Research Article Boltzmann s Six-Moment One-Dimensional Nonlinear System Equations with the Maxwell-Auzhan Boundary Conditions

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1 Hndw Publshng Corporon Journl of Appled Mhemcs Volume 216, Arcle ID , 8 pges hp://dx.do.org/1.1155/216/ Reserch Arcle Bolzmnn s Sx-Momen One-Dmensonl Nonlner Sysem Equons wh he Mxwell-Auzhn Boundry Condons A. Skbekov 1 nd Y. Auzhn 2 1 Reserch Lborory Appled Modelng Ol nd Gs Felds, Kzkh-Brsh Techncl Unversy, 59 Tole B Sree, Almy 5, Kzkhsn 2 Al-Frb Kzkh Nonl Unversy, 71 Al-Frb Avenue, Almy 54, Kzkhsn Correspondence should be ddressed o A. Skbekov;.skbekov@kbu.kz Receved 2 Februry 216; Revsed 1 June 216; Acceped 5 June 216 Acdemc Edor: Peer G. L. Lech Copyrgh 216 A. Skbekov nd Y. Auzhn. Ths s n open ccess rcle dsrbued under he Creve Commons Arbuon Lcense, whch perms unresrced use, dsrbuon, nd reproducon n ny medum, provded he orgnl work s properly ced. We prove exsence nd unqueness of he soluon of he problem wh nl nd Mxwell-Auzhn boundry condons for nonsonry nonlner one-dmensonl Bolzmnn s sx-momen sysem equons n spce of funcons connuous n me nd summble n squre by spl vrble. In order o obn pror esmon of he nl nd boundry vlue problem for nonsonry nonlner one-dmensonl Bolzmnn s sx-momen sysem equons we ge he negrl equly nd hen use he sphercl represenon of vecor. Then we obn he nl vlue problem for Rcc equon. We hve mnged o obn prculr soluon of hs equon n n explc form. 1. Inroducon In cse of one-om gs ny mcroscopc sysem durng process of s evoluon o n equlbrum se psses 3 sges: nl rnson perod, descrbed n erms of full funcon of sysem dsrbuon; he knec perod, by mens of oneprl dsrbuon funcon; nd he hydrodynmc perod, by mens of fve frs momens of he dsrbuon funcon. In knec regme he behvor of rrefed gs n spce of me nd velocy s descrbed by Bolzmnn s equon. I s known fromgsdynmcshnmosofheencouneredproblems here s no need o use deled mcroscopc gs descrpon wh help of he dsrbuon funcon. Therefore s nurl o look for less deled descrpon usng mcroscopc hydrodynmc vrbles (densy, hydrodynmc velocy, emperure, ec.). As hese vrbles re defned n erms of momens of he dsrbuon funcon, we re fced wh he problemofnlyznghevrousmomensofbolzmnn s equon. Noe h Bolzmnn s momen equons re nermede beween Bolzmnn (knec heory) nd hydrodynmc levels of descrpon of se of he rrefed gs nd form clss of nonlner prl dfferenl equons. Exsence of such clss of equons ws noced by Grd [1, 2] n He obned he momen sysem by expndng he prcle dsrbuon funcon n Herme polynomls ner he locl Mxwell dsrbuon. Grd used Cresn coordnes of veloces nd Grd s momen sysem conned s coeffcens such unknown hydrodynmc chrcerscs lke densy, emperure, verge speed, nd so forh. Formulon of boundry condons for Grd s sysem s lmos mpossble, s he chrcersc equons for vrous pproxmons of Grd s hyperbolc sysem conn unknown prmeers lke densy, emperure, nd verge speed. However, 13- nd 2-momen Grd equons re wdely used n solvng mny problems of he knec heory of gses nd plsm. Bolzmnn s equon s equvlen o n nfne sysem of dfferenl equons relve o he momens of he prcle dsrbuon funcon n he complee sysem of egenfuncons of lnerzed operor. As rule we lm sudy by fne momen sysem of equons becuse solvng nfne sysem of equons does no seem o be possble. Fne sysem of momen equons for specfc sk wh cern degree of ccurcy replces Bolzmnn s equon. I s necessry, lso roughly, o replce he boundry condons for he prcle dsrbuon funcon by number of mcroscopc condons for he momens; h s, here rses he problem of boundry condons for fne sysem of

2 2 Journl of Appled Mhemcs equons h pproxme he mcroscopc boundry condons for Bolzmnn s equon. The queson of boundry condons for fne sysem of momen equons cn be dvded no wo prs: how mny condons mus be mposed nd how hey should be prepred. From mcroscopc boundry condons for Bolzmnn s equon here cn be obned n nfne se of boundry condons for ech ype of decomposon. However, he number of boundry condons s deermned no by he number of momen equons; h s, s mpossble, for exmple, o ke s mny boundry condons s equons, lhough he number of momen equons ffecs he number of boundry condons. In ddon, he boundry condons mus be conssen wh he momen equons nd he resulng problem mus be correc. Boundry condon problems rse n he followng sks: (1) momen boundry condons n rrefed gs slp-flow problems; (2) defnon of boundry condons on he surfces of sremlned rrefed gs; (3) predcon erospce erodynmc chrcerscs of rcrf very hgh speeds nd hgh ludes nd so forh. In work [3] we hve obned he momen sysem whch dffers from Grd s sysem of equons. We used sphercl velocy coordnes nd decomposed he dsrbuon funcon no seres of egenfuncons of he lnerzed collson operor [4, 5], whch s he produc of Sonne polynomls nd sphercl funcons. The resulng sysem of equons, whch correspond o he prl sum of seres nd whch we clled Bolzmnn s momen sysem of equons, s nonlner hyperbolc sysem n relon o he momens of he prcles dsrbuon funcon. The srucure of Bolzmnn s momen sysem of equons correspondsohesrucureofbolzmnn sequon;nmely, he dfferenl pr of he resulng sysem s lner n relon o he momens of he dsrbuon funcon nd nonlnery s ncluded s momens of collson negrl [6]. The lnery of dfferenl pr of Bolzmnn s momen sysem of equons smplfes he sk of formulon of he boundry condons. In work [3] homogeneous boundry condon for prcles dsrbuon funcon ws pproxmed nd proved he correcness of nl nd boundry vlue problem for nonlner nonsonry Bolzmnn s momen sysem of equons n hree-dmensonl spce. In work [7] he nl nd boundry vlue problem for one-dmensonl nonsonry Bolzmnn s equon wh boundry condons of Mxwell ws pproxmed by correspondng problem for Bolzmnn s momen sysem of equons. The boundry condons for Bolzmnn s momen sysem of equons were clled Mxwell-Auzhn condons. In work [8] sysemc nonperurbve dervon of herrchy of closed sysems of momen equons correspondng o ny clsscl heory hs been presened. Ths pper s fundmenl work where closed sysems of momen equons descrbe rnson regme. Moreover, hydrodynmcl model s used o descrbe chrge rnspor n generc compound semconducor. Compound semconducors hve found wde use n he mcroelecronc ndusry.theevoluonequonformcroscopcvrbless obned by kng momens of he rnspor equon [9 13]. The sudy of vrous problems for Bolzmnn s momen sysem of equons s n mporn nd cul sk n he heory of rrefed gs nd oher pplcons of he momen sysem equons. The correcness of nl nd boundry problems for Bolzmnn s momen sysem of equons wh Mxwell-Auzhn boundry condons s beng suded for he frs me. 2. Exsence nd Unqueness of he Soluons of Inl nd Boundry Vlue Problem for Sx-Momen One-Dmensonl Bolzmnn s Sysem of Equons wh Mxwell-Auzhn Boundry Condons In hs secon we prove he exsence nd unqueness of soluons of he nl nd boundry vlue problem for sx-momen one-dmensonl Bolzmnn s sysem of equons wh Mxwell-Auzhn boundry condons n spce of funcons, connuous n me nd summble n squre by spl vrble. Noe h he heorem of he exsence of globl soluon n me of he nl nd boundry vlue problem for 3-dmensonl nonlner Bolzmnn s equon wh boundry condons of Mxwell s proved n work [14]. We wre n n expnded form sysem of one-dmensonl Bolzmnn s momen equons n he kh pproxmon, whch corresponds o decomposon of he prcle dsrbuon funcon by egenfuncons of he lnerzed collson operor f nl + 1 α x [l ( 2 (n+l+1/2) (2l 1)(2l + 1) f n,l 2 (n+1) (2l 1)(2l + 1) f n+1,l)+(l+1) ( 2 (n+l+1/2) (2l + 1)(2l + 3) f n,l+1 2n (2l + 1)(2l + 3) f n,l+1)] = I nl, 2n+l=,1,...,k, where he momens I nl of nonlner collson operor re expressed hrough coeffcens of Tlm nd Klebsh-Gordon [15, 16] nd hve he form I nl = N 3 L 3 n 3 l 3 : l nl : l N 3 L 3 n 3 l 3 :l n 1 l 1 n 2 l 2 :l ( l 1l 2 l )(σ l 3 σ ) f n1 l 1 f n2 l 2, where N 3 L 3 n 3 l 3 :l n 1 l 1 n 2 l 2 :l re generlzed coeffcens of Tlm, (l 1 l 2 /l) re Klebsh-Gordon coeffcens, α = 1/ Rθ s he consn, R s Bolzmnn s consn, nd θ s he del gs emperure. (1) (2)

3 Journl of Appled Mhemcs 3 For generlzed coeffcens of Tlm here exss ble [15] for ech vlue of qunum number ξ = 2n + l from o 6. Moreover, here s progrm on IBM for clculon of generlzed coeffcens of Tlm. If n (1) 2n + l kes vlues from o 3, we ge Bolzmnn s momen sysem equons n he hrd pproxmon. We wre n n expnded form: f 2 f + 1 f 1 α x =, + 1 α x ( 2 3 f f f 11) =J 2, f 1 f α x ( 2 3 f f 11) =, + 1 α x (f f f 1) =, f α x 5 f 2 =J 3, f α x ( f f 1) =J 11, x (, ), >, (3) (Aw +Bu ) x= = 1 β (Aw+ Bu + ) x= where (1 β) η F, [, T], αβ π (Aw Bu ) x= = 1 β (Aw+ +Bu + ) x= A= 1 α ( + 2 ( 3 (1 β) η F, [, T], αβ π B= 1 ( α π 1 ( 3 J 1 (u, w) =(,J 2,), ), ) ) 3 2 ) (6) (7) (8) where f =f (, x) nd f 1 =f 1 (,x),...,f 11 =f 11 (, x) re he momens of he prcle dsrbuon funcon; J 2 = (σ 2 σ )(f f 2 f 2 1 / 3)/2, J 3 = (1/4)(σ 3 +3σ 1 4σ )f f 3 +(1/4 5)(2σ 1 +σ 3σ 3 )f 1 f 2,ndJ 11 = (σ 1 σ )(f f 11 + (1/2) 5/3f 1 f 1 ( 2/ 15)f 1 f 2 ) re momens of he collson negrl, where σ, σ 1, σ 2,ndσ 3 re he Fourer coeffcens of he cross secon expnson by he Legendre polynomls. The frs, hrd, nd fourh equons of sysem (3) correspond o mss conservon lw, momenum conservon lw, nd energy conservon lw, respecvely. We sudy he correcness of nl nd boundry vlue problem for sx-momen one-dmensonl Bolzmnn s sysem equons A J 2 (u, w) =(,J 3,J 11 ), u=(f,f 2,f 1 ), w = (f 1,f 3,f 11 ), F=( 1 4 2, 1 8 6, ). s he rnspose mrx; B s he posve defne mrx; u (x) = (f (x), f 2 (x), f 1 (x)) nd w (x) = (f 1 (x), f 3 (x), f 11 (x)) re gven nl vecor-funcons; w +,u + re vecor momens of fllng o he boundry prcle dsrbuon funcon; w,u re vecor momens of reflecng from he boundry prcle dsrbuon funcon. Equon (4) s vecor mrx form of sysem equons (3). I s possble o check hrough drec clculons h u +A w x =J 1 (u, w) w u +A x =J 2 (u, w), u = =u (x), w = =w (x), (, T], x (, ), x [, ], (4) (5) de A 1 = de ( A A ) =, (9) nd mrx A 1 hs hree posve nd he sme number of negve nonzero egenvlues. From (6)-(7) follows h he number of boundry condons on he lef nd rgh ends of nervl (, ) s equl o he number of posve nd negve egenvlues of he mrx A 1. Thus sysem (4) s symmerc hyperbolc nonlner prl dfferenl equons sysem. Le us show h J 2

4 4 Journl of Appled Mhemcs s sgn-nondefned squre form. I s esy o check h f f 2 (f 2 1 / 3) = (CU, U),whereU = (u, w) nd C= ( 1. (1) 3 ) ( ) Egenvlues of mrx C re /2, / 3,,,,nd1/2. Therefore J 2 s sgn-nondefned squre form. Smlrly, we cn show h J 3 nd J 11 re lso sgn-nondefned squre forms. For problem (4) (7) he followng heorem kes plce. Theorem 1. If U =(u (x), w (x)) L 2 [, ],henproblem (4) (7) hs unque soluon n domn [, ] [, T] belongng o he spce C([, T]; L 2 [, ]);moreover U L 2 [,] r 1 C[,T] C 1 ( U L 2 [,] r 1 ()), (11) where C 1 s consn ndependen from U nd T M( U L 2 [,] r 1 ()) ), r 1 () s he prculr soluon of Rcc equon (15) n n explc form. Proof. Le U L 2 [, ]. Le us prove esmon (11). We mulply he frs equon n sysem (4) by u nd he second equon by w nd perform negron from o : 1 d 2 d [(u, u) + (w, w)] dx + [(A w u x,u)+(a, w)] dx x = [(J 1,u)+(J 2,w)]dx. Afer negron by prs we ge 1 d 2 d [(u, u) + (w, w)] dx + (u,aw ) x= (u,aw ) = x= [(J 1,u)+(J 2,w)]dx. (12) (13) Tkng no ccoun boundry condons (6)-(7) we rewre equly (13) n he followng form: 1 d 2 d [(u, u) + (w, w)] dx + (Bu,u ) x= +(F 1,u ) x= = [(J 1 (u, w),u)+(j 2 (u, w),w)]dx, (14) where F 1 = ((1 β)η/αβ π)f. Le us use he sphercl represenon [17] of vecor U(, x) = r()ω(, x), whereω(, x) = (ω 1 (, x), ω 2 (, x)), r() = U(, ) L 2 [,], nd ω L 2 [,] = 1.Furhermore ssume h ω(, x) C 2 [, T],wherehevlueofTwedefne below. Subsung he vlues u = r()ω 1 (, x) nd w = r()ω 2 (, x) no (14) we hve h where dr d +rp() =r2 Q () f(), (15) P () = (Bω 1,ω 1 ) x= +(Bω 1,ω 1 ) x= + 1 β [(Aω+ 2,ω 1 ) x= +(Bω+ 1,ω 1 ) x= + (Bω + 1,ω 1 ) x= (Aω+ 2,ω 1 ) x= ], Q () = [(J 1 (ω 1,ω 2 ),ω 1 )+(J 2 (ω 1,ω 2 ),ω 2 )] dx, f () = (F 1,ω 1 ) x= + (F 1,ω 1 ) x=. Le us sudy (15) wh n nl condon (16) r () = U = U L 2 [,]. (17) Le r 1 () be prculr soluon of Rcc equon (15). Then he generl soluon of (15) hs he form r () =r 1 () + exp ( (2Q (τ) r 1 (τ) P(τ))dτ)[C Q (τ) τ exp ( (2Q (ξ) r 1 (ξ) P(ξ))dξ)dτ]. (18) +(Bu,u ) x= 1 β ((Aw+ Bu + ),u ) x= + 1 β ((Aw+ +Bu + ),u ) x= +(F 1,u ) x= Hence, kng no consderon nl condon (17) we obn

5 Journl of Appled Mhemcs 5 r () =r 1 () + exp ( 1 (2Q (τ) r 1 (τ) P(τ))dτ)[ U r 1 () τ Q (τ) exp ( (2Q (ξ) r 1 (ξ) P(ξ))dξ)dτ]. (19) If R() Q(τ) exp( τ(2q(ξ)r 1(ξ) P(ξ))dξ)dτ, hen he second erm of r() s bounded [, + ) under he condon h dfference U r 1 () s posve. Furher we suppose dfference U r 1 () s posve number. Le R() >. We denoe by T 1 he momen of me whch 1 U r 1 () T 1 =. τ Q (τ) exp ( (2Q (ξ) r 1 (ξ) P(ξ))dξ)dτ (2) Then r() r 1 () s bounded [, T], wheret<t 1 nd T 1 M(1/( U r 1 ())). Anlyzngsrucureofr 1 () (see (31)) from equly (2) we obn Q() > for ny [,T]. Now we found one prculr soluon of (15). The nonlner Rcc equon cn lwys be reduced o secondorder lner ordnry dfferenl equon [18] where y +f 1 () y +f 2 () y=, (21) y () = exp ( Q (τ) r (τ) dτ), f 1 () = Q () P () Q (), Q () f 2 () =Q() f (). (22) Soluon of hs equon wll led o soluon of r() = y /yq() of he orgnl Rcc equon. By subsuon y () =z() exp ( 1 2 f 1 (τ) dτ) (23) wewre(21)nhefollowngform[18]: z () +() z () =, (24) where () = f 2 () (1/2)f 1 () (1/4)f2 1 (). If nsed of f 1 () nd f 2 () we subsue vlues from (22), hen we ge () =Qf() 2(Q2 P Q Q+Q 2 )+(QP Q ) 2 4Q 2. In work [19] ws proven h funcon z () =1+ k=1 (25) b k () (26) s he soluon of (24), where b 1 () = b k () = ξ (τ) dτ dξ, ξ (τ) b k (τ) dτ dξ (27) nd he seres on he rgh hnd sde of (26) unformly converge. Then y=(1+ k=1 b k ()) exp ( 1 2 f 1 (τ) dτ), r 1 () = y yq = z () (1/2) z () f 1 () z () Q () s prculr soluon of Rcc equon (15). Snce z () =,vlueof r 1 () = z () (1/2) z () f 1 () z () Q () (28) = f 1 () 2Q (). (29) Now nsed of f 1 () we subsue s vlue P() Q ()/Q() nd ge r 1 () = (P()Q() Q ())/2Q 2 (). The vlue of Q() wh =s Q () = ((J 1 (ω 1,ω 2 ),ω 1 )+(J 2 (ω 1,ω 2 ),ω 2 )) dx = (3) >, snce Q() > for ny [, T]. An expnson form of he funcon r 1 () = 1 Q () [ 1 z () + Q () P () Q () ]. 2Q () (τ) z (τ) dτ (31) I s no dffcul o prove usng mmede subsuon h funcon r 1 () ssfes Rcc equon (15). From equly (19) we obn Hence r () r 1 () C( U r 1 ()). (32) U (, ) L 2 [,] r 1 () C( U r 1 ()), U L 2 [,] r 1 C[,T] C( U r 1 ()). (33)

6 6 Journl of Appled Mhemcs We ke s nervl of he soluon of problem (15) (17) segmen [; 1/( U r 1 ())], snce he negrnd Q(τ) exp( τ (2Q(ξ)r 1(ξ) P(ξ))dξ) s bounded. Hence [, T] (where T T 1 ) pror esmon (11) kes plce. Now we prove he exsence of soluon for (4) (7) usng Glerkn mehod. Le {V l (x)} l=1 be bss n spce L2 [, ], where he dmenson of vecor V l (x) s equl o he dmenson of vecor U.Forechm we defne n pproxme soluon U m of (4) (7) s follows: U m = m =1 c m () V (x), (34) (( U m U +A m 1 x ),V (x))dx = (J (U m ),V (x))dx, =1, m, (, T], (35) U m = =U m (x), x R, (36) (Aw m Bu m ) x= = 1 β (Aw+ m ±Bu+ m ) x=± ± (1 β) η αβ π F, (37) where U m s he orhogonl proecon n L 2 of he funcon U on he subspce, spnned by V 1,...,V m, J(U m )=(J 1 (u m,w m ),J 2 (u m,w m )). (38) We represen V (x) n form V (x) = (V (1), V(2) ),where +(F 1 +(F 1 ) x= ] = ((A V(2) (J ( ) +(F x= 1 ) x= x, V(1) m =1 V(1) )+(A c m V ),V )dx, x, V(2) )) dx} =1, m, (, T] (4) c m () =d m, = 1, m, (41) where d m s he h componen of U m. We mulply (35) by c m () nd sum over from 1 o m: (( U m U +A m 1 x ),U m)dx = (J (U m ),U m )dx. (42) Wh help of he bove shown rgumens we now prove h r m () s bounded n some me nervl [, T m ],where U m (, x) = r m ()ω m (, x), T m M(( U m r m1 ()) ), T m T m,nd V (1) =(V 1, V 2, V 3 ), V (2) =(V 4, V 5, V 6 ). (39) U m L 2 [,] r m1 C[,T] C 2 ( U L 2 [,] r 1 ()), (43) The coeffcens c m () re deermned from he equons m =1 { dc m d +(BV (1) +(BV (1) (V, V )dx+c m [(BV (1) + 1 β ((AV+(2) 1 β ((AV+(2) + 1 β ((AV+(2) 1 β (AV+(2) ) x= +(BV (1) ) x= +BV +(1) BV +(1) +BV +(1) BV +(1) ),V (1) ) x= ),V (1) ) x= ),V (1) ) x= ) x= ) x= ) +(F x= 1 ) x= where C 2 s consn nd ndependen from m nd r m1 () s prculr soluon of Rcc equon bou r m (). Problem (4)-(41) represens Cuchy problem for he ordnry sysem of dfferenl equons. Exsence of he soluon of problem (4)-(41) follows from heory of ordnry sysem of dfferenl equons (Pcrd s exsence nd unqueness heorem) [2]. Hence he exsence of he soluon of problem (34) (37) follows. Thus, ccordng o esmon (43) he sequence {U m r m1 } of pproxme soluons of problem (5) (7) s unformly bounded n funcon spce C([, T]; L 2 [, ]).Moreover,he homogeneous sysem of equons τe + (1/α)Aξ wh respec o τ, ξ hs only rvl soluon. Then follows from resuls n [21] h U m r m1 U r 1 s week n C([, T]; L 2 [, ]) nd J(U m, ) J(U)s week n C([, T]; L 2 [, ]) s m. Furher cn be shown wh sndrd mehods h lm elemen s wek soluon of problem (5) (7). The unqueness of soluon of problem (5) (7) s proved by conrdcon. Le problem (5) (7) hve wo dfferen soluons u 1,w 1 nd u 2,w 2. We denoe hem gn by u=u 1 u 2

7 Journl of Appled Mhemcs 7 nd w=w 1 w 2. Then wh respec o new vlues of u nd w we obn he followng problem: u +A w x =J 1 (u 1,w 1 ) J 1 (u 2,w 2 ) w u +A x =J 2 (u 1,w 1 ) J 2 (u 2,w 2 ), u = =, w = =, (Aw +Bu ) x= = 1 β (Aw+ Bu + ) x=, x (, ) (44) (45) (Aw Bu ) x= = 1 β (Aw+ +Bu + ) x=. (46) We prove h soluon of problem (44) (46) s rvl. Hence he unqueness of he soluon of problem (5) (7) follows. Usng he mehod whch ws menoned before we ge (see equly (14)) 1 d 2 d [(u, u) + (w, w)] dx + (Bu,u ) x= +(Bu,u ) x= 1 β ((Aw+ Bu + ),u ) x= + 1 β ((Aw+ +Bu + ),u ) x= = [((J 1 (u 1,w 1 ) J 1 (u 2,w 2 )), u) +((J 2 (u 1,w 1 ) J 2 (u 2,w 2 )), w)] dx. (47) Le us rnsform he negrnd on he rgh sde of he equly s ((J 1 (u 1,w 1 ) J 1 (u 2,w 2 )), u) = σ 2 σ 2 [(f,1 f 2,1 f2 1,1 3 ) (f,2f 2,2 f2 1,2 3 )] (f 2,1 f 2,2 )= σ 2 σ [(f 2,1 f 2,1 f,2 f 2,2 ) 1 3 (f2 1,1 f2 1,2 )] (f 2,1 f 2,2 ) = σ 2 σ 2 [(f,1 f,2 )f 2,1 +f,2 (f 2,1 f 2,2 ) 1 3 (f 1,1 f 1,2 )(f 1,1 +f 1,2 )] (f 2,1 f 2,2 ); ((J 2 (u 1,w 1 ) J 2 (u 2,w 2 )), u) = ((J 3 (u 1,w 1 ) J 3 (u 2,w 2 )), (f 3,1 f 3,2 )) + ((J 11 (u 1,w 1 ) J 11 (u 2,w 2 )), (f 11,1 f 11,2 )) = ( 1 4 (σ 3 +3σ 1 4σ )[(f,1 f,2 )f 3,1 +f,2 (f 3,1 f 3,2 )] (2σ 1 +σ 3σ 3 )[(f 1,1 f 1,2 )f 2,1 +f 1,2 (f 2,1 f 2,2 )]) (f 3,1 f 3,2 )+(σ 1 σ ) ((f,1 f,2 )f 11,1 +f,2 (f 11,1 f 11,2 ) [(f 1,1 f 1,2 )f 1,1 +f 1,2 (f 1,1 f 1,2 )] 2 15 [(f 1,1 f 1,2 )f 2,1 +f 1,2 (f 2,1 f 2,2 )]) (f 11,1 f 11,2 ). (48) Once gn we use sphercl represenon of u=r()w 1 (, x) nd w=r()w 2 (, x), wherer() = U(, ) L 2 [,]. Then concernng r() we obn new nl vlue problem dr d +rp() =r2 Q 1 (), (49) r () =, (5) where P() hshesmevluesn(15)nd Q 1 () = σ 2 σ 2 [(ω,1 ω,2 )ω 2,1 +ω,2 (ω 2,1 ω 2,2 ) 1 3 (ω 1,1 ω 1,2 )(ω 1,1 +ω 1,2 )] (ω 2,1 ω 2,2 )dx+ 1 4 (σ 3 +3σ 1 4σ ) [(ω,1 ω,2 )ω 3,1 ++ω,2 (ω 3,1 ω 3,2 )] (ω 3,1 ω 3,2 )dx (2σ 1 +σ 3σ 3 ) [(ω 1,1 ω 1,2 )ω 2,1 +ω 1,2 (ω 2,1 ω 2,2 )] (ω 3,1 ω 3,2 )dx+(σ 1 σ ) [(ω,1 ω,2 )ω 11,1 +ω,2 (ω 11,1 ω 11,2 )+ 1 2

8 8 Journl of Appled Mhemcs 5 2 [(ω 1,1 ω 1,2 )ω 1,1 +ω 1,2 (ω 1,1 ω 1,2 )] 2 15 [(ω 1,1 ω 1,2 )ω 2,1 +ω 1,2 (ω 2,1 ω 2,2 )] (ω 11,1 ω 11,2 )] dx. The generl soluon of (49) s r () = exp ( P (τ) dτ) [C τ Q 1 (τ) exp ( P (ξ) dξ) dτ]. (51) (52) Soluon of (49), whch s ssfyng homogeneous condon (5), s rvl; h s, r() =. Hence, U C([,T];L 2 [,]) = nd u 1 =u 2 nd w 1 =w 2. The heorem s proved. Compeng Ineress The uhors declre h hey hve no compeng neress. References [1] H. Grd, On he knec heory of rrefed gses, Communcons on Pure nd Appled Mhemcs,vol.2,no.4,pp , [2] H. Grd, Prncple of he knec heory of gses, n Thermodynmcs of Gses,vol.12ofHndbuch der Physk, pp , Sprnger, Berln, Germny, [3] A. Skbekov, Inl-Boundry Vlue Problems for he Bolzmnn s Momen Sysem Equons Gylym, Almy,22. [4] C. Cercgnn, Theory nd Applcon of he Bolzmnn Equon, Insuo d Memc, Mlno, Ily, [5] M. N. Kogn, Dynmc of Rrefed Gs,Nuk,Moscow,Russ, [6] K. Kumr, Polynoml expnsons n knec heory of gses, Annls of Physcs,vol.37,no.1,pp ,1966. [7] A. Skbekov nd Y. Auzhn, Boundry condons for he onedmensonl nonlner nonsonry Bolzmnn s momen sysem equons, Journl of Mhemcl Physcs, vol.55, Arcle ID 12357, 214. [8] C. D. Levermore, Momen closure herrches for knec heores, Journl of Sscl Physcs,vol.83,no.5-6,pp , [9] G. Mscl nd V. Romno, A hydrodynmcl model for holes n slcon semconducors: he cse of non-prbolc wrped bnds, Mhemcl nd Compuer Modellng, vol.53,no.1-2, pp , 211. [1] G. Mscl nd V. Romno, A non prbolc hydrodynmcl subbnd model for semconducors bsed on he mxmum enropy prncple, Mhemcl nd Compuer Modellng,vol. 55, no. 3-4, pp , 212. [11] V. D. Cmol nd V. Romno, 2DEG-3DEG chrge rnspor model for MOSFET bsed on he mxmum enropy prncple, SIAMJournlonAppledMhemcs,vol.73,no.4,pp , 213. [12] G. Alì, G. Mscl, V. Romno, nd R. C. Torcso, A hydrodynmcl model for covlen semconducors wh generlzed energy dsperson relon, Europen Journl of Appled Mhemcs,vol.25,no.2,pp ,214. [13] V. D. Cmol nd V. Romno, Hydrodynmcl model for chrge rnspor n grphene, JournlofSsclPhyscs, vol. 157, no. 6, pp , 214. [14] S. Mschler, Knec equons wh Mxwell boundry condons, Annles Scenfques de l Ecole Normle Supereure, vol. 43,no.5,pp ,21. [15] V.G.NeudchnndU.F.Smrnov,Nucleon Assocon of Esy Kernel,Nuk,Moscow,Russ,1969. [16] M. Moshnsky, The Hrmonc Oscllor n Modern Physcs: from Aoms o Qurks,196. [17] S. I. Pokhozhev, On n pproch o nonlner equon, Dokldy Akdem Nuk SSSR,vol.247,pp ,1979. [18] E. Kmke, Dfferenlglechungen Lösungsmehoden und Lösungen, I. Gewohulche Dfferenlglechungen, B.G.Teubner, Lepzg, Germny, [19] A. Tungrov nd D. K. Akhmed-Zk, Cuchy problem for one clss of ordnry dfferenl equons, Inernonl Journl of Mhemcl Anlyss, vol.6,no.14,pp , 212. [2] M. Tenenbum nd H. Pollrd, Ordnry Dfferenl Equons, HrperndRow,NewYork,NY,USA,1963. [21] L. Trr, Compensed compcness nd pplcons o prl dfferenl equons, n Proceedngs of he Non-Lner Anlyss nd Mechncs, Hero-W Symposum, R.J.Knops, Ed., vol. 4 of Reserch Noes n Mh,pp ,1979.

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Mhemcs Journl of Volume 214 Hndw Publshng

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Research Article Oscillatory Criteria for Higher Order Functional Differential Equations with Damping

Research Article Oscillatory Criteria for Higher Order Functional Differential Equations with Damping Journl of Funcon Spces nd Applcons Volume 2013, Arcle ID 968356, 5 pges hp://dx.do.org/10.1155/2013/968356 Reserch Arcle Oscllory Crer for Hgher Order Funconl Dfferenl Equons wh Dmpng Pegung Wng 1 nd H