!"$#%& ' ( )+* , -/.%021354.76 L 1@M2N-POQ* =7RST= C )7.:63H.%O+9[M];_^ Z 8 .7Ue1f6gOC ?5.76c;!9dOQ* 8 13@4 A UNVOW1KO+9X= 3Y1-EZ =;>M_=E` .73KOG=7RhZ =7R .73:9<;>=7? FOW1VOa. 1VOB5.7C B5=DC 1VOB5.7C 1E; FGBH=I6J.K; 1VO+9[MV; 1354 SG*]\ b 379/?5.76c;!9dOQ* 1VO+9[MV; 1354 FOW1KO+9<;iO+9[M];_^b 3:9j\ kh.7)Y6l1E;]m!1 n NVO- 9 3H. A B5. A B5. 8 [email protected];>. Z =4 9.E4 Z =4 9.E4 k NC 021C 9 6 .EM!O OW1- m =7R OB:9<; 8 6 =D)-/.7C 6 =D)-/.7C 9 6 . M!O 3_?5.:6c; . .76 9[MH1-, =7R , 47;>=I6UKO+9X= 3 =7R , 47;>=I6UKO+9X= 3 6 = )-<.7C 6 = )-<.7C -=6 9dOBC ; U-/.V; If !"$#%& (u + a)t + (c1(x)u − D1ux)x = 0, at = γ(φ(x)u − a), R I = 6 13@4 0 ≤ x ≤ L, 0 ≤ t ≤ T 9 3:9dO+9 1- MK= 354 9dO+9X= 3 ; u(0, t) u(L, t) + βux (L, t) u(x, 0) a(x, 0) = = = = ;]N) $.EM!O Oa= ):= N3@4I1f6 * µ(t), 0 ≤ t ≤ T 0, 0 ≤ t ≤ T 0, 0 ≤ x ≤ L 0, 0 ≤ x ≤ L • 1 6 . ; M51-/. 4 _ f M =D3@MK.:3VO6l1KO+9X=D3 ; =7R u(x, t), a(x, t) O6l13 ;]U7=I6gOa.E4 1354 154:;>=6):.E4 ] ; N)E;iOW135M_.^G6 .V] ; U • <9 ; O 6l13 ;]U7=I6gO ?5.:-/=Mf9dOQ*]^ 4 9 N ! ; 9X=D3 MK=Y\ c1(x) D . MY9X.73KO 1 3@4 1 U7=;!9dO+9 ?@. (NV0 1 M_ = . M 9X.73KO Y • β <9 ; 1 l6 1VOa. Ue1f6l1C . Oa.:6 13@4 γ ?e1Y6g*9 3 5 1 4:;>=6UKO+9X=D3 f U 6 = 7 U .:6gO* 9 3 C . 4 9 NC 1 φ(x) =7R(OB5. ;]UY1KO+9 1--j* N354.76-j*V\ (u + a)t + (c1 (x)u − D1 ux)x at u(0, t) u(L, t) + βux (L, t) u(x, 0) a(x, 0) 9 [email protected] OBY1KO Uf6 =D)-/.:C )+* 9; 1 !"$# ;>= -NVO+9[=D3 13@4 0, γ(φ(x)u − a), µ(t), 0 ≤ t ≤ T 0, 0 ≤ t ≤ T 0, 0 ≤ x ≤ L 0, 0 ≤ x ≤ L u(x, t) 9/?5.73 = = = = = = Oa= OBH. 4 9 6 .EM!O .0715MO 4I1KOW1^:;]N5M7B 1E; g(x) = u(x, T ), 0 ≤ x ≤ L, Oa= 4.Oa.76C 9 3H. OB5. MK=. Mf9X.:3VO R N3@M!O+9X= 3 φ(x), x ∈ [0, L]. ' . =DC ("( 9dO OB5. If Oa.:6C ; c0(x) 13@4 ut Oa= = )VOW1:9 3 c1(x)ux + at = D1uxx at = γ(φ(x)u − a) R =I6 0 ≤ x ≤ L, 0 ≤ t ≤ T 13@4 ;]N) $.EM!O Oa= ):= N3@4I1f6 * 9 3:9dO+9 1- MK= 354 9dO+9X= 3 ; B5. 6J. M!O 6 .V;]N-PO+9 3 OBY1KO 9/?5.73 Uf6 = )-<.7C 9 3_?5.:6 ;>. UY6 =D)-/.7C u(x, t) 9 [email protected] )+* 9; 1 4.Oa.:6C 9 [email protected] 93 9 35. MK=. 9; ;>= -NVO+9[=D3 ^ 13@4 Oa= OB5. .%021@M!O 4 9j\ Mf9X.:3VO(R$N35MO+9X=D3 41KOW1 g(x) = u(x, T ), 0 ≤ x ≤ L, Oa= u(0, t) = µ(t), 0 ≤ t ≤ T u(L, t) + βux (L, t) = 0, 0 ≤ t ≤ T a(x, 0) = 0, 0 ≤ x ≤ L. A φ(x), x ∈ [0, L], φ ∈ C[0, L] 13@4 R =I6 φ(x) > 0 x ∈ [0, L]. c1(x) ∈ C[0, L]. A B5. M_=D3 ;iOW13KO ; µ ∈ C[0, L] 13@4 D1 , γ µ 9<; 1354 β 1f6 . 35=D3]\ 3H. 1KO+9 ?@. 1354 1 µ ∈ C [0, L], µ(0) = 0 9<; 0 µ U:=;!9dO+9 ?5. =D3 [0, T ]. U:=; 9 O+9 ?5. = 3 (0, T ]. R R =6 µ0(t) − γ 0t e−γ(t−τ )µ0(τ ) dτ ≥ 0, 0≤t≤ T. " , ;a;%NC . UY1Y6 1C .%Oa.76GM_=D3@4 9 O+9X=D3 ; \ 8 Be1E; 1 N379N5. A BH.:3 4 9 6 .EM!O Uf6 = )-<.7C 3H= 3%\ 35. I1VO+9/?5. ; = -NKO+9X= 3 ` 9dOB u(x, t), a(x, t) ux, uxx , at ` B5.76 . C[QT ] QT = [0, L] × [0, T ]. ∈ 3 Oa. D6l1VOa. K Oa= = )VW O 1:9 3 1354 N ; . Be1KO O R =6 1 -- a(x, t) = γφ(x) RD` . ; N) ;iO+9dONVOa. ] . Ne1KO+9X=D3 OBH. OBH. (x, t) ∈ QT , Z t 0 9 3KOa= c1 (x)ux + γφ(x)u − γ φ(x) Z t ^@` . =D)KOW1:9 3 0 Z −uxx + c(x)ux + dφ(x)u − γdφ(x) t d = γ/D1, 13@4 e−γ(t−τ ) u(x, τ ) dτ = 0. 0 R =D--<=E` 1-/.:3VO 3H. ; OBe1KO OB5. Oa= OB5. Uf6 = )-/.:C MH13 35=E` .%0 U-/=9dO 4 9 T.76 .73KO+9 1- .DNe1VO+9X= 3 3H= 3%\ 35. I 1VO+9/?:9dOQ* OBH.E=6 .7C 13@4 4 9 6 .EMO ^ UY6J= )-/.7C Uf6 = U7.:6gO+9X.V; Oa= e−γ(t−τ ) u(x, τ ) dτ = D1uxx. k = ` G -/.O 13@4 1 OB79<; .DNe1VO+c(x) 9X= 3 ):= . MKc= 1C (x)/D .V; O e−γ(t−τ ) u(x, τ ) dτ. 1354 2 9 3:9dO+9 1-MK= 354 9dO+9X= 3 Uf6 =7?5. N379 DN5.735.V;a; ^ 1354 M7-[179 C 9 3VOa. 6 =Y\ =7R OB79<; OB5. . N:9 ?V\ 9; .%09<;iOa.:35M_.^ ; =2R OB5. L =6 -[1KOa.76 6J.]R .:6 .73@MK.^ 6J. 1Y6 4 9 3 Z dφ(x)u−γdφ(x) t e −γ(t−τ ) 1 Oa.76C u(x, τ ) dτ = dφ(x) 0 BY1KOT4= ST= 354 9dO+9X= 3 1544 9 O+9X=D3e1-KUY1Y6l1C *:9X.:-<4:; " R 1 C , ;a;%NC 9; 1 . 1V09 C ; = -NKO+9X= 3 Oa= R =6 Z t e−γ(t−τ ) ut(x, τ ) dτ. 0 .Oa.76MK= 354 9dO+9X= 3 ; *:9X.:-<4 NC UY1Y6 1C u, a OBH.:3 1 3@4 ut ∈ C[QT ] 93 UY6 9 35MY9 U-/. .%Oa.76GM_=D3@4 9 O+9X=D3 ; OBH. 4 9 6 .EMO \ UY6 =D)-/.7C ^ (x, t) ∈ QT µ0(t)k(x) ≤ ut(x, t) ≤ max µ0(t), ` B5.76 . 0≤t≤T 9<; OBH. ;>=D-NKO+9X= 3 Oa= k(x) −v 00 + c(x)v 0 + ; 1VO+9<; R *:9 3 ):=DN354I1f6 * MK= 3@4 9dO+9[=D3 ; dφ(x)v = 0 1354 1 k(L) + βk0 (L) = 0. ST= 354 9dO+9X= 3 " .%O 4 9 6 . M!O *:9X.:-<4:; , ;a;%NC 1 . M_=DC k(0) = UY1Y6 9<;>=D3 UY1Y6 1C u(x, t; φi), a(x, t; φi) Uf6 = )-<.7C )7. M_=66 .V;]U:=D3@4 9 3 R =I6 φ1 (x) ≤ φ2(x) x ∈ [0, L], R =I6 ut(x, t; φ2 ) (x, t) ∈ QT Uf6 9 3@Mf9 U-/. .%Oa.76GM_=D3@4 9 O+9X=D3 ; 1 ;>= -NVO+9[=D3 Oa= OB5.73 Oa= \ OB5. R φi, i = 1, 2. ut(x, t; φ1 ) ≥ ' ("( 6J. M!O 9/?5.73 OBe1VO Uf6 = )-<.7C 9<; u(x, t) 9 [email protected] 1 )+* ;>= -NKO+9X=D3 ^ 13@4 Oa= OB5. .%021@M!O 41KOW1 g(x) = u(x, T ), 0 ≤ x ≤ L, Oa= 4.Oa.76C [0, L], O+9X=D3 ; 9 3H. 9 [email protected] OB5. MK=. 93 Mf9X.:3VO 4 9j\ . C R N3@M!O+9X= 3 1m . OBH.K;>. φ(x), x ∈ 1 ;a;]NC U%\ 1354 2 g(x) ∈ C [0, L] O+9X= 3 g(0) = µ(T ) 1354 g(x) 9<; 1 U:=;!9dO+9 ?5. R N35M \ g(L) + βg 0(L) = 0. 13@4 00 0 g (x) − c(x)g (x) ∈ C[0, L] R N3@M!O+9X= 3 = 3 9<; 1 U:=; 9 O+9 ?5. [0, L]. . MH13 -/.7C )+* ; .E. 13 1UUf6 =I1@M2B ;]N)E;iO+9dONKO+9 3 g(x) OB79<; .DNe1VO+9X= 3 R = 6 13@4 φ(x) 9 UY1KOa. NE;!9 3 1 OB5. 0@. 4 U:=9 3KO R =I6C N-[1 =DU:.76l1KOa=6 ST.:6 OW1:9 3-j* 9. Aφ = φ, e 9<; 1 0f.E4 φ 00 0 ):=eOB )7= N354I1f6 * e φ(x) 9<; 1 RT 0 9<; 1 φ(x) ^ 1 0@. 4 U:=9 3KO =7R g (x)−c(x)g (x) = d F9 3@MK. d Z N ;>. .73KO φ(x) ;>=D-j?9 3 9 3KOa= 1f6 D NC R =6 φ = Aφ ≡ Oa= Oa= ^DOBH.:3 OB:9<; ) * 4. ( 35. 4. ( 3:9 3 1 t (x, τ ; φ) dτ ; =D- V > N O+9X= 3 U:= 9 V 3 O =7R Oa= A. ;>=D-j?9 3 kh.0_O g 00(x) − c(x)g 0(x) e−γ(T −τ ) u Uf6 = )%\ 13VO+9[M \ 13@4 3H= 3- 9 35.51f6 , O BH. .DNY1KO+9X= 3 ST= [email protected];>.7-j*]^ 9 R A, T 0 e)−c(x)ux(x, T ; φe). e−γ(T −τ ) ut(x, τ ; φ) dτ = uxx(x, T ; φ 13@4 ; 1KO+9<; R * OB5. e u(x, T ; φ) M_=D3@4 9dO+9X= 3E;_^ ; = e > g(x) = u(x, T ; φ) g(x) ;>= -NVO+9[=D3 Oa= OB5. 9 3_?5.76c;>. UY6J= )-/.7C ; 1C . OBe1KO n O R = --/=E` ; R$6 =DC A B5._=I6 .7C OBY1KO R =6 1 U:=:;!9dO+9/?5. φ(x), g 00(x) − c(x)g 0(x) ≥ RT ≡ h(x) Aφ = R T −γ(T −τ ) −γ(T −τ ) ut(x, τ ; φ) dτ µM dτ d 0 e d 0 e g 00(x) − c(x)g 0(x) ` B5.76 . µ0(t). µM = max0≤t≤T 4. ( 3H. . E = {φ(x) ∈ C[0, L] | φ(x) ≥ h(x), x ∈ [0, L]}. A B5.73 OBH. ;>.O MK= 3KO+9 3N5= NE; 1354 =7R U:=:;!9dO+9 ?@. R N3@M!O+9X= 3E; 93 E ⊂ P, A B6 = N D BH= NKO OB5. R =D--<=E` 9 3 ^ ` . N ;>. OBH. C[0, L]. N3:9 R =6C 3H=I6C ; 93 13@4 A B5. k·k C[0, L] C[QT ]. =DU:.76l1KOa=6 9<; ; 179[4 Oa= )7. 9 Ri^R =6 R$N3@M+\ A O+9X=D3 ; 1354 9 3 OBH. 4= C 1:9 3 =2R ` 9 OB φ(x) Ψ(x) A R =6 ` . Be1K?@. φ(x) ≤ Ψ(x) x ∈ [0, L], Aφ(x) ≤ R =6 U:.76l1VOa=I6 9<; x ∈ [0, L]. A AΨ(x) 9 R 9dO 9<; Uf6 .EM_=DC ; .:6gO ; MK= C UY1@M!O OBY1KO ;>.O ; 9R OB5. Ue15MOV^1354 =6 4.:6 , ` C 1UE; .7-- mY35=E` =DU:.76l1KOa=6 =DU:.:6 1KOa=6, ):=DN354.E4 C A 9<; ;>.O ; 9 3KOa= OB5._=I6 .7C 1 ;$\ C =D35=eOa= 35. 13@4 3 1U ; OB5. 35= 3%\$.7C 9 3KOa.76 ?e1- UVOQ* 9 3KOa= 0 0 [u0 , u ] = {u | u0 ≤ u ≤ u } 9dO ;>.7-jR ^OB5.73 OB5. ;>.DNH.:35M_.V; =2R 9dOa.76l1KOa.V; =6 {Anu0} M = [email protected] . Oa= 0@. 4 U:= 9 3VO ; =7R K {Anu0} A. n n 9dOB B:9[M2B ; .%O ; A 9<; 1 C A BH. ` . MH13 = U:.76l1VOa=I6 MK= 3VO+9 3N5=DN ; 13@4 1UE; ):=DN354.E4 B5._=I6J.:C R =I6 OB5. 9 35= -/= 2 * Oa.76C " )+* ` OB:9<; 9<; N ; . 4 .09<;iOa.73@MK. ;>.O ; Oa= ;]BH= ` C = 3H=2Oa=D35. ;]BH= ` R " OBH. =7R ;>= -NVO+9[=D3 ; 41KOW1 MK= 354 9dO+9X= 3E; .V;a; 1f6 * 1354 Uf6 =D)-/.:C .09 ; O ` 1 B:9[M2B 8 R OBY1KO Oa= \ Be1K?@. = U7.:6l1VOa=I6 m .%* Oa= .Oa.76 M_=D3@4 9dO+9X= 3E; ; 1VO+9<; .E4 ^ OB5.73 1f6 . Mf9X.:3VO U:=:;!9dO+9/?5. MK= 354 9dO+9X= 3 1 R =I6 ;>= -NKO+9X=D3 OBH.E=6 .7C \ 13@4 1 35.EM \ OBH. 9<; 9 3_?5.76c;>. OBe1VO OB5.76 . R$N35MO+9X=D3 φ0(x) ∈ C[0, L] Aφ0(x) ≤ φ0(x), 0 ≤ x ≤ L. 9<; M2N ;a;!9X=D3 ;]N UY1Y6l1C 9 [email protected] . N:9[MK= 3KO+9 3N5= NE; 9 3KOa= A : P → E BY1 ; =7R Uf6 ==7R 1 Aφ = φ, ;>= -NKO+9X=D3 ;>= R =I6 OW1m . ^ φ(x) φ0 = φ. ` . Be1K?@. 1-6 .51@4e* n ST= [email protected];>.7-j*]^ ;%NUU7=;>. OBe1VO U:=:;!9dO+9/?5. M_=D3KO+9 3NH= NE; R N35MO+9X= 3 ; 1KO+9<; .V; OB5. 9 3H. Ne1- d9 OQ* φ (x) * 0 ` . Be1K?@. OBY1KO R =I6 1 3_* Aφ U:0 =:(x) ;!9dO+9 ?@≤ . φ0 (x).13@4 S D = 3 ;>.DNH.:3VO-j*]^ φ(x) x ∈ [0, L], h(x) ≤ Aφ(x). OBH. C =D35=eOa= 3:9[Mf9dOQ* =7R 9 C U- 9X.K; OBY1KO C 1U ; OB5. A A =6 4.:6 9 3VOa.:6g?21- I = {φ(x) ∈ C[0, L] | h(x) ≤ φ(x) ≤ φ0(x), 0 ≤ x ≤ L} 9 3VOa= 9dO ;>.7- R * A BH.E=6 .7C OB5. = U7.:6l1VOa=I6 A:I→ 9<; 1 C = 3H=2Oa= 3H. M_=DC UY1@M!O =DU:.76l1KOa=6 O R = --/= ` ; I OBY1KO OBH. ; . N5.73@MK. =7R 9dOa.:6l1VOa.K; φ0, Aφ0 , A2φ0 , . . . MK= [email protected] .K; Oa= 1 0f.E4 U:=9 3VO =7R ` B:9[M2B 1E; ` . φ A, BY1_?5. ;>._.:3 9 3 OB5. 9 3KO6J=4DN5MO+9X= 3 =2R OB79 ; ;>.EMO+9X=D3I^ C N ; O ; 1VO+9 ;iR * u(x, T ; φ) = g(x), 0 ≤ x ≤ L. , MK= C UNKOW1VO+9X= 3e1-- * N ;>.%R$N-RQ1@M!O I R OBH. 9 [email protected];>. ; = -NKO+9X= 3^HOB5.73 OBH. ;>.DNH.:35M_. M = [email protected] .K; Oa= 1 ;>=D- NVO+9X= 3 =2R K ;%N@M2B 35. MH13 U 6 = )-<.7C f =7R 9dOa.76l1KOa.V ; BY1 ; 1 h, Ah, A2h, . . . R$N3@M!O+9X= 3 ; φ (x) MK= 3_?5.76 . n Oa= un(x, T ) = u(x, T ; φn) N3:9 R =6C -j* =D3 *].%O u0(x, t) = u(x, t; φ0 ) [0, L], Oa.:354:; Oa= ` 9dOB A BNE; OBH. kφn − φ0 k = n ∞ n. 9 [email protected];>. Uf6 = )-<.7C 9 ; 35=eO ;iOW1)-/. OBY1KO MK= 3E;iO6N5MO n D" R =6 OBH. −uxx+p(x)ux+q(x)u−γq(x) 4. 35. 4. 35.E4 ; *H; Oa.:C Z t =D3 QT e−γ(t−τ ) u(x, τ ) dτ = F (x, t) 0 u(0, t) = µ(t), 0 ≤ t ≤ T u(L, t) + βux(L, t) = 0, 0 ≤ t ≤ T U (x, t) = .%O Z t e−γ(t−τ ) u(x, τ ) dτ. 0 1 354 ):. ;iOa.7U ;!9K.V; 9 3 13@4 ∆xOa= = L/M ):. N ;>.E4 9 3 ∆t OB5= . 4 T9 ;>/N M76 .O+9H1KO+9X=D3 =7R OB79 ; x Uf6 =D)-/.:C t ^` B5.76 . 13@4 1f6 . U:=:;!9dO+9 ?@. 9 3KOa. .76c; . ?9X.` 4 135M N1 ; 13 .%?@=D- NVO+9X= 3e1f6 * ;i*H;$\ Oa.7C 9 3 OB5. N3me35=E` 3E ; 1354 A B5. .%?@=D- NVO+9X= 3e1f6 * u-/.@1f6 .76 U.) * OB5. =D) ;>.76 ?e1KO+9X= 3 1E;]U:.EMO =2R 9<; C 1@4. M2 U OBY1KO U (x, t) = Z t−∆t e −γ(t−τ ) u(x, τ )dτ + 0 = e−γ∆tU (x, t − ∆t) + Z Z t e−γ(t−τ )u(x, τ )dτ t−∆t t e−γ(t−τ )u(x, τ ) dτ. t−∆t n n F".O 1354 tn = n∆t, 8 n. (=3H. 1, 2, . . . N xk = k∆x, pk = p(k∆x), qk = q(k∆x), k = 1, 2, . . . M. µn = µ(n∆t), Fk,n = F (k∆x, n∆t), uk,n ≈ u(xk , tn) = 1354 u(k∆x, n∆t) U 135≈ U (x , t ) = U (k∆x, n∆t) b ;>. M_.73KOa.76 .E4 (6 ;iO k,n 4 ;>.EMK=k 3@4 n 4 9 .:6J.:35M_.V; Oa= 4 9<; 7 M 6J.%O+9K. 4 9<; 7 M 6J.%O+9K. ∆x − 1 + pk 2 1354 1 O 6l1U7. K = 9[4I1= )VOW1:9 3 . n 2 n Fk,n + γe −γ∆t . Ne1VO+9[=D3 ; )7= N354I1f6 * 1VO MK= 354 9dO+9X= 3 ; uM +1,n. ;!9 3 D -/. U- 9 Mf9dO 93 O+9 C C . \ OB Oa= ;>= -j?5. 1Y6 M7B79 3 13@4 O+9 C R =6G?e1-N5.V; Oa= 9C . =7R ∆x uk,n− 1 − pk 2 o .7- 9 C C U- 9[MY9dO \ 1VO OB5. .%0 U- 9[MY9dO-j* R =6 9 3e1KOa. 9<; .OB5=4^h` 93 ;]UY1@MK. OB5. \ OB n 35=4.V; bG; . Oa.76C ; 3H= ` ;>= -j?5. 4 B:9[M7B n . n = 1, 2, . . . N. -/.%[email protected] ^I;>=D- ?5. u Oa= ∆t uk,n−1 Uk,n−1 + 2 13@4 6 =D)-/.:C ;iOa.7U (n − 1) o .OB5=4 ∆t −γ∆t e uk,n−1 + uk,n . 1@M_M_=DN23VOGR =I6 O5 B . 9 3KOa.76 9X=I6 (xk , tn ), k = 1, . . . , M )+* qk γ∆t 1− 2 C Uk,n = e−γ∆t Uk,n−1 + BH.K; . 13@4 2 uk−1,n+ 2 + (∆x) qk (∆x) A u0,n 9<; 9 [email protected] .0K\ U, u - 9 3H.@1f6 ;i*H;iOa.7C -/.%[email protected] ^ OBH.:3 n 1VO OBH. \ OB U n N ;>. n -/.]?5.7 uk+1,n = 2/γ 1354 C 93 Oa= .%[email protected] 9<; ;>.EM_=D3@4 =I6 ; OW1):9 - 9dOQ* ∆x < 2/ kpk MK= [email protected] .K; 6J=I6c; U:=:;>. OB5. 9 OBC OBH. M513 ):. ;iOa.:U M_=DC ; 9 V.K;_^ R =D--<=E` . 4 .0_O6l1U:=D-[1KO+9X= 3 =7R OB5. )+* Oa= ; 93 D -/. 1 *9X.7-<4 1 9dOB ;iOa.7U A O6N35M51VO+9[=D3 ;iOa.7U A OBe1KO 1354 =2R =I6 4.76 1- \ BY1-jR ; Oa.:U 9[M7BY1Y6 47;>= 3 C .%OBH=4 T. MH1--OBe1VO . 9dO .0715MOh4I1KOW1 Aφ = L 9 6c;iOV^ RT d 0 e−γ(T −τ )ut(x,τ ;φ) dτ 4 9<; M76J.%O+9K. OBH. 4I1VOW1 R$N35MO+9X=D3 13@4 1f6 DNC .73KO g ?9 1 R N35MO+9X= 3 φ gk = g(k∆x), k = 0, 1, . . . M + 1, 13@4 . M513 4 9/RJ\ φk = φ(k∆x), k = 1, 2, . . . , M. R .:6J.:3VO+9 1VOa. . Ne1KO+9X=D3 Oa= =D)KOW179 3 OBe1KO v(x, t) = 1-d;>= ; 1VO+9 ; .K;T;%N@M2B 13 . DNY1KO+9X= 3^ Oa= .OB5.76 ut(x, t) ` 9dOB ):= N3@4I1f6 * MK= 354 9dO+9X= 3 ; 13@4 v(0, t) = µ0(t) ; = OBe1KO OBH. R$N3@M!O+9[=D3 v(L, t) + βvx(L, t) = 0, R M513 )7. 1UUf6 =E0K\ V (x, t) =) 0t e−γ(t−τ ) ut(x, τ ) dτ 9 C 1KOa.E4 )+* OB5. = NKOUNKO =7R OB5. 1- =6 9dOBC ^ 1UU- 9X.E4 Oa= 6J.K;]N-PO+9 3 9 3 1UUf6 =E09 C 1KOa. 35=4. v(x, t), ?e1-NH.K; Vk,N . n g 00 (x)−c(x)g 0(x) , ; ;]NC .:6 \ B5.76 .%R =6 .^eOB5. 1VO ;iOa.7U ∆t < 1- =I6 9 OBC ;!9K.K;K^ ;>= R =DN6gOB ` ;!9K.V; UNKOa.E4 OBH. ; = -NKO+9X= 3 U7= ` .:6 ; =I6 4.:6 9 3 ^ 13@4 6 .K; O6 9[MO+9X= 3E; 9 [email protected] 1f6 D NC 13 M26 .O+9K. 4 B:9[M2B 1 9<; R =6 ;!9 C 6 9 OBC U-/. ^ ?5.76c;>. R =6 0f.E4 .:35M_. MK= 3@4 9dO+9[=D3 ; .OB5=4 ^ ` MK= C Uf6 = )-<.7C A UNVO+9 3 =2R U:=9 3KO -<.O b ;>. S =6 = --[1f6 * B:9[M2B ` . OBH. Oa.:6C 1 ; 9 Oa.76l1VO+9[=D3 ; = -NKO+9X= 3 4. ( 3H. 4 =2R Oa= OB5. 93 Φ0 ) * 4.]?:9<;>. Oa= 9 379dO+9 1- N5.V;a; OBH. h(x) 6 .DN79 6 .K; B:9<; k = 1, 2, . . . M. ` B:[9 M7B 1f6 . 4.%Oa.76C 9 3H. 4 gM +1, g0, 4 9<; M76J.%O+9H1VO+9[=D3 NE;>. )+* R =6 C 9 [email protected] 4 9; \ dVk,N (∆x)2 ):= N3@4I1f6 * 1 OBH. ∆x )g )g − 2g + (1 − c (1 + ck ∆x k−1 k k k+1 2 2 ?e1- 9[4 ?e1-N5. OBH. Φ = (φ1 , . . . , φM ), =DU:.:6 1KOa=6 9<; (A M Φ ) k = ` .73KO 1-=Y\ OBH. 9 3]\ )7. OB5. A BH.:3 N3VO+9 - M_=D3_?5.:6 \ n # .O L = 2, T = 2, γ = β = 1, c(x) = 1 + cos(3x)/3, φ(x) = 2 − x/2 + x2 sin(3x)/3 1 3@4 ` B5.76 . 1f6l1C \ µ(t) = (1 − e−αt )/α, α > 0. .Oa.76 MK= 354 9dO+9X= 3 α = 0.2 O+9X=D3 Oa= 4.:C ` 9dOB .:6 . Oa= OB5. - 9 35.51f6 6 1KOa. 9dOa.76l1KO+9X=D3I^` 0f.E4 U7=9 3KO 1Y6 1C 9R .%Oa.76 . Oa= =6 4.:6 1@M_M2N6l15M%* A BH.:3 =D)KOW179 3 =2R N ; . OB79<; 9 3_?5.76c;>. 9 3:9dO+9 1- D NH.K; ; ; .%O =7R . M_=D3_?5.:6 .73@MK. =2R BH.:6J. OB5. 9dOa.76l1KO+9X=D3 4I1VOW1 R$N3@M+\ 15MEM2N6l15M%* OBY13 ; = -NKO+9X= 3 φ(x) ;>._. 13 OB5. 9 3]\ 93 OW13]\ OB5. Oa= M_=DC UNKOa. 9; OBH. R N35MO+9X= 3 1UUY6 =E09 C 1KOa.7-j* R N3VO+9 -1):=DNKO OB5. 4 9<; M76J.%O+9H1VO+9[=D3 Oa= Oa= 1- =I6 9 OBC UY6J= )-/.7C R =I6 OBH. =2R N3H.%0_O6l1U:= -[1KOa.E4 φ0 (x) = h(x) B:9 DBH.:6 1 Uf6 = )-<.7C ; = -NKO+9X= 3 1-=6 9dOBC .%0 U:.EMOa.E4 ?5.76c;>. α0 < 0.2031. bG;>. OB5. .0EO6l1U]\ ∆x = ∆t = 0.01. 1354 g(x) ; 1KO+9<; .E4 1Y6J. ; 1KO+9<; .E4 9<; =D-[1KOa.E4 OBH. \ MK= 354 9dO+9X= 3 ; ;iOW1-- OB .:66 =I6hMH1NE;>.K; n Example 5.1 iterates with α=0.2. 2 Iterate 1 Iterate 2 Iterate 3 Iterate 4 Exact Solution 1.8 1.6 1.4 φ(x) 1.2 1 0.8 0.6 0.4 0.2 0 ,GO 0 0.2 0.4 OB5. R = N6gOB 0.6 0.8 1 x 9dOa.76l1KO+9X= 3^OB5. 1.2 35=6C 1.4 =2R 1.6 OB5. 1.8 .:66 =I6 9<; A B5. C = 3H=2Oa= 3H. ):.7Be1K?9X=6 . kg − φ4 k = 0.0021. =2R OBH. =DU:.:6 1KOa=6 9<; .%?9[4.73KO B5.73 ` . Be1-j?5. 4 A )7=2OB 13@4 OBH. (3Y1-e.:66J=I6 ` 1 ; 4.EM26 .@1E;>.E4 ∆x ∆t, n 2 )+* 1 R1@M!Oa=I6 =7R 1-=I6 9dOBC 1):=DNKO 9; R = N6 ^ MK= 3 ( 6C ;>.EM_=D3@4 93 OBY1KO =I6 4.:6 1@M_M7N6l1KOa. OB5. Fixed point iterates for Example 5.1 with α=3.0. 2.5 Iterate 1 Iterate 2 Iterate 3 Iterate 4 Exact Solution 2 φ(x) 1.5 1 0.5 0 0 0.2 1Y6l1C 0, OBH. R =I6 0.4 0.8 .Oa.76MK= 354 9dO+9X= 3 0≤t≤T U79 M!ON6J. R =I6 OB5. 0.6 C ^ RQ179 -d; OBe1VO = 35=eOa= 3H. OB:9<; ^ 1 x 1.2 1.4 1.6 1.8 2 R t −γ(t−τ ) 0 0 µ (t)−γ 0 e µ (τ ) dτ ≥ R =6 . ;>._. R$6 =DC α=3 M_=D3@4 9 O+9X=D3 Uf6 =DU:.:6 OQ* =7R 9; A 6J.@1- -j* 35._. 4. 4 . MO =2R 3H=9<;>. A BH.K; . . E .EMO ; 1Y6 . ;iO6 = 3 F9 35M_. =DU:.76l1KOa=6 9<; M_=DC UY1@M!OV^ = 3H. 35._. 47; 1 6J. @\ A N-[1Y6 9H1VO+9X= 3 ; O6l1KOa. 2* 8 9<; M26 .%O+9H1VO+9X= 3 =7R OBH. 4.%R \ 9 379 O+9X=D3 )+* ` 1+* =2R 9<; 9dO ;>.:-jR 1 6J. @\ =7R N-[1Y6 9 H1VO+ 9X= 3 ;iO6l1KA Oa. 2* ` 9dOB 6J. N-[1Y6 9 1KO+9X= 3 Ue1f6l1C \ .Oa.76 F9 3@MK. OB5. =DU:.:6 1KOa=6 9<; 35=D3- 9 35.51Y6 ^ 13 ∆x. A .021@M!O 13Y1-j*H; 9 ; =7RhOBH. .766 =6 9<; 4 9 M2N-PO B5=E` .%[email protected] ^ =D35. .%0 U:.EMO ; OBe1VO OB5. Uf6 9 3@Mf9 Ue1- ;>=DN6 MK. =7R 4 9 \ M2N-POQ* 9<; 9 3 OB5. 4 9<; M26 .%O+9 H1VO+9X= 3 =2R OB5. 3NC .76l1VOa=I6 93 , ;!9 C U-/. A 1+* -/=6 ;>.:6 9[.V; 13e1-j*H;!9<; ;]BH= ` ; OBY1K O 9 R(OBH. 35= 9<;>. -/.]?5.:-79<; ):=DN354.E4 )+* OB5.73 OB5. δ, 4 9<; M76J.%O+9 H1VO+9[=D3 .766 =6 9<; ):=DN3@4. 4 )+* 1 R$N3@M!O+9X= 3 =7R OBH. R =6C δ c1 δ 2 + c (∆x) + 2 (∆x)2 ∆x R =I6 ;]N:9dOW1)-/. MK= 3 ; OW13KO ; 4.:U:.73@4 9 3 = 3 c1 , c 2 1 3@4 OBH. ;>.EM_=D3@4 1354 OB:9 6 4 4.:6 9 ?e1KO+9 ?5.K; 2 = R c(x). A B:9<; =6 4.:6 9 3 .%0 Uf6 .K; ;!9X= 3 ∆x 1KO 6 . N-[1f6 9H1KO+9X= 3 1 NE;>. A 9 meB5=D35=2? .DNY1KO+9X= 3 =2R 41KOW1 ;>=DC 93 ;>=6gO OBH. 9 3:9 C 9K. 4 ut -/.@154 9 3 Oa= 6 . D N- 1f6 9H1VO+9[=D3 ϕ = Aφ. . C 13@4 ` B:9[M2B f U 6 =7?:9[4.V; 1/4 ∆x = (c2 δ) , ;iO6l1VOa. e* , -POa.:63e1KOa.7-j*]^5= 5 3 . M_=DN-<4 = N6 MH1-<M7N-[1KO+9X= 3E; NE;>. 9<; c(x) =7R ;]C A B:9<; =D3 ` = N-<4 ; 9 3:9 MH13VO- * ==eOB:9 3 4 9<; M26 .%O+9H1VO+9X= 3 ` OB5. 1-jOa.76 = U7.:6l1VOa=I6 OBH. 6 =D35. R =6C . =6 ( -PO6l1VO+9[=D3 MK= N- 4 .7C U-/=H* 93 =7RGOB5. n a , ;!9 C =2R U-/. OBH. 4. 6 ._. b ;>. 4I1VOW1 OBH. 9<; ; 1C . MH1-<M7N-[1KO+9X= 3^ 3 ?:9X.%` U-/.^ = 3H. R =I6 1-/= 3 ;%C ` ==eOB5.E4 =2R(1 Oa.:3VOB M7-/.51Y6- * 9<; 1 N ; 93 OB5.73 13 = UVO+9 C 13@4 F9 3@MK. OB5. 4I1VOW1 ` 9dOB OBe1KO A 1f6 . OB79<; .%021C 1Y6J. -[1@M2m@9 3 .:66 =I6 UNKOW1VO+9[=D3 ; M_=DC UY1Y6 .^e` . ; Ne1f6 .K; 9dOV^fOBH.:3 A \ 9 -- NE;iO6l1VOa. 4^ -/.51 ;iO 9 1-WOa= = 3 δ = 0.0001. S =DC 1 BH. 9 1-;%C R =6 O 1UU- 9X. 4 U-/=2O$Oa.E4 U:= -j*35=DC 6J. N-[1Y6 9 1KO+9X= 3 B5. 6J. N-[1Y6 9 1KO+9X= 3 A = ∆x = 0.01. 9 3@4 9[M51VOa.K; ;]NU:.:6 9[=6 4= 9 3 15M \ 35=9<;>. ` . ∆x = 0.1 U:= -j*35=DC B:9 DBH.:6 ^ MH1- \ 1KO+9X= 3^ O6 9 1-G13@4 .%021@M!O ;>=D-NKO+9X= 3 4. D 6J.E. . 4 9 ; O6 9 )NKOa.E4 NC O ;>= C U-<. 1 -j* 1- ?21- NH. )+* Oa==Dm = UVO+9 C 9 3_R =6C 071C UY6J. MK. 4 9 3 ∆x ≈ 0.1. 9 1- =7R 1@44 . OBH. 1-=6 9dOBC ;%N-PO ):. 93 ; D NY1Y6J.K; -<.51 ; O N379/R =I6C OB5. 4.76 9 ?21VO+9 ?@. 9dOB 1 ; 1354 4 9<; M2N ;a;!9X=D3 ∆x = 0.05 1 U:= -j*35= C .Oa.:6 ; 1354 .0U:.EM!O ; .K;iO 9 - -(;]N 1 [−δ/2, δ/2]. .%Oa.76Oa= 93 6l1354=DC =7R OB5. UY1Y6 1C ` 4I1KOW1 9 3KOa.76 ?21- 4= .0715MO-j* M2N6 1@M* OBH. Oa= Ue1f6l1C OBH. 9<; Oa= U:=9 3KO ; M2N-[1VOa. 3H=9<;>. 1UUf6 = 15M7B 6 . \ ==eOB]\ Fifth iterates Fifth iterate, polynomial smoothed data 2 2.2 Dx = 0.01 Solution 1.8 2 1.6 1.8 Dx = 0.05 Dx = 0.1 Solution 1.4 1.6 1.4 φ(x) φ(x) 1.2 1 1.2 0.8 1 0.6 0.8 0.4 0.6 0.2 0.4 0 0 0.5 ∆x = 0.01 6l1UB 1 x ` 1.5 1 ; Oa== 0.2 2 )e154 Oa= 0 ):. 0.5 UNKO 1 x 93 1.5 6c;iO OB5. 2 $ . 1f6 . 6J. M!O 8 OBH. A = 13@4 .735.76l1 -"Uf6 ;>.E. ` OB5. 4.:6 9/?e1KO+9 ?5. A B5. -/.7C 9 R1 R =6 OBY1KO C ;>=DN6 MK. ;!9[4. 93 u ; 8 = 3 9\ 13e1- * ;!9<; =7R M_=D335.EM!O+9[=D3 ; Oa= N@M2B =7R = N6G13Y1-j*H; 9 ; Oa.:6C =2R .735.76l1-8 OB5. N6 U:=I6 OW13KO ; 35=eOa. BY13@4 Oa= 9<; t) 6 =4N@MK. 3VfO(x, U-[1@MK. 4 1 O+9 C M_=66 .V;]U:=D3@4 9 3 =DU:.76l1KOa=6R =6OB5. 9 3_?5.:6c; . Uf6 = )%\ 9<; B5.V;>. R =I6C 9dOa.76l1KO+9X=D3 4 9 6 . M!O 9 [email protected];>. ; g 00(x) − c(x)g 0(x) − ut(x, T ; φ) RT d 0 e−γ(T −τ ) ut(x, τ ; φ) dτ 3Y1KON6 1--j* ; M7BH.:C UY6 =D)-/.7C Uf6 = )-<.7C .V; 1354 R =I6 .K;iO 15M35=2? 4.76 9 ?9 3 ;>=D-NKO+9X= 3 Oa= ;]N 0f.E4 . = )VOW1:9 3 −D1uxx + c1 (x)ux + at = −ut at = γ(φ(x)u − a) φ = Aφ ≡ A 9C =D)-/.:C B_*]^` . 6 9 DBKO 6 = )-/.:C BY1_?5. MH1f66 9X.K; OB6J= N B =D3 6 .K; .@1f6 M2B 3_?5.:6c; . 1354 MK= 3VO+9 3N:9 3 I U:=9 3KO 9dOa.:6l1VO+9X= 3 OQ*U:. OB5. R =6"OB5.
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