015840+ 2Φ1015840

Φ

= 0

120585 [

(1198882minus 1198862) 120585Φ1015840Ψ1015840

ΦΨ

+

1198862(3Φ1015840+ 120585Φ10158401015840)

Φ

+

1198882120585Ψ10158401015840minus (1198882+ 1198862)Ψ1015840

Ψ

] + 1198862= 0

(34)

Without loss of generality we can take the following usefultransformation

120585 = exp [120579]

119889Ψ

119889120585

= exp [minus120579]

119889Ψ

119889120579

1198892Ψ

1198891205852

= exp [minus2120579] (

1198892Ψ

1198891205792

minus

119889Ψ

119889120579

)

(35)

Then the system of ODEs (34) transforms to

Ψ

Ψ

+

Φ + Φ

Φ

= 0 (36)

(1198882minus 1198862) ΦΨ

ΦΨ

+

1198862(2Φ + Φ)

Φ

+

1198882Ψ minus (2119888

2+ 1198862) Ψ

Ψ

+ 1198862= 0

(37)

Equation (36) can be written in the following form

Ψ = minus

Ψ

Φ

(Φ + Φ) (38)

Abstract and Applied Analysis 5

From the above equation if we substitute Ψ in (37) we canobtain the following form

(1198882minus 1198862) [

ΦΨ

ΦΨ

minus

Φ

Φ

] + (21198862minus 1198882)(

Φ

Φ

)

minus (21198882+ 1198862)(

Ψ

Ψ

) + 1198862= 0

(39)

Equation (39) is a nonlinear ODE which is very difficult tosolve However it is worth noting that this equation is easyto solve when 119888 = 119886 In this case we can integrate (39) andobtain the following

Φ (120579) = 1198861Ψ3(120579) exp [minus120579] (40)

where 1198861is an arbitrary constant of integration Substituting

(40) in (36) we have the following ODE of the function Ψ

only

Ψ(4Ψ minus 3Ψ) + 6Ψ2= 0 (41)

The general solution of the above equation is

Ψ (120579) = 1198863(1198862+ exp [

3120579

4

])

25

(42)

where 1198862and 1198863are arbitrary constants of integration Now

by using (40) and the inverse of transformations (35) and (33)we can find the solution as follows

119860 (119905) = 119886 (119905 + 119887)

119891 (119909 119905) = 1198863(1198862+ [

exp [119886119909]

119905 + 119887

]

34

)

25

119892 (119909 119905) = 11988611198863(119905 + 119887) (119886

2+ [

exp [119886119909]

119905 + 119887

]

34

)

65

(43)

It is observed from (43) and (3) that the line element (2) canbe written in the following form

1198891199042= 1198891199052minus 1198862(119905 + 119887)

21198891199092minus 1198892(119905 + 119887)

2

times (1198862+ [

exp [119886119909]

119905 + 119887

]

34

)

45

1198891199102

minus 1199022(119905 + 119887)

4(1198862+ [

exp [119886119909]

119905 + 119887

]

34

)

125

1198891199112

(44)

where 119886 119887 119889 = 1198861198863 119902 = 119886119886

11198863 and 119886

2are arbitrary constants

Case 4 When 1198881

= 1198883

= 0 the similarity variable andsimilarity functions can be written as follows

120585 = 119886119909 + 119887119905 119891 (119909 119905) = Ψ (120585)

119892 (119909 119905) = Φ (120585) exp [119888119909]

(45)

where 119886 = 1198884 119887 = minus119888

2 and 119888 = 119888

51198882are arbitrary constants In

this case we have119860(119905) = 1198887 Substituting transformations (45)

in field equations (15) leads to the following system of ODEs

119886Ψ10158401015840

Ψ

+

119886Φ10158401015840+ 119888Φ1015840

Φ

= 0 (46)

(11988721199032minus 1198862)Φ1015840Ψ1015840

ΦΨ

+

119886 (2119888Φ1015840+ 119886Φ10158401015840)

Φ

+

11988721199032Ψ10158401015840minus 119886119888Ψ

1015840

Ψ

+ 1198882= 0

(47)

Equation (46) can be written in the following form

Ψ10158401015840

= minus

Ψ

119886Φ

(119886Φ10158401015840+ 119888Φ1015840) (48)

From the above equation if we substitute Ψ in (47) we canobtain the following form

(11988721199032minus 1198862) [

Φ1015840Ψ1015840

ΦΨ

minus

Φ10158401015840

Φ

] + (21198862minus 11988721199032)(

119888Φ1015840

119886Φ

)

minus 119886119888(

Ψ1015840

Ψ

) + 1198882= 0

(49)

Equation (49) is a nonlinear ODE which is very difficult tosolve However it is worth noting that this equation is easyto solve when 119886 = 119887119903 In this case we can integrate (49) andobtain the following

Φ (120585) = 1198861Ψ (120585) exp [minus

119888120585

119887119903

] (50)

where 1198861is an arbitrary constant of integration Substituting

(50) in (46) we have the following ODE of function Ψ only

2119887119903Ψ10158401015840

= 119888Ψ (51)

The general solution of the above equation is

Ψ (120585) = 1198863+ 1198862exp [

119888120585

2119887119903

] (52)

where 1198862and 1198863are arbitrary constants of integration Now

by using (50) and (33) we can find the solution as follows

119860 (119905) = 119903

119891 (119909 119905) = 1198863+ 1198862exp [

119888 (119905 + 119903119909)

2119887119903

]

119892 (119909 119905) = 1198861exp [minus

119888119905

119903

] (1198863+ 1198862exp [

119888 (119905 + 119903119909)

2119887119903

])

(53)

It is observed from (53) and (3) that the line element (2) canbe written in the following form

1198891199042= 1198891199052minus 11990321198891199092minus 1199032(1198863+ 1198862exp [

119888 (119905 + 119903119909)

2119887119903

])

2

times (1198891199102+ 1198862

1exp [minus

2119888119905

119903

] 1198891199112)

(54)

where 119903 119888 119887 1198861 1198862 and 119886

3are arbitrary constants

6 Abstract and Applied Analysis

5 Physical Properties of the Model

The field equations (15)ndash(16) constitute a system of fivehighly nonlinear differential equations with seven unknownsvariables 119860 119891 119892 119901 120588 120583 and 120573 The symmetries giveone condition (24) for function 119860 Therefore one physicallyreasonable condition amongst these parameters is required toobtain explicit solutions of the field equations Let us assumethat the density 120588 and the pressure 119901 are related by barotropicequation of state

119901 = 120582120588 0 le 120582 le 1 (55)

51 ForModel (44) Using (43) in the Einstein field equations(16) with taking into account condition (55) the expressionsfor density 120588 pressure 119901 magnetic permeability 120583 anddisplacement field 120573 are given by

120588 (119909 119905) =

9

5120594 (1 minus 120582)

times[

[

51198862+ [exp [119886119909] (119905 + 119887)]

34

(119905 + 119887)2(1198862+ [exp [119886119909] (119905 + 119887)]

34

)

]

]

119901 (119909 119905) =

9120582

5120594 (1 minus 120582)

times[

[

51198862+ [exp [119886119909] (119905 + 119887)]

34

(119905 + 119887)2(1198862+ [exp [119886119909] (119905 + 119887)]

34

)

]

]

120583 (119909 119905) =

1205941198652

12(119909)

3119886211988621198892(119905 + 119887)

2(1198862+ [

exp [119886119909]

119905 + 119887

]

34

)

15

1205732(119909 119905) =

2

15

[

[

(51198862(5 + 13120582)

+ 2 (2 + 7120582) [

exp [119886119909]

119905 + 119887

]

34

)

times ((120582 minus 1) (119905 + 119887)2

times(1198862+ [

exp [119886119909]

119905 + 119887

]

34

))

minus1

]

(56)

For the line element (44) using (4) (8) and (9) we have thefollowing physical properties The volume element is

119881 = 119886119889119902(119905 + 119887)4(1198862+ [

exp [119886119909]

119905 + 119887

]

34

)

85

(57)

The expansion scalar which determines the volume behaviorof the fluid is given by

Θ =

2

5

[

[

101198862+ 7[exp [119886119909] (119905 + 119887)]

34

(119905 + 119887) (1198862+ [exp [119886119909] (119905 + 119887)]

34

)

]

]

(58)

The nonvanishing components of the shear tensor 120590119895119894 are

1205901

1=

[exp [119886119909] (119905 + 119887)]34

minus 51198862

15 (119905 + 119887) (1198862+ [exp [119886119909] (119905 + 119887)]

34

)

1205902

2= minus

1

30

[

[

101198862+ [exp [119886119909] (119905 + 119887)]

34

(119905 + 119887) (1198862+ [exp [119886119909] (119905 + 119887)]

34

)

]

]

1205903

3=

41198862+ 7[exp [119886119909] (119905 + 119887)]

34

6 (119905 + 119887) (1198862+ [exp [119886119909] (119905 + 119887)]

34

)

1205904

4= minus

4

15

[

[

101198862+ 7[exp [119886119909] (119905 + 119887)]

34

(119905 + 119887) (1198862+ [exp [119886119909] (119905 + 119887)]

34

)

]

]

(59)

Hence the shear scalar 120590 is given by

1205902= (3500119886

2

2+ 4630119886

2[exp [119886119909] (119905 + 119887)]

34

+1607[exp [119886119909] (119905 + 119887)]32

)

times (900(119905 + 119887)2(1198862+ [exp [119886119909] (119905 + 119887)]

34

)

2

)

minus1

(60)

The model does not admit acceleration and rotation since119894= 0 and 120596

119894119895= 0 We can see that

1205904

4

Θ

= minus

2

3

(61)

which is a constant of proportional We found also that

lim119905rarrinfin

1205902

Θ

=

5radic35

2

= 0 (62)

and this means that there is no possibility that the universemay got isotropized in some later time that is it remainsanisotropic for all times

52 ForModel (54) Using (53) in the Einstein field equations(16) with taking into account condition (55) the expressions

Abstract and Applied Analysis 7

for density 120588 pressure 119901 magnetic permeability 120583 anddisplacement field 120573 are given by

120588 (119909 119905) =

1198882

1205941199032(120582 minus 1)

(

1198882exp [119888 (119905 + 119903119909) 2119903] minus 119888

3

1198882exp [119888 (119905 + 119903119909) 2119903] + 119888

3

)

119901 (119909 119905) =

1205821198882

1205941199032(120582 minus 1)

(

1198882exp [119888 (119905 + 119903119909) 2119903] minus 119888

3

1198882exp [119888 (119905 + 119903119909) 2119903] + 119888

3

)

120583 (119909 119905) =

1205941198652

12(119909)

119888311988821199032

(1198882exp [

119888 (119905 + 119903119909)

2119903

] + 1198883)

minus1

1205732(119909 119905) =

21198882

31199032(120582 minus 1)

times (

1198883(1 + 120582) minus 2120582119888

2exp [119888 (119905 + 119903119909) 2119903]

1198883+ 1198882exp [119888 (119905 + 119903119909) 2119903]

)

(63)

For the line element (54) using (4) (8) and (9) we have thefollowing physical properties The volume element is

119881 = 11988811199033 exp [minus

119888119905

119903

] (1198883+ 1198882exp [

119888 (119905 + 119903119909)

2119903

])

2

(64)

The expansion scalar which determines the volume behaviorof the fluid is given by

Θ = minus

1198881198883

119903

(1198883+ 1198882exp [

119888 (119905 + 119903119909)

2119903

])

minus1

(65)

The nonvanishing components of the shear tensor 120590119895119894 satisfy

1205901

1

Θ

= minus

1

3

1205902

2

Θ

= minus(

1

3

+

1198882

21198883

exp [

119888 (119905 + 119903119909)

2119903

])

1205903

3

Θ

=

2

3

+

1198882

21198883

exp [

119888 (119905 + 119903119909)

2119903

]

1205904

4

Θ

= minus

2

3

(66)

Hence the shear scalar 120590 is given by

1205902=

1198882

41199032+

111198882

31198882

361199032

(1198883+ 1198882exp [

119888 (119905 + 119903119909)

2119903

])

minus2

(67)

The model does not admit acceleration and rotation since119894= 0 and 120596

119894119895= 0

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah under Grant no130-016-D1434 The author therefore acknowledges withthanks DSR technical and financial support

References

[1] A Krasinski Inhomogeneous Cosmological Models CambridgeUniversity Press Cambridge UK 1997

[2] G Lyra ldquoUber eine Modifikation der Riemannschen Geome-trierdquoMathematische Zeitschrift vol 54 pp 52ndash64 1951

[3] D K Sen ldquoA static cosmological modelrdquo Zeitschrift fur PhysikC Particles and Fields vol 149 pp 311ndash323 1957

[4] D K Sen ldquoOn geodesics of a modified Riemannian manifoldrdquoCanadian Mathematical Society vol 3 pp 255ndash261 1960

[5] D K Sen and K A Dunn ldquoA scalar-tensor theory of gravitationin a modified Riemannian manifoldrdquo Journal of MathematicalPhysics vol 12 pp 578ndash586 1971

[6] W D Halford ldquoCosmological theory based on Lyrarsquos geometryrdquoAustralian Journal of Physics vol 23 no 5 pp 863ndash870 1970

[7] S S De and F Rahaman Finsler Geometry of Hadrons and LyraGeometry Cosmological Aspects Lambert Academic Publish-ing Saarbrucken Germany 2012

[8] A Pradhan I Aotemshi and G P Singh ldquoPlane symmetricdomain wall in Lyra geometryrdquo Astrophysics and Space Sciencevol 288 no 3 pp 315ndash325 2003

[9] A Pradhan and A K Vishwakarma ldquoA new class of LRSBianchi type-I cosmological models in Lyra geometryrdquo Journalof Geometry and Physics vol 49 no 3-4 pp 332ndash342 2004

[10] A Pradhan and S S Kumhar ldquoPlane symmetric inhomoge-neous perfect fluid universe with electromagnetic field in Lyrageometryrdquo Astrophysics and Space Science vol 321 no 2 pp137ndash146 2009

[11] A Pradhan and P Ram ldquoA plane-symmetric magnetizedinhomogeneous cosmological models of perfect fluid distribu-tion with variable magnetic permeability in Lyra geometryrdquoInternational Journal of Theoretical Physics vol 48 no 11 pp3188ndash3201 2009

[12] A Pradhan H Amirhashchi and H Zainuddin ldquoA new classof inhomogeneous cosmological models with electromagneticfield in normal gauge for Lyras manifoldrdquo International Journalof Theoretical Physics vol 50 no 1 pp 56ndash69 2011

[13] P Anirudh A Singh and R S Singh ldquoA plane-symmetric mag-netized inhomogeneous cosmological model of perfect fluiddistribution with variable magnetic permeabilityrdquo RomanianReports of Physics vol 56 no 1-2 pp 297ndash307 2011

[14] A Pradhan and A K Singh ldquoAnisotropic Bianchi type-I stringcosmological models in normal Gauge for Lyras manifoldwith constant deceleration parameterrdquo International Journal ofTheoretical Physics vol 50 no 3 pp 916ndash933 2011

[15] R Casana C AM deMelo andBM Pimentel ldquoSpinorial fieldand Lyra geometryrdquoAstrophysics and Space Science vol 305 no2 pp 125ndash132 2006

[16] F Rahaman B C Bhui and G Bag ldquoCan Lyra geometryexplain the singularity free as well as accelerating UniverserdquoAstrophysics and Space Science vol 295 no 4 pp 507ndash513 2005

[17] R Bali and N K Chandnani ldquoBianchi type-I cosmologicalmodel for perfect fluid distribution in Lyra geometryrdquo Journal of

8 Abstract and Applied Analysis

Mathematical Physics vol 49 no 3 Article ID 032502 8 pages2008

[18] R Bali andN K Chandnani ldquoBianchi typeV barotropic perfectfluid cosmological model in Lyra geometryrdquo InternationalJournal ofTheoretical Physics vol 48 no 5 pp 1523ndash1533 2009

[19] S Kumar andC P Singh ldquoAn exact Bianchi type-I cosmologicalmodel in Lyrarsquos manifoldrdquo International Journal of ModernPhysics A vol 23 no 6 pp 813ndash822 2008

[20] A K Yadav A Pradhan and A Singh ldquoLRS Bianchi type-II massive string cosmological models in general relativityrdquoRomanianReports of Physics vol 56 no 7-8 pp 1019ndash1034 2011

[21] V U M Rao T Vinutha and M V Santhi ldquoBianchi type-Vcosmological model with perfect fluid using negative constantdeceleration parameter in a scalar tensor theory based on LyraManifoldrdquo Astrophysics and Space Science vol 314 no 1ndash3 pp213ndash216 2008

[22] R Zia and R P Singh ldquoBulk viscous inhomogeneous cosmo-logical models with electromagnetic field in Lyra geometryrdquoRomanian Journal of Physics vol 57 no 3-4 pp 761ndash778 2012

[23] A T Ali and F Rahaman ldquoNew class of magnetized inho-mogeneous Bianchi type-I cosmological model with variablemagnetic permeability in Lyra geometryrdquo International Journalof Theoretical Physics vol 52 no 11 pp 4055ndash4067 2013

[24] A T Ali ldquoNew generalized Jacobi elliptic function rationalexpansion methodrdquo Journal of Computational and AppliedMathematics vol 235 no 14 pp 4117ndash4127 2011

[25] A T Ali and E R Hassan ldquoGeneral Expa-function methodfor nonlinear evolution equationsrdquo Applied Mathematics andComputation vol 217 no 2 pp 451ndash459 2010

[26] M F El-Sabbagh and A T Ali ldquoNew exact solutions for (3+1)-dimensional Kadomtsev-Petviashvili equation and generalized(2+1)-dimensional Boussinesq equationrdquo International Journalof Nonlinear Sciences andNumerical Simulation vol 6 no 2 pp151ndash162 2005

[27] M F El-Sabbagh and A T Ali ldquoNew generalized Jacobi ellipticfunction expansion methodrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 9 pp 1758ndash17662008

[28] L V Ovsiannikov Group Analysis of Differential Equationstranslated by Y Chapovsky edited by W F Ames AcademicPress New York NY USA 1982

[29] G W Bluman and S Kumei Symmetries and DifferentialEquations in Applied Sciences Springer New York NY USA1989

[30] N H Ibragimov Transformation Groups Applied to Mathemat-ical Physics D Reidel Dordrecht The Netherlands 1985

[31] P J Olver Applications of Lie Groups to Differential Equationsvol 107 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 1993

[32] A T Ali A K Yadav and S R Mahmoud ldquoSome planesymmetric inhomogeneous cosmological models in the scalar-tensor theory of gravitationrdquoAstrophysics and Space Science vol349 no 1 pp 539ndash547 2014

[33] A T Ali ldquoNew exact solutions of the Einstein vacuumequationsfor rotating axially symmetric fieldsrdquoPhysica Scripta vol 79 no3 Article ID 035006 2009

[34] S K Attallah M F El-Sabbagh and A T Ali ldquoIsovectorfields and similarity solutions of Einstein vacuum equationsfor rotating fieldsrdquo Communications in Nonlinear Science andNumerical Simulation vol 12 no 7 pp 1153ndash1161 2007

[35] K SMekheimer S Z Husseny A T Ali andA E Abo-ElkhairldquoLie point symmetries and similarity solutions for an electricallyconducting Jeffrey fluidrdquo Physica Scripta vol 83 no 1 ArticleID 015017 2011

[36] A K Raychaudhuri Theoritical Cosmology Oxford SciencePublications Oxford UK 1979

[37] A Lichnerowicz Relativistic Hydrodynamics and Magneto-Hydro-Dynamics W A Benjamin New York NY USA 1967

[38] A Pradhan and P Mathur ldquoInhomogeneous perfect fluiduniverse with electromagnetic field in Lyra geometryrdquo Fizika Bvol 18 no 4 pp 243ndash264 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

where 119888119894 119894 = 1 2 5 are arbitrary constants and the

function 119860(119905) must equal

119860 (119905) = 11988861205852(119905) exp [minus119888

1int

119889119905

1205852(119905)

] (24)

Therefore 119860(119905) becomes

119860 (119905) = 1198886(1198883119905 + 1198884)1minus(11988811198883)

if 1198883

= 0

119860 (119905) = 1198887exp [minus

1198881

1198884

119905] if 1198883= 0

(25)

where 1198886and 1198887= 11988841198886are arbitrary constants

4 Invariant Solutions

The characteristic equations corresponding to the symme-tries (23) are given by

119889119909

1198881119909 + 1198882

=

119889119905

1198883119905 + 1198884

=

119889119891

0

=

119889119892

1198885119892

(26)

By solving the above system we have the following four cases

Case 1 When 1198881

= 0 and 1198883

= 0 the similarity variable andsimilarity functions can be written as follows

120585 =

119909 + 119886

(119905 + 119887)119888 119891 (119909 119905) = Ψ (120585)

119892 (119909 119905) = (119909 + 119886)119889Φ (120585)

(27)

where 119886 = 11988821198881 119887 = 119888

41198883 119888 = 119888

11198883 and 119889 = 119888

51198881are

arbitrary constants In this case 119860(119905) = 119902(119905 + 119887)1minus119888 where

119902 = 11988861198881minus119888

3 Substituting the transformations (27) in the field

(15) leads to the following system of ODEs

120585Ψ10158401015840+ Ψ1015840

Ψ

+

120585Φ10158401015840+ (1 + 119889)Φ

1015840

Φ

= 0 (28)

(119888211990221205852minus 1) 120585Φ

1015840Ψ1015840

ΦΨ

+

2119889Φ1015840+ 120585Φ10158401015840

Φ

+

119888211990221205853Ψ10158401015840minus [119889 + 2119888 (1 minus 2119888) 119902

21205852]Ψ1015840

Ψ

=

119889 (1 minus 119889)

120585

(29)

If one solves the system of second order NLPDEs (28)-(29)one can obtain the exact solutions of the original Einstein fieldequations (15) corresponding to reduction (30)

Case 2 When 1198881

= 0 and 1198883

= 0 the similarity variable andsimilarity functions can be written as follows

120585 = (119909 + 119886) exp [119887119905] 119891 (119909 119905) = Ψ (120585)

119892 (119909 119905) = (119909 + 119886)119888Φ (120585)

(30)

where 119886 = 11988821198881 119887 = minus119888

11198884 and 119888 = 119888

51198881are arbitrary

constants In this case 119860(119905) = 119889 exp[119887119905] where 119889 = 1198887

Substituting transformations (30) in the field equations (15)leads to the following system of ODEs

120585Ψ10158401015840+ Ψ1015840

Ψ

+

120585Φ10158401015840+ (1 + 119888)Φ

1015840

Φ

= 0 (31)

(119887211988921205852minus 1) 120585Φ

1015840Ψ1015840

ΦΨ

+

2119888Φ1015840+ 120585Φ10158401015840

Φ

+

119887211988921205853Ψ10158401015840+ (4119887211988921205852minus 119888)Ψ

1015840

Ψ

=

119888 (1 minus 119888)

120585

(32)

If one solves the system of second order NLPDEs (31)-(32)one can obtain the exact solutions of the original Einstein fieldequations (15) corresponding to reduction (30)

Case 3 When 1198881

= 0 and 1198883

= 0 the similarity variable andsimilarity functions can be written as follows

120585 =

exp [119886119909]

119905 + 119887

119891 (119909 119905) = Ψ (120585)

119892 (119909 119905) = Φ (120585) exp [119886119909]

(33)

where 119886 = 11988831198882and 119887 = 119888

41198863are arbitrary constants In this

case we have 119860(119905) = 119888(119905 + 119887) where 119888 = 1198886 Substituting the

transformations (33) in the field equations (15) leads to thefollowing system of ODEs

120585Ψ10158401015840+ Ψ1015840

Ψ

+

120585Φ10158401015840+ 2Φ1015840

Φ

= 0

120585 [

(1198882minus 1198862) 120585Φ1015840Ψ1015840

ΦΨ

+

1198862(3Φ1015840+ 120585Φ10158401015840)

Φ

+

1198882120585Ψ10158401015840minus (1198882+ 1198862)Ψ1015840

Ψ

] + 1198862= 0

(34)

Without loss of generality we can take the following usefultransformation

120585 = exp [120579]

119889Ψ

119889120585

= exp [minus120579]

119889Ψ

119889120579

1198892Ψ

1198891205852

= exp [minus2120579] (

1198892Ψ

1198891205792

minus

119889Ψ

119889120579

)

(35)

Then the system of ODEs (34) transforms to

Ψ

Ψ

+

Φ + Φ

Φ

= 0 (36)

(1198882minus 1198862) ΦΨ

ΦΨ

+

1198862(2Φ + Φ)

Φ

+

1198882Ψ minus (2119888

2+ 1198862) Ψ

Ψ

+ 1198862= 0

(37)

Equation (36) can be written in the following form

Ψ = minus

Ψ

Φ

(Φ + Φ) (38)

Abstract and Applied Analysis 5

From the above equation if we substitute Ψ in (37) we canobtain the following form

(1198882minus 1198862) [

ΦΨ

ΦΨ

minus

Φ

Φ

] + (21198862minus 1198882)(

Φ

Φ

)

minus (21198882+ 1198862)(

Ψ

Ψ

) + 1198862= 0

(39)

Equation (39) is a nonlinear ODE which is very difficult tosolve However it is worth noting that this equation is easyto solve when 119888 = 119886 In this case we can integrate (39) andobtain the following

Φ (120579) = 1198861Ψ3(120579) exp [minus120579] (40)

where 1198861is an arbitrary constant of integration Substituting

(40) in (36) we have the following ODE of the function Ψ

only

Ψ(4Ψ minus 3Ψ) + 6Ψ2= 0 (41)

The general solution of the above equation is

Ψ (120579) = 1198863(1198862+ exp [

3120579

4

])

25

(42)

where 1198862and 1198863are arbitrary constants of integration Now

by using (40) and the inverse of transformations (35) and (33)we can find the solution as follows

119860 (119905) = 119886 (119905 + 119887)

119891 (119909 119905) = 1198863(1198862+ [

exp [119886119909]

119905 + 119887

]

34

)

25

119892 (119909 119905) = 11988611198863(119905 + 119887) (119886

2+ [

exp [119886119909]

119905 + 119887

]

34

)

65

(43)

It is observed from (43) and (3) that the line element (2) canbe written in the following form

1198891199042= 1198891199052minus 1198862(119905 + 119887)

21198891199092minus 1198892(119905 + 119887)

2

times (1198862+ [

exp [119886119909]

119905 + 119887

]

34

)

45

1198891199102

minus 1199022(119905 + 119887)

4(1198862+ [

exp [119886119909]

119905 + 119887

]

34

)

125

1198891199112

(44)

where 119886 119887 119889 = 1198861198863 119902 = 119886119886

11198863 and 119886

2are arbitrary constants

Case 4 When 1198881

= 1198883

= 0 the similarity variable andsimilarity functions can be written as follows

120585 = 119886119909 + 119887119905 119891 (119909 119905) = Ψ (120585)

119892 (119909 119905) = Φ (120585) exp [119888119909]

(45)

where 119886 = 1198884 119887 = minus119888

2 and 119888 = 119888

51198882are arbitrary constants In

this case we have119860(119905) = 1198887 Substituting transformations (45)

in field equations (15) leads to the following system of ODEs

119886Ψ10158401015840

Ψ

+

119886Φ10158401015840+ 119888Φ1015840

Φ

= 0 (46)

(11988721199032minus 1198862)Φ1015840Ψ1015840

ΦΨ

+

119886 (2119888Φ1015840+ 119886Φ10158401015840)

Φ

+

11988721199032Ψ10158401015840minus 119886119888Ψ

1015840

Ψ

+ 1198882= 0

(47)

Equation (46) can be written in the following form

Ψ10158401015840

= minus

Ψ

119886Φ

(119886Φ10158401015840+ 119888Φ1015840) (48)

From the above equation if we substitute Ψ in (47) we canobtain the following form

(11988721199032minus 1198862) [

Φ1015840Ψ1015840

ΦΨ

minus

Φ10158401015840

Φ

] + (21198862minus 11988721199032)(

119888Φ1015840

119886Φ

)

minus 119886119888(

Ψ1015840

Ψ

) + 1198882= 0

(49)

Equation (49) is a nonlinear ODE which is very difficult tosolve However it is worth noting that this equation is easyto solve when 119886 = 119887119903 In this case we can integrate (49) andobtain the following

Φ (120585) = 1198861Ψ (120585) exp [minus

119888120585

119887119903

] (50)

where 1198861is an arbitrary constant of integration Substituting

(50) in (46) we have the following ODE of function Ψ only

2119887119903Ψ10158401015840

= 119888Ψ (51)

The general solution of the above equation is

Ψ (120585) = 1198863+ 1198862exp [

119888120585

2119887119903

] (52)

where 1198862and 1198863are arbitrary constants of integration Now

by using (50) and (33) we can find the solution as follows

119860 (119905) = 119903

119891 (119909 119905) = 1198863+ 1198862exp [

119888 (119905 + 119903119909)

2119887119903

]

119892 (119909 119905) = 1198861exp [minus

119888119905

119903

] (1198863+ 1198862exp [

119888 (119905 + 119903119909)

2119887119903

])

(53)

It is observed from (53) and (3) that the line element (2) canbe written in the following form

1198891199042= 1198891199052minus 11990321198891199092minus 1199032(1198863+ 1198862exp [

119888 (119905 + 119903119909)

2119887119903

])

2

times (1198891199102+ 1198862

1exp [minus

2119888119905

119903

] 1198891199112)

(54)

where 119903 119888 119887 1198861 1198862 and 119886

3are arbitrary constants

6 Abstract and Applied Analysis

5 Physical Properties of the Model

The field equations (15)ndash(16) constitute a system of fivehighly nonlinear differential equations with seven unknownsvariables 119860 119891 119892 119901 120588 120583 and 120573 The symmetries giveone condition (24) for function 119860 Therefore one physicallyreasonable condition amongst these parameters is required toobtain explicit solutions of the field equations Let us assumethat the density 120588 and the pressure 119901 are related by barotropicequation of state

119901 = 120582120588 0 le 120582 le 1 (55)

51 ForModel (44) Using (43) in the Einstein field equations(16) with taking into account condition (55) the expressionsfor density 120588 pressure 119901 magnetic permeability 120583 anddisplacement field 120573 are given by

120588 (119909 119905) =

9

5120594 (1 minus 120582)

times[

[

51198862+ [exp [119886119909] (119905 + 119887)]

34

(119905 + 119887)2(1198862+ [exp [119886119909] (119905 + 119887)]

34

)

]

]

119901 (119909 119905) =

9120582

5120594 (1 minus 120582)

times[

[

51198862+ [exp [119886119909] (119905 + 119887)]

34

(119905 + 119887)2(1198862+ [exp [119886119909] (119905 + 119887)]

34

)

]

]

120583 (119909 119905) =

1205941198652

12(119909)

3119886211988621198892(119905 + 119887)

2(1198862+ [

exp [119886119909]

119905 + 119887

]

34

)

15

1205732(119909 119905) =

2

15

[

[

(51198862(5 + 13120582)

+ 2 (2 + 7120582) [

exp [119886119909]

119905 + 119887

]

34

)

times ((120582 minus 1) (119905 + 119887)2

times(1198862+ [

exp [119886119909]

119905 + 119887

]

34

))

minus1

]

(56)

For the line element (44) using (4) (8) and (9) we have thefollowing physical properties The volume element is

119881 = 119886119889119902(119905 + 119887)4(1198862+ [

exp [119886119909]

119905 + 119887

]

34

)

85

(57)

The expansion scalar which determines the volume behaviorof the fluid is given by

Θ =

2

5

[

[

101198862+ 7[exp [119886119909] (119905 + 119887)]

34

(119905 + 119887) (1198862+ [exp [119886119909] (119905 + 119887)]

34

)

]

]

(58)

The nonvanishing components of the shear tensor 120590119895119894 are

1205901

1=

[exp [119886119909] (119905 + 119887)]34

minus 51198862

15 (119905 + 119887) (1198862+ [exp [119886119909] (119905 + 119887)]

34

)

1205902

2= minus

1

30

[

[

101198862+ [exp [119886119909] (119905 + 119887)]

34

(119905 + 119887) (1198862+ [exp [119886119909] (119905 + 119887)]

34

)

]

]

1205903

3=

41198862+ 7[exp [119886119909] (119905 + 119887)]

34

6 (119905 + 119887) (1198862+ [exp [119886119909] (119905 + 119887)]

34

)

1205904

4= minus

4

15

[

[

101198862+ 7[exp [119886119909] (119905 + 119887)]

34

(119905 + 119887) (1198862+ [exp [119886119909] (119905 + 119887)]

34

)

]

]

(59)

Hence the shear scalar 120590 is given by

1205902= (3500119886

2

2+ 4630119886

2[exp [119886119909] (119905 + 119887)]

34

+1607[exp [119886119909] (119905 + 119887)]32

)

times (900(119905 + 119887)2(1198862+ [exp [119886119909] (119905 + 119887)]

34

)

2

)

minus1

(60)

The model does not admit acceleration and rotation since119894= 0 and 120596

119894119895= 0 We can see that

1205904

4

Θ

= minus

2

3

(61)

which is a constant of proportional We found also that

lim119905rarrinfin

1205902

Θ

=

5radic35

2

= 0 (62)

and this means that there is no possibility that the universemay got isotropized in some later time that is it remainsanisotropic for all times

52 ForModel (54) Using (53) in the Einstein field equations(16) with taking into account condition (55) the expressions

Abstract and Applied Analysis 7

for density 120588 pressure 119901 magnetic permeability 120583 anddisplacement field 120573 are given by

120588 (119909 119905) =

1198882

1205941199032(120582 minus 1)

(

1198882exp [119888 (119905 + 119903119909) 2119903] minus 119888

3

1198882exp [119888 (119905 + 119903119909) 2119903] + 119888

3

)

119901 (119909 119905) =

1205821198882

1205941199032(120582 minus 1)

(

1198882exp [119888 (119905 + 119903119909) 2119903] minus 119888

3

1198882exp [119888 (119905 + 119903119909) 2119903] + 119888

3

)

120583 (119909 119905) =

1205941198652

12(119909)

119888311988821199032

(1198882exp [

119888 (119905 + 119903119909)

2119903

] + 1198883)

minus1

1205732(119909 119905) =

21198882

31199032(120582 minus 1)

times (

1198883(1 + 120582) minus 2120582119888

2exp [119888 (119905 + 119903119909) 2119903]

1198883+ 1198882exp [119888 (119905 + 119903119909) 2119903]

)

(63)

For the line element (54) using (4) (8) and (9) we have thefollowing physical properties The volume element is

119881 = 11988811199033 exp [minus

119888119905

119903

] (1198883+ 1198882exp [

119888 (119905 + 119903119909)

2119903

])

2

(64)

The expansion scalar which determines the volume behaviorof the fluid is given by

Θ = minus

1198881198883

119903

(1198883+ 1198882exp [

119888 (119905 + 119903119909)

2119903

])

minus1

(65)

The nonvanishing components of the shear tensor 120590119895119894 satisfy

1205901

1

Θ

= minus

1

3

1205902

2

Θ

= minus(

1

3

+

1198882

21198883

exp [

119888 (119905 + 119903119909)

2119903

])

1205903

3

Θ

=

2

3

+

1198882

21198883

exp [

119888 (119905 + 119903119909)

2119903

]

1205904

4

Θ

= minus

2

3

(66)

Hence the shear scalar 120590 is given by

1205902=

1198882

41199032+

111198882

31198882

361199032

(1198883+ 1198882exp [

119888 (119905 + 119903119909)

2119903

])

minus2

(67)

The model does not admit acceleration and rotation since119894= 0 and 120596

119894119895= 0

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah under Grant no130-016-D1434 The author therefore acknowledges withthanks DSR technical and financial support

References

[1] A Krasinski Inhomogeneous Cosmological Models CambridgeUniversity Press Cambridge UK 1997

[2] G Lyra ldquoUber eine Modifikation der Riemannschen Geome-trierdquoMathematische Zeitschrift vol 54 pp 52ndash64 1951

[3] D K Sen ldquoA static cosmological modelrdquo Zeitschrift fur PhysikC Particles and Fields vol 149 pp 311ndash323 1957

[4] D K Sen ldquoOn geodesics of a modified Riemannian manifoldrdquoCanadian Mathematical Society vol 3 pp 255ndash261 1960

[5] D K Sen and K A Dunn ldquoA scalar-tensor theory of gravitationin a modified Riemannian manifoldrdquo Journal of MathematicalPhysics vol 12 pp 578ndash586 1971

[6] W D Halford ldquoCosmological theory based on Lyrarsquos geometryrdquoAustralian Journal of Physics vol 23 no 5 pp 863ndash870 1970

[7] S S De and F Rahaman Finsler Geometry of Hadrons and LyraGeometry Cosmological Aspects Lambert Academic Publish-ing Saarbrucken Germany 2012

[8] A Pradhan I Aotemshi and G P Singh ldquoPlane symmetricdomain wall in Lyra geometryrdquo Astrophysics and Space Sciencevol 288 no 3 pp 315ndash325 2003

[9] A Pradhan and A K Vishwakarma ldquoA new class of LRSBianchi type-I cosmological models in Lyra geometryrdquo Journalof Geometry and Physics vol 49 no 3-4 pp 332ndash342 2004

[10] A Pradhan and S S Kumhar ldquoPlane symmetric inhomoge-neous perfect fluid universe with electromagnetic field in Lyrageometryrdquo Astrophysics and Space Science vol 321 no 2 pp137ndash146 2009

[11] A Pradhan and P Ram ldquoA plane-symmetric magnetizedinhomogeneous cosmological models of perfect fluid distribu-tion with variable magnetic permeability in Lyra geometryrdquoInternational Journal of Theoretical Physics vol 48 no 11 pp3188ndash3201 2009

[12] A Pradhan H Amirhashchi and H Zainuddin ldquoA new classof inhomogeneous cosmological models with electromagneticfield in normal gauge for Lyras manifoldrdquo International Journalof Theoretical Physics vol 50 no 1 pp 56ndash69 2011

[13] P Anirudh A Singh and R S Singh ldquoA plane-symmetric mag-netized inhomogeneous cosmological model of perfect fluiddistribution with variable magnetic permeabilityrdquo RomanianReports of Physics vol 56 no 1-2 pp 297ndash307 2011

[14] A Pradhan and A K Singh ldquoAnisotropic Bianchi type-I stringcosmological models in normal Gauge for Lyras manifoldwith constant deceleration parameterrdquo International Journal ofTheoretical Physics vol 50 no 3 pp 916ndash933 2011

[15] R Casana C AM deMelo andBM Pimentel ldquoSpinorial fieldand Lyra geometryrdquoAstrophysics and Space Science vol 305 no2 pp 125ndash132 2006

[16] F Rahaman B C Bhui and G Bag ldquoCan Lyra geometryexplain the singularity free as well as accelerating UniverserdquoAstrophysics and Space Science vol 295 no 4 pp 507ndash513 2005

[17] R Bali and N K Chandnani ldquoBianchi type-I cosmologicalmodel for perfect fluid distribution in Lyra geometryrdquo Journal of

8 Abstract and Applied Analysis

Mathematical Physics vol 49 no 3 Article ID 032502 8 pages2008

[18] R Bali andN K Chandnani ldquoBianchi typeV barotropic perfectfluid cosmological model in Lyra geometryrdquo InternationalJournal ofTheoretical Physics vol 48 no 5 pp 1523ndash1533 2009

[19] S Kumar andC P Singh ldquoAn exact Bianchi type-I cosmologicalmodel in Lyrarsquos manifoldrdquo International Journal of ModernPhysics A vol 23 no 6 pp 813ndash822 2008

[20] A K Yadav A Pradhan and A Singh ldquoLRS Bianchi type-II massive string cosmological models in general relativityrdquoRomanianReports of Physics vol 56 no 7-8 pp 1019ndash1034 2011

[21] V U M Rao T Vinutha and M V Santhi ldquoBianchi type-Vcosmological model with perfect fluid using negative constantdeceleration parameter in a scalar tensor theory based on LyraManifoldrdquo Astrophysics and Space Science vol 314 no 1ndash3 pp213ndash216 2008

[22] R Zia and R P Singh ldquoBulk viscous inhomogeneous cosmo-logical models with electromagnetic field in Lyra geometryrdquoRomanian Journal of Physics vol 57 no 3-4 pp 761ndash778 2012

[23] A T Ali and F Rahaman ldquoNew class of magnetized inho-mogeneous Bianchi type-I cosmological model with variablemagnetic permeability in Lyra geometryrdquo International Journalof Theoretical Physics vol 52 no 11 pp 4055ndash4067 2013

[24] A T Ali ldquoNew generalized Jacobi elliptic function rationalexpansion methodrdquo Journal of Computational and AppliedMathematics vol 235 no 14 pp 4117ndash4127 2011

[25] A T Ali and E R Hassan ldquoGeneral Expa-function methodfor nonlinear evolution equationsrdquo Applied Mathematics andComputation vol 217 no 2 pp 451ndash459 2010

[26] M F El-Sabbagh and A T Ali ldquoNew exact solutions for (3+1)-dimensional Kadomtsev-Petviashvili equation and generalized(2+1)-dimensional Boussinesq equationrdquo International Journalof Nonlinear Sciences andNumerical Simulation vol 6 no 2 pp151ndash162 2005

[27] M F El-Sabbagh and A T Ali ldquoNew generalized Jacobi ellipticfunction expansion methodrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 9 pp 1758ndash17662008

[28] L V Ovsiannikov Group Analysis of Differential Equationstranslated by Y Chapovsky edited by W F Ames AcademicPress New York NY USA 1982

[29] G W Bluman and S Kumei Symmetries and DifferentialEquations in Applied Sciences Springer New York NY USA1989

[30] N H Ibragimov Transformation Groups Applied to Mathemat-ical Physics D Reidel Dordrecht The Netherlands 1985

[31] P J Olver Applications of Lie Groups to Differential Equationsvol 107 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 1993

[32] A T Ali A K Yadav and S R Mahmoud ldquoSome planesymmetric inhomogeneous cosmological models in the scalar-tensor theory of gravitationrdquoAstrophysics and Space Science vol349 no 1 pp 539ndash547 2014

[33] A T Ali ldquoNew exact solutions of the Einstein vacuumequationsfor rotating axially symmetric fieldsrdquoPhysica Scripta vol 79 no3 Article ID 035006 2009

[34] S K Attallah M F El-Sabbagh and A T Ali ldquoIsovectorfields and similarity solutions of Einstein vacuum equationsfor rotating fieldsrdquo Communications in Nonlinear Science andNumerical Simulation vol 12 no 7 pp 1153ndash1161 2007

[35] K SMekheimer S Z Husseny A T Ali andA E Abo-ElkhairldquoLie point symmetries and similarity solutions for an electricallyconducting Jeffrey fluidrdquo Physica Scripta vol 83 no 1 ArticleID 015017 2011

[36] A K Raychaudhuri Theoritical Cosmology Oxford SciencePublications Oxford UK 1979

[37] A Lichnerowicz Relativistic Hydrodynamics and Magneto-Hydro-Dynamics W A Benjamin New York NY USA 1967

[38] A Pradhan and P Mathur ldquoInhomogeneous perfect fluiduniverse with electromagnetic field in Lyra geometryrdquo Fizika Bvol 18 no 4 pp 243ndash264 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

From the above equation if we substitute Ψ in (37) we canobtain the following form

(1198882minus 1198862) [

ΦΨ

ΦΨ

minus

Φ

Φ

] + (21198862minus 1198882)(

Φ

Φ

)

minus (21198882+ 1198862)(

Ψ

Ψ

) + 1198862= 0

(39)

Equation (39) is a nonlinear ODE which is very difficult tosolve However it is worth noting that this equation is easyto solve when 119888 = 119886 In this case we can integrate (39) andobtain the following

Φ (120579) = 1198861Ψ3(120579) exp [minus120579] (40)

where 1198861is an arbitrary constant of integration Substituting

(40) in (36) we have the following ODE of the function Ψ

only

Ψ(4Ψ minus 3Ψ) + 6Ψ2= 0 (41)

The general solution of the above equation is

Ψ (120579) = 1198863(1198862+ exp [

3120579

4

])

25

(42)

where 1198862and 1198863are arbitrary constants of integration Now

by using (40) and the inverse of transformations (35) and (33)we can find the solution as follows

119860 (119905) = 119886 (119905 + 119887)

119891 (119909 119905) = 1198863(1198862+ [

exp [119886119909]

119905 + 119887

]

34

)

25

119892 (119909 119905) = 11988611198863(119905 + 119887) (119886

2+ [

exp [119886119909]

119905 + 119887

]

34

)

65

(43)

It is observed from (43) and (3) that the line element (2) canbe written in the following form

1198891199042= 1198891199052minus 1198862(119905 + 119887)

21198891199092minus 1198892(119905 + 119887)

2

times (1198862+ [

exp [119886119909]

119905 + 119887

]

34

)

45

1198891199102

minus 1199022(119905 + 119887)

4(1198862+ [

exp [119886119909]

119905 + 119887

]

34

)

125

1198891199112

(44)

where 119886 119887 119889 = 1198861198863 119902 = 119886119886

11198863 and 119886

2are arbitrary constants

Case 4 When 1198881

= 1198883

= 0 the similarity variable andsimilarity functions can be written as follows

120585 = 119886119909 + 119887119905 119891 (119909 119905) = Ψ (120585)

119892 (119909 119905) = Φ (120585) exp [119888119909]

(45)

where 119886 = 1198884 119887 = minus119888

2 and 119888 = 119888

51198882are arbitrary constants In

this case we have119860(119905) = 1198887 Substituting transformations (45)

in field equations (15) leads to the following system of ODEs

119886Ψ10158401015840

Ψ

+

119886Φ10158401015840+ 119888Φ1015840

Φ

= 0 (46)

(11988721199032minus 1198862)Φ1015840Ψ1015840

ΦΨ

+

119886 (2119888Φ1015840+ 119886Φ10158401015840)

Φ

+

11988721199032Ψ10158401015840minus 119886119888Ψ

1015840

Ψ

+ 1198882= 0

(47)

Equation (46) can be written in the following form

Ψ10158401015840

= minus

Ψ

119886Φ

(119886Φ10158401015840+ 119888Φ1015840) (48)

From the above equation if we substitute Ψ in (47) we canobtain the following form

(11988721199032minus 1198862) [

Φ1015840Ψ1015840

ΦΨ

minus

Φ10158401015840

Φ

] + (21198862minus 11988721199032)(

119888Φ1015840

119886Φ

)

minus 119886119888(

Ψ1015840

Ψ

) + 1198882= 0

(49)

Equation (49) is a nonlinear ODE which is very difficult tosolve However it is worth noting that this equation is easyto solve when 119886 = 119887119903 In this case we can integrate (49) andobtain the following

Φ (120585) = 1198861Ψ (120585) exp [minus

119888120585

119887119903

] (50)

where 1198861is an arbitrary constant of integration Substituting

(50) in (46) we have the following ODE of function Ψ only

2119887119903Ψ10158401015840

= 119888Ψ (51)

The general solution of the above equation is

Ψ (120585) = 1198863+ 1198862exp [

119888120585

2119887119903

] (52)

where 1198862and 1198863are arbitrary constants of integration Now

by using (50) and (33) we can find the solution as follows

119860 (119905) = 119903

119891 (119909 119905) = 1198863+ 1198862exp [

119888 (119905 + 119903119909)

2119887119903

]

119892 (119909 119905) = 1198861exp [minus

119888119905

119903

] (1198863+ 1198862exp [

119888 (119905 + 119903119909)

2119887119903

])

(53)

It is observed from (53) and (3) that the line element (2) canbe written in the following form

1198891199042= 1198891199052minus 11990321198891199092minus 1199032(1198863+ 1198862exp [

119888 (119905 + 119903119909)

2119887119903

])

2

times (1198891199102+ 1198862

1exp [minus

2119888119905

119903

] 1198891199112)

(54)

where 119903 119888 119887 1198861 1198862 and 119886

3are arbitrary constants

6 Abstract and Applied Analysis

5 Physical Properties of the Model

The field equations (15)ndash(16) constitute a system of fivehighly nonlinear differential equations with seven unknownsvariables 119860 119891 119892 119901 120588 120583 and 120573 The symmetries giveone condition (24) for function 119860 Therefore one physicallyreasonable condition amongst these parameters is required toobtain explicit solutions of the field equations Let us assumethat the density 120588 and the pressure 119901 are related by barotropicequation of state

119901 = 120582120588 0 le 120582 le 1 (55)

51 ForModel (44) Using (43) in the Einstein field equations(16) with taking into account condition (55) the expressionsfor density 120588 pressure 119901 magnetic permeability 120583 anddisplacement field 120573 are given by

120588 (119909 119905) =

9

5120594 (1 minus 120582)

times[

[

51198862+ [exp [119886119909] (119905 + 119887)]

34

(119905 + 119887)2(1198862+ [exp [119886119909] (119905 + 119887)]

34

)

]

]

119901 (119909 119905) =

9120582

5120594 (1 minus 120582)

times[

[

51198862+ [exp [119886119909] (119905 + 119887)]

34

(119905 + 119887)2(1198862+ [exp [119886119909] (119905 + 119887)]

34

)

]

]

120583 (119909 119905) =

1205941198652

12(119909)

3119886211988621198892(119905 + 119887)

2(1198862+ [

exp [119886119909]

119905 + 119887

]

34

)

15

1205732(119909 119905) =

2

15

[

[

(51198862(5 + 13120582)

+ 2 (2 + 7120582) [

exp [119886119909]

119905 + 119887

]

34

)

times ((120582 minus 1) (119905 + 119887)2

times(1198862+ [

exp [119886119909]

119905 + 119887

]

34

))

minus1

]

(56)

For the line element (44) using (4) (8) and (9) we have thefollowing physical properties The volume element is

119881 = 119886119889119902(119905 + 119887)4(1198862+ [

exp [119886119909]

119905 + 119887

]

34

)

85

(57)

The expansion scalar which determines the volume behaviorof the fluid is given by

Θ =

2

5

[

[

101198862+ 7[exp [119886119909] (119905 + 119887)]

34

(119905 + 119887) (1198862+ [exp [119886119909] (119905 + 119887)]

34

)

]

]

(58)

The nonvanishing components of the shear tensor 120590119895119894 are

1205901

1=

[exp [119886119909] (119905 + 119887)]34

minus 51198862

15 (119905 + 119887) (1198862+ [exp [119886119909] (119905 + 119887)]

34

)

1205902

2= minus

1

30

[

[

101198862+ [exp [119886119909] (119905 + 119887)]

34

(119905 + 119887) (1198862+ [exp [119886119909] (119905 + 119887)]

34

)

]

]

1205903

3=

41198862+ 7[exp [119886119909] (119905 + 119887)]

34

6 (119905 + 119887) (1198862+ [exp [119886119909] (119905 + 119887)]

34

)

1205904

4= minus

4

15

[

[

101198862+ 7[exp [119886119909] (119905 + 119887)]

34

(119905 + 119887) (1198862+ [exp [119886119909] (119905 + 119887)]

34

)

]

]

(59)

Hence the shear scalar 120590 is given by

1205902= (3500119886

2

2+ 4630119886

2[exp [119886119909] (119905 + 119887)]

34

+1607[exp [119886119909] (119905 + 119887)]32

)

times (900(119905 + 119887)2(1198862+ [exp [119886119909] (119905 + 119887)]

34

)

2

)

minus1

(60)

The model does not admit acceleration and rotation since119894= 0 and 120596

119894119895= 0 We can see that

1205904

4

Θ

= minus

2

3

(61)

which is a constant of proportional We found also that

lim119905rarrinfin

1205902

Θ

=

5radic35

2

= 0 (62)

and this means that there is no possibility that the universemay got isotropized in some later time that is it remainsanisotropic for all times

52 ForModel (54) Using (53) in the Einstein field equations(16) with taking into account condition (55) the expressions

Abstract and Applied Analysis 7

for density 120588 pressure 119901 magnetic permeability 120583 anddisplacement field 120573 are given by

120588 (119909 119905) =

1198882

1205941199032(120582 minus 1)

(

1198882exp [119888 (119905 + 119903119909) 2119903] minus 119888

3

1198882exp [119888 (119905 + 119903119909) 2119903] + 119888

3

)

119901 (119909 119905) =

1205821198882

1205941199032(120582 minus 1)

(

1198882exp [119888 (119905 + 119903119909) 2119903] minus 119888

3

1198882exp [119888 (119905 + 119903119909) 2119903] + 119888

3

)

120583 (119909 119905) =

1205941198652

12(119909)

119888311988821199032

(1198882exp [

119888 (119905 + 119903119909)

2119903

] + 1198883)

minus1

1205732(119909 119905) =

21198882

31199032(120582 minus 1)

times (

1198883(1 + 120582) minus 2120582119888

2exp [119888 (119905 + 119903119909) 2119903]

1198883+ 1198882exp [119888 (119905 + 119903119909) 2119903]

)

(63)

For the line element (54) using (4) (8) and (9) we have thefollowing physical properties The volume element is

119881 = 11988811199033 exp [minus

119888119905

119903

] (1198883+ 1198882exp [

119888 (119905 + 119903119909)

2119903

])

2

(64)

The expansion scalar which determines the volume behaviorof the fluid is given by

Θ = minus

1198881198883

119903

(1198883+ 1198882exp [

119888 (119905 + 119903119909)

2119903

])

minus1

(65)

The nonvanishing components of the shear tensor 120590119895119894 satisfy

1205901

1

Θ

= minus

1

3

1205902

2

Θ

= minus(

1

3

+

1198882

21198883

exp [

119888 (119905 + 119903119909)

2119903

])

1205903

3

Θ

=

2

3

+

1198882

21198883

exp [

119888 (119905 + 119903119909)

2119903

]

1205904

4

Θ

= minus

2

3

(66)

Hence the shear scalar 120590 is given by

1205902=

1198882

41199032+

111198882

31198882

361199032

(1198883+ 1198882exp [

119888 (119905 + 119903119909)

2119903

])

minus2

(67)

The model does not admit acceleration and rotation since119894= 0 and 120596

119894119895= 0

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah under Grant no130-016-D1434 The author therefore acknowledges withthanks DSR technical and financial support

References

[1] A Krasinski Inhomogeneous Cosmological Models CambridgeUniversity Press Cambridge UK 1997

[2] G Lyra ldquoUber eine Modifikation der Riemannschen Geome-trierdquoMathematische Zeitschrift vol 54 pp 52ndash64 1951

[3] D K Sen ldquoA static cosmological modelrdquo Zeitschrift fur PhysikC Particles and Fields vol 149 pp 311ndash323 1957

[4] D K Sen ldquoOn geodesics of a modified Riemannian manifoldrdquoCanadian Mathematical Society vol 3 pp 255ndash261 1960

[5] D K Sen and K A Dunn ldquoA scalar-tensor theory of gravitationin a modified Riemannian manifoldrdquo Journal of MathematicalPhysics vol 12 pp 578ndash586 1971

[6] W D Halford ldquoCosmological theory based on Lyrarsquos geometryrdquoAustralian Journal of Physics vol 23 no 5 pp 863ndash870 1970

[7] S S De and F Rahaman Finsler Geometry of Hadrons and LyraGeometry Cosmological Aspects Lambert Academic Publish-ing Saarbrucken Germany 2012

[8] A Pradhan I Aotemshi and G P Singh ldquoPlane symmetricdomain wall in Lyra geometryrdquo Astrophysics and Space Sciencevol 288 no 3 pp 315ndash325 2003

[9] A Pradhan and A K Vishwakarma ldquoA new class of LRSBianchi type-I cosmological models in Lyra geometryrdquo Journalof Geometry and Physics vol 49 no 3-4 pp 332ndash342 2004

[10] A Pradhan and S S Kumhar ldquoPlane symmetric inhomoge-neous perfect fluid universe with electromagnetic field in Lyrageometryrdquo Astrophysics and Space Science vol 321 no 2 pp137ndash146 2009

[11] A Pradhan and P Ram ldquoA plane-symmetric magnetizedinhomogeneous cosmological models of perfect fluid distribu-tion with variable magnetic permeability in Lyra geometryrdquoInternational Journal of Theoretical Physics vol 48 no 11 pp3188ndash3201 2009

[12] A Pradhan H Amirhashchi and H Zainuddin ldquoA new classof inhomogeneous cosmological models with electromagneticfield in normal gauge for Lyras manifoldrdquo International Journalof Theoretical Physics vol 50 no 1 pp 56ndash69 2011

[13] P Anirudh A Singh and R S Singh ldquoA plane-symmetric mag-netized inhomogeneous cosmological model of perfect fluiddistribution with variable magnetic permeabilityrdquo RomanianReports of Physics vol 56 no 1-2 pp 297ndash307 2011

[14] A Pradhan and A K Singh ldquoAnisotropic Bianchi type-I stringcosmological models in normal Gauge for Lyras manifoldwith constant deceleration parameterrdquo International Journal ofTheoretical Physics vol 50 no 3 pp 916ndash933 2011

[15] R Casana C AM deMelo andBM Pimentel ldquoSpinorial fieldand Lyra geometryrdquoAstrophysics and Space Science vol 305 no2 pp 125ndash132 2006

[16] F Rahaman B C Bhui and G Bag ldquoCan Lyra geometryexplain the singularity free as well as accelerating UniverserdquoAstrophysics and Space Science vol 295 no 4 pp 507ndash513 2005

[17] R Bali and N K Chandnani ldquoBianchi type-I cosmologicalmodel for perfect fluid distribution in Lyra geometryrdquo Journal of

8 Abstract and Applied Analysis

Mathematical Physics vol 49 no 3 Article ID 032502 8 pages2008

[18] R Bali andN K Chandnani ldquoBianchi typeV barotropic perfectfluid cosmological model in Lyra geometryrdquo InternationalJournal ofTheoretical Physics vol 48 no 5 pp 1523ndash1533 2009

[19] S Kumar andC P Singh ldquoAn exact Bianchi type-I cosmologicalmodel in Lyrarsquos manifoldrdquo International Journal of ModernPhysics A vol 23 no 6 pp 813ndash822 2008

[20] A K Yadav A Pradhan and A Singh ldquoLRS Bianchi type-II massive string cosmological models in general relativityrdquoRomanianReports of Physics vol 56 no 7-8 pp 1019ndash1034 2011

[21] V U M Rao T Vinutha and M V Santhi ldquoBianchi type-Vcosmological model with perfect fluid using negative constantdeceleration parameter in a scalar tensor theory based on LyraManifoldrdquo Astrophysics and Space Science vol 314 no 1ndash3 pp213ndash216 2008

[22] R Zia and R P Singh ldquoBulk viscous inhomogeneous cosmo-logical models with electromagnetic field in Lyra geometryrdquoRomanian Journal of Physics vol 57 no 3-4 pp 761ndash778 2012

[23] A T Ali and F Rahaman ldquoNew class of magnetized inho-mogeneous Bianchi type-I cosmological model with variablemagnetic permeability in Lyra geometryrdquo International Journalof Theoretical Physics vol 52 no 11 pp 4055ndash4067 2013

[24] A T Ali ldquoNew generalized Jacobi elliptic function rationalexpansion methodrdquo Journal of Computational and AppliedMathematics vol 235 no 14 pp 4117ndash4127 2011

[25] A T Ali and E R Hassan ldquoGeneral Expa-function methodfor nonlinear evolution equationsrdquo Applied Mathematics andComputation vol 217 no 2 pp 451ndash459 2010

[26] M F El-Sabbagh and A T Ali ldquoNew exact solutions for (3+1)-dimensional Kadomtsev-Petviashvili equation and generalized(2+1)-dimensional Boussinesq equationrdquo International Journalof Nonlinear Sciences andNumerical Simulation vol 6 no 2 pp151ndash162 2005

[27] M F El-Sabbagh and A T Ali ldquoNew generalized Jacobi ellipticfunction expansion methodrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 9 pp 1758ndash17662008

[28] L V Ovsiannikov Group Analysis of Differential Equationstranslated by Y Chapovsky edited by W F Ames AcademicPress New York NY USA 1982

[29] G W Bluman and S Kumei Symmetries and DifferentialEquations in Applied Sciences Springer New York NY USA1989

[30] N H Ibragimov Transformation Groups Applied to Mathemat-ical Physics D Reidel Dordrecht The Netherlands 1985

[31] P J Olver Applications of Lie Groups to Differential Equationsvol 107 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 1993

[32] A T Ali A K Yadav and S R Mahmoud ldquoSome planesymmetric inhomogeneous cosmological models in the scalar-tensor theory of gravitationrdquoAstrophysics and Space Science vol349 no 1 pp 539ndash547 2014

[33] A T Ali ldquoNew exact solutions of the Einstein vacuumequationsfor rotating axially symmetric fieldsrdquoPhysica Scripta vol 79 no3 Article ID 035006 2009

[34] S K Attallah M F El-Sabbagh and A T Ali ldquoIsovectorfields and similarity solutions of Einstein vacuum equationsfor rotating fieldsrdquo Communications in Nonlinear Science andNumerical Simulation vol 12 no 7 pp 1153ndash1161 2007

[35] K SMekheimer S Z Husseny A T Ali andA E Abo-ElkhairldquoLie point symmetries and similarity solutions for an electricallyconducting Jeffrey fluidrdquo Physica Scripta vol 83 no 1 ArticleID 015017 2011

[36] A K Raychaudhuri Theoritical Cosmology Oxford SciencePublications Oxford UK 1979

[37] A Lichnerowicz Relativistic Hydrodynamics and Magneto-Hydro-Dynamics W A Benjamin New York NY USA 1967

[38] A Pradhan and P Mathur ldquoInhomogeneous perfect fluiduniverse with electromagnetic field in Lyra geometryrdquo Fizika Bvol 18 no 4 pp 243ndash264 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

5 Physical Properties of the Model

The field equations (15)ndash(16) constitute a system of fivehighly nonlinear differential equations with seven unknownsvariables 119860 119891 119892 119901 120588 120583 and 120573 The symmetries giveone condition (24) for function 119860 Therefore one physicallyreasonable condition amongst these parameters is required toobtain explicit solutions of the field equations Let us assumethat the density 120588 and the pressure 119901 are related by barotropicequation of state

119901 = 120582120588 0 le 120582 le 1 (55)

51 ForModel (44) Using (43) in the Einstein field equations(16) with taking into account condition (55) the expressionsfor density 120588 pressure 119901 magnetic permeability 120583 anddisplacement field 120573 are given by

120588 (119909 119905) =

9

5120594 (1 minus 120582)

times[

[

51198862+ [exp [119886119909] (119905 + 119887)]

34

(119905 + 119887)2(1198862+ [exp [119886119909] (119905 + 119887)]

34

)

]

]

119901 (119909 119905) =

9120582

5120594 (1 minus 120582)

times[

[

51198862+ [exp [119886119909] (119905 + 119887)]

34

(119905 + 119887)2(1198862+ [exp [119886119909] (119905 + 119887)]

34

)

]

]

120583 (119909 119905) =

1205941198652

12(119909)

3119886211988621198892(119905 + 119887)

2(1198862+ [

exp [119886119909]

119905 + 119887

]

34

)

15

1205732(119909 119905) =

2

15

[

[

(51198862(5 + 13120582)

+ 2 (2 + 7120582) [

exp [119886119909]

119905 + 119887

]

34

)

times ((120582 minus 1) (119905 + 119887)2

times(1198862+ [

exp [119886119909]

119905 + 119887

]

34

))

minus1

]

(56)

For the line element (44) using (4) (8) and (9) we have thefollowing physical properties The volume element is

119881 = 119886119889119902(119905 + 119887)4(1198862+ [

exp [119886119909]

119905 + 119887

]

34

)

85

(57)

The expansion scalar which determines the volume behaviorof the fluid is given by

Θ =

2

5

[

[

101198862+ 7[exp [119886119909] (119905 + 119887)]

34

(119905 + 119887) (1198862+ [exp [119886119909] (119905 + 119887)]

34

)

]

]

(58)

The nonvanishing components of the shear tensor 120590119895119894 are

1205901

1=

[exp [119886119909] (119905 + 119887)]34

minus 51198862

15 (119905 + 119887) (1198862+ [exp [119886119909] (119905 + 119887)]

34

)

1205902

2= minus

1

30

[

[

101198862+ [exp [119886119909] (119905 + 119887)]

34

(119905 + 119887) (1198862+ [exp [119886119909] (119905 + 119887)]

34

)

]

]

1205903

3=

41198862+ 7[exp [119886119909] (119905 + 119887)]

34

6 (119905 + 119887) (1198862+ [exp [119886119909] (119905 + 119887)]

34

)

1205904

4= minus

4

15

[

[

101198862+ 7[exp [119886119909] (119905 + 119887)]

34

(119905 + 119887) (1198862+ [exp [119886119909] (119905 + 119887)]

34

)

]

]

(59)

Hence the shear scalar 120590 is given by

1205902= (3500119886

2

2+ 4630119886

2[exp [119886119909] (119905 + 119887)]

34

+1607[exp [119886119909] (119905 + 119887)]32

)

times (900(119905 + 119887)2(1198862+ [exp [119886119909] (119905 + 119887)]

34

)

2

)

minus1

(60)

The model does not admit acceleration and rotation since119894= 0 and 120596

119894119895= 0 We can see that

1205904

4

Θ

= minus

2

3

(61)

which is a constant of proportional We found also that

lim119905rarrinfin

1205902

Θ

=

5radic35

2

= 0 (62)

and this means that there is no possibility that the universemay got isotropized in some later time that is it remainsanisotropic for all times

52 ForModel (54) Using (53) in the Einstein field equations(16) with taking into account condition (55) the expressions

Abstract and Applied Analysis 7

for density 120588 pressure 119901 magnetic permeability 120583 anddisplacement field 120573 are given by

120588 (119909 119905) =

1198882

1205941199032(120582 minus 1)

(

1198882exp [119888 (119905 + 119903119909) 2119903] minus 119888

3

1198882exp [119888 (119905 + 119903119909) 2119903] + 119888

3

)

119901 (119909 119905) =

1205821198882

1205941199032(120582 minus 1)

(

1198882exp [119888 (119905 + 119903119909) 2119903] minus 119888

3

1198882exp [119888 (119905 + 119903119909) 2119903] + 119888

3

)

120583 (119909 119905) =

1205941198652

12(119909)

119888311988821199032

(1198882exp [

119888 (119905 + 119903119909)

2119903

] + 1198883)

minus1

1205732(119909 119905) =

21198882

31199032(120582 minus 1)

times (

1198883(1 + 120582) minus 2120582119888

2exp [119888 (119905 + 119903119909) 2119903]

1198883+ 1198882exp [119888 (119905 + 119903119909) 2119903]

)

(63)

For the line element (54) using (4) (8) and (9) we have thefollowing physical properties The volume element is

119881 = 11988811199033 exp [minus

119888119905

119903

] (1198883+ 1198882exp [

119888 (119905 + 119903119909)

2119903

])

2

(64)

The expansion scalar which determines the volume behaviorof the fluid is given by

Θ = minus

1198881198883

119903

(1198883+ 1198882exp [

119888 (119905 + 119903119909)

2119903

])

minus1

(65)

The nonvanishing components of the shear tensor 120590119895119894 satisfy

1205901

1

Θ

= minus

1

3

1205902

2

Θ

= minus(

1

3

+

1198882

21198883

exp [

119888 (119905 + 119903119909)

2119903

])

1205903

3

Θ

=

2

3

+

1198882

21198883

exp [

119888 (119905 + 119903119909)

2119903

]

1205904

4

Θ

= minus

2

3

(66)

Hence the shear scalar 120590 is given by

1205902=

1198882

41199032+

111198882

31198882

361199032

(1198883+ 1198882exp [

119888 (119905 + 119903119909)

2119903

])

minus2

(67)

The model does not admit acceleration and rotation since119894= 0 and 120596

119894119895= 0

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah under Grant no130-016-D1434 The author therefore acknowledges withthanks DSR technical and financial support

References

[1] A Krasinski Inhomogeneous Cosmological Models CambridgeUniversity Press Cambridge UK 1997

[2] G Lyra ldquoUber eine Modifikation der Riemannschen Geome-trierdquoMathematische Zeitschrift vol 54 pp 52ndash64 1951

[3] D K Sen ldquoA static cosmological modelrdquo Zeitschrift fur PhysikC Particles and Fields vol 149 pp 311ndash323 1957

[4] D K Sen ldquoOn geodesics of a modified Riemannian manifoldrdquoCanadian Mathematical Society vol 3 pp 255ndash261 1960

[5] D K Sen and K A Dunn ldquoA scalar-tensor theory of gravitationin a modified Riemannian manifoldrdquo Journal of MathematicalPhysics vol 12 pp 578ndash586 1971

[6] W D Halford ldquoCosmological theory based on Lyrarsquos geometryrdquoAustralian Journal of Physics vol 23 no 5 pp 863ndash870 1970

[7] S S De and F Rahaman Finsler Geometry of Hadrons and LyraGeometry Cosmological Aspects Lambert Academic Publish-ing Saarbrucken Germany 2012

[8] A Pradhan I Aotemshi and G P Singh ldquoPlane symmetricdomain wall in Lyra geometryrdquo Astrophysics and Space Sciencevol 288 no 3 pp 315ndash325 2003

[9] A Pradhan and A K Vishwakarma ldquoA new class of LRSBianchi type-I cosmological models in Lyra geometryrdquo Journalof Geometry and Physics vol 49 no 3-4 pp 332ndash342 2004

[10] A Pradhan and S S Kumhar ldquoPlane symmetric inhomoge-neous perfect fluid universe with electromagnetic field in Lyrageometryrdquo Astrophysics and Space Science vol 321 no 2 pp137ndash146 2009

[11] A Pradhan and P Ram ldquoA plane-symmetric magnetizedinhomogeneous cosmological models of perfect fluid distribu-tion with variable magnetic permeability in Lyra geometryrdquoInternational Journal of Theoretical Physics vol 48 no 11 pp3188ndash3201 2009

[12] A Pradhan H Amirhashchi and H Zainuddin ldquoA new classof inhomogeneous cosmological models with electromagneticfield in normal gauge for Lyras manifoldrdquo International Journalof Theoretical Physics vol 50 no 1 pp 56ndash69 2011

[13] P Anirudh A Singh and R S Singh ldquoA plane-symmetric mag-netized inhomogeneous cosmological model of perfect fluiddistribution with variable magnetic permeabilityrdquo RomanianReports of Physics vol 56 no 1-2 pp 297ndash307 2011

[14] A Pradhan and A K Singh ldquoAnisotropic Bianchi type-I stringcosmological models in normal Gauge for Lyras manifoldwith constant deceleration parameterrdquo International Journal ofTheoretical Physics vol 50 no 3 pp 916ndash933 2011

[15] R Casana C AM deMelo andBM Pimentel ldquoSpinorial fieldand Lyra geometryrdquoAstrophysics and Space Science vol 305 no2 pp 125ndash132 2006

[16] F Rahaman B C Bhui and G Bag ldquoCan Lyra geometryexplain the singularity free as well as accelerating UniverserdquoAstrophysics and Space Science vol 295 no 4 pp 507ndash513 2005

[17] R Bali and N K Chandnani ldquoBianchi type-I cosmologicalmodel for perfect fluid distribution in Lyra geometryrdquo Journal of

8 Abstract and Applied Analysis

Mathematical Physics vol 49 no 3 Article ID 032502 8 pages2008

[18] R Bali andN K Chandnani ldquoBianchi typeV barotropic perfectfluid cosmological model in Lyra geometryrdquo InternationalJournal ofTheoretical Physics vol 48 no 5 pp 1523ndash1533 2009

[19] S Kumar andC P Singh ldquoAn exact Bianchi type-I cosmologicalmodel in Lyrarsquos manifoldrdquo International Journal of ModernPhysics A vol 23 no 6 pp 813ndash822 2008

[20] A K Yadav A Pradhan and A Singh ldquoLRS Bianchi type-II massive string cosmological models in general relativityrdquoRomanianReports of Physics vol 56 no 7-8 pp 1019ndash1034 2011

[21] V U M Rao T Vinutha and M V Santhi ldquoBianchi type-Vcosmological model with perfect fluid using negative constantdeceleration parameter in a scalar tensor theory based on LyraManifoldrdquo Astrophysics and Space Science vol 314 no 1ndash3 pp213ndash216 2008

[22] R Zia and R P Singh ldquoBulk viscous inhomogeneous cosmo-logical models with electromagnetic field in Lyra geometryrdquoRomanian Journal of Physics vol 57 no 3-4 pp 761ndash778 2012

[23] A T Ali and F Rahaman ldquoNew class of magnetized inho-mogeneous Bianchi type-I cosmological model with variablemagnetic permeability in Lyra geometryrdquo International Journalof Theoretical Physics vol 52 no 11 pp 4055ndash4067 2013

[24] A T Ali ldquoNew generalized Jacobi elliptic function rationalexpansion methodrdquo Journal of Computational and AppliedMathematics vol 235 no 14 pp 4117ndash4127 2011

[25] A T Ali and E R Hassan ldquoGeneral Expa-function methodfor nonlinear evolution equationsrdquo Applied Mathematics andComputation vol 217 no 2 pp 451ndash459 2010

[26] M F El-Sabbagh and A T Ali ldquoNew exact solutions for (3+1)-dimensional Kadomtsev-Petviashvili equation and generalized(2+1)-dimensional Boussinesq equationrdquo International Journalof Nonlinear Sciences andNumerical Simulation vol 6 no 2 pp151ndash162 2005

[27] M F El-Sabbagh and A T Ali ldquoNew generalized Jacobi ellipticfunction expansion methodrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 9 pp 1758ndash17662008

[28] L V Ovsiannikov Group Analysis of Differential Equationstranslated by Y Chapovsky edited by W F Ames AcademicPress New York NY USA 1982

[29] G W Bluman and S Kumei Symmetries and DifferentialEquations in Applied Sciences Springer New York NY USA1989

[30] N H Ibragimov Transformation Groups Applied to Mathemat-ical Physics D Reidel Dordrecht The Netherlands 1985

[31] P J Olver Applications of Lie Groups to Differential Equationsvol 107 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 1993

[32] A T Ali A K Yadav and S R Mahmoud ldquoSome planesymmetric inhomogeneous cosmological models in the scalar-tensor theory of gravitationrdquoAstrophysics and Space Science vol349 no 1 pp 539ndash547 2014

[33] A T Ali ldquoNew exact solutions of the Einstein vacuumequationsfor rotating axially symmetric fieldsrdquoPhysica Scripta vol 79 no3 Article ID 035006 2009

[34] S K Attallah M F El-Sabbagh and A T Ali ldquoIsovectorfields and similarity solutions of Einstein vacuum equationsfor rotating fieldsrdquo Communications in Nonlinear Science andNumerical Simulation vol 12 no 7 pp 1153ndash1161 2007

[35] K SMekheimer S Z Husseny A T Ali andA E Abo-ElkhairldquoLie point symmetries and similarity solutions for an electricallyconducting Jeffrey fluidrdquo Physica Scripta vol 83 no 1 ArticleID 015017 2011

[36] A K Raychaudhuri Theoritical Cosmology Oxford SciencePublications Oxford UK 1979

[37] A Lichnerowicz Relativistic Hydrodynamics and Magneto-Hydro-Dynamics W A Benjamin New York NY USA 1967

[38] A Pradhan and P Mathur ldquoInhomogeneous perfect fluiduniverse with electromagnetic field in Lyra geometryrdquo Fizika Bvol 18 no 4 pp 243ndash264 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

for density 120588 pressure 119901 magnetic permeability 120583 anddisplacement field 120573 are given by

120588 (119909 119905) =

1198882

1205941199032(120582 minus 1)

(

1198882exp [119888 (119905 + 119903119909) 2119903] minus 119888

3

1198882exp [119888 (119905 + 119903119909) 2119903] + 119888

3

)

119901 (119909 119905) =

1205821198882

1205941199032(120582 minus 1)

(

1198882exp [119888 (119905 + 119903119909) 2119903] minus 119888

3

1198882exp [119888 (119905 + 119903119909) 2119903] + 119888

3

)

120583 (119909 119905) =

1205941198652

12(119909)

119888311988821199032

(1198882exp [

119888 (119905 + 119903119909)

2119903

] + 1198883)

minus1

1205732(119909 119905) =

21198882

31199032(120582 minus 1)

times (

1198883(1 + 120582) minus 2120582119888

2exp [119888 (119905 + 119903119909) 2119903]

1198883+ 1198882exp [119888 (119905 + 119903119909) 2119903]

)

(63)

For the line element (54) using (4) (8) and (9) we have thefollowing physical properties The volume element is

119881 = 11988811199033 exp [minus

119888119905

119903

] (1198883+ 1198882exp [

119888 (119905 + 119903119909)

2119903

])

2

(64)

The expansion scalar which determines the volume behaviorof the fluid is given by

Θ = minus

1198881198883

119903

(1198883+ 1198882exp [

119888 (119905 + 119903119909)

2119903

])

minus1

(65)

The nonvanishing components of the shear tensor 120590119895119894 satisfy

1205901

1

Θ

= minus

1

3

1205902

2

Θ

= minus(

1

3

+

1198882

21198883

exp [

119888 (119905 + 119903119909)

2119903

])

1205903

3

Θ

=

2

3

+

1198882

21198883

exp [

119888 (119905 + 119903119909)

2119903

]

1205904

4

Θ

= minus

2

3

(66)

Hence the shear scalar 120590 is given by

1205902=

1198882

41199032+

111198882

31198882

361199032

(1198883+ 1198882exp [

119888 (119905 + 119903119909)

2119903

])

minus2

(67)

The model does not admit acceleration and rotation since119894= 0 and 120596

119894119895= 0

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah under Grant no130-016-D1434 The author therefore acknowledges withthanks DSR technical and financial support

References

[1] A Krasinski Inhomogeneous Cosmological Models CambridgeUniversity Press Cambridge UK 1997

[2] G Lyra ldquoUber eine Modifikation der Riemannschen Geome-trierdquoMathematische Zeitschrift vol 54 pp 52ndash64 1951

[3] D K Sen ldquoA static cosmological modelrdquo Zeitschrift fur PhysikC Particles and Fields vol 149 pp 311ndash323 1957

[4] D K Sen ldquoOn geodesics of a modified Riemannian manifoldrdquoCanadian Mathematical Society vol 3 pp 255ndash261 1960

[5] D K Sen and K A Dunn ldquoA scalar-tensor theory of gravitationin a modified Riemannian manifoldrdquo Journal of MathematicalPhysics vol 12 pp 578ndash586 1971

[6] W D Halford ldquoCosmological theory based on Lyrarsquos geometryrdquoAustralian Journal of Physics vol 23 no 5 pp 863ndash870 1970

[7] S S De and F Rahaman Finsler Geometry of Hadrons and LyraGeometry Cosmological Aspects Lambert Academic Publish-ing Saarbrucken Germany 2012

[8] A Pradhan I Aotemshi and G P Singh ldquoPlane symmetricdomain wall in Lyra geometryrdquo Astrophysics and Space Sciencevol 288 no 3 pp 315ndash325 2003

[9] A Pradhan and A K Vishwakarma ldquoA new class of LRSBianchi type-I cosmological models in Lyra geometryrdquo Journalof Geometry and Physics vol 49 no 3-4 pp 332ndash342 2004

[10] A Pradhan and S S Kumhar ldquoPlane symmetric inhomoge-neous perfect fluid universe with electromagnetic field in Lyrageometryrdquo Astrophysics and Space Science vol 321 no 2 pp137ndash146 2009

[11] A Pradhan and P Ram ldquoA plane-symmetric magnetizedinhomogeneous cosmological models of perfect fluid distribu-tion with variable magnetic permeability in Lyra geometryrdquoInternational Journal of Theoretical Physics vol 48 no 11 pp3188ndash3201 2009

[12] A Pradhan H Amirhashchi and H Zainuddin ldquoA new classof inhomogeneous cosmological models with electromagneticfield in normal gauge for Lyras manifoldrdquo International Journalof Theoretical Physics vol 50 no 1 pp 56ndash69 2011

[13] P Anirudh A Singh and R S Singh ldquoA plane-symmetric mag-netized inhomogeneous cosmological model of perfect fluiddistribution with variable magnetic permeabilityrdquo RomanianReports of Physics vol 56 no 1-2 pp 297ndash307 2011

[14] A Pradhan and A K Singh ldquoAnisotropic Bianchi type-I stringcosmological models in normal Gauge for Lyras manifoldwith constant deceleration parameterrdquo International Journal ofTheoretical Physics vol 50 no 3 pp 916ndash933 2011

[15] R Casana C AM deMelo andBM Pimentel ldquoSpinorial fieldand Lyra geometryrdquoAstrophysics and Space Science vol 305 no2 pp 125ndash132 2006

[16] F Rahaman B C Bhui and G Bag ldquoCan Lyra geometryexplain the singularity free as well as accelerating UniverserdquoAstrophysics and Space Science vol 295 no 4 pp 507ndash513 2005

[17] R Bali and N K Chandnani ldquoBianchi type-I cosmologicalmodel for perfect fluid distribution in Lyra geometryrdquo Journal of

8 Abstract and Applied Analysis

Mathematical Physics vol 49 no 3 Article ID 032502 8 pages2008

[18] R Bali andN K Chandnani ldquoBianchi typeV barotropic perfectfluid cosmological model in Lyra geometryrdquo InternationalJournal ofTheoretical Physics vol 48 no 5 pp 1523ndash1533 2009

[19] S Kumar andC P Singh ldquoAn exact Bianchi type-I cosmologicalmodel in Lyrarsquos manifoldrdquo International Journal of ModernPhysics A vol 23 no 6 pp 813ndash822 2008

[20] A K Yadav A Pradhan and A Singh ldquoLRS Bianchi type-II massive string cosmological models in general relativityrdquoRomanianReports of Physics vol 56 no 7-8 pp 1019ndash1034 2011

[21] V U M Rao T Vinutha and M V Santhi ldquoBianchi type-Vcosmological model with perfect fluid using negative constantdeceleration parameter in a scalar tensor theory based on LyraManifoldrdquo Astrophysics and Space Science vol 314 no 1ndash3 pp213ndash216 2008

[22] R Zia and R P Singh ldquoBulk viscous inhomogeneous cosmo-logical models with electromagnetic field in Lyra geometryrdquoRomanian Journal of Physics vol 57 no 3-4 pp 761ndash778 2012

[23] A T Ali and F Rahaman ldquoNew class of magnetized inho-mogeneous Bianchi type-I cosmological model with variablemagnetic permeability in Lyra geometryrdquo International Journalof Theoretical Physics vol 52 no 11 pp 4055ndash4067 2013

[24] A T Ali ldquoNew generalized Jacobi elliptic function rationalexpansion methodrdquo Journal of Computational and AppliedMathematics vol 235 no 14 pp 4117ndash4127 2011

[25] A T Ali and E R Hassan ldquoGeneral Expa-function methodfor nonlinear evolution equationsrdquo Applied Mathematics andComputation vol 217 no 2 pp 451ndash459 2010

[26] M F El-Sabbagh and A T Ali ldquoNew exact solutions for (3+1)-dimensional Kadomtsev-Petviashvili equation and generalized(2+1)-dimensional Boussinesq equationrdquo International Journalof Nonlinear Sciences andNumerical Simulation vol 6 no 2 pp151ndash162 2005

[27] M F El-Sabbagh and A T Ali ldquoNew generalized Jacobi ellipticfunction expansion methodrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 9 pp 1758ndash17662008

[28] L V Ovsiannikov Group Analysis of Differential Equationstranslated by Y Chapovsky edited by W F Ames AcademicPress New York NY USA 1982

[29] G W Bluman and S Kumei Symmetries and DifferentialEquations in Applied Sciences Springer New York NY USA1989

[30] N H Ibragimov Transformation Groups Applied to Mathemat-ical Physics D Reidel Dordrecht The Netherlands 1985

[31] P J Olver Applications of Lie Groups to Differential Equationsvol 107 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 1993

[32] A T Ali A K Yadav and S R Mahmoud ldquoSome planesymmetric inhomogeneous cosmological models in the scalar-tensor theory of gravitationrdquoAstrophysics and Space Science vol349 no 1 pp 539ndash547 2014

[33] A T Ali ldquoNew exact solutions of the Einstein vacuumequationsfor rotating axially symmetric fieldsrdquoPhysica Scripta vol 79 no3 Article ID 035006 2009

[34] S K Attallah M F El-Sabbagh and A T Ali ldquoIsovectorfields and similarity solutions of Einstein vacuum equationsfor rotating fieldsrdquo Communications in Nonlinear Science andNumerical Simulation vol 12 no 7 pp 1153ndash1161 2007

[35] K SMekheimer S Z Husseny A T Ali andA E Abo-ElkhairldquoLie point symmetries and similarity solutions for an electricallyconducting Jeffrey fluidrdquo Physica Scripta vol 83 no 1 ArticleID 015017 2011

[36] A K Raychaudhuri Theoritical Cosmology Oxford SciencePublications Oxford UK 1979

[37] A Lichnerowicz Relativistic Hydrodynamics and Magneto-Hydro-Dynamics W A Benjamin New York NY USA 1967

[38] A Pradhan and P Mathur ldquoInhomogeneous perfect fluiduniverse with electromagnetic field in Lyra geometryrdquo Fizika Bvol 18 no 4 pp 243ndash264 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

Mathematical Physics vol 49 no 3 Article ID 032502 8 pages2008

[18] R Bali andN K Chandnani ldquoBianchi typeV barotropic perfectfluid cosmological model in Lyra geometryrdquo InternationalJournal ofTheoretical Physics vol 48 no 5 pp 1523ndash1533 2009

[19] S Kumar andC P Singh ldquoAn exact Bianchi type-I cosmologicalmodel in Lyrarsquos manifoldrdquo International Journal of ModernPhysics A vol 23 no 6 pp 813ndash822 2008

[20] A K Yadav A Pradhan and A Singh ldquoLRS Bianchi type-II massive string cosmological models in general relativityrdquoRomanianReports of Physics vol 56 no 7-8 pp 1019ndash1034 2011

[21] V U M Rao T Vinutha and M V Santhi ldquoBianchi type-Vcosmological model with perfect fluid using negative constantdeceleration parameter in a scalar tensor theory based on LyraManifoldrdquo Astrophysics and Space Science vol 314 no 1ndash3 pp213ndash216 2008

[22] R Zia and R P Singh ldquoBulk viscous inhomogeneous cosmo-logical models with electromagnetic field in Lyra geometryrdquoRomanian Journal of Physics vol 57 no 3-4 pp 761ndash778 2012

[23] A T Ali and F Rahaman ldquoNew class of magnetized inho-mogeneous Bianchi type-I cosmological model with variablemagnetic permeability in Lyra geometryrdquo International Journalof Theoretical Physics vol 52 no 11 pp 4055ndash4067 2013

[24] A T Ali ldquoNew generalized Jacobi elliptic function rationalexpansion methodrdquo Journal of Computational and AppliedMathematics vol 235 no 14 pp 4117ndash4127 2011

[25] A T Ali and E R Hassan ldquoGeneral Expa-function methodfor nonlinear evolution equationsrdquo Applied Mathematics andComputation vol 217 no 2 pp 451ndash459 2010

[26] M F El-Sabbagh and A T Ali ldquoNew exact solutions for (3+1)-dimensional Kadomtsev-Petviashvili equation and generalized(2+1)-dimensional Boussinesq equationrdquo International Journalof Nonlinear Sciences andNumerical Simulation vol 6 no 2 pp151ndash162 2005

[27] M F El-Sabbagh and A T Ali ldquoNew generalized Jacobi ellipticfunction expansion methodrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 9 pp 1758ndash17662008

[28] L V Ovsiannikov Group Analysis of Differential Equationstranslated by Y Chapovsky edited by W F Ames AcademicPress New York NY USA 1982

[29] G W Bluman and S Kumei Symmetries and DifferentialEquations in Applied Sciences Springer New York NY USA1989

[30] N H Ibragimov Transformation Groups Applied to Mathemat-ical Physics D Reidel Dordrecht The Netherlands 1985

[31] P J Olver Applications of Lie Groups to Differential Equationsvol 107 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 1993

[32] A T Ali A K Yadav and S R Mahmoud ldquoSome planesymmetric inhomogeneous cosmological models in the scalar-tensor theory of gravitationrdquoAstrophysics and Space Science vol349 no 1 pp 539ndash547 2014

[33] A T Ali ldquoNew exact solutions of the Einstein vacuumequationsfor rotating axially symmetric fieldsrdquoPhysica Scripta vol 79 no3 Article ID 035006 2009

[34] S K Attallah M F El-Sabbagh and A T Ali ldquoIsovectorfields and similarity solutions of Einstein vacuum equationsfor rotating fieldsrdquo Communications in Nonlinear Science andNumerical Simulation vol 12 no 7 pp 1153ndash1161 2007

[35] K SMekheimer S Z Husseny A T Ali andA E Abo-ElkhairldquoLie point symmetries and similarity solutions for an electricallyconducting Jeffrey fluidrdquo Physica Scripta vol 83 no 1 ArticleID 015017 2011

[36] A K Raychaudhuri Theoritical Cosmology Oxford SciencePublications Oxford UK 1979

[37] A Lichnerowicz Relativistic Hydrodynamics and Magneto-Hydro-Dynamics W A Benjamin New York NY USA 1967

[38] A Pradhan and P Mathur ldquoInhomogeneous perfect fluiduniverse with electromagnetic field in Lyra geometryrdquo Fizika Bvol 18 no 4 pp 243ndash264 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of