Proceedings of he rd IMT-GT Regional Conference on Mahemaics Saisics and Applicaions Universii Sains Malaysia ANALYSIS ON () + () () = G( ( ) ()) Jessada Tanhanch School of Mahemaics Insie of Science Sranaree Universiy of Technology Nakhonrachasrima 0000 Thailand e-mail: jessada@s.ac.h Absrac. Eqaion () + () () = G( ( ) ()) is a delay parial differenial eqaion wih an arbirary fncional G. This delay parial differenial eqaion is more general han () + () () = G( ( ) ) which has been applied grop analysis o find represenaions of analyical solions []. Applicaion of grop analysis o he eqaion and grop classificaion of represenaions of solions where G = g( () ( ) ) + () g and are arbirary fncions are presened in he aricle. Inrodcion Consider delay parial differenial eqaion (DPDE) wih delay > 0 (.) () + () () = G( ( ) ()) For he simpliciy noaion will be sed o denoe ( ) denoes () and mean firs parial derivaives of wih respec o and respecively. Eqaion (.) can be simply wrien as (.) + = G( ). Eqaion (.) is similar o opf or inviscid Brgers' eqaion []. owever eqaion (.) has a delay erm which makes he eqaion difficl o be solved []. Applicaions of delay differenial eqaions can be fond in [56]. The represenaions of solions for he pariclar case of eqaion (.) (.) = G( ). has been fond []. These solions were obained by applying grop analysis mehod [789] o he eqaion. Grop analysis also classifies eqaion (.) w.r.. symmeries ino wo cases arbirary fncional G and G = k : - For arbirary fncional G ( ) The solion is = f( C C) where f is an arbirary fncion C C are arbirary consans. The solion redces eqaion (.) ino a fncional ordinary differenial eqaion (FODE) G( f( θ + C) ) f '( θ) = Cf () θ C where θ = C C. - For pariclar fncional G ( ) = k where k is an arbirary consan. For his case eqaion (.) has wo possible forms of represenaion of solions i.e.. = ( + C) f( ) where f is an arbirary fncion C is an arbirary consan. This solion redces he eqaion ino delay ordinary differenial eqaion (DODE) f [ ] '( θ) = kf( ) f( ). C. ( ) 5 C = e f ( + C) e where 5 f is an arbirary fncion C C are arbirary consans. 5 C5 By his solion eqaion (.) can be simplified o FODE Cf 5 ( φ) kf( e φ) f '( θ) = where f( φ) C5φ 5 φ = ( + C) e C. In his aricle grop analysis is applied o find symmeries of eqaion (.) which is more general han (.). owever for he sake of simpliciy eqaion (.) is considered for he case G = g( () ( ) ) + () only where g and are arbirary fncions. Classificaion of he eqaion wih respec o grops of symmeries admied by he eqaion are presened in he following secions.
Applicaions of grop analysis o delay differenial eqaions By he heory of grop analysis a symmery of eqaion (.) is defined as he ransformaion ϕ : Ω Δ Ω which ransforms a solion of he differenial eqaion o a solion of he same eqaion where Ω is a se of variables ( ) and Δ R is a symmeric inerval wih respec o zero. Variable ε is considered as a parameer of ransformaion ϕ which ransforms variable ( ) o new variable ( ) of he same space. Le ϕ( ;) ε be denoed by ϕ ε( ). The se of fncions ϕ forms a one-parameer ransformaion grop ε of space Ω if he following properies hold [789]: () ϕ 0( ) = ( ) for any ( ) Ω; () ϕε ( ϕ ( ) ) ( ) ε = ϕ ε + for any ε ε ε ε + ε Δ and ( ) Ω; () if ϕ ε( ) = ( ) for any ( ) Ω hen ε = 0. The oher noaions = ϕ ( ; ε) = ϕ ( ; ε) = ϕ ( ; ε) are sed as he same meaning as ϕ ε( ) = ( ). The ransformed variable wih delay erm and is derivaives are defined by = ( ) and = / = / respecively. Consider DPDE (.) F ( ) = 0. [6] shows he derivaive of an eqaion wih he ransformed variables and derivaives wih respec o parameer ε vanishes if he ransformaion is symmery : (.) F ( ) = XF( ) 0. (.) ε ε= 0(.) The operaor X is defined by X = ( ζ ξ η) + ( ζ ξ η ) + ζ + ζ where ϕ ϕ ϕ ξ( ) = ( ;0) η( ) = ;0)( ζ ) = ( ;0) ε ε ε ξ = ξ( ) η = η( ) ζ = ζ( ) ζ = D ( ζ ξ η) ζ = D( ζ ξ η) D = + + + + + D = + + + + + The operaor X is is called a canonical Lie-Bäcklnd infiniesimal generaor of a symmery. Eqaion (.) is called a deermining eqaion (DME). Since XF( ) 0 we say ha he operaor X is (.) admied by eqaion (.) or eqaion (.) admis he operaor X. Lie's heory [789] shows he generaor is one-o-one corresponden o he symmery. This generaor is also eqivalen o an infiniesimal generaor [8] (.) X = ξ + η + ζ. Finding and solving he deermining eqaion The DME for = G( ) can be fond by leing F = G( ) and sbsie i ino eqaion (.) (.) X( G( ) ) 0. = G( ) By leing = G so ( ) = G + G = G + G and = G where where G = G( ) = ( ) = ( ) = ( ). Ths DME (.) becomes G [ ( η η ) + ξ ξ] + [ ( η + η+ ηg) ( ξ + ξ+ ξg) + ζ] (.). G( η + η + ηg) + G [ G ( η η) ζ ] Gζ + ζ + ζ + ζg 0 By he heory of eisence of a solion of a delay differenial eqaion he iniial vale problem has a pariclar solion corresponding o a pariclar iniial vale. Becase iniial vales are arbirary variables and heir derivaives can be considered as arbirary elemens. Since every ransformed-solion () is a solion of eqaion (.) he DME ms be idenical o zero. Ths if DME (.) is wrien as a polynomial of variables and heir derivaives he coefficiens of hese variables in he eqaions ms vanish. In order o solve a DME
one solves he several eqaions of hese coefficiens. This mehod is called spliing he DME. Unknown fncions ξ η and ζ can be obained from his process. By spliing eqaion (.) wih respec o one obains G ( η η ) ξ ξ + 0. Since eqaion (.) is considered as a DPDE i is assmed G 0. The eqaion is simplified o (.) ( η η ) + ξ ξ 0 By he assmpion ξ and η depend on variables while ξ and η depend on if one differeniaes eqaion (.) w.r.. he derivaive becomes η ξ 0. Spliing he eqaion w.r. implies ξ = 0 and η = 0 which means ξ and η do no depend on. By he similar srcre of ξ and ξ and η and η boh ξ and η depend on only variables and. Eqaion (.) can be spli again w.r.. which implies ξ () = ξ() and η () = η(). The condiions obained mean ξ and η are periodic fncions w.r.. wih period i.e. (.) ξ( ) = ξ() η( ) = η(). Again spliing he DME w.r.. one ges (.5) ζ = ξ + ξ ( η + η) ζ = ξ + ξ ( η + η ). Sbsie ξηζ and ζ ino he DME and differeniae i wih respec o ξ( [ G + G ]) + ξ ( [ G + G ]) + (.6) η( G + G G ) + η ( G + G ( ) G + G ) 0 ere if we consider eqaion (.6) as ξa + ξb+ ηc+ ηd 0 which may be wrien in a vecor form as (.7) ξ ξ η η ABCD 0 where A= [ G + G ] B= [ G + G ] C= G + G G and D= G G ( + ) G + G we are able o classify eqaion (.) as he followings.. The kernel of admied Lie grops The se of symmeries which are admied for any fncional appeared in he eqaion is called a kernel of admied generaors. Assme eqaion (.6) is valid for any fncional G. Since G G G G vary arbirarily he se spanned by ABCD has dimension. Ths ξ ξ η η ms be a zero vecor i.e. all of ξ ξ η η vanish. This implies ξ and η are consans and ζ is zero. Le ξ and η be denoed by C and C respecively. The infiniesimal generaor admied by eqaion (.) is X = C + C. By he heory from grop analysis he characerisic eqaions d d d = = imply ξ η ζ = f ( C C) is a represenaion of a solion. I redces eqaion (.) ino FODE G( f() θ f( θ + C) ) f '( θ) = Cf () θ C where θ = C C.. Eension of he kernel Eensions are symmeries for he pariclar fncional G. ere for he sake of simpliciy case A=0 is considered only. For his case i implies (.8) G ( ) = g ( ) + () where g is an arbirary fncion of sch ha g 0 (or g ' 0) and is an arbirary fncion of variable. Eqaion (.6) is redced ino he form (.9) ξ ( ) g'' + η( g' [ ] g'' ) + η ( ( ) g'' + [ ] g' ) 0. Eqaion (.9) can be considered as a vecor form ξ η η 0 where = ( ) g'' C ABC. Le V be he se spanned by vecor ABC. B = g' [ ] g'' and = ( ) g'' + [ ] g' All possible cases which make eqaion (.9) valid are considered according o he dimension of V. A
.. dim V =. This condiion means vecor ξ η η ms be a zero vecor i.e. ξ η η vanish. Ths he DME is simplified o '( ) ξ'() + ξ''() = 0. The derivaive of he DME w.r.. is ''( ) ξ '( ) = 0. - Case ''( ) = 0 ere ( ) = + is a solion of he eqaion where are arbirary consans. owever by he arbirariness of fncion g can be omied. The DME is ξ'( ) + ξ''( ) = 0 which has ξ = Ce + C as a solion where C C are arbirary consans. The periodic condiion (.) of ξ implies ( ) Ce + C = Ce + C. The condiion is valid for = 0 or C = 0. For his case ξ ms be a consan. - Case ''( ) 0. The eqaion immediaely implies ξ is a consan. Boh wo cases show eqaion G ( ) = g ( ) + () admis X = ξ + η where ξ and η are arbirary consans and g are arbirary fncions. For dim V = i has he same solion wih he kernel case... dim V =. This condiion means here eiss a consan vecor αβγ 0 which is orhogonal o se ABC A B C V i.e. αβγ = α + β + γ = 0. By changing of variable z = ( ) he eqaion is derived o z( α β + γz) g'' + ( β + γz) g' + γ( zg'' + g' ) = 0. Spliing he eqaion w.r.. we have (.0) γ ( zg '' + g ') = 0 (.) z( α β + γz) g'' + ( β + γz) g' = 0. / - Case γ 0. Solving eqaion (.0) makes gz () = Cz + C where C C are arbirary consans. Eqaion (.) is simplified o ( / ) [ ] 5 0 C α + β z + γz =. By he arbirariness of z α + β and γ ms vanish. This case conradics o he assmpion γ 0. - Case γ = 0. Eqaion (.5) is redced o (.) z( α β) g'' + βg' = 0 If α β = 0 (or α = β) i makes β g ' = 0. This case conradics o he condiion αβγ 0 is no zero and g ' 0. Condiion α β 0 (or α β) will be considered only. For he condiions γ = 0 α β eqaion (.) is considered ino wo cases : Case β α β = i.e. α = β. The above condiion α β implies α 0. The eqaion can be redced o zg '' + g ' = 0 which has a solion gz () = Clnz+ C where C is a nonzero arbirary consans C is a consan. owever he consan C can be omied becase of he arbirariness of. Sbsie g ino he DME and differeniae i w.r.. he eqaion calclaed is C [ η ξ + η + η ] = 0. Since C 0 and nknown fncions ξ and η depend on ( ) he eqaion can be spli w.r.. and which implies η = η() and ξ() = η'() + ξ() where ξ is an arbirary fncion of. Sbsie boh obained fncions ino he DME : (.) [ η'''( ) η''( ) '( ) ] + [ η''( ) η'( ) '( ) ] Cη'( ) + ξ''( ) '( ) ξ'( ) = 0. Since nknown fncions ξ η do no depend on hen η'''( ) η''( ) '( ) = 0. This can be considered ino sbcases '( ) = 0 and '( ) 0. () '( ) = 0 i.e. is a consan. Then η '''( ) = 0. The periodic condiion implies η is only a consan. The DME is simplified o ξ ''( ) = 0. ξ is also a consan by he periodic condiion. This sbcase shows = C ln( ) + admis he generaor ξ + η where ξη are arbirary consans. () '( ) 0. The mied derivaive of DME (.) w.r.. and shows η ''( ) ''( ) = 0. This can be considered ino wo sbcases ''( ) = 0 and ''( ) 0. - ''( ) = 0. I implies = + where are arbirary consans and 0. The derivaive of eqaion (.) w.r.. is ( η'''( ) η''( )) = 0. Ths is solion is η = C + C + C5e. By he periodic condiion C and C ms idenical o zero i.e. 5 η is a consan. The DME is redced o ξ ''( ) ξ '( ) = 0 which has a solion ξ = C6 + C7e. Also he periodic condiion of ξ implies C 7 = 0. Then = Cln( ) + + admis he generaor C + C. 6
- ''( ) 0. The eqaion implies η ''( ) = 0 i.e. wih he periodic condiion η is a consan only. The DME is redced o ξ ''() '( ) ξ '() = 0. Differeniae he eqaion w.r.. ''( ) ξ '( ) = 0 i implies ξ is a consan. All above cases shows = f ( η ξ) where ξη are arbirary consans and f is arbirary fncion is solion of = C ln( ) + ( ) where C is a nonzero arbirary consan and is an arbirary fncion of. Case β α β. Le β δ =. ence he solion of eqaion (.) is g = C α ( ) δ+ + C β where C is a nonzero arbirary consans C is a consan. owever he consan C can be omied becase of he arbirariness of. Spliing he DME eqaion w.r.. shows δ (.) C ( δ + ( ) ) δξ ( δ) η ( δ( )) η + + + = 0. Since ξ and η depend on ( ) hen eqaion (.) can be spli w.r.. and. I implies ( δ) η = 0 and ( + δ) η = 0. The arbirariness of δ implies η = 0 i.e. η = η(). Eqaion (.) is simplified o (.5) δξ + ( δ) η '( ) = 0. Case δ 0. () If δ = eqaion (.5) shows ξ = 0 i.e. ξ = ξ(). The DME is redced o ξ''( ) η''( ) η'( ) ( ) + [ η'( ) ξ'( ) ] '( ) = 0. The second derivaive of DME w.r.. implies [ η'( ) ξ'( ) ] '''( ) = 0. If '''( ) 0 hen η'( ) ξ'( ) = 0. Spliing he eqaion w.r.. shows ξ and η are consans. Ths he solion of eqaion = C ln( ) + ( ) is also = f ( η ξ). Sppose '''( ) = 0. This means = + + where are arbirary consans. The derivaive of he DME w.r.. is [ ξ'() + η''() + η'() ] = 0. (a) = 0 and = 0. The periodic condiion implies η is a consan. The DME is redced o ξ ''( ) = 0. So ξ is also a consan. (b) = 0 b 0. The eqaion shows η = C + Ce. The periodic condiion redces erm Ce which makes η a consan. The DME is redced o ξ''( ) ξ( ) = 0 also ξ ms be a consan. (c) 0. Then η''( ) + η'( ) ξ '( ) =. The DME is simplified o η'''( ) + λη'( ) = 0 where λ = ( ) ( C + ). If λ = 0 his shows η '''( ) = 0. ξ can be only a consan and ξ is a consan also. λ λ If λ < 0 η = C + Ce + C5e. The periodic propery of η implies C and C vanish and 5 ξ ms be also a consan. If λ > 0 η = C + C cosλ + C5 sin λ. By he periodic condiion i is considered ino wo cases : - π. In his case C and C ms vanish and i implies 5 ξ o be a consan. λ - π =. ere ξ is eqal o η = C6 + C7 cosλ + C8 sin λ where C 6 is an arbirary consan λ C + λ C 5 5 7 C + λ C = C C 8 =. By he condiion (.5) ζ = λ ( C8 C5) cosλ + ( C7 + C) sin λ The solion of eqaion = C( ) + + + can be fond from he characerisic eqaions i.e. pd pd = e qe d + ( ψ( )) F where C5 cosλ C sinλ p = λ C8 cosλ C7 sinλ q = λ C6 + C7 cosλ + C8 sin λ ψ() = d C + C cosλ + C sinλ C + C cosλ + C sinλ C + C cosλ + C sin λ 5 5 5
F is an arbirary fncion C C C C5 C6 are arbirary consans 0 λ C7 C8 are he consans which were defined in his secion. () If δ eqaion (.5) implies δ ξ = η'( ) + ξ( ) where ξ δ () is an arbirary fncion of. Sbsie ξ ino he DME and differeniae i w.r.. boh and we obain δ ''( ) η ''( ) = 0. δ This may be considered ino wo sbcases. (a) ''( ) 0. This implies η ''( ) = 0. Similar o he previos case η is a consan. The DME is redced o ξ''() '( ) ξ'() = 0. The derivaive of he eqaion w.r.. shows ''( ) ξ '( ) = 0 which means ξ is a consan. This shows boh ξ and η are consans. (b) ''( ) = 0. Then = + where are arbirary consans. Derivaive of he DME w.r.. is δ η''( ) η'( ) = 0. δ If δ = he derivaive of he DME w.r.. gives s η'''() η''() = 0. Similar o he previos case η is a consan. The DME is also ξ''( ) ξ'( ) = 0 and is solion is a consan. If δ. For arbirary consan and periodic propery η ms be a consan. The DME is ξ''( ) ξ'( ) = 0 and is solion is a consan. δ+ Boh sbcases show eqaion = C( ) + ( ) for δ has he same solion wih he kernel case. Case δ = 0. Eqaion (.5) shows η '( ) = 0 i.e. η is a consan. The DME is redced o ( ) ξ '( )( ξ + ξ) + ξ + ξ + ξ = 0. In order o classify a solion of DPDE we have o analyze by he following cases : () ξ = 0. The DME is simplified o ξ''( ) '( ) ξ'( ) = 0. Is derivaive w.r.. is ''( ) ξ '( ) = 0. If ''( ) = 0 hen = + where are arbirary consans. The DME is ξ''( ) ξ'( ) = 0. Wih he periodic condiion ξ can be only a consan. If ''( ) 0 hen ξ '( ) = 0 which show ξ is also a consan. () ξ 0. The hird derivaive of he DME w.r.. can be rewrien as (.6) ξ () ''' + ( ) + ( ) = 0. ξ The derivaives of eqaion (.6) w.r.. and give d ξ () ( ) = 0 and d ξ d ξ () ( ) = 0. We d ξ consider he problem ino wo sbcases : - () ( ) = 0 hen '''( ) = 0 which makes = + + saisfying eqaion (.6). The second derivaive of he DME w.r.. is ( ξ ξ ) = 0. If = 0 hen ξ( ) = ξ() + ξ(). The derivaive of he DME w.r.. shows ξ '( ) = 0 i.e. ξ is a consan. The DME is ξ ''( ) ξ '( ) + ξ = 0. Le λ = ( ) ξ. Wih he periodic condiion fncion ξ can be fond according o λ : (a) λ 0. ξ ms be a consan. Then he DME is ξ = 0. If 0 hen ξ = 0. Ths = C( ) + admis he same generaor and has he same solion wih he kernel case. owever if = 0 and ξ 0 hen = C + C where C C are arbirary consans admis ( ξ + ξ) + η + ξ and has a solion = ( ξ + ξ) F ( ηln( ξ + ξ) ξ) where F is an arbirary fncion. This solion redces he eqaion (.7) ino an FODE F' ( χ) [ ηf ( χ) ] + [ F ( χ) ] = CF ( χ) + CF ( χ+ ξ) where ξ 0 η are arbirary consans and χ = ηln( ξ + ξ) ξ. (b) λ < 0. Le ρ = λ/. Then ξ = e ( C cos ρ + C sin ρ). Wih he periodic condiion ms be idenical o zero ξ > 0 and π =. Then eqaion = C( ) + 0 admis λ ( ξ + C cos ρ + C sin ρ) + η + ( ξ C ρsin ρ + C ρcos ρ)
where ρ = ξ C C are arbirary consans. This case is oo complicaed o find an eac form of a solion. If 0 hen ξ( ) = ξ() e + ξ(). The derivaive of DME w.r.. is ξ'( ) = 0 which means ξ '( ) = 0 or ξ is a consan. DME is simplified o e ( ξ''() ξ'() + ξ() ) = 0. λ = ( ). Wih he periodic condiion fncion ξ can be fond according o λ : (a) λ 0. ξ ms be a consan. If > 0 hen ξ = 0 and DME vanishes i.e. he solion form is no differen o he kernel case. If = 0 hen ξ is any consan. The DME is ξ e = 0. If 0 he eqaion has he similar solion wih he previos case. On he oher hand = 0 implies = C( ) + admis ( ξe + ξ) + η + ξe. This means η ξ e = ( ξe + ξ) F ln is a ξ ξe + ξ solion of he eqaion and redces he FPDE o F( Θ) - F( Θ+ ) + [ ] ξf( Θ) F'( Θ ) = C where ηf( Θ) - η ξe Θ= ln and C is an arbirary consan. ξ ξe + ξ (b) Le ρ = λ/. Then ξ = e ( C cos ρ + C sin ρ). Wih he periodic condiion ms be idenical o zero > 0 and π =. Then eqaion = C( ) + + 0 λ admis ( ( ) ) e Ccos ρ + C sin ρ + ξ + η + e ( Ccos ρ C sin ρ) ρ( Csin ρ C cos ρ) + + + where ρ = and C C are arbirary consans. This case is oo complicaed o find an eac form of a solion. - () ( ) 0.I means ξ / ξ is a consan which has a solion ξ = ψ( + K) where K is a consan () and ψ is an arbirary fncion. Sbsie ξ ino eqaion (.6) hen ( K + ) ( ) + '''( ) = 0. The eqaion has a solion ( ) = ( K+ )ln( K+ ) + + + where are consans. The DME is simplified o [ ψ '' ψ '] + [ kψ '' ( + K) ψ '] + K ψ '' + [ K( + ) ] ψ ' = 0. Spliing he eqaion w.r.. and shows ψ is a consan. This case has he same solion wih he kernel case... dim V =. ere ABC can be represened by ABC = αβγ φ( ) where αβγ are arbirary consans which αβγ 0 0 0 and φ is a nonconsan fncion. The sysem of eqaions corresponding o he vecor is (.7) ( ) g'' = αφ( ) (.8) g' ( ) g'' = βφ( ) (.9) ( ) g'' + g' g' = γφ( ). - Case α 0. Eqaion (.9) can be derived from eqaion (.7) and (.8) ino [(α + β) + ( α β) γ] φ = 0 Since φ is no idenical o zero hen is coefficien ms vanish and implies α = β = γ = 0. This conradics o he assmpion. - Case α = 0. ere g '' = 0 which implies g = C( ) + C where C C are arbirary consans. Sbsie g ino eqaion (.8) C = βφ( ) is obained. If β = 0 i implies C = 0 and g is a consan which is invalid. Also if β does no vanish he eqaion implies φ is a consan fncion which also conradics o he assmpion. This proves ha case dim V = is invalid... dim V =0. ABC can be consider as a consan vecor αβγ i.e. (.0) ( ) g'' = α (.) g' ( ) g'' = β
(.) ( ) g'' + g' g' = γ where αβγ are arbirary consans. Sbsie eqaion (.0) ino eqaion (.) i leads o g ' = α + β and g '' = 0. Sbsie boh vales ino eqaion (.) he eqaion is redced o ( α + β) ( α + β) = γ. By he arbirariness of and α + β vanishes which makes g ' = 0. I conradics o he assmpion. This case is invalid also. Conclsion Solions of eqaion = C( ) + + + = C + C = C( ) + and = C( ) + + are presened in he aricle. For oher forms of eqaion = g( ) + ( ) where g are arbirary fncions he solion is = f ( η ξ) where f is an arbirary fncion and ξη are arbirary consans. 5 Acknowledgemen This work is sppored by Comission On igher Edcaion and Thailand Research Fnd gran nmber MRG8805 and Sranaree Universiy of Technology. Ahor is deeply indebed o Prof.Sergey Meleshko for his nmeros help. I wold like o epress my sincere hank o him and Ass.Prof.Dr.Apichai emalin. 6 References [] J. D. Logan An inrodcion o nonlinear parial differenial eqaions New York: John Wiley & Sons 999. [] R. D. Driver Ordinary and Delay Differenial Eqaions New York: Springer-Verlag 977. [] J. Tanhanch Symmery Analysis on () + () () = G( ( ) ) Proceeding of IMT-GT 006 Regional Conference on Mahemaics Saisics and Applicaions. Universii Sains Malaysia Pla Pinang Malaysia Vol. I - Pre Mahemaics ISBN : 98-9-86-9 7-. [] J. Tanhanch and S. V. Meleshko On definiion of an admied Lie grop for fncional differenial eqaions Commnicaion in Nonlinear Science and Nmerical Simlaion 9 (00) 7-5. [5] J. Tanhanch and S. V. Meleshko Applicaion of grop analysis o delay differenial eqaions. Nonlinear acosics a he beginning of he s cenry. Moscow: Moscow Sae Universiy 00 pp. 607-60. [6] J. Tanhanch Applicaion of Grop Analysis o Fncional Differenial Eqaions Ph.D. Thesis Nakhonrachasrima Thailand: Sranaree Universiy of Technology 00. [7] L. V. Ovsiannikov Grop Analysis of Differenial Eqaions New York: Academic Press 98. [8] N.. Ibragimov Elemenrary Lie Grop Analysis and Ordinary Differenial Eqaions London: John Wiley \& Sons Ld 999. [9] N.. Ibragimov Lie grop analysis of differenial eqaions Vol.-. Florida: CRC Press. 99.