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Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 848163 4 pageshttpdxdoiorg1011552013848163

Research ArticleA Note on Four-Dimensional Symmetry Algebras andFourth-Order Ordinary Differential Equations

A Fatima1 Muhammad Ayub2 and F M Mahomed1

1 Centre for Differential Equations Continuum Mechanics and Applications School of Computational and Applied MathematicsUniversity of the Witwatersrand Johannesburg 2050 South Africa

2 Department of Mathematics COMSATS Institute of Information Technology Abbottabad 22060 Pakistan

Correspondence should be addressed to A Fatima emanfatima81yahoocom and F M Mahomed fazalmahomedwitsacza

Received 12 November 2012 Accepted 24 December 2012

Academic Editor Mehmet Pakdemirli

Copyright copy 2013 A Fatima et alThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We provide a supplementation of the results on the canonical forms for scalar fourth-order ordinary differential equations (ODEs)which admit four-dimensional Lie algebras obtained recently Together with these new canonical forms a complete list of scalarfourth-order ODEs that admit four-dimensional Lie algebras is available

1 Introduction

The integrability of scalar ordinary differential equations(ODEs) by use of the Lie symmetry method depends ontheir symmetrical Lie algebra if the Lie algebra is solv-able and of sufficient dimension There exists two differentapproaches to the integrability of differential equations usingLie point symmetries One is the direct method in whichLie point symmetries are utilized to perform integrability bysuccessive reduction of order of the equation using idealsof the algebra The other approach is the canonical formmethod if the equations are classified into different typesaccording to the canonical forms of the corresponding Liealgebra

Lie [1] classified scalar second-order ODEs into fourtypes on the basis of their admitted two-dimensional Liealgebras and also performed integration of the representativeequations corresponding to the canonical forms of the two-dimensional symmetry algebra Therefore scalar second-order ODEs can be integrated by using canonical variableswhichmap the symmetry generators to Liersquos canonical formsAlso here one can use successive reduction of order by usingideals of the symmetry algebra (see eg Olver [2]) TheNoether equivalence approach for Lagrangians correspond-ing to scalar second-order ODEs is discussed in Kara et al[3]

The canonical forms for scalar third-order ODEs thatadmit three symmetries were obtained by Mahomed andLeach [4] Then Ibragimov and Nucci [5] provided the inte-grability of these canonical forms

In their paper Cerquetelli et al [6] constructed realiza-tions in the plane of four-dimensional Lie algebras listed byPatera andWinternitz [7]Moreover the classification of sub-algebras of all real Lie algebras of dimensionle4was discussedin [7] The construction of Cerquetelli et al [6] was basedon the three-dimensional subalgebras provided in [7] Theyinvoked the realizations of three-dimensional Lie algebrasin the plane derived earlier by Mahomed and Leach [8] forthis purpose They then determined the fourth-order ODEsadmitting the realizations of the obtained four-dimensionalalgebras as their Lie symmetry algebra Finally they providedthe route to the integration of the classified fourth-orderODEs However the derived realizations of four-dimensionalLie algebras need supplementation in the light of recent workby Popovych et al [9] Recently these authors constructed acomplete set of inequivalent realizations of real Lie algebras ofdimension not greater than four in vector fields in the space ofan arbitrary (finite) numbers of variables We use the resultsof [9] to complete the classification of fourth-order ODEs interms of their four-dimensional algebras presented in [6]

Apart from scalar fourth-order ODEs arising in thesymmetry reductions of partial differential equations such as

2 Journal of Applied Mathematics

the linear wave equation in an inhomogeneous medium (see[10]) they occur prominently as model equations in the formof the static Euler-Bernoulli beam (see eg [11]) and Emden-Fowler equations Such equations have been investigated forsymmetry properties in [12 13]

Firstly we provide a comparison of the results of [9] andthat of [6] related to the realizations of four-dimensional Liealgebras as vector fields in the plane Then we list the newcanonical forms of scalar fourth-order ODEs which possessfour-dimensional algebras

2 Comparison of the Results of [6 9]

We show here that the results on realizations of four-dimensional algebras in the plane given in [6] are a specialcase of the corresponding set of realizations given in [9] Wemake a comparison of the lists of realizations given in [6]and [9] It should be remarked that in general a result ofclassification of realizations may contain errors of two typesnamely

(i) missing of some inequivalent cases and(ii) mutually equivalent cases

In the following comparison some first type of errorsexist in [6] Five cases are missing There are some othercases which can be combined in a compact form and alsosome arbitrary parameters and functions need modificationaccording to the results of [9] related to realizations in theplane Below we keep the notations of both on the left handside the notations of [6] and on right hand side that of [9]However for the final results and further utilization we keepthe notations of [9]

21 Four-Dimensional Algebras We use the nomenclature ofPatera and Winternitz [7] in the naming of the algebras suchas 4119860

1Thuswe do not provide a table of the abstract algebras

of dimension four as this is easily availableHere the 119873119906 refers to the realizations given in the work

[6] and 119877 to that of [9]

(i) 119873119906(41198601) sim (4119860

1) (1198901= 1198902 1198902= 1198903 1198903= 1198901 1198904= 1198904)

119873119906(41198601) sim 119877(4119860

1 11) (119905 = 119910 119909 = 119909)

(ii) 119873119906(1198602oplus21198601) sim (119860

21oplus21198601) (1198901= 1198902 1198902= minus1198901 1198903=

11989031198904= 1198904) No realization exists in (1+1)-dimension

(iii) 119873119906(21198602) sim (2119860

21) (1198901

= 1198902 1198902

= minus1198901 1198903

= 1198904

1198904= minus1198903)

119873119906(21198602) sim 119877(2119860

21 5) (119905 = 119909 119909 = 119910) whereas

119877(211986021

7) is missing in [6](iv) 119873119906(119860

31oplus1198601) sim (119860

31oplus1198601) No realization exists in

(1 + 1)-dimension(v) 119873119906(119860

32oplus 1198601 119891(119909) = 0) sim (119860

32oplus 1198601)

119873119906(11986032

oplus 1198601 119891(119909) = 0) sim 119877(119860

32oplus 1198601 9) (119905 = 119910

119909 = minus119909)

(vi) 119873119906(11986033

oplus1198601) sim (119860


(1 + 1)-dimension

(vii) 119873119906(11986034

oplus 1198601 119891(119909) = 0) sim (119860

119886

34oplus 1198601 119886 = minus1)

119873119906(11986034

oplus 1198601 119891(119909) = 0) sim 119877(119860

119886

34oplus 1198601 9 119886 = minus1)

(119905 = 119910 119909 = 1199092)

(viii) 119873119906(119860119886

35oplus1198601 0 lt |119886| lt 1 119891(119909) = 0) sim (119860

119886

34oplus

1198601 |119886| le 1 119886 = 0 plusmn1)

119873119906(119860119886

35oplus 1198601 0 lt |119886| lt 1 119891(119909) = 0) sim 119877(119860

119886

34oplus

1198601 9) (119905 = 119910 119909 = 119909

1minus119886)

(ix) 119873119906(11986036

oplus 1198601 119891(119909) = 0) sim (119860

119887

35oplus 1198601 119887 = 0)

119873119906(11986036

oplus 1198601 119891(119909) = 0) sim 119877(119860

119887

35oplus 1198601 8 119887 = 0)

(119905 = 119910 119909 = (1199092minus 1)12

)

(x) 119873119906(119860119887

37oplus1198601 119887 gt 0 119891(119909) = 0) sim (119860

119887

35oplus1198601 119887 gt 0)

119873119906(119860119887

37oplus 1198601 119887 gt 0 119891(119909) = 0) sim 119877(119860

119887

35oplus 1198601 8

119887 gt 0) (119905 = 119910 119909 = 119909)

(xi) 119873119906(11986038

oplus 1198601 119891(119909) = 1) sim (Sl(2R) oplus 119860

1) (1198901= 1198901

1198902= 1198902 1198903= minus1198903 1198904= 1198904)

119873119906(11986038

oplus 1198601 119887 gt 0 119891(119909) = 1) sim 119877(Sl(2R) oplus

1198601 9) (119905 = 119910 119909 = 119909) whereas 119877(Sl(2 R) oplus 119860

1 8))

is missing in [6](xii) 119873119906(119860

39oplus 1198601) sim (So(3) oplus 119860

1) No realization exists

in (1 + 1)-dimension(xiii) 119873119906(119860

41 119891(119909) = 0) sim (119860

41)

119873119906(11986041

119891(119909) = 0) sim 119877(11986041

8) (119905 = 119910 119909 = 119909)

(xiv) 119873119906(119860119887

42 119887 = 0 1) sim (119860

119887

42 119887 = 0 1) (119890

1= 1198901 1198902= minus1198902

1198903= minus1198903 1198904= 1198904)

119873119906(119860119887

42) sim 119877(119860

119887

42 8) (119905 = 119890

(119887minus1)119909119910 119909 = 119890

(119887minus1)119909)

(xv) 119873119906(1198601

42) sim (119860

119887

42 119887 = 1) No realization exists in (1 +

1)-dimension(xvi) 119873119906(119860

43 119891(119909) = 0) sim (119860

43)

119873119906(11986043

119891(119909) = 0) sim 119877(11986043

8) (119905 = 119910 119909 = 119909)(xvii) 119873119906(119860

44 119891(119909) = 0) sim (119860

44)

119873119906(11986044

119891(119909) = 0) sim 119877(11986044

7) (119905 = 119910 119909 = 119909)

(xviii) 119873119906(119860119886119887

45 minus1 le 119886 lt 119887 lt 1 119886119887 = 0) sim (119860

119886119887

45 minus1 le

119886 lt 119887 lt 119888 = 1 119886119887119888 = 0) (1198901

= 1198902 1198902

= 1198903 1198903

= 1198901

1198904= 1198904)

119873119906(119860119886119887

45 minus1 le 119886 lt 119887 lt 1 119886119887 = 0) sim 119877(119860

119886119887

45 7 minus1 le

119886 lt 119887 lt 119888 = 1 119886119887119888 = 0 119887 gt 0 if 119886 = minus1) (119905 = 119909119886minus1

119910 +

119909119886 ln(1|119909|) 119909 = ln |119909|)

(xix) (119860119886119886

45 minus1 le 119886 lt 1 119886 = 0) sim (119860

11119886minus1

45 minus1 le 119886 lt 1

119886 = 0) (1198901

= 1198903 1198902

= 1198902 1198903

= 1198901 1198904

= 1198904) No

realization exists in (1 + 1)-dimension(xx) 119873119906(119860

1198861

45 minus1 le 119886 lt 1 119886 = 0) sim (119860

11119886

45 minus1 le 119886 lt 1

119886 = 0) (1198901

= 1198901 1198902

= 1198903 1198903

= 1198902 1198904

= 1198904) No

realization exists in (1 + 1)-dimension(xxi) 119873119906(119860

11

45) sim (119860

119886119887

45 119886 = 119887 = 119888 = 1) (119890

1= 1198902 1198902= 1198903

1198903= 1198901 1198904= 1198904)

119873119906(11986011

45) sim 119877(119860

119886119887

45 10 119886 = 119887 = 119888 = 1) (119905 = 119910

119909 = 119909)(xxii) 119873119906(119860

119886119887

46 119886 = 0 119887 ge 0) sim (119860

119886119887

46 119886 gt 0) (119890

1= 1198901

1198902= 1198903 1198903= minus1198902 1198904= 1198904)

119873119906(119860119886119887

46 119886 = 0 119887 ge 0) sim 119877(119860

119886119887

46 6 119886 gt 0) (119905 =

119910(1 + 1199092)minus12

119890(119886minus119887)tanminus1119909

119909 = tanminus1119909)

Journal of Applied Mathematics 3

Table 1

Lie algebra 119873 Realizations (generators)41198601

11 120597119905 119909120597119905 119896(119909)120597

119905 ℎ(119909)120597

119905

211986021

5 120597119905 119905120597119905 120597119909 119909120597119909

lowast7 120597119905 119905120597119905+ 119909120597119909 119909120597119905 minus119909120597

119909

11986032

oplus 1198601

9 120597119905 119909120597119905 119905120597119905minus120597119909 119890minus119909

120597119905

119860119886

34oplus 1198601 |119886| le 1 119886 = 0 1 9 120597

119905 119909120597119905 119905120597119905+ (1 minus 119886)119909120597

119909 |119909|1(1minus119886)

120597119905

119860119887

35oplus 1198601 119887 ge 0 8 120597

119905 119909120597119905 (119887 minus 119909)119905120597

119905minus (1 + 119909

2)120597119909 radic1 + 119909

2119890minus119887 arctan 119909

120597119905

Sl(2 R) oplus 1198601

lowast8 120597119905 119905120597119905+ 119909120597119909 1199052120597119905+ 2119905119909120597

119909 119909120597119909

9 120597119905 119905120597119905 1199052120597119905 120597119909

11986041

8 120597119905 119909120597119905 (12) 119909

2120597119905 minus120597119909

119860119887

42 119887 = 0 8 120597

119905 119909120597119905 (1 (1 minus 119887)) 119909 ln |119909| 120597

119905 119887119905120597

119905+ (119887 minus 1)119909120597

119909 119887 = 1

11986043

8 120597119905 119909120597119905 minus119909 ln |119909| 120597

119905 119905120597119905+ 119909120597119909

11986044

7 120597119905 119909120597119905 (12) 119909

2120597119905 119905120597119905minus120597119909

119860119886 119887 119888

45 minus1 le 119886 lt 119887 lt 119888 = 1 119886119887119888 = 0

7 120597119905 119890(119886minus119887)119909

120597119905 119890(119886minus1)119909

120597119905 119886119905120597

119905+120597119909 119887 gt 0 if 119886 = minus1

10 120597119905 119909120597119905 120601(119909)120597

119905 119905120597119905 12060110158401015840(119909) = 0 119886 = 119887 = 119888 = 1

119860119886 119887

46 119886 gt 0 6 120597

119905 119890(119886minus119887)119909 cos 119909120597

119905 minus119890(119886minus119887)119909 sin119909120597

119905 119886119905120597

119905+120597119909

11986047

5 120597119905 119909120597119905 minus120597

119909 (2119905 minus (12) 119909

2)120597119905+ 119909120597119909

119860119887

48 |119887| le 1

5 120597119905 120597119909 119909120597119905 (1 + 119887)119905120597

119905+ 119909120597119909

lowast7 120597119905 119909120597119905 minus120597

119909 (1 + 119887)119905120597

119905+ 119887119909120597

119909 119887 = plusmn 1

119860410

lowast6 120597119905 120597119909 119905120597119905+ 119909120597119909 119909120597119905minus 119905120597119909

7 120597119905 119909120597119905 119905120597119905 minus119905119909120597

119905minus (1 + 119909

2)120597119909

(xxiii) 119873119906(11986047

) sim (11986047

)119873119906(119860

47) sim 119877(119860

47 5) (119905 = 119910 + (119909

24)(1 minus 2 log |119909|)

119909 = 119909)

(xxiv) 119873119906(11986048

) sim (119860119887

48 |119887| le 1 119887 = minus1)

119873119906(11986048

) sim 119877(119860119887

48 |119887| le 1 5 119887 = minus1) (119905 = 119910

119909 = 119909)

(xxv) 119873119906(119860119887

49 0 lt |119887| lt 1) sim (119860

119887

48 |119887| le 1 119887 = plusmn 1 0)

119873119906(119860119887

49) sim 119877(119860

119887

48 5) (119905 = 119910 119909 = 119909) whereas 119877

(119860119887

48 7 119887 = plusmn 1 0) is missing in [6]

(xxvi) 119873119906(1198601

49) sim (119860

119887

48 119887 = 1)

119873119906(1198601

49) sim 119877(119860

1

48 5) (119905 = 119910 119909 = 119909)

(xxvii) 119873119906(1198600

49) sim (119860

119887

48 119887 = 0)

119873119906(1198600

49) sim 119877(119860

0

48 5) (119905 = 119910 119909 = 119909) whereas 119877

(119860119887

48 7 119887 = 0) is missing in [6]

(xxviii) 119873119906(119860410

) sim (119860119886

49 119886 = 0) No realization exists in

(1 + 1)-dimension

(xxix) 119873119906(119860119886

411 119886 gt 0) sim (119860

119886

49 119886 gt 0) No realization

exists in (1 + 1)-dimension

(xxx) 119873119906(119860412

) sim (119860410

)119873119906(119860

412) sim 119877(119860

410 7) (119905 = 119910 119909 = 119909) whereas

119877(119860410

6) is missing in [6]

Remarks for Table 1

(i) 41198601 1 119909 ℎ(119909) and 119896(119909) form a linearly independent

set

Remarks for Tables 1 and 2 In both tables we have

(i) ℎ 119896 and 120601 are arbitrary functions with specifiedconditions mentioned in the corresponding realiza-tions

(ii) 119886 119887 and 119888 are parameters and arbitrary constantswhose range and values are mentioned in each of therealizations

(iii) lowast these are the cases of realizations which aremissingin [6]

Remarks for Table 2

(i) lowast these are the canonical forms of fourth-orderODEswhich are missing in [6] Only these are given herewith their corresponding algebras and realizations


Table 2

Lie algebra 119873 Realizations and equations

211986021

lowast7 120597119905 119905120597119905+ 119909120597119909 119909120597119905 minus119909120597

119909

sdotsdotsdotsdot

119909=10

minus

153

2

+2

1199092

119891(119909

minus

3119909

2

)

Sl(2R) oplus 1198601

lowast8 120597119905 119905120597119905+ 119909120597119909 1199052120597119905+ 2119905119909120597

119909 119909120597119909

sdotsdotsdotsdot

119909= minus2

119909+

1

1199093(

2

2minus 119909)

2

119891(1199092

((22) minus 119909)

32)

119860119887

48 |119887| le 1

lowast7 120597119905 119909120597119905 minus120597119909 (1 + 119887) 119905120597

119905+ 119887119909120597

119909 119887 = plusmn 1

sdotsdotsdotsdot

119909=10

minus

153

2

+ (1minus3119887)(1minus119887)

(4minus2119887)(1minus119887)

119891((

4minus

32

5)(

3)

(1minus2119887)(119887minus1)

)

119860410

lowast6 120597119905 120597119909 119905120597119905+ 119909120597119909 119909120597119905minus 119905120597119909

sdotsdotsdotsdot

119909=10

(1 + 2)minus

1523

(1 + 2)2+

3

(1 + 2)2119891(

(1 + 2)

2

minus 3)

3 Concluding Remarks

In this contribution we have supplemented the work [6] forthe canonical forms of scalar fourth-order ODEs and haveobtained four new forms as listed in Table 2The integrabilityof these equations has the same route as the others which arediscussed at length in [6]

Acknowledgments

A Fatima gratefully acknowledges the financial supportand scholarship from School of Computational and AppliedMathematics University of the Witwatersrand

References

[1] S Lie ldquoClassification und Integration von gewohnlichen Dif-ferentialgleichungen zwischen x y die eine Gruppe von Trans-formationen gestattenrdquoMathematische Annalen vol 8 no 9 p187 1888

[2] P J Olver Applications of Lie Groups to Differential Equationsvol 107 Springer New York NY USA 2nd edition 1993

[3] A H Kara F M Mahomed and P G L Leach ldquoNoetherequivalence problem for particle Lagrangiansrdquo Journal of Math-ematical Analysis and Applications vol 188 no 3 pp 867ndash8841994

[4] F M Mahomed and P G L Leach ldquoNormal forms for third-order equationsrdquo in Proceedings of the Workshop on FiniteDimensional Integrable Nonlinear Dynamical Systems P G LLeach and W H Steeb Eds p 178 World Scientic Johannes-burg South Africa 1988

[5] N H Ibragimov and M C Nucci ldquoIntegration of thirdorder ordinary differential equations by lies method equationsadmitting three-dimensional lie algebrasrdquo Lie Groups andTheirApplications vol 1 p 4964 1994

[6] T Cerquetelli N Ciccoli and M C Nucci ldquoFour dimensionalLie symmetry algebras and fourth order ordinary differentialequationsrdquo Journal of Nonlinear Mathematical Physics vol 9supplement 2 pp 24ndash35 2002

[7] J Patera and PWinternitz ldquoSubalgebras of real three- and four-dimensional Lie algebrasrdquo Journal of Mathematical Physics vol18 no 7 pp 1449ndash1455 1977

[8] FMMahomed and P G L Leach ldquoLie algebras associated withscalar second-order ordinary differential equationsrdquo Journal ofMathematical Physics vol 30 no 12 pp 2770ndash2777 1989

[9] R O Popovych V M Boyko M O Nesterenko and M WLutfullin ldquoRealizations of real low-dimensional Lie algebrasrdquoJournal of Physics A vol 36 no 26 pp 7337ndash7360 2003

[10] G Bluman and S Kumei ldquoOn invariance properties of the waveequationrdquo Journal of Mathematical Physics vol 28 no 2 pp307ndash318 1987

[11] S P TimoshenkoTheory of Elastic Stability McGraw-Hill NewYork NY USA 2nd edition 1961

[12] A H Bokhari F M Mahomed and F D Zaman ldquoSymmetriesand integrability of a fourth-order Euler-Bernoulli beam equa-tionrdquo Journal of Mathematical Physics vol 51 no 5 Article ID053517 2010

[13] F I Leite P D Lea and T Mariano ldquoSymmetry and integrabil-ity of a fourth-order Emden-Fowler equationrdquo Technical ReportCMCC-UFABC1-13 2012

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Stochastic AnalysisInternational Journal of


the linear wave equation in an inhomogeneous medium (see[10]) they occur prominently as model equations in the formof the static Euler-Bernoulli beam (see eg [11]) and Emden-Fowler equations Such equations have been investigated forsymmetry properties in [12 13]

Firstly we provide a comparison of the results of [9] andthat of [6] related to the realizations of four-dimensional Liealgebras as vector fields in the plane Then we list the newcanonical forms of scalar fourth-order ODEs which possessfour-dimensional algebras

2 Comparison of the Results of [6 9]

We show here that the results on realizations of four-dimensional algebras in the plane given in [6] are a specialcase of the corresponding set of realizations given in [9] Wemake a comparison of the lists of realizations given in [6]and [9] It should be remarked that in general a result ofclassification of realizations may contain errors of two typesnamely

(i) missing of some inequivalent cases and(ii) mutually equivalent cases

In the following comparison some first type of errorsexist in [6] Five cases are missing There are some othercases which can be combined in a compact form and alsosome arbitrary parameters and functions need modificationaccording to the results of [9] related to realizations in theplane Below we keep the notations of both on the left handside the notations of [6] and on right hand side that of [9]However for the final results and further utilization we keepthe notations of [9]

21 Four-Dimensional Algebras We use the nomenclature ofPatera and Winternitz [7] in the naming of the algebras suchas 4119860

1Thuswe do not provide a table of the abstract algebras

of dimension four as this is easily availableHere the 119873119906 refers to the realizations given in the work

[6] and 119877 to that of [9]

(i) 119873119906(41198601) sim (4119860

1) (1198901= 1198902 1198902= 1198903 1198903= 1198901 1198904= 1198904)

119873119906(41198601) sim 119877(4119860

1 11) (119905 = 119910 119909 = 119909)

(ii) 119873119906(1198602oplus21198601) sim (119860

21oplus21198601) (1198901= 1198902 1198902= minus1198901 1198903=

11989031198904= 1198904) No realization exists in (1+1)-dimension

(iii) 119873119906(21198602) sim (2119860

21) (1198901

= 1198902 1198902

= minus1198901 1198903

= 1198904

1198904= minus1198903)

119873119906(21198602) sim 119877(2119860

21 5) (119905 = 119909 119909 = 119910) whereas

119877(211986021

7) is missing in [6](iv) 119873119906(119860

31oplus1198601) sim (119860


(1 + 1)-dimension(v) 119873119906(119860

32oplus 1198601 119891(119909) = 0) sim (119860

32oplus 1198601)

119873119906(11986032

oplus 1198601 119891(119909) = 0) sim 119877(119860

32oplus 1198601 9) (119905 = 119910

119909 = minus119909)

(vi) 119873119906(11986033

oplus1198601) sim (119860


(1 + 1)-dimension

(vii) 119873119906(11986034

oplus 1198601 119891(119909) = 0) sim (119860

119886

34oplus 1198601 119886 = minus1)

119873119906(11986034

oplus 1198601 119891(119909) = 0) sim 119877(119860

119886

34oplus 1198601 9 119886 = minus1)

(119905 = 119910 119909 = 1199092)

(viii) 119873119906(119860119886

35oplus1198601 0 lt |119886| lt 1 119891(119909) = 0) sim (119860

119886

34oplus

1198601 |119886| le 1 119886 = 0 plusmn1)

119873119906(119860119886

35oplus 1198601 0 lt |119886| lt 1 119891(119909) = 0) sim 119877(119860

119886

34oplus

1198601 9) (119905 = 119910 119909 = 119909

1minus119886)

(ix) 119873119906(11986036

oplus 1198601 119891(119909) = 0) sim (119860

119887

35oplus 1198601 119887 = 0)

119873119906(11986036

oplus 1198601 119891(119909) = 0) sim 119877(119860

119887

35oplus 1198601 8 119887 = 0)

(119905 = 119910 119909 = (1199092minus 1)12

)

(x) 119873119906(119860119887

37oplus1198601 119887 gt 0 119891(119909) = 0) sim (119860

119887

35oplus1198601 119887 gt 0)

119873119906(119860119887

37oplus 1198601 119887 gt 0 119891(119909) = 0) sim 119877(119860

119887

35oplus 1198601 8

119887 gt 0) (119905 = 119910 119909 = 119909)

(xi) 119873119906(11986038

oplus 1198601 119891(119909) = 1) sim (Sl(2R) oplus 119860

1) (1198901= 1198901

1198902= 1198902 1198903= minus1198903 1198904= 1198904)

119873119906(11986038

oplus 1198601 119887 gt 0 119891(119909) = 1) sim 119877(Sl(2R) oplus

1198601 9) (119905 = 119910 119909 = 119909) whereas 119877(Sl(2 R) oplus 119860

1 8))

is missing in [6](xii) 119873119906(119860

39oplus 1198601) sim (So(3) oplus 119860

1) No realization exists

in (1 + 1)-dimension(xiii) 119873119906(119860

41 119891(119909) = 0) sim (119860

41)

119873119906(11986041

119891(119909) = 0) sim 119877(11986041

8) (119905 = 119910 119909 = 119909)

(xiv) 119873119906(119860119887

42 119887 = 0 1) sim (119860

119887

42 119887 = 0 1) (119890

1= 1198901 1198902= minus1198902

1198903= minus1198903 1198904= 1198904)

119873119906(119860119887

42) sim 119877(119860

119887

42 8) (119905 = 119890

(119887minus1)119909119910 119909 = 119890

(119887minus1)119909)

(xv) 119873119906(1198601

42) sim (119860

119887

42 119887 = 1) No realization exists in (1 +

1)-dimension(xvi) 119873119906(119860

43 119891(119909) = 0) sim (119860

43)

119873119906(11986043

119891(119909) = 0) sim 119877(11986043

8) (119905 = 119910 119909 = 119909)(xvii) 119873119906(119860

44 119891(119909) = 0) sim (119860

44)

119873119906(11986044

119891(119909) = 0) sim 119877(11986044

7) (119905 = 119910 119909 = 119909)

(xviii) 119873119906(119860119886119887

45 minus1 le 119886 lt 119887 lt 1 119886119887 = 0) sim (119860

119886119887

45 minus1 le

119886 lt 119887 lt 119888 = 1 119886119887119888 = 0) (1198901

= 1198902 1198902

= 1198903 1198903

= 1198901

1198904= 1198904)

119873119906(119860119886119887

45 minus1 le 119886 lt 119887 lt 1 119886119887 = 0) sim 119877(119860

119886119887

45 7 minus1 le

119886 lt 119887 lt 119888 = 1 119886119887119888 = 0 119887 gt 0 if 119886 = minus1) (119905 = 119909119886minus1

119910 +

119909119886 ln(1|119909|) 119909 = ln |119909|)

(xix) (119860119886119886

45 minus1 le 119886 lt 1 119886 = 0) sim (119860

11119886minus1

45 minus1 le 119886 lt 1

119886 = 0) (1198901

= 1198903 1198902

= 1198902 1198903

= 1198901 1198904

= 1198904) No

realization exists in (1 + 1)-dimension(xx) 119873119906(119860

1198861

45 minus1 le 119886 lt 1 119886 = 0) sim (119860

11119886

45 minus1 le 119886 lt 1

119886 = 0) (1198901

= 1198901 1198902

= 1198903 1198903

= 1198902 1198904

= 1198904) No

realization exists in (1 + 1)-dimension(xxi) 119873119906(119860

11

45) sim (119860

119886119887

45 119886 = 119887 = 119888 = 1) (119890

1= 1198902 1198902= 1198903

1198903= 1198901 1198904= 1198904)

119873119906(11986011

45) sim 119877(119860

119886119887

45 10 119886 = 119887 = 119888 = 1) (119905 = 119910

119909 = 119909)(xxii) 119873119906(119860

119886119887

46 119886 = 0 119887 ge 0) sim (119860

119886119887

46 119886 gt 0) (119890

1= 1198901

1198902= 1198903 1198903= minus1198902 1198904= 1198904)

119873119906(119860119886119887

46 119886 = 0 119887 ge 0) sim 119877(119860

119886119887

46 6 119886 gt 0) (119905 =

119910(1 + 1199092)minus12

119890(119886minus119887)tanminus1119909

119909 = tanminus1119909)


Table 1


11 120597119905 119909120597119905 119896(119909)120597

119905 ℎ(119909)120597

119905

211986021

5 120597119905 119905120597119905 120597119909 119909120597119909

lowast7 120597119905 119905120597119905+ 119909120597119909 119909120597119905 minus119909120597

119909

11986032

oplus 1198601

9 120597119905 119909120597119905 119905120597119905minus120597119909 119890minus119909

120597119905

119860119886

34oplus 1198601 |119886| le 1 119886 = 0 1 9 120597

119905 119909120597119905 119905120597119905+ (1 minus 119886)119909120597

119909 |119909|1(1minus119886)

120597119905

119860119887

35oplus 1198601 119887 ge 0 8 120597

119905 119909120597119905 (119887 minus 119909)119905120597

119905minus (1 + 119909

2)120597119909 radic1 + 119909

2119890minus119887 arctan 119909

120597119905

Sl(2 R) oplus 1198601

lowast8 120597119905 119905120597119905+ 119909120597119909 1199052120597119905+ 2119905119909120597

119909 119909120597119909

9 120597119905 119905120597119905 1199052120597119905 120597119909

11986041

8 120597119905 119909120597119905 (12) 119909

2120597119905 minus120597119909

119860119887

42 119887 = 0 8 120597

119905 119909120597119905 (1 (1 minus 119887)) 119909 ln |119909| 120597

119905 119887119905120597

119905+ (119887 minus 1)119909120597

119909 119887 = 1

11986043

8 120597119905 119909120597119905 minus119909 ln |119909| 120597

119905 119905120597119905+ 119909120597119909

11986044

7 120597119905 119909120597119905 (12) 119909

2120597119905 119905120597119905minus120597119909

119860119886 119887 119888

45 minus1 le 119886 lt 119887 lt 119888 = 1 119886119887119888 = 0

7 120597119905 119890(119886minus119887)119909

120597119905 119890(119886minus1)119909

120597119905 119886119905120597

119905+120597119909 119887 gt 0 if 119886 = minus1

10 120597119905 119909120597119905 120601(119909)120597

119905 119905120597119905 12060110158401015840(119909) = 0 119886 = 119887 = 119888 = 1

119860119886 119887

46 119886 gt 0 6 120597

119905 119890(119886minus119887)119909 cos 119909120597

119905 minus119890(119886minus119887)119909 sin119909120597

119905 119886119905120597

119905+120597119909

11986047

5 120597119905 119909120597119905 minus120597

119909 (2119905 minus (12) 119909

2)120597119905+ 119909120597119909

119860119887

48 |119887| le 1

5 120597119905 120597119909 119909120597119905 (1 + 119887)119905120597

119905+ 119909120597119909

lowast7 120597119905 119909120597119905 minus120597

119909 (1 + 119887)119905120597

119905+ 119887119909120597

119909 119887 = plusmn 1

119860410

lowast6 120597119905 120597119909 119905120597119905+ 119909120597119909 119909120597119905minus 119905120597119909

7 120597119905 119909120597119905 119905120597119905 minus119905119909120597

119905minus (1 + 119909

2)120597119909

(xxiii) 119873119906(11986047

) sim (11986047

)119873119906(119860

47) sim 119877(119860

47 5) (119905 = 119910 + (119909

24)(1 minus 2 log |119909|)

119909 = 119909)

(xxiv) 119873119906(11986048

) sim (119860119887

48 |119887| le 1 119887 = minus1)

119873119906(11986048

) sim 119877(119860119887

48 |119887| le 1 5 119887 = minus1) (119905 = 119910

119909 = 119909)

(xxv) 119873119906(119860119887

49 0 lt |119887| lt 1) sim (119860

119887

48 |119887| le 1 119887 = plusmn 1 0)

119873119906(119860119887

49) sim 119877(119860

119887

48 5) (119905 = 119910 119909 = 119909) whereas 119877

(119860119887


(xxvi) 119873119906(1198601

49) sim (119860

119887

48 119887 = 1)

119873119906(1198601

49) sim 119877(119860

1

48 5) (119905 = 119910 119909 = 119909)

(xxvii) 119873119906(1198600

49) sim (119860

119887

48 119887 = 0)

119873119906(1198600

49) sim 119877(119860

0

48 5) (119905 = 119910 119909 = 119909) whereas 119877

(119860119887

48 7 119887 = 0) is missing in [6]

(xxviii) 119873119906(119860410

) sim (119860119886


(1 + 1)-dimension

(xxix) 119873119906(119860119886

411 119886 gt 0) sim (119860

119886



(xxx) 119873119906(119860412

) sim (119860410

)119873119906(119860

412) sim 119877(119860

410 7) (119905 = 119910 119909 = 119909) whereas

119877(119860410


Remarks for Table 1


set





Remarks for Table 2



Table 2


211986021

lowast7 120597119905 119905120597119905+ 119909120597119909 119909120597119905 minus119909120597

119909

sdotsdotsdotsdot

119909=10

minus

153

2

+2

1199092

119891(119909

minus

3119909

2

)


lowast8 120597119905 119905120597119905+ 119909120597119909 1199052120597119905+ 2119905119909120597

119909 119909120597119909

sdotsdotsdotsdot

119909= minus2

119909+

1

1199093(

2

2minus 119909)

2

119891(1199092

((22) minus 119909)

32)

119860119887

48 |119887| le 1

lowast7 120597119905 119909120597119905 minus120597119909 (1 + 119887) 119905120597

119905+ 119887119909120597

119909 119887 = plusmn 1

sdotsdotsdotsdot

119909=10

minus

153

2

+ (1minus3119887)(1minus119887)

(4minus2119887)(1minus119887)

119891((

4minus

32

5)(

3)

(1minus2119887)(119887minus1)

)

119860410

lowast6 120597119905 120597119909 119905120597119905+ 119909120597119909 119909120597119905minus 119905120597119909

sdotsdotsdotsdot

119909=10

(1 + 2)minus

1523

(1 + 2)2+

3

(1 + 2)2119891(

(1 + 2)

2

minus 3)



Acknowledgments


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 1


11 120597119905 119909120597119905 119896(119909)120597

119905 ℎ(119909)120597

119905

211986021

5 120597119905 119905120597119905 120597119909 119909120597119909

lowast7 120597119905 119905120597119905+ 119909120597119909 119909120597119905 minus119909120597

119909

11986032

oplus 1198601

9 120597119905 119909120597119905 119905120597119905minus120597119909 119890minus119909

120597119905

119860119886

34oplus 1198601 |119886| le 1 119886 = 0 1 9 120597

119905 119909120597119905 119905120597119905+ (1 minus 119886)119909120597

119909 |119909|1(1minus119886)

120597119905

119860119887

35oplus 1198601 119887 ge 0 8 120597

119905 119909120597119905 (119887 minus 119909)119905120597

119905minus (1 + 119909

2)120597119909 radic1 + 119909

2119890minus119887 arctan 119909

120597119905

Sl(2 R) oplus 1198601

lowast8 120597119905 119905120597119905+ 119909120597119909 1199052120597119905+ 2119905119909120597

119909 119909120597119909

9 120597119905 119905120597119905 1199052120597119905 120597119909

11986041

8 120597119905 119909120597119905 (12) 119909

2120597119905 minus120597119909

119860119887

42 119887 = 0 8 120597

119905 119909120597119905 (1 (1 minus 119887)) 119909 ln |119909| 120597

119905 119887119905120597

119905+ (119887 minus 1)119909120597

119909 119887 = 1

11986043

8 120597119905 119909120597119905 minus119909 ln |119909| 120597

119905 119905120597119905+ 119909120597119909

11986044

7 120597119905 119909120597119905 (12) 119909

2120597119905 119905120597119905minus120597119909

119860119886 119887 119888

45 minus1 le 119886 lt 119887 lt 119888 = 1 119886119887119888 = 0

7 120597119905 119890(119886minus119887)119909

120597119905 119890(119886minus1)119909

120597119905 119886119905120597

119905+120597119909 119887 gt 0 if 119886 = minus1

10 120597119905 119909120597119905 120601(119909)120597

119905 119905120597119905 12060110158401015840(119909) = 0 119886 = 119887 = 119888 = 1

119860119886 119887

46 119886 gt 0 6 120597

119905 119890(119886minus119887)119909 cos 119909120597

119905 minus119890(119886minus119887)119909 sin119909120597

119905 119886119905120597

119905+120597119909

11986047

5 120597119905 119909120597119905 minus120597

119909 (2119905 minus (12) 119909

2)120597119905+ 119909120597119909

119860119887

48 |119887| le 1

5 120597119905 120597119909 119909120597119905 (1 + 119887)119905120597

119905+ 119909120597119909

lowast7 120597119905 119909120597119905 minus120597

119909 (1 + 119887)119905120597

119905+ 119887119909120597

119909 119887 = plusmn 1

119860410

lowast6 120597119905 120597119909 119905120597119905+ 119909120597119909 119909120597119905minus 119905120597119909

7 120597119905 119909120597119905 119905120597119905 minus119905119909120597

119905minus (1 + 119909

2)120597119909

(xxiii) 119873119906(11986047

) sim (11986047

)119873119906(119860

47) sim 119877(119860

47 5) (119905 = 119910 + (119909

24)(1 minus 2 log |119909|)

119909 = 119909)

(xxiv) 119873119906(11986048

) sim (119860119887

48 |119887| le 1 119887 = minus1)

119873119906(11986048

) sim 119877(119860119887

48 |119887| le 1 5 119887 = minus1) (119905 = 119910

119909 = 119909)

(xxv) 119873119906(119860119887

49 0 lt |119887| lt 1) sim (119860

119887

48 |119887| le 1 119887 = plusmn 1 0)

119873119906(119860119887

49) sim 119877(119860

119887

48 5) (119905 = 119910 119909 = 119909) whereas 119877

(119860119887


(xxvi) 119873119906(1198601

49) sim (119860

119887

48 119887 = 1)

119873119906(1198601

49) sim 119877(119860

1

48 5) (119905 = 119910 119909 = 119909)

(xxvii) 119873119906(1198600

49) sim (119860

119887

48 119887 = 0)

119873119906(1198600

49) sim 119877(119860

0

48 5) (119905 = 119910 119909 = 119909) whereas 119877

(119860119887

48 7 119887 = 0) is missing in [6]

(xxviii) 119873119906(119860410

) sim (119860119886


(1 + 1)-dimension

(xxix) 119873119906(119860119886

411 119886 gt 0) sim (119860

119886



(xxx) 119873119906(119860412

) sim (119860410

)119873119906(119860

412) sim 119877(119860

410 7) (119905 = 119910 119909 = 119909) whereas

119877(119860410


Remarks for Table 1


set





Remarks for Table 2



Table 2


211986021

lowast7 120597119905 119905120597119905+ 119909120597119909 119909120597119905 minus119909120597

119909

sdotsdotsdotsdot

119909=10

minus

153

2

+2

1199092

119891(119909

minus

3119909

2

)


lowast8 120597119905 119905120597119905+ 119909120597119909 1199052120597119905+ 2119905119909120597

119909 119909120597119909

sdotsdotsdotsdot

119909= minus2

119909+

1

1199093(

2

2minus 119909)

2

119891(1199092

((22) minus 119909)

32)

119860119887

48 |119887| le 1

lowast7 120597119905 119909120597119905 minus120597119909 (1 + 119887) 119905120597

119905+ 119887119909120597

119909 119887 = plusmn 1

sdotsdotsdotsdot

119909=10

minus

153

2

+ (1minus3119887)(1minus119887)

(4minus2119887)(1minus119887)

119891((

4minus

32

5)(

3)

(1minus2119887)(119887minus1)

)

119860410

lowast6 120597119905 120597119909 119905120597119905+ 119909120597119909 119909120597119905minus 119905120597119909

sdotsdotsdotsdot

119909=10

(1 + 2)minus

1523

(1 + 2)2+

3

(1 + 2)2119891(

(1 + 2)

2

minus 3)



Acknowledgments


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 2


211986021

lowast7 120597119905 119905120597119905+ 119909120597119909 119909120597119905 minus119909120597

119909

sdotsdotsdotsdot

119909=10

minus

153

2

+2

1199092

119891(119909

minus

3119909

2

)


lowast8 120597119905 119905120597119905+ 119909120597119909 1199052120597119905+ 2119905119909120597

119909 119909120597119909

sdotsdotsdotsdot

119909= minus2

119909+

1

1199093(

2

2minus 119909)

2

119891(1199092

((22) minus 119909)

32)

119860119887

48 |119887| le 1

lowast7 120597119905 119909120597119905 minus120597119909 (1 + 119887) 119905120597

119905+ 119887119909120597

119909 119887 = plusmn 1

sdotsdotsdotsdot

119909=10

minus

153

2

+ (1minus3119887)(1minus119887)

(4minus2119887)(1minus119887)

119891((

4minus

32

5)(

3)

(1minus2119887)(119887minus1)

)

119860410

lowast6 120597119905 120597119909 119905120597119905+ 119909120597119909 119909120597119905minus 119905120597119909

sdotsdotsdotsdot

119909=10

(1 + 2)minus

1523

(1 + 2)2+

3

(1 + 2)2119891(

(1 + 2)

2

minus 3)



Acknowledgments


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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