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Chu(Jng 3: MQl s(/ bai loan quy v~ phuang trinh tfch chi;ip CHu'dNG 3: " , ", ~ M9TSOBAITOANqUYVE 1-IIUdNG TIIINII rrtcll CII~I Qua tdnh giai quye't nhi@u b~ti loan c1iav~t ly, ky thu~t nhu' cac b~ti to 0) vI trang thl!c te Jlhi~t dQ eung nhli t6c (1Qbien thie n clla nbi~ t (1Qtbeo khong gian va thai gian khong th@ tang den va h 0, 38 >0 saGcho voi x ~ thl I I u(x) Vi u bi ch~n tren R nen ta d~t M = I E u(O)1 < lIulloo Tll' (3.2) SHYfa: Voi >0 cO'dinh, no E N cho \i n no ta c6: r f (x)dx < ~ 2M ~xl>/j II ChU'cIng 3: Mell s{/ hdi loan quy v€ phU'(fng lrll1h ([eh eh~lp Suyra: II~XJfl1(x).u(x)dx - U(o)!=1£:'[11(x).u(x)dx -£:fl1 (X).U(O)d~S; s;r:f" (x)lu(x)- u(O)ldx = f f"(x)lu(x)- u(O)ldx+ IxJ:58 + f (, (x)lu(x)- u(O)ldx< ~ f f"(x)dx + M f fn (x)dx < Ixl>8 S 8 S f (x)dx+M =-.I+-=s -if) n 2M 2 r: f" (x)u(x)dx = u(O) J!~XJ 86' d~ 1.2 Hams6r(x,t,~,1:)= -(X-~)2 thoa phu'dng tr)nh truy€n nhiet exp 2~n(t - 't) [ 4(t - 1:) ] (3.1) va c1~ng th((c: o2r or o(i 7); ClllJ'ng minh: o2r Ta clul'ngminh: or -1 -= ax -1 exp ox 4.Jn(t -1:)2 -1 : , 1- 4vn(t-1:)2 ~ [ or = ~ at = 2.Jn { [ { 2(t-1)2 -1 ~ ] - - ) '" [ 4(t -1:) ] exp [ _(X-~)2 (X-~)2 J exp 2(t - 1:) [ 4(t -1:) J} _(X-~)2 4(t - 1:) ] exp ~ix - ~)2 + -~ (X-~)2 ,[ 4vn(t -1:)2 o2r D[ => =ax Ut _ (X_!: 4(t -1:) (x- ~)22(t - 1) -1 or = - at -(X-_~)2 ~= = oX2 (x-~).exp ,~ 4vn(t-1:)2 o2r (x - ~)2 exp =-(x -:~ [ 4(t-1:) ] )t-1 4(t-1) _(X-~)2 exp 2(t'-1:) ] [ 4(t-1:) ] V~y r (hc'SaplU(dng (dnh (rUY€llllhi~( (3.1) - 4(t-1:) ]j Chuang 3: M(jl sf)' bel; loan quy v€ phuang lrlnh tfch dl(lP a2r ar Ta ch((ng lTIinh: a~ ==- at ar x ~ - - == a~ exp [ 4(t - t) ] 4-Fn(t - t)3/2 a2r -== a~2 -exp 4l;;(t-t)3/2' { == exp I ~ 4\111:(t-t)2 ar ==~ at t 2(t-t)2 I a2r [ ~ exp [ 4(t-1) - II J (X-~)2 ]{ 1- 2(t-~) } ar a1 == a~ I (X-~)2 ]{ 2(t-t) 4(t-t) _(X-~)2 4\111:(t-1)2 ===> _(X-~)2 ]} r == _(X-~)2 _(X-~)2 (X-~)2 + ex p[ 4(t-t) [ 4(t-t) ] 2(t-t) -(X-~)2 -(X-~)2 (X-~)2 1 cxp exp ~ [ 4(t - t) ] [ 4(t - t)2 ] [ 4(t -1) ] ~ 2J11: - (x - ~)2 Menh d~ 3.1.3 Bai loan nhi9t gia td dfiu co nghi9lTIdliQcxac djnh bdi cong th('(c: ~2J;t fOOexp- u(x, t) == [ -00 (x 4t ~)2 X E R, t > v(~)d~; CluIng min/1l D?t r(x, tf ~f 1:) == (x-=-~~ exp [ 4(t 1:) ] 2~n(t - 1:) Voi (x, t) c6 (1jnh,ham s6 r xac djnh vOl: - 00 < ~ f < Ft 11I > cis )- Tli (3.7) va (3.8) ta SHYfa: 11 )- co LI 'Ly lu~n tl(dng t~(nhu' tren Ll, ta cling co ke't que! tre11L, nhl( san: > f ds ~ n ~ Cf) L, + Tren L2 : 'T -t = 0, - 11 < ~< n, n2 = (0;-1) II ( OCJ 11= -JU(~,O).r(x,t,~,O)d~ = -J2-fnt ~)2 exp [ - X4~ ] V(~)'X[-II,II](~)d~ Do ham s6 v bj ch~n, ap d\l11gd1nh ly hQi t~1b1 ch~n (1.5, Chl(dng 1) tren R, do: _(X-~)2 _(X-~)2 (~) = exp - 4t 'X[-I1I1](~)' n = 1,2,3, f(~) = g(~) = exp 4t [ ] [ [II ta dl(c;Jc: + f< F,n2 > ds ~ I - _(X-~)2 +OCJ fexp 2-vnt -OCJ , L2 [ ] 4t v(~)d~ n ~ 00 + TrenL4: 'T=to' -> -n tadtiQc, +00 Jds-+ J (t-t) -00 2~n L, khl n-+oo - (x - ~)2 exp- j [ 4(t-to) () u(~,to)d~ Nhti v~y, tli (3,6) ta suy fa: +00 f "ex -002-v71:t { (~)2 x4t v B ~t t-to =-, n +oo {-4(t-to) (x - ~)2] ;?' -2 [ x-~ =-,ta -ex 71: { , n luTI -e n-,oo-00 71: fJ¥ ] { n(x - ~)2 - u(~,t )d~ j 11 11 -nt2 -e 71: iu(x-2tl't ~)dt111 f~ u(~,t )d~= n -00 taco: -nt2 iu(x-2t[,t -)dt] Suyra: - -ex f~~.171: 2-00 +OO Apd~1l1gkStquacuab6d~3,l.l, +oo 11(~,to)d~ ' ): ~ f ,u(~,to)d~=- - n(x - , ) n l d tidc: +OO -00 +oo I ex I ",:,,: ' B 01 tJlenso:t ~)2 suyra: -002~71:(t-to) ] ( f ex - x{ 4(t-to) -002~71:(t-to) ,v(~)d~= f +00 u(x,t)= =u(x,t) n - (x Sex 71:t-00 4t +00 ; { ~)2 ] v(~)d~; x E R,t > (3.9) Bay 1a ham phan b6 nhi~t dQ (j V1tri x t~i thai di€m t > cua b?ti tmll1 nhi~t gia trj dfiu, Bay giG ta xet bai tmln nhi~t ngl(c,1cthai gian: a2u au ax-2 - [ u(x,l) Tt( cong th((c (3,9), thay t at ' = u(x), X E R, t > TIm vex) = lI(X,O)? = 1, ta dl(QCphu'dng trtnh tich pilau Fredholm Im~i (fin v): +00 ):2 - - ( x-C;) 2';; -!exp [ ] ,v(~)d~ = HeX),x E R (3.10) Chu'(jng 3;' MQ[ so' bili loan quy v~ phuong [dnh tfch ch(ip E>~t K, (x) = 2}; exp(_x: ) Khi phuong trlnh (3.10) duQc vie't dtfoi cI~lI1g tich ch~p: (Kl*v) (x) = U (x) 3.2 Bai tocm nhh~t 16 khoan thorn Ta xem 16 khoan tren m~t da't nhti 1a nlia cl5n nhi~t dai vo Iwn va t~li di0m khoan ling voi tQa dO x = 0, Ngudi ta mu6n xac dinh nhi~t dO t?i t?i m?t c1a't(ling voi x = 0) vao thai di~111 t nao Do nhi€u lac dOng eua moi tnidng hen ngoai qua (HItl11ake't qua thu du'Qc d~fa tren tinh loan tr~e tie'p se co sai sO',VI the' ngtfdi ta xae dinh nhi~t dO d vi tri x> t?i thdi di0m t, san SHYra gia tri c~n tIm Truoe lien, ta xet b~tiloan nhi~t gia tri bien: Cho bie't u(O,t) = v(t), TIm phan b6 nhi~t u(x,t) t?i vi tri x >0 vao thai c1i~mt B~d loan khong ma't Hnh t6ng quat ne'u gia sli ding t?i mQi vi tri x >0 dell c6 nhi~t dO bfing t?i thdi di0m t =0, B~titmln nhi~t gia tfi bien co phuong trlnh: au a2u -, ax2 at u(x,O) = , u(O, t) = v(t); x> t >0 (3.11) x> t > 0, ? TIm u ( x, t) ? au au 1" , sox a caCUlill " , I TtfOn g tti n 1lU (I b al loan n hlet gm tn (1au, ta g m tllet u, - ,'A" ' ;1; '? 'X ' ax bi ch~n, Ngoai fa, ta md rOng ham sO'u(x,t) bfing each b6 sung: u(x,t) = 0, x < 0, t >0 Khi c16:ham u xac dinh tren R x R+ B6 c1~3.2.1 E>~t r(x, t,~, T)= 2~1t(t - T) exp - (x - ~)2 [ 4(t - T) ] (-CX)O ] ':> , 2(t-T) t J = aT x +1 + _(X+~)2 ] 2(t-T) 2(t-T) _(X+~)2 - ] (X-_~)2 expo [ 4(t-T) s (X-~)2 - J ':> aG =:>-=-a~2 ) 4( t - T)l 4(t-T)2 ][1 ] 4(t-T) 4(t - T) l - + sexp, _(X-~)2 [ - + ] (X+~)2 4(t-T) { exp 2(t-T) _(X-~)2 _(X+~)2 - [ 4(t-T) ~ (X+~)2 [ [ (x - ~)2 - 2(t - T) (X-~)2 4(t-T) exp 2(t-T)2 - [ { 4Fn(t-T)% ] -(X-~)2 exp ] 4(t - T) _(x+~)2 [ { 4FnCt-T)% = [ { (x - 02 + } -(x+~)2 +(x+~)exp ] 4(t-T) - (x - ~) - exp [ ] '2(t-T) ] +exp [ 4(t-T) -(X-~)2 ~ (x-~),exp { 4FnCt-T)2 x+~ - ]} Chuang 3: M6t seJbd(todn quy vi phu'ang trinh tfch ch(zp Xet ham Green cho bai loan Direchlet: G(x, t,~, 1) = r(x, t,~, T)- r(x, t,-~, 1) Lffy U(~,1 ) la mOt nghi~m cua phuong trlnh (3,11) (3.12) CO'dinh (x,t), x> , t> 0, Xet tntong vect(i F: F(~,1) = u(~, 1)~G(X, ( , a~ t,~, 1) - G(x, t,~, 1)~ a~ u(~, 1); u(~, 1).G(X, t,~, T) ] (0< ~ ds Nho b6 d€ 3,2.1 va (3.12) ta co: , a aG au dlVF(~ 1)=u -G' a~ [ a~ a~ ] a(uG) + a1 = u a2G -G a2u + a(uGl a~ a~ ch =- u aG - G au + a(uG) a1 a1 a1 =0 (3.14) l' (Hlnh 2) -> -> ~ x Ky hi~u nhl! tren Hinh 2, ta co: 11 ~ ChLCang3: M()[ sf)' bili [Dan quy v~ phLCang trinh tfch ch4p -> > f fds+ fds un s, s) > f< F, n + > > ds + S, f< F, 114 (3.15) > ds S4 Tli (3.13), (3.14) va (3.15) ta suy fa: > L f cIs =0 (3.16) i=1 s; > Tren 51: T = 0, < ~< n, 111 = (0; -1) Vi u(~,O) =0 nen ta co: f< F,l~ > ds = s, - I u(~,O).G(x, t,~,O)d~ = (3.17) > < T < to, Tren 52: ~ =11, 112 = (1; 0) CG a fds= u(I1,T) (x,t,n,T)dT-1)G(x,t,n,T)-u(n,T)ch= a; a; > r" r" S2 = A I " x-n '+V7T i ex ) -(x-n)2 : (t-T)2 r" { 4(t-T) -(x-n)2 2[;; 1)) -exl-~t-T [ )cl ( un, T T+ A I ] '+V7T " x+n i ) (t-T)2 aIr" -(x+n)2 ~ ex! { 4(t-T) ] -(x+n)2 -_exl 4(t-T) ] -u(n,T)ch-+ -=)) a; 2J7T) ft-T LIen,T)dT f 4(t-T) ] a -u(n,T)clT a; 5ti dl,ll1gket qua oil tlnh tren Ll chling ll1jnh m~nh c1~3.1.3, ta SHYfa: > Jds 5, -~Okhjn-~oo (3.18) -> Tren 53: T = to, < ~ < n , 113 = (0;1) > f< F,n] > ds = - [u(~, to)G(x, t,~, to)d~ = S, = [u(~, to)r(x, t,~, to)d~ - [u(C; to)r(x, t,~ to)d~ Thea ket qUC!oil tlnh tren L4 chling minh ll1~nh c1~3.1.3, ta co: _Chu'o'ng3: M()l so' bcii loan guy v~ pluiong tdnh tlch ch(ip 11 11 ~ U( c"to) f~ 0fl1(~,to)l(x,t,~,to)d~= -n2 -1-0') -0') 4(t-to) ~ J d~ (~ ) uc,t -x-) [- n,n ](~)d~ ~ n ( t' toO) exp [ 4((t - t~ ) ] X f = -(x-c,) ,exp [ net-to) - JU(~,lo)r(X, M~c khac: I,-~, 10)d~ = ~ HeX,t) -112 net-to) J Jl(~, 10) n exr [- - )- 00 (x ~ ~)2 d~ 4(t to) ] J: -I- _(X+~)2 d f ] [ = _Cf)2~n(t-to) exp 4(t-to) X[-n.n](~)~~ O') u(~,to) -IO') u(~,to) f ~ _(X+~)2 exp [ 4(t - to) ]d _CI)2~n(t- to) IV vat t ' kl1J n -~ ,:?, n 2' oo oo 00 ' K c1 to =- 1, va c1 01 bJen soK t l = -,X + ~ talidc: - + u(~,to) f _002~n(t-to) ;_(X+~)2 J: - + exp -c, exp [ 4(t to) ]d 2-00 n f~ +oo ~ fe 11 = ~ c, -n -l1li -00 -11(X+~)2 [ - J ~ U(c"t )d c, n u(2t,-x,t -)dt, n Apch;ll1g k€t qua ci'1ab6c1€3,l.I,tac1l(Qc: +OO , IHn I~-en 11-'DO -DO I -1112 (VJ (-x) < 0) 'u(2t,-x,t )dt,=u(-x,t)=0 7t n -> V~y : (3.19) I< F,n, > ds -+ u(x,t) n-+ co s) Tren 54: ~ = 0, -> < 1" < to , I" Ids=s, -, 114=(-1;0) au l"au(O,1") , fu(O,1") (X,t,O,'t:)d1" + a~ f U(x,t,O,1")ch a~ t , v I U (x, t, 0,1")d 1"= 0, ta soy - I " au(O,1") , I" au IU(O, 1")-(x, aJ: ° c, I" (j(x, t, ,1")d 1"= a~ x2 xv(1") t,O, 1")d1"= J-~ ~exp I °2\i7t(t-1")2- - (h [ ( t - 1") ] Chuang 3: M()[ so' bai loan quy vi phuang [dnh tich ch4p x - 2Fn [ - oo X2 v ('t ) 00 (t-'t)2 ('t)d't [ ] - exp [ 4(t-'t) ] X o,t_.! 11 Ap d~lllg dinh 19 hQi tv bi ch~n, ta c6: x ro vCt) 11) ex 2- ;n ) - (t-1)2 " V ] d1~- - K€t hQp tli (3.17) o€n (3.20), J i - n) v( 't) (t - 't)2 { 4(t { 4(t-'t) - X2 3exl (t - 1f x :1 ex ~ v(1) ! 2- ;n ) $4 HeX,t) = ! (t-1)2 X I J < F ,n4>cs~ 1 2- ;n) -x vCt) ' J ch kIll n~Cf) ' - 1) ]d 1: khIn~Cf) (3.20) ta SHY ra: -x exp d't, x > 0, t > [ 4( t - 't) ] (3.2l) Day 1a ham phan b6 tinh nhi~t dQ d vi tri x >0 VaG thai c1i@mt clJa bai tmln nhi~t gia tri bien Bay gia ta xet bai loan nhi~t 16 khoan tham do: a2u ax au at' u(x,O)= ; x> 0, t> x>O t>0 u(1,t) = net); Tli cong thltc (3.21), thay x ,TIm vet) = tI(O,t)? = 1, ta Ol(\jCphl(Ong trlnh rich phan Fredholm lo (3.22) -1 .exp(-);t> ~ 4t , ] 2t2Fn t va md rQng ham s6 v voi v (t) ;t sO = t s O Khi d6 phl(Ong trlnh (3.22) dl(QC viet du'oi d f< F, n + > > ds + S, f< F, 114 (3. 15) > ds S4 Tli (3. 13) , (3. 14) va (3. 15) ta suy fa: > L f cIs =0 (3. 16) i=1 s; > Tren 51: T = 0, < ~< n, 111 = (0; -1)... 11EN} Ki hi~u 30 la bien clla mien md Q Ap d\l1lgdjnh ly divergence cho tHrong vectd F tren Q, ta co: , f divF(~, 1)d~d1 =,Xlf < n (3. 13) F(~, 1), n(~, 1) > ds Nho b6 d€ 3, 2.1 va (3. 12) ta co:... f< F,l~ > ds + f< F,t;: > ds + f< F,t~:> cis (3. 6) aD LI L2 L] -> Tiy (3. 4), (3. 5) va (3. 6) ta SHYfa: I J < F(~,T),11;(~,T)> cis= ;=1L; L''I Chu''ong 3: M()t sc/ b!!i loan quy v~ phuong trlnh rich