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J 3dg'LMc-J 450O?Z6RK1=04SLM?X^ 45:B3 J OP43K0L¶OHReOd66D3P6 45:-021GcX^G: OP4 81v?d6Ó4ÕO-?P4 6RK3A3dg`<6D1 45:eOP4 45:j3 0O43 1§3;J 4KLM?X^ L´6 LM? c-?;L|476 1@X1GHëcJ 3 K3;0c-?KL¡4 4SLMJ 3-, / LI@B3;? 45:B3 81vq?B8A3;?450bO4SL[1v? 1 J OP43K0L¶OHÆ3K?P43K0LM?B^ 45:B3 8jOA@KL¡4R< O-?Xg 45:B3 ?-c-J FK3;0C6 < O-?Bg <`\ 3 O0u3 S0(t), 41 gl3]43;0#J L?j3 45:j3 O-Fd6D1h04SVL[1v? q 0OP43 S(0) FeOd6D3Zg 1G? O r 6ËL?B^GHI3 J 3BOZ6]c-023;J 3;?4 , S(t0 ) < < éæêìdíWî êð-ñ-ò ìdó ;õ-è ð-ò=î é ð)õ / LI@B3;? S(0) S0(t) < O-?Xg < gl3]43;0#J LM?B3 < , 1 V q 61vH@X3 45:KL´6A<}FeαOHTO-?X83 0O43P6 1S(t 80:e)O?X^t03 O->?Bg 01GF4ÕOKLM? \ :KLT8:h<n1v? 6D354²4SLM?X^ V dS = qS (t) − qS(t) − aS(t) 0 dt :OZ6 61vHæc4SL[1v? (q + a)/V = α, S(t) = S(0)e−αt + Z t 0 e−α(t−r)S0(r) dr. õ'öËìZêCø5ì êð)ñòæìdó Kõè ð-ò=î é ð)õ / L|@B3K? / |L @B3K? < < < O?Xg < <Ügl3]43;0#J L ?j3 S(0) S0(t) V q S(t0 ) t0 > 0 , 1 6D1vH@B3 45:;L´6A< ?B143 45:O4 α S(t0 ) = S(0)e−αt0 + Z t 0 0 e−αxS0(t0 − x) dx. >? ^3;?B3;0O-HM< |L L´6 A8 1G?4SLM?-cj1vcZ6< B? 1v??B3d^hOP4SLI@B3 f (x) O?Xg ?B1G?Zs3;021 1G? < 45:B3;? 45:B3 J G O -;LM?X^ [0, t0 ] G : α → S(t0 ) = S(0)e−αt0 + Z t 0 0 e−αx f (x) dx ´L 6 c-?;LI1h0J H 81v?P4SL?cB1GcZ6A<n6Ó450LT845H J 1G?B141G?B3 gl3Q Rt 8023BOZ6LM?B^ O-?Bg :B3;023ËQ G : [0, ∞) → (0, 00 f (x) dx]. 1h0u3-<45:B3 LM?@X3;0C6D3 W021GF-HI3KJ 1;Æ6D1GH@lLM?X^ : L´6 = G(α) \ 3;HëHK1l63Zg<v45:j1vcB^G: 3]±)K3;0LMJ 3;?4ÕOHëH \ 3 0#c-? 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L 6 dø î8;ñ-ò ìB< 54 :eOP4 ´L 6A< L¡4 @ OW0L[36 81v?P4SLM?-cB1GcZ6RH \ L¡45: 45B: 3 ' ^ L|@B3K? eWO 0OJ 53 43K0t6 1; 45:B3 W021GF-HI3KJ , 0u1vF-HI3;J ä à (7 v 6 , 45:eO4 L´6 ?B1e4 \ 3;HæH QÕK1l6D3dg L´6 8BOHëHI3dg " åå Þ >? ?-cJ ; 3 0LT8BOH O-?eOHj6L´6 \ 3 O0u3 ²0u3Zý"cB3;?45H j8 O-cRQ 4SL[1G?B3dg 41 J O-pÎ3 6]c-023 45:O4 O W021GF-HI3KJ L´6 \ 3;HæH ;1l6D3dg FK3] 1=023 \ 3 g3P6LT^G? 61vHæc4SL[1v? O-H´^1=0L¡45:-J 6d, 9 w 354Pþ 6 023]45c0#? 41 6D1GJ 3 1 1Gc-0Ü35±;OJ HI36d, f 3 6ÎOS\ 3BOW0#H¶LT3;045:eOP4 LM?A@B3K0C63 W021vFH´3;J 6 ?B3A3Zg ?j14 FK3 H¶LM?B3BO0 W021"83P66D3P6d, y 1v?j3]45:j3KHI3P66A< H¶LM?B3BO0 021GF-HI3;J 6 O023 O 1"8jO-H K1%LM?P4 1; LM?A@B3;0C6D3 45:B3A1h0P, > H¶L?j3XOW0 LM?A@B3;0C6D3 W021GF-HI3KJ 6 W023P6D3K?P4 0u1vF-HI3;J 6<'\ 3 8jO-? 1v?H LMJ OB^WQ LM?j3 :j1Z\ J cB8;: J 1h023 g÷.L : 8c-H 4 ?j1v?-H¶LM?B3BOW0 0u1vF-HI3;J 6 J LT^G:P4 FK3-, 6 O 8jOZ6D3 LM? K1%LM?P4P<Æ81v?d6LTg3;0 V±O-J -HI3 Ax = #; J c-H 4SLM-H¶LT8BOP4SL[1v? 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"SRã!" à $ ¦ G 6 ã o 1Z\ j8 O-? \ 3 ]3 ±A450LT8BOP43 v 1 c-0t6D3KH@B36 °021vJ 45:B3P6D3 0u1vFRQ IH 3;J 6<IL 1v ? H 1h045:B3 ú 6 LMJ -HI35ü 67Z6Ó43KJ Ax = b? ª âI" ( "$'&Æ* f 3 cd6D3 O? 1GH g 450LT8;pKx HI3BOZ6Ó4 69ýGcO023P6A< \ :KLT8: Ü?Xg;6 45:B3 45:eO4 J LM?KLMJ L sP3645:B3 6L´s3 1;45:B3 x 0u36LTg"ceO-Hj#69ýGcO023dgh :KL´645c-0#?d6 1Gc4}41 kb − Axk2 . F;3 3dýGc;L|@OH´3;?4 41 61vH@lLM?B^ 45:j3 õlðeêó ;ò`ì0A7B)î éæð-õjø AT Ax = AT b, O 6Ój6Ó43;J \ :KLT8: L´6 ^GcO0O?43A3Zg 41 :OA@B3 O 6D1vHæc]Q S4 L[1G?, Vc-045:j3K0_<\ 3 8jO-? 6D3A3 45:O4 LI :eOd6 Ax = b ;# õ T 61vHæc4SL[1v?=< 45:B3;? 3]@X3;0¯ 61vHæc4SL[1v? 41 45:B3 ?j1h0 Q J O-H 3dýGcO4SL[1v?d6 L 6 O 61vHæc4SL[1v? 41 :KL´6 Ax = b. 450L8p 3]±A43;?XgK6 41 J 1h0u3 O-FZ6450OX84 H¶LM?B3BO0 1vK3;0OP41h0C6 1; 3Zý"ceOP4SLT1G?Z6 cd6LM?X^ 45:B3 81v?X83;4 1 K Kx = y ú OXg 1%LM?P4²ü 1v;3K0OP41h0t6 \ :KLT8: -HTOË 45:B3 O04 1; ( K∗ O 450O?Z6RK1K6D3 J O450L¡± T A . Ø) £ $ "SU Ph"$h)(5(* f 3 ú 023Z^"c-HTOW0L´s3]ü 45:j3 W021vFH´3;J , ³ OP45:B3;0 54 :O-? ^'L|@B3 45:j3 ^3;?B3;0O-H÷45:j3d1=0¯R<Æ\ 3þHæHLMHæHëcd6Ó450OP43 L¡4 FS 1v?j3 eOW04SLT8;cHTO0 pXLM?Bg 1; 023d^Gc-HTOW0L´sjO4SL[1G?h<Ü8BOHæH´3dg V é F ð-õð'ö 023Z^"c-HTOW0L´sjO4SL[1v?h, r ?B3 L?P45021"g"cX836 O 0u3Z^XQ cHTO0L´sjOP4SL[1v? eOW0OJ 3543K0 LM? 6]cX8: O \ OS 45:eOP4 α>0 6]J OHëH ^'L|@X3 cd6 O W021GF-HI3KJ 45:O4 L´6 ú 8;HI1K6D3]ü 41 α 45:j3 1=0LT^'LM?eOH, >? 45:B3 8BOd6D3 1 45:B3 ?B1=0#J OH3Zý"ceORQ 4SL[1G?Z6A<`1G?B3 8BO? 6R:j1Z\ 45:eOP4 J LM?;LJ L´sWLM?X^ 45:B3 J 1"gWQ L 3dg 0236ËLg"ceOH kb − Axk2 + α kxk2 HI3BOXg;641 45:B3 H~LM?j3XOW0=67Z6Ó43KJ T A A + αI x = AT b, \ :B3;023 L´6 45:j3 Lgl3K?P4SL¡4 J OP450L¡±', r ?B3 8BO? 6]:B1d\ I 45:j3 8A1' 3 : 8WLT3;?4 J OP450L¡± T L´6 OH\ OSj6 ?B1G?RQ A A + αI 6ËL?B^GcHOW0, :j3K023]1=023<`45:j3 W021GF-HI3KJ :OZ6 O c-?;Lý"cB3 61vHæc4SL[1v?h, f :eOP4 6R:j1vc-H´g \ 3 g1 OFK1vcP4 :KL´6 L´6 α? 1G?B3 1 45:B3 J 1h0u3 ²c?Xg=O-J 3K?P4ÕO-H°O-?Xg LM?450L^"cKLM?X^= W021GF-HI3KJ 6 1; LM?A@B3;0C6D3 45:j3d1=0¯P, w354Pþ 6 O-?O-HZs3 1G?B3 1 1Gc-0 6ËLJ -HI3 6Ój6Ó43;J 6 1h0 L?d6LT^G:P4P<V6qOS " 1 1 1 1 #" x1 x2 # = " 1 1 # Ø >?A@eO0L~OF-HR<`1Gc-0 LM?--cP4 g=O4ÕO 1h0 45:j3 ML ?A@B3K0C63 W021vF]Q HI3;J < :eOZ6 3K0#0u1h0 LM? L¡4P< 6qOS \ 3 :eOA@B3 (1, 1), (1 + 1=0 ghOP4ÕO L?d6Ó43BOXgn, w354 δ1 , 1 + δ 2 ) :B3 023d^GcHOW0L´s3dg 6Ój6Ó43;J FK3d81vJ 36 δ = δ1 + δ2. " 2+α 2 2 2+α #" x1 x2 # = \ :KLT8: :OZ6 c?KLTýGcj3 61vHæc4SL[1v? " x1 x2 # = " 2+α 2 2 2+α " 2+δ 2+δ #−1 # (2+δ) = (2+δ) " 1 1 # = " 1 1 # 2+δ 4+α r dF 6D3;0¯@X3 45:eOP4 L|}45:j3 LM?--cP4 3;0#021=0 \ 3;023 O-HæHj\ 3 \ 1vcH´g :OA@B3 41 g1 L´6 H´354 O-?Xδg \ 3 \ 0, 1vc-H´g ^354 45:j3 @eO-H¶LTg 6D1vHæc4SL[1G? 1 α → a 0 c4 ^'L|@B3K? 45:O4 45:B3 (1, 1). LM?-c4 3;0#021=0L 6 ?j14 s3;2021n <d4ÕO-pBL?B^ 45:B3 H¶LJ L¡4 OZ6 ^÷L|@X3P6 cZ6 O \ 1=0C6D3 O-W021d±LMJ O4SL[1v? 41 O 61vαHæc→ 4SL[1v0? 45:O-? \ 3 \ 1vc-H´g 145:j3K0 \ L´6D3 ^l3]4 FË 8:B1'1l6LM?X^ α≈ 2δ. B: 3;023 OW023 J O?A ýGcj36Ó4SL[1G?Z6 :B3;023-< 3-,ë^`,< :B1d\ g1 \ 3 pe?j1Z\ LM? ^l3K?j3K0OH \ :O4 45:B3 FK3P6Ó4 8:B1'L83 1; 0u3Z^"c-HTO0L sZO4SL[1v? O0OJ 3543;0 L´6A< L| O-?ç :KL´6 O?Xg 1e45:B3;0 L´66]cB3P6 \ 1vc-H´g FK3 45:B3 6RcF 3Z84 J O4²43;0 1 O ( 81vc0C6D3 LM? LM?@X3;0C6D3 45:B3A1=0¯P, Ø " 1 1 # SãW'D" åX"Y[Z * f 3 ^l3]4 45:KL´6 1=0 °023d3 / 1h0 3XOB8;: êì\?7] ;+ò Bê#é_^eìdè 021GF-HI3;J , f 3 8BO?-?j14 :B1GK3 41 :eOA@B3 64ÕO-FKLMH¶L¡4 1=045:B3 c-?023Z^"c-HTOW0L´s3Zg 021GF-HI3;J Kx = y, 3]@B3K? L| L 6 \ 3;HëHng3Ü?j3Zg °OH 45:B1vcB^G: 6Ó4ÕOFKLMH¶L¡4 K −1 :O-K3K?d6 LM? 6D1GJ 3 8BOd6D36Î], o}1d\ 3]@X3;0_< \ 3 :eOA@B3 41 H´1'1vp 41 LM?N?KL¡43 g÷LJ 3;?Z6L[1v?O-H3]±O-J -HI3P6 6RcB8;: Od6 1vc0 V±O-J -HI3 1GK3K0bO41=0 L´6 LM?43d^G0OP4SLT1G?e5< K \ :B3;023 L¡4 8BO? FK3 6R:j1Z\ ? 45:eOP4 _g÷LI{}3;023;?4SL¶ORQ K −1 4SL[1G?e 3]±lL´6Ó476 F-cP4 L´6 ?B1e4 81v?P4SLM?-cB1GcZ6A<3R@B3;? 45:B1GcX^G: L´6d, UK3d8L `8jO-HæHR<W\ 3 :OA@B3 K K : C[0, 1] → C[0, 1] R @KL¶O 45:B3 0#c-HI3 L 6 O 1v?B3ËQ 41WQ²1v?j3 Kf (x) = 0x f (y) dy °c-?B854SL[1v?h, N 3BOZ6Rc023 6L´s3 FË 45:j3 6]c- ?B1=0#J x kf k = sup |f (x)| 0≤x≤1 6 1 45:eO4 45:j3 ú 8HI1l6D3;?B3P66Óü 1 O-?Bg L´6 f (x) g(x) 43;0#J LM?B3dg FS 45:B3 ?-c-J FK3;0 :B3;? 1v?j3 kf − gk . 6]:B1d\ 45:eO4 45:j3 1GK3K0bO41=0 L 6 81v?P4SLM?-cB1GcZ6 LM? K 63K?d6D3 45:eOP4 L| O?Xg OW023 8H´1K6D3< 45:j3K? f (x) g(x) OW023 O?Xg Kf (x) Kg(x). Ø= g 3ËQ 8BO? 45:B3 61 w354 45:B3 0 O?X^l3 1 B: 3;? K. 1v?B3ËQ 41WQ²1v?j3), a c4 L¡4 L´6 ?j14 81v?P4SLM?-cB1GcZ6d, 1h0 8A1G?Z6LTgl3K045:B3 ²c?X84SLT1G? R = K(C[0, 1]), L´6 O-H¡6D1 K −1 : R → C[0, 1] x gε(x) = ε sin 2 , ε f 3 :OA@B3 \ :B3;023 , Un1 ε > 0. kgεk = kgε − 0k ≤ ε 1h0 R6 J O-HæH L´6 8HI1l6D3 41 45:j3 s3;021 ²c?X84SLT1G?, ε, gε(x) 354P< x ε K −1gε(x) = gε(x)0 = 2 cos 2 ε ε 1 x = cos 2 ε ε 6 1 45:eO4 −1 1 <61 54 :eO4 F 3d81vJ 3 6 K −1 K gε gε = ε K #O0 ²021GJ s3;021 Od6 o 3K?B8A3 ´L 6 ?j14 O ε → 0. K −1 81v?P4SLM?-cB1GcZ6 1GK3;0O41=0, B: 3P6D3 8jO-H´8c-HTO4SL[1G?Z6 0O;L´6D3 O ýGcB3P6Ó4SL[1v?hx o}1d\ g1 \ 3 ú 023Z^"c-HTOW0L´s3]ü 45:B3 g'L|{}3;023K?P4SL¶O4SL[1v? 1vK3;0O41=0ç :j3K0u3 L´6 O Lp:B1G?B1;@ 023d^GcHOW0L´sjOP4SLT1G?h< Fc4 \ 3 \ O?4 41 6]:B1d\ O-?B1e45:B3;0)6D1GJ 35\ :O46LMJ -HI3K0÷O-021hOX8:h<d\ :KLT8: ?;LT8A3;H LMHæHæcZ6Ó450bO43P6 45:j3 3R{}3d85476 1; ?B1'L 6Ó g=O4ÕOh, Uc-K1K6D3 \ 3 8BO? 6qO-J -HI3 ± P O 4 3dýGcO-HæH 6ROX83Zg ?j1"g3P6 O ?Xg xk = kh, k = 0, 1, . . . , n h = 1/n. Ø, r 81vc0C6D3 45:B3;023 L´6 ?j1%L´6D3 LM? 45:B3 ghOP4ÕOh<-6D1 LM?Z6Ó43BOXg 1 \ 3 ^354 \ L¡45: 3BOX8: F = F (x ), F +δ |δ | ≤ δ, 6ÎOËP, k f 3þHæHcZk63 83K?P43K0u3Zg k g'L|{}k3K0u3K?B8A3P6 41 O-k021d±PQ LMJ O43 45:B3 g3;0L|@OP4SLI@B3 @OHëcj36 Fk0 = F 0(xk ), k = f 3 p?B1d\ °021vJ 8jO-H´8c-HæcZ6 45:eO4 1, . . . , n − 1. Fk+1 − Fk−1 = Fk0 + O(h2 ) 2h o 3K023 ú ü J 3BO-?Z6 0u1vcX^":-H ú O ýGceO?4SL¡4 B? 1 FKLT^WQ O(t) ^l3K0h45:eO-? 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Ü?Xg @OHæcB3P6 1; 45:B3 eOW0OJ 3543K0t6 45:O4 J LM?KLQ 2 J L´s3 45:B3 6Rc-J 1;G6DýGceOW023P6 1;Ü023P6LTgGcO-H¡6 (k) x1(tk ) − x1 0u36RcH 4SL?B^ ²021GJ 3XOB8;: g=O4ÕO K1'LM?4 ð =îj ï =î ò kì Aødî ø AIBjêìÎø J 3]45:j1"l # g ; OBg-g O 023d^GcHOW0L´sjOP4SLT1G? 43;0#J 41 ^l3]4 45:B3 ï`ìdm õ Kòæ_é ^eìdè ò Mì ødî ø AIJBêìqø J 3]45:j1"gn,M >? O?A 8jOZ6D3<"45:KL´6 6Ój6Ó43;J L´6 éæèlìZõî gé EmKñ-ò ìB, ` ® ã ( ù Dz L´6Ó4SLM?X84 023BO-H=0u1%1476q o 3K023 45:B3 ^l3K?j3K0OH 6 1vHæc4SL[1v? ´L 6 x1 (t) = d11 eλ1 t + d12 eλ2 t x2 (t) = d21 eλ1 t + d22 eλ2 t f 3 O-H¡6D1 \ B: 3;023 OW023 45:B3 3eLT^3;?A@eO-Hæcj36 1 λ1 , λ 2 A. :OA@B3 45:eOP4 λ1 λ2 = bc λ1 + λ2 = −(a + b + c) 6 1 45:eOP4 FK145: OW023 ?B1v?]QÎK1K6L¡4SLI@B3-, :cZ6A< λ ,λ 1vc0'1Gc45c4766R:j1v1c-H´g 26RcM: 8A3 41 g3543K0#J LM?B3 d , λ1 , d12 O?Xg :-cZ6 L´6 g3543;0#J LM?j3Zg LM? W11 0LM?X8WL Q λ . x (t) HI3 O?X2g O-HæH°c-045:j3K10 1vcP45-c476 OXgg ?j145:KLM?B^ 41 1vc0 p?B1d\ HI3dg-^3 F-cP4 81vM? Ü0#J O4SL[1v? 1;'45:j3 1h0#J 1; x (t). :KL´6 LM?1=0#J OP4SL[1v? 8A1GJ 3P6 8;HI1l63 41 gl3]43;0#J L?;1LM?X^ 45:j3 83An 1 : 8L[3;?476 1 L? @lL[35\ 1; 45:B3 OFK1;@B3 4#\ 1 A 3dýGcO4SL[1v?d6d, o}1d\ 3]@B3K0_<;45:j363 3dýGcO4SL[1v?d6 O0u3 LM?d6RcAuQ 8L[3;?4 41 g3543K0#J LM?B3 OHëH45:023A3 1 :j3K0u3Q a, b, c. 1h0u3-<W45:KL´6 67Z6Ó43;J L´6 õð÷î éëèlìZõî oé EKñòæì OZ6 L¡4 6Ó4ÕO-?BgKp6 , á 76 " ) (;* 9 +B > L ?;L¡4SL~OH=81v?Bg'L¡4SL[1v?d6 OW023 O @O;LMHOF-HI3<ÜcZ63 54 :B3;J Od645:B3 0C6Ó4Ü4#\ 1 ghO4ÕO K1%LM?P476d, :B3;? 6qO-J -HI3 54 :-023A3 J 1=023 1vcP54 -c476A, 6 OFK1;@B3<v\ 3 j8 O-? g3543K0#J LMB? 3 x1 (t) = d11 eλ1 t + d12 eλ2 t 35±)-H¶LT8L¡45HP, z |L {}3K0u3K?P4SL~OP43 54 :;L´6 6D1GHæc4SL[1v? 1=0 x1(t) O?Xg 3R@eO-HæcO43 PO 4 41 v 1 FP4ÕOKLM? t=0 d11 λ1 +d12λ2 = x01(0) = −(a+c)x1(0)+bx2(0) B: 3 1v?H 4SLMJ 3 45:;L 6 3Zý"ceO4SL[1G? \ 1vc-H´g ?j14 OXgg ?j3]\ LM?1=0#J OP4SL[1v? L´6 \ :j3K? FK1e45: O-?Bg OW023 x1(0) x2(0) sP3K021 1h0 1 6qOJ 3 J OX^"?KL¡45cXgl3 O?Xg 1v-;1l6L¡43 6ËL^"?, y3eL¡45:B3;01;}45:j36D3 8A1G?Xg'L¡4SL[1v?d6 ^'L|@X3 -:AZ6LT8jO-HhLM?;L|4SL¶OH 81v?Bg'L¡4SL[1v?d6d, :KL´6 \ LMHæHh6D1vH@B3 45:j3 W021vFH´3;J O?Xg LM? 45:;L´6 81v?435±A4P<"45:B3 67Z6Ó43KJ L 6 éæèìdõ÷î éoEKñò ìX, mh UO-J -HI3 45:j3 G 1 c45c476 \ L¡45: LM?-c4 °c-?B854SL[1v? ? O -W021G0L¶OP43 L?-cP4P<Æ3-,ë^`,< u(t) = [u2(t)]. 1=0 0, 1=0 0 ≤ t < 1 u2(t) = 1, 1≤t<∞ 1=0 J 1l64 LM?-c476< 023BO-HæHe \ LMHæHKL[3KH´g 6RcM: W8 L[3K?P4 ML ?RQ 1h0J PO 4SLT1G? 41 LTg3;?4SL| 45:B3 67Z6Ó43KJ , ( h1 ? n±=K1G?B3;?4SL¶O-Hv/ 021d\ 45: N 1"gl3KH U c-K1K6D3 O K 1vc-HTO4SL[1v? °c-?B854SL[1v? 8jO-? FK3 J 1"gWQ u(t) 3;HI3Zg FS 45:j3 ]3 ±)K1v?j3K?P4SL¶O-H^"021Z\ 45: HTOq\ du = ru dt \ :B3;023 Ý r = r(t) L´6 O ?j1v?]Q8A1G?Z6Ó4ÕO?4 ^"021Z\ 45: 0bO43-, à 'å á * / L|@B3K? 45:j3 J 1"gl3KHM<lL , 3-,<^G021d\ 45: #! h& ä ! q 0bO43 ²c-?B854SL[1G? <;8A1GJ c43 45B: 3 K1vc-HTO4SL[1v? ^G021d\ 45: r(t) °c-?B854SL[1v? " u(t). ¦ $ % "!#( S4 L[1G? $ u(t), ! à 'vå á * / L|@X3;? 54 :B3 K 1vc-HTO4SL[1v? ²c?X8SQ 81vJ -cP43 45:B3 ^"20 1d\ 45: 0O43 ²c-?B854SL[1G? r(t). >? 45:;L´6 8jOZ6D3<G45:B3 LM?A@B3K0t6D3 021GF-HI3;J r(t) = 6D3d3;J 6 3BOZ67x du/dt d = ln u(t). u dt `9 f U 1 O-HæH\ 3 :OA@B3 4 1 g1 L´6 4ÕOpq3 45:j3 ?O45c0O-H IH 1)^ n 1 45:j3 g=O4ÕO O-?Bg ÷ g LI{}3;023;?4SL¶O43 45:O4 ²c-?B854SL[1G? 1=0 g÷L|{3;023;?4SL¶O43 O-?Xg ' g L|@KLTg3 FË 45:j3 °c-?B854SL[1v?h, :O4 81vcH´g FK3 6LMJ -HI3;0_<0LT^G:P4Õç f u0 1v?X^K Ü1Gc pe?B1d\ \ :O4 45:B3 8BOP48;: L´6 ²021GJ w3d8545c023 +)x ?-c-J 3;0LT8BOH g'L|{}3;023K?P4SL¶O4SL[1v? 1 g=O4ÕO \ L¡45: 3K0021h0t6 LM? L¡4 L 6 J cX8: J 1h0u3 g'L: 8c-H 4 45:O-? L¡4 6D1vc?Xg;6A< O?Xg 4ÕO-pBL?B^ 45:B3 HI1-^ 1 45:B3 ghOP4ÕO Ü0C64 81vcH´g J Opq3 45:KLM?X^;6 3R@B3K? \ 1h0C63), > v\ 3 :eOA@B3 4SLMJ 3< \ 3þHæHG:eO@X3 6D1GJ 3 °c-? \ L¡45: 45:;L 6 3]±O-J -HI3 HTO43;0 FË \ OS 1 45:j3 N V w a / > y c-J g3;0L|@÷, `< Û "SR Ý ä "$6v%$ $ àä P å¶ãr#"æà $ b à G 6 å B: 3 g÷LM023Z84 0u1vF-HI3;J :B3;023 L 6 O J 1"g3;H1=0 O K1v]Q cHTO4SL[1v? gl3K?d6L¡4 ²c?X84SL[1v? \ :B3;023 v = v(x, t) t≥0 L´6 4SLMJ 3 O-?Bg L´6 O 698jO-HI3Zg 6L´s3 8HOd66 x, 0 ≤ x ≤ 1 @eOW0L¶O-F-HI3-, :;L?p 1; Od6 O pBL?Bg 1; O0bO-J 3]43;0 x 45:O4 ý"ceO-?P4SL 36 6D1vJ 3 8;:O0OB8543;0L´6Ó4SLT8 1; 45:B3 K1v]Q 3 ;023;J 3KJ Q cHTO4SL[1v?h, ÷1=0 3]±O-J -HI3< 8A1Gc-H´g FK3 OX^l! x F;3K0 45:O4 \ 3 :eO@X3 698jO-HI3Zg 6D1 45:eOP4 L? 698BOHI3Zg x ?c-J FK3;0C6 L 6 45:B3 OX^3 1 O ?j3]\ FK1=0#? O-?Bg L6 0 1 45:j3 HTO02^3P6Ó4 OB^3 OB8;:;L[3R@B3Zg FË O?A J 3;J FK3;0§1 45:B3 ;1v-cHTOX83), ?B1e45:B3;0%35±O-J -HI3 L´6 45:O4 81vcH´g 023;RQ x 0u36D3;?4 698jO-HI3dg 6L´s3-, f :eO43]@B3K0v45:j3 8:eOW0OX843;0L´6Ó4SLT8 1 45:;L´6 eOW0OJ 3543K0 < \ 3 OZ6 6Rc-J 3 45:eOP4 J 3KJ FK3;0C6 1 45:B3 K1G-cHOB8A3 021-^G023P66 6Ó43XOBg'LMH LM? 4SLMJ 3 ²021GJ 8HTOZ66 6L´s3 41 8;HTOd66 6L´s3 O?Xg O-HI1G?X^ 45:B3 \ OS 0 1 ^÷L|@X3 F;L0 45: 41 ?j3]\ FK1h0?Z6 1;Ü8;HTOZ6 6 6ËL sP3 OB8d81=0ûg'LM?X^ 0 41 6D1GJ 3 8HTOZ66 6L´s3 FKLM045: 0O43 °c-?B854SL[1v?h, Ø :B3 FeOd6LT8 eOW0O-J 3]43;0 ²c?X84SL[1v?Z6 1=0 45:KL´6 J 1"gl3KHx L 6}45:B3 ;1v-cHTO4SL[1v? g3;?Z6L¡4 1 }J ; 3 J FK3;0C6 1 45:j3 6RK3d8L[3P6 \ L¡45: 8;HTOd66 6ËL sP3 O 4 4SL J 3 • v(x, t) x • g(x) • µ(x) • p(x) • ψ(x) 4SLMJ 3 t. L´6 O 6L´s3 g3;K3;?Xg3;?4 ^G021d\ 45: 0O43-, L 6 O 6L´s3 g3;K3K?Bg3;?4 J 1=04ÕO-H¶L¡4 0OP43), L´6 O 6L´s3 g3;K3;?Xgl3K?P4 FKLM045: 0OP43), L´6 45:j3 LM?;L|4SL¶OH;K1vc-HTO4SL[1v? g'L´6Ó450LMF-cP4SL[1v? OP4 t = 0. FeOHTO-?X83 HTOS\ O-?O-Hj6ËL 6 KLT3;H´gK6 J 1"gl3KH;3Zý"ceO4SL[1G?Z6dx vt + (gv )x = −µv, 0 ≤ x ≤ 1, t ≥ 0 g(0)v(0, t) = Z 1 0 p(s)v(s, t) ds, t ≥ 0 v(x, 0) = ψ(x), 0 ≤ x ≤ 1 h Ý " !#h& ! à 'å á * / L|@B3K? O-?Bg g(x), µ(x), p(x) 81vJ -cP43 45:B3 ;1v-cHTO4SL[1v? g3;?Z6ËL|4# ²c-?B85S4 L[1G? $ Ψ(x), v(x, t). ¦ $ % "!#( $ ! à 'vå á * / L|@B3K? OHæH F cP4 1G?B3 1 45:B3 °c-?B854SL[1v?d6 O?Xg O?Xg 6RcKL¡4©Q g(x), µ(x), p(x) Ψ(x), 8AG 1 J c43 45:j3 023ËQ OF-HI3 g=O4ÕO 6qOJ H~LM?B^ 1; v(x, t), J OKLM?;L?B^ eOW0O-J 3]43;0 ²c?X84SL[1v?, K : L´6 L´6 1v?B^1%LM?B^ 02363XOW0û8: 1%LM?45H \ L¡45: H´35±O-?Xg=O0 ( z 3;?Z6L[1;@, o}3;023 L´6 O 6qOJ H~LM?B^ 1; \ e : OP4 \ 3þ¶@B3 g'L´6²Q 81;@B3;023Zg 6D1 OW0, ³3d^hOW0ûg'LM?B^ 45:j3 g÷LM02Z3 84 W021GF-HI3KJ x +), § @B3;? 45:B3 g÷LM023Z84 021GF-HI3;J ´L 6 ?j1v?P450L|@lL¶O-H, r ?B3 :OZ6 41 g1 O H~L¡4²45HI3 \ 1=0#p 41 R6 :j1Z\ L¡4 L´6 O \ ; 3 HæH Q K1K6D3dg W021vFHI3KJ ÓL¡4 L´6d,M mV, y cJ 3;0LT8jO-HK81vJ -cP4ÕO4SL[1v? 1 54 :B3 ÷ g L0u3Z84 W021vF]Q HI3;J L´6 OH|61 ?j1v?450LI@KL¶O-HM< Fc4 \ 3 :eO @X3 O? O-RQ O023;?45H gl3Z83K?P4 J 3]45:j1"gn, h ³3d^hOW0ûg÷L?B^ LM?@X3;0C6D3 W021vFH´3;J 6dx +), Uc-K1K6D3 L 6 O 8A1G?Z6Ó4ÕO?4P< g(x) = g T = 1/g O?Xg OHëHhO0OJ 3543;0C6 35±W8A3;4 OW023 ^÷L|@X3;?, ?RQ µ gl3K0 6D1GJ 3 0u3XOd6D1v?O-F-HI3 0u36Ó450L84SL[1v?d6A< L´6 µ(x) c?KLTýGcj3KH g3543;0#J LM?j3Zg FS 45:B3 g=O4ÕO v(1, t), 0 ≤ O-?Bg Y 1vH 43;0#0O LM?43d^G0OHh3Zý"ceOP4SLT1G? t≤T Z x 1 µ(s) ds = g ln 1−x u(1, g ) φ(x) , 0 ≤ x ≤ 1. mV, Uc-K1K6D3 L´6 O 8A1G?Z6Ó4ÕO?4 O ?Bg O-HæHGeO]Q g(x) = g 0OJ 3543;0C6 35±W8A3;4 OW023 ^'L|@B3K?h, > ? ^3;?B3;0O-HM< q(x) 8jO-?-?j14 F;3 c?KLTýGcj3KH g3543K0J LM?Bd3 g FS ;õ#T q(x) J 3XOd6Rc-0u3KJ 3K?P476 1; v(x, t).
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