"!#%$'&)(*
+ , . ,"/ 021%3547698;:=<;>?A@B3;0C6D3 E 021GF-HI3KJ 6 LM? 54 :B3 N OP45:RQ
3;J O 4SLT8BOH9UV8WLT3;?X83P6A<ZY L[3]\ 3ZX^ Q_Y 3K0#HTOB^`<'a 0bO c-?d698:\ 3eLT^`<
f L[3P6RFeOBg3;?h< +jiilkV,
mn, N , o O-?pq3 O?Xg r , UV8;:j3K0ts3K0 < >?@X3;0C6D3 E 0u1vFRQ
HI3;J 6 w`LT^G:P4x y c-J 3;0TL 8BOH z L|{}3K0u3K?P4SL~OP4SL[1v?h< J 3;0,
N O45:, N 1v?P45:-HR<GY 1GHj+
l m=
+X5<`+AmQlmn+),
kn, ," LM0C698:h< ? >?450u1"gGcB854SL[1v? 4 1 N O45:j3KJ O 4SLT8jO-H
:B3A1h0 1; >?A@B3K0C63 E 021GF-HI3KJ 6A<
U 0L?B^3;0Q_Y 3K0 HOB^`<
y3]\ 1h0#p;< +ji-i`,
, , N Odsc-J ghOW0_< ? > ?P45021"gGcB854SL[1G? 4 1 N O45:j3Q
J O4SLT8jO-HdE :AZ6L[1vHI1-^ O ?Bg a L[1vHI1)^eR<mv?Xg g`,<h. O-J Q
FW0LTg-^l3 ?KL|@X3;0C6L¡4 E 0u3 66 j+ ii-i,
¢ £ ¤ ¥§¦©¨ ª
¢ «
¤ ¬ ª ª
¥ ª ® ¤ £ ª *
+), a Od6LT8 >Cg3BOZ6 O-?Bg :B3A1h0¯
°Oe Un1GJ 3 V±;OJ HI36
Fe
/ 3;?B3;0OHUn3]4²4SLM?B^
_8B :B3 ³3d^Gc-HTOW0L´sjO4SL[1G? >tg3BO
_gh µeUO-J -HI3 . O-H´8c-HTO4SL[1v?d6
mV, --H¶LT8jO4SL[1v?d6 41 a L[1vHI1)^e N 1%g3;H|6
°Oe
E :eOW0OJ OB8A1GpXLM?B354SLT8 N 1%g3;H
Fe ? n±=K1G?B3;?4SL¶O-Hv/ 021d\ 45: N 1"gl3KH
_8B
UL´s3 z 3KK3;?Xgl3K?P4 En1G-c-HTOP4SLT1G? N 1"g3;H
·R¸K¹º[»¼ ½_¾¿ À ÁX ÃPÁÄ ÅÆ¸ÇnÈ"ÉËÊ ÌË¿ÎͺÐÏPÑXÒÓ¹½º[Ê ¿ ÔS¿ÕÍ¯Ê Öº ½b¹×
Ø
¥ ª ® ¤ £ ª
¢ ß
Ù * Ú Û Ü¦#®
¦Ý ª Û
Û ¨ Ý
¤ ¬ ª Þ
f
àá ª â-ãvá ävå )(
: O4 OW023 \ 3 4ÕO-HæpXLM?B^ O FK1vcP4Õç èéëêìZíXî ïlêð)ñòæìdó
´L 6 45:j3 6D1=04 ;
1 45:;LM?X^ \ 3 54 0OBg'L¡4SL[1v?O-HæH 45:;L?p OFK1vcP4
ML ? J OP45:B3;J PO 4SLT8P6dx
ô cB3P6Ó4SL[1v?
? éæõ÷öìdêuø5ì ïlêð)ñòæìdó
−→
?d6Ó\ 3K0
HI1%1Gp]6 H¶LMpÎ3 45:KL´6dx
ô cB3P6Ó4SL[1v?
←−
?d6Ó\ 3K0
ª â ãvá ävå ÙÆù :B3 ^hOJ 3 HOSR3Zg v
1 ? Y 6R:j1Z\
ú B3d1GeOW0ûgüSx ^÷L|@X3;? 45:B3 O?Z6Ó\ ;
3 0_<X6qOS 54 :j3 ýGcj364SLT1G?,
845ceO-HæHR<§45:KL´6 698;:j3KJ O4SLT8 g1%3P6R?hþ4 ý"cKL¡43 B8 O45c023
45:j3 023XOHÿ§O@X1=0 1 LM?A@B3K0t6D3 021GF-HI3;J 6d, > 4 R6 :j1vcH g
HI1%1Gp J 1=023 H¶LpÎ3
ô cj36Ó4SL[1G?
←−
-W021d±LMJ O43B ?d6Ó\ 3;0
ª â ãvá ävå ù :B3 ^hOJ 3 HOSR3Zg 1v? Y
ú f :j3d3;Hh1 1h045c?B35üS,
æé êìdíWî ïlêð)ñòæìdó
R1Gc 6RK3;HæHvL¡4,
6R:j1Z\
L| 6D1GJ 3A1v?j3 6qOSj6 45:j3 J 3P66qOB^3<
õ öËìZêCø5ì ïlêð)ñòæìdó L|v6D1GJ 3A1v?j3 6R;3KHæH¡6 45:B3 J 3P66qOX^l3
'
°O-W021d±LMJ O43;H e5<GR1vc 6qOË L¡4,
o 3K023 OW023 6D1GJ 3 ó =
îlìZó =
î éæí
KòV35±;OJ HI36dx
ª â-ãvá ävå ù En1vH-?B1GJ L¶O-H`021%1476A,
æé êìdíWîïêð)ñ-ò ìdó
0u1%1476d,
^'L|@B3K? O K1vH-?B1GJ L¶O-HM<Ü?Bg OHëHKL|476
õ öËìZêCø5ì l
'
ï êð)ñò ìZó ^'L|@B3K? O H~L´6Ó4 1 021%1e476A< ?Xg O
;1vH)?j1vJ ¶L O-Hl\ L|45: 53 ±;OB8545H 45:j1l6D3 20 1'1476d,
¨ à ù :KL´6 L´6 1v?j3 ;
1 45:B1K6D3 0OW023B 6ËL|45cO4SL[1v?d6 LM?
\ :KLT8: 45:B3 LM?A@B3K0C63 W20 1vFH´3;J L 6 J cX8: 3XOd6L[3K0 45:O-?
45:j3 g'LM023d854 021GF-HI3;J ª â-ãvá ävå O?Xg
x
ù :B3
J O450L¡±
<
@X3d841h0
m×n
A n×1
@X3d8541=0
6ÎO4SL´67b
m×1
éæêìdíWî ïêð-ñ-ò ìdó
ª â-ãvá ävå ^'L|@B3K?
õ'öËìZêCø5ì ïêð-ñ-ò ìdó
Ax = b.
b
A, x
^'L|@B3K?
81vJ -c43
A, b,
b.
81vJ -cP43
x.
ù Dz L|{}3;023K?P4SL¶O4SL[1v?
?Xg 45:B3
æé êìdíWî ïêð-ñ-ò ìdó ^'L|@B3K?
f (x) ∈ C[0, 1],
LM?Bg3Ü?KL¡43 LM?43d^G0OH
F (x) =
õ'öËìZêCø5ì ïêð)ñ-ò ìdó
Ü?Xg
1
C [0, 1],
Z x
^'L|@B3K?
0
f (y) dy.
F (0) = 0
O-?Bg
F (x) ∈
F 0 (x).
D ³3d8jO-HæHv45:eOP4
L´6 54 :B3 D6 354 1 OHëH
C[0, 1] C 1[0, 1]
81v?P4SLM?-cB1GcZ6 ²c-?B854SL[1G?Z6 °c-?B854SL[1v?d6 \ ¡L 45: A8 1G?4SLM?-c]Q
1GcZ6 gl3K0LI@eO4SL|@B3X 1v? 45:B3 LM?43;0¯@eO-H
[0, 1].
ª â-ãvá ävå ù DEÆO0OJ 3543;0 LTg3;?4SL`8jO4SL[1v? >#x O-?
3BOZ6Ó W021vFHI3KJ , z LT^3P6Ó4SLMF-HI3 J OP43K0L¶OHReOd66D3P645:-021GcX^G:
O g'LT^3P6Ó4SL|@B3 6Ój6Ó43;J °O 8jOA@KL¡4 1;@X1GHëcJ 3
\ L¡45:
V
81v?B8A3;?450bO4SL[1v?
1; J OP43K0L~OH;3K0 c-?KL¡4 @B1vHæc-J 3
S(t)
LM? 45:j3 67Z6Ó43;J OP4 4SLMJ 3
O?Xg c-?;L|1=0#J O?Xg LM?RQ
t
64ÕO-?B3A1vcd6 J L|±lLM?X^`, :B3 J O43;0L¶O-H L 6 O-Fd6D1h0FK3Zg
LM?P41 45:B3 6Ój6Ó43;J O4 O 0O43 W021vK1=04SL[1v?O-HÆ41 45:B3
81v?B8A3;?450bO4SL[1v? \ L¡45: O-FZ61h0#P4SL[1v? #OX841h0
:B3
a.
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-? LM?41
45:j3 W021vFHI3KJ 45:O4 35±=;3K0LMJ 3K?P4ÕO-HJ 3XOd6Rc-0u3KJ 3K?P476
J LT^G:P4 FK3 OP4 45:B3 3;?Xg"K1%LM?P4 1h0 3R@B3K? #O-HæH§1vcP476LTg3
45:j3 0bO-?X^l3
45:B3 c--;3K0 3;?Xg< 45:eO4 L´6d,M
(0, kf k1]
y1v?B3545:B3;HI36 6A< 45:B3 LM?A@B3K0t6D3 021GF-HI3;J O-K3BOW0C6 41
F;3 J O45:j3KJ O4SLT8jO-HæH 6Ó4ÕO-FH´3 \ L¡45: 023P6RK3d84 41 45:B3
J 3BOZ6Rc023KJ 3;?4
O-?Xg
h = S(t0 )
f (x) = S0(t0 − x).
Û
Ëã!#" ( #" & ãvå $ "!#( ä h
& #"&% à $ ¦ $ % "!#(5 $ ! à 'lÞ
å á ¦ *÷f 3 6D3d3 54 :eOP4 45:B3 ² c?X84SL[1v?
cZ6Ó4 gl3
?B3dg
G(
8jO-? FK3 \ 0 L¡4²43K? ZO 6
G(α) = S(t0 ) = X,
\ :B3;023 ?B1d\ \ 3 45:;L?p 1
Od6 O ê)Kõ-èlð)ó ö*Bê#é+;ñ-ò ì
X
\ :KLT8: L´6 1GF4ÕO;L?j3Zg FË ;3K0¯1h0J LM?X^ 45:B3 35±=;3K0LQ
X
J 3;?4 O-?Xg 4ÕOpXLM?X^ 45:j3 J 3BOZ6]c-023;J 3;?4 OP4 45:j3 '±X3Zg
4SLMJ 3
, V021vJ 45:;L´6 K1%LM?P4 1;@lL[35\ <\ 3 6D3d3 45:eOP4
t
α
L´6 OH|61 0 O 0O?Xg1GJ @eOW0L¶O-F-HI3 6LM?X83
L6
α = G−1(X)
O °c-?X84SL[1v? 1; O 0O-?Bg1vJ @OW0L¶O-FH´3 O?Xg 45:B3;023]1=023
L¡476D3;H O 0O?Xgl1vJ @eO0L¶OF-HI3),
y 1Z\ \ 3 8BO? 6]K3XOp 1; 6qOJ H~LM?B^ O-?Xg 6Ó4ÕOP4SL´6Ó4SLT8P6
6]cX8: Od6 45:j3 J 3BO-? @OHëcj3 1; O 6qOJ H´3<%3548-, :;L´6
FW0LM?X^;6 45:j3 3;?4SLM023 6Rc-F 3d854 1 6Ó4ÕO4SL´6Ó4SLT8R6 41 FK3BO0
(
1G? 45:j3 L?@X3;0C6D3 W021vFHI3KJ 1 LM?43;023P6Ó4, f 3 6R:j1vcH g
?j143 45:eOP4 45:KL´6 6Ó4ÕOP4SL 64SL8jO-H 45:KLM?-pBL?B^ L´6 K1K66LMF-HI3
1G?-H F;3Z8jO-cd6D3 45:j3 O0bO-J 3]43;041 FK3 Lgl3K?P4SL3Zg L 6
O 698jO-HTO0_, f 3 81vc-H´g OH¡6D1 FW0LM?X^ 6Ó4ÕOP4SL´6Ó4SLT8P6 41 FK3BO0
1G? 45:j3 021GF-HI3;J LI 45:j3 eOW0O-J 3]43;0C6 LM? ýGcB3P6Ó4SL[1v?
\ 3;023 J c-H 4SLM-HI3 698jO-HTOW0C6A< \ :KLT8: 81vc-H´g F;3 OZ6 6D3KJ Q
FHI3Zg LM?41 O 6LM?B^GHI3 0O-?Bg1vJ @B3d8541=0,
,
ª â ãvá ävå - ù DEOW0OJ 3543K0 LTgl3K?P4SL.8BOP4SLT1G? >C>x O
:O02g3;0 W021GF-HI3KJ ,M o 3XOP4 ÿv1Z\ 6 LM? O 6Ó43BOXge 6Ó4ÕOP43
45:021vcB^G: O? LM?Z6]c-HTO43dg LM?-:j1vJ 1)^l3K?j3d1GcZ6 021%g \ L¡45:
O p?B1d\ ? :j3XOP4 6D1Gc-0û83 O?Xg 45:B3 43KJ K3;0O45c023 :j3KH´g
OP4 s3K0u1 O4 45:j3 3;?XgG;1%LM?476d, ?Bg3;0 J 1"gl364 023Q
6450LT854SL[1v?d6A<j45:j3 43KJ K3;0O45c023 ²c-?B854SL[1G?
1vFK3]j6
u(x)
45:j3 HOq\
− k(x)u
0 0
= f (x), 0 < x < 1
\ ¡L 45: F;1vc-?BghOW0¯ 8A1G?Xg÷L|4SL[1G?Z6
u(0) = 0 = u(1),
45:j3K0J OH%81v?BgGcB854SL|@KL|4#
O-?Bg
k(x), 0 ≤ x ≤ 1
f (x)
gl3]43;0#J LM?B3dg FË 45:B3 :j3XOP4 6D1Gc-0û83),
æé êìdíWî
?Xg
êð)ñòæìdó
^÷LI@B3;? eOW0OJ 3543K0t6
k(x), f (x),
u(x) = u(x; k).
õ öËìZêCø5ì
'
1
u(x),
ê ð)ñòæìdó
Ü?Xg
^÷L|@X3;?
f (x)
O-?Bg J 3BOZ6Rc023KJ 3;?4
k(x).
:B3 J O-KLM?X^ °021vJ eOW0O-J ]3 43;0 63]4 ÓLM?--cP476q 41
6 1vHæc4SL[1v?d6 1vcP45-cP476q L´6 ?B1e4 O ~H LM?j3XOW0h°c-?B854SL[1v?=<hL , 3-,<
O?Xg L¡476 LM?]Q
u(x; k1 + k2) 6= u(x; k1 ) + u(x; k2),
@B3;0C6D3 J OS ?j14 3R@B3K? FK3 gl
3 Ü?j3Zg / 61
L´6 ?B1e4
?j3Z83P66qOW0LMH LTgl3K?P4S.L VO-F-HI3 ²0u1vJ O-? 1vcP45k(x)
-c4706 1
Û
2
"$h"! ãvå 3#" $ 4
f
: O4 J O-pÎ36 L?@X3;0C6D3 0u1vF-HI3;J 6 O-?A g'L|{}3K0u3K?P4ܲ021GJ
g÷LM023Z84 W021vFHI3KJ 6qç >? 61vJ 3 8jOZ636A< 45:B3;023 023BO-HæH
L´6R?=þ 4 J cB8;: g'L|{}3;023K?B8A3-, f :O4 J OP45:B3;J OP4SL8WL¶O-?d6
p?B1d\ O?Xg HI1@X3 L´6 45:B3 pBL?Bg 1W021vFHI3KJ g3Ü?j3Zg FË
o OXghOJ OW0ûg O0u1vc-?Bg 45:B3 FK3d^'LM?-?;LM?X^ 1; 45:B3 4#\ 3;?RQ
4SL[3545: 83K?P45c-0¯P, 5 åå ÞËäà (576 ä ! à 'vå á L´6 8;:O0Q
OB8543;0L´s3dg FË 45:-0u3d3 W021v;3K04SL[3P6dx
+-, :j3 W021GF-HI3KJ
:OZ6 O 61vHæc4SL[1v?h,
mn, :j3 61vHæc4SL[1v? L´6 c?KLTýGcj3),
kn, :B3 6D1GHëcP4SL[1v? 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? FS 45:B3
J O450L¡±
8BO?
b
m×n
A
F;3 45:j1vcB^G:4 1; OZ6 O H¶LM?B3BOW01v;3K0OP41h0 023BOXg ú H¶LM?B3BO0
°c-?B854SL[1v?PüÓ
>? 45:;L 6 HO?X^"ceOX^l3-<45:B3
n
m
K :R →R .
g÷LM023Z84 W021GF-HI3KJ L´6 41 ?Xg
^÷L|@X3;?
y = Kx,
A, x,
O?Xg 45:j3 LM?A@B3K0t6D3 W021vFHI3KJ L 6 41 ?Xg
^÷L|@X3;?
x,
K, y
O?Xg
Kx = y.
w LMpq35\ L´6D3<"V±O-J -HI3 _ g'L|{}3K023;?4SL¶OP4SLT1G?e L´6 O H¶LM?B3BO0
W021GF-HI3KJ , :B3;023 L´6 O 8Zc 641vJ O0¯ \ OË 1;35±)023P66²Q
LM?B^ 45:KL´6 3]±O-J -HI3 LM? 45B: 3 HTO-?X^"ceOB^3 1; ú LM?43d^G0OH
3dýGcO4SL[1v?d6d,Tü z 3 Ü?j3
FS
k(x, y)
Ø=<
k(x, y) =
61 45:O4
(
L|
0,
|L 0 ≤ x < y ≤ 1
1,
0≤y≤x≤1
Rx
F (x) = 0 f (y) dy
Z x
F (x) =
0
8jO-? FK3 35±=W023P66D3dg Od6
k(x, y)f (y) dy =
Z 1
0
k(x, y)f (y) dy
K
: L´6 L´6 O >Vð)ò îìdê#ê= éæõîì@?"ê=Kòhì0A7B)î éæð-õ ðDCîlì E)êuøAî
F éæõ-è L|45:B3 HTOZ64 43;0#J
L´6 g3;HI3]43dg O-?Xg O GVêìdè!lð)òæó
éæõîì@?"ê=Kò)ìHAIJ=î é ð)õ ðDCîlì E-êuødî F éëõè L|Ü45:B3 J LTggGHI3
43;0#J L 6 g3;HI3]43dgn, z 3;?B1e43 FË
45:B3 1GK3K0bO41=0
K
45:O4 J O-d6
41
:B3 -c-?B8;: H~LM?j3 L´6 45:B3
f (x)
F (x).
6ÎO-J 3 OZ6 LM? V±O-J -HI3
<e45:eO4 L´6A<;6D1vH@KL?B^
Ax = b)
45:j3 H¶LM?B3BOW0 1vK3;0OP41h0 3dýGcO4SL[1v?
1h0
L´6
Kf = F
F
45:j3 g'LM023d84 W021GF-HI3KJ < O-?Bg 6D1vH@KL?B^ L|4 1h0
L´6 45:B3
f
LM?@X3;0C6D3 W021vFH´3;J ,
Un1 \ :O4Pþ~6 54 :j3 °cZ66Îç :B3 g'LM023d84 W021vFH´3;J 1;
81vJ -cP4SLM?X^
²021GJ
L´6 3XOd67 O?Xg 45:B3
F
F = Kf
61vHæc4SL[1v? 41 45:j3 LM?A@B3K0t6D3 W 021vFH´3;J L´6
f = K −1F,
0L^":4Õç
ØØ
f
u0 1v?X^K HæH§1
ý"cKLM023;J 3;?476 #O-HæH
ú 6LMJ H´35ü LM?A@B3K0C63
O H~LM?j3XOW0
Ax = b
o XO g=O-J O0ûgþ 6 \ 3KHæH QÎK1K6D3Zg"?B3P66 20 3Q
FS 45:B3 \ OSj6ËLgl3-< 3]@X3;? 1=0 45B: 3
\ L¡54 :
0u1vF-HI3;J 1 6D1vH@KLM?X^ 1=0
x
67Z6Ó43;J ,
+-, :j3 6Ój6Ó43;J
"
1 1
1 1
#"
1 1
1 1
#"
1 1
1 1
#"
:OZ6 ?B1 6D1vHæc4SL[1G?L
x1
x2
#
x1
x2
#
x1
x2
#
=
"
=
"
0
1
#
0
0
#
0
ε
#
mn, :j3 6Ój6Ó43;J
"
:OZ6 L?MÜ?KL¡43;H J O?A 6D1GHëcP4SL[1v?Z6A,
kn, :j3 6Ój6Ó43;J
"
=
"
: OZ6 ?B1 61vHæc4SL[1v? 1h0
O-?Bg LM?NÜ?;L¡43KH J O-?
ε =
6 0
1h0
6D1 6D1GHëcP4SL[1v?Z6 g 1 ?Be
1 4 @eO0 8A1G?4SLM?-cj1vcZ6]H
ε = 0,
\ L¡45: 45:j3 eOW0O-J 3]43;0
ε.
)Ø ¤
O
74QP å~ã ! "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-? O 81v?d6Ó4ÕO-?P4 J cH4SLMH´3 1 1h0"6]cN: 8 L[3;?45H
t
6]J OHëH ,Tü f :eOP4 \ 3 KíXî`BKòæòaT 8BOH´8;cHOP43 L´6
t
c0
F
k =
Fk+1 + δk+1 − (Fk−1 + δk−1)
2h
Fk+1 − Fk−1
δk+1 − δk−1
=
+
2h
2h
δ
0
2
= Fk + O(h ) + O( )
h
K
: LM?-p 1 1=0 OZ6 45:B3 023d^GcHOW0L´sjOP4SLT1G? eOW0O-J 3]43;0
h
n
0u3Z8jO-HæH 45:O4
, >? 45:B3 LTg3BO-H \ 1=0#H´g 1
h = 1/n)
35±OX84 ghO4ÕO O-?Xg 8jO-H´8c-HTO4SL[1G?Z6A< \ 3 \ 1Gc-H´g 6LMJ -H
HI354
41 1vF4ÕO;LM? 3]@B3K0 FK3]4²43;0 O-?d6Ó\ 3;0C6d,
h→0
)Ø 1
>? 45:j3 LMJ K3;0¯3d854 023BO-HÕ\ 1h0#H´g<W45:j3 8jO-H´8c-HTO43dg 43K0J
6]:B1d\ 6 8;HI3BO0#H 45:eOP4 L¡4 L´6 ?B1e4 \ L 63 41 HI3]4
h → 0
LM? 45:j3 6ËL|45cO4SL[1v? 1; O ?B1G?Zs3;021 6ÎO-J -H¶LM?X^ 3;0#021=0
δ.
³ OP45:B3;0\ 3 6R:B1Gc-H´g 450¯ 41 FO-HTO-?B8A3 45:B3 3;0#021=0-43K0#J 6
LM? 6RcX8: O \ OS 45:eO4 45:B3 45:B3A1h02354SLT8BOHh3;0#021=0
O(h2 )
W023dg1GJ LM?eOP436A, f 3 J LT^":4 450¯
O(h2 ) ≥ O( hδ ),
\ :KLT8: 6RcB^-^3P6Ó476 1h0}61vJ 3 81v?d6Ó4ÕO-?P4
45:O4 2
c,
h ≥
5
4
e
:
O
4
´
L
A
6
<
K
:
´
L
6
L
6
O
#
O
J
M
L
~
H
¶
L
W
O
0
I
H
P
3
66D1v?
cδ ,
1/3 .
h
≥
cδ
(
)
LMh? 3;HI3KJ 3;?4ÕOW0¯ ?-cJ 3;0LT8BOHBO-?O-Hj6ËL 6 \ :B3;? 1G?B3 450¯j6
41 OX8A81vc-?P4 1=0 45:j3 LJ eOB854 1; Ü?KL¡43 W023d8L´6L[1v?
8jO-H´8c-HTO4SL[1G?Z6 1v? ?c-J 3K0L8jO-H=g'L|{}3K0u3K?P4SL~OP4SL[1v?,
r 81vc-0t6D3-< 45:B3;023 OW023 145:j3K0 O-W021OB8;:j36A, r ?B3
B@ 3;0¯ cZ6D3]²cHLTg3BO L 6 41 O--W021d±LMJ O43;H LM?43;0#K1GHOP43
45:j3 ghOP4ÕO \ L|45: 61vJ 3 6D1=04 1; 6RJ 1%145: 8;c0¯@X3 H¶LpÎ3
O 6R-H¶LM?B3B O-?Bg 45:j3K? g'L|{}3K023;?4SL¶OP43 45:B3 6RJ 1'145:
8c-0@X3-, @B3;0¯ 8:eOW0#J L?B^ W02363K?P4ÕO4SL[1v? 81vJ eOW0Q
LM?B^ 8A3;?43;023dg g'L|{}3;023K?B8A3P6 O?Xg ?O45c-0bO-HÎ8;cFKLT8 6]-H¶LM?B3P6
\ OZ6 ^'L|@B3K? LM? O 023d8A3;?4 J 3K0L8jO-? N O45: N 1v?P45:-HR<
@B1vH +j
l`< R3XOW0 m=
-
v+), >?43;023P6Ó4SLM?X^GH 3;?B1GcX^":h< O
8jOW023R°c-H O?eO-HZ6L´6 ^'L|@B36 45:B3 6qO-J 3 6D1=04 1; HI1Z\ 3K0
OZ6 \ 3 6qOq\ O-FK1@X3-,
F;1vc-?BgK6 1G? 6Ó43; 6L´s3
h
=Ø 9
¥ ª ® ¤ £ ª }* Û $
¢ ¥¢ 2 ß b ¢ Ý ª ¥
Û
$ Oã
$
¥Æ¦#® Û ¤ ¦#¢ ¨
! á ã & àdce" $
#"
&
b
¤ ¢
Ú ¦tÞ
à v
6 å
U c-K1K6D3 O g"0#cX^ L 6 LM?45021"g"cX83Zg LM?41 45:B3 F-HI1%1"g
645023XOJ 1=0 O J 354ÕO-F;1vH¶L¡43 L´6 OHë0u3XOBg LM? 45:j3 F-HI1%1"g
645023XOJ <-O?Xg L´6 45:B3;?X83 35±8:eO?X^3dg \ L¡45: O? 1h02^hO-?
\ L¡45:KLM? 45:B3 FK1"ge 45:-021GcX^": FK1vc?Xg'LM?B^ J 3;J FW0O-?j36A,
f 3þ HëHKOd66Rc-J 3 45:eOP4 45:B3 3]±W8:eO-?B^3 0O43P6 F;3]4#\ 3A3K?
45:j363 4#\ 1 HI1"8jO-HI3P6 O023 021GK1=04SL[1v?eOHÕ41 45:B3 6D1Gc-0û83
1ÜJ 3]4ÕOFK1vH¶L¡43 FK3eLM?B^ 450O-?d6RK1=043dg OX8021K6645:B3 J 3;J Q
FW0O?B3P6 O-?Bg 45:eOP4 45:j3 0bO43 8A1'
3 : 8WLT3;?476 O0u3 ?j1v?]Q
?j3Z^=O4SL|@B3 81v?d6Ó4ÕO-?P476d, o}3K0u3 L´6 O 698:B3;J OP4SLT8 1=0
45:j3 OX84SLT1G? L´6 4SLMJ 3X]x
t
i (t)
i1(t)
2
a
(1)
Blood
b
(2)
Organ
c
`<
f 3 OW023 LM?43;023643Zg LM? 45:j3 8A1G?X83K?P450O4SL[1v?d6
1;
xi(t)
54 :j3 J 354ÕO-FK1GH~L¡43 LM? 3XOB8;: 8A1GJ O045J 3;?4
9y 1e43)x
i.
@B1vHæcJ 3P6 :eO@X3 FK3A3K? LM?B8A1=0#K1=0O43dg LM?41 45:j3 0OP43
?c-J FK3;0C6A<'\ :KLT8: O023 íð)õ-íìdõ÷î ê=)î éæð-õ 0bO43P6 1; 35±Q
8:eO?X^l3),M :j363 ²c-?B854SL[1G?Z6 O023 ^1;@B3;0#?B3dg FË O
g÷L|{3;023;?4SL¶O-H÷3dýGcO4SL[1v? 1 45:B3 1=0#J >#þ J O-K3;?XgWQ
LM?B^ 45:j3 1G4SL[1v?O-HLM?-c4 LM? eOW023;?45:B3P6D3P6d,M
"
x1(t)
x2(t)
#0
=
"
−(a + c) b
a
−b
#"
x1(t)
x2(t)
#
+
"
i1(t)
i2(t)
K
: L´6 L´6 O? 3]±O-J -HI3 1; O ú 8A1GJ O0 45J 3;?4 J 1"gl3KH,Tü
r ?j3 8A1Gc-H´g LMJ OX^÷L?j3 J 1h023 81vJ eOW045J 3K?P476A<ÜJ 1=023
45:O-? 1v?B3 gG0#cB^ 1=0 J 354ÕO-FK1GH¶L|43 c-?Xgl3K0 81v?d6LTg3;0ORQ
4SL[1G? O-?Bg LM?-c476 LM?41 6D1vJ 3 1;45:j3 81vJ eOW045J 3K?P476
45:O4 \ 1vcH´g ?B14 F;3 023;W02363K?P4ÕO43dg FS 0OP43 8A1%3]2Q
8L[3;?476A< F-cP4 FS LM?-c4 °c-?B854SL[1v?d6d,
Ø
#!
¨ àËã!#"æà $h( à ! ® àá äã ! Pá "
$
b
à v
6 å*
x(t)0 =
Ax(t) + B w(t)
• x(t) = (x1 (t), x2(t))
#"æà
$ ,
• y(t) =
ä
P • C
h
1 0
i
fPv$'& #"æà $ ,
L´6 45:B3 à
"
P ä P x1(t)
x2(t)
$ ä
P ( ,
• u(t) =
ä
P • B
"
1 0
0 1
fPv$'& #"æà $ ,
L 6 45:B3
"
$ ä
P #
= C x(t)
L´645:B3 à
P Þ
( ãvá ävåX" $4 á ã! ! ¶" â ,
• w(t) = (u1(t), u2(t))
"
L 6 45:B3 !#)( äà $( fPv$'& Þ
#"
L´6 O (
i1(t)
i2(t)
6 "( !
#
à & à $ ! à åå 76
= B w(t)
"g' P "æà
L´6 45:B3
$ á ã! ! ¶" â ,
h
"
$ Þ
f
L¡45: O HæHË1;G45:KL´6 B? 14ÕOP4SL[1v?h<X45:j3 ^1;@B3;0#?KLM?X^ g -?eOJ Q
TL 8R6 1 54 :;L´6 81vJ eWO 045J 3K?P4 J 1"gl3KHB8BO? FK3 ^'|L @B3K? OZ6
x0(t) = Ax(t) + B w(t)
Ý " !#h& ä ! à 'å á * / L|@X3;? 54 :j3 J 1"g3;HM< 81vJ -cP43
54 :j3 L?-cP4 ² c?X84SLT1G?
O?Xg 1vc45c4
u(t) = B w(t)
°c-?B854SL[1v?
y(t) = C x(t).
¦ $ % "!#( $ ! à 'vå á * / L|@X3;? 45:B3 LM?--cP476 O?Xg 1vc4©Q
c476A<Æ81vJ -c 43 45B: 3 J PO 450L¡±
A.
y 143-x :KL´6 LM?@X3;0C6D3 W021vFH´3;J L´6 O 6RK3d8L¶OH 6D1h0 4
1 O0bO-J 3]43;0LTg3;?4SL`8jO4SL[1v? W021vFH´3;J <-\ :j3K0u3 45:B3
O0bO-J 3]43;0C6 O0u3 8A1G?Z6Ó4ÕO?476d, :B3 ^1hO-HL´6 41 023BO-HæH
LTgl3K?P4SLI 45:B3;J °O4 HI3XOd6Ó4 L? W0LM?X8WLM-HI3B5<6D1 6RpXLM 023d^WQ
cHTO0L´sjOP4SL[1v? eOW0OJ 3543K0t6A< ?B3BOW0#FË 6D1vHæc4SL[1G?Z6A< 3]48,
> p?B1d\ HI3dg-^3 1 45:j3 LM?-c476 O-?Xg 1vcP45-cP476 L´6 6RMc : Q
8WL[3K?P4 41 81vJ -cP43 45:j3 O0OJ 3543;0C6 35±)-H¶LT8L¡45H ÓL?
ã 'å ,
W0LM?X8WLM-HI3B5<\ 3 6qOË 45:j3 67j643KJ L´6 " 6v%$ #"&i h
w 354Pþ 6 35±;OJ ML ?B3 v
1 c03]±O-J -HI3 \ L¡45:B1Gc4 LM?--cP476q LM?
gl3]4ÕO;LMH`O-?Bg 6d3 3 |L \ 3 8jO-? gl36ËL^"? O? 3]±)K3;0LMJ 3;?4
45:O4 :j3KHæZ6 cd6 LTl
g 3K?P4SLI 45:j3 67Z6Ó43;J , o 3K023 1vc0
6Ój6Ó43;J L´6
"
x1(t)
x2(t)
#0
=
"
−(a + c) b
a
−b
O?Xg 1vc0 1vcP45-cP4 ²c-?B854SL[1G? L´6
y(t) =
h
1 0
i
"
x1 (t)
x2 (t)
#
#"
x1(t)
x2(t)
#
.
= x1(t).
f 3 j8 O-H´8c-HTO43 45:B3 8:eOW0OX843;0L´6Ó4SLT8 K1vH-?B1GJ L¶O-H1;
Od6
A
−(a + c) − s
b
p(s) = a
−b − s
= s2 + (a + b + c)s + bc
\ :KLT8: :OZ6 g÷L´698;0LJ LM?eO?4
(a + b + c)2 − 4cb = (b − c)2 + a2 + 2a(b + c).
K
: L´6 43KHæH¡6 cZ6 45:eO4 45:j3 3eLT^l3K?@OHëcj36 1;
O0u3 023ËQ
A
;3XOP43Zg L|
O-?Bg
O-?Bg 1e45:B3;0\ L´6D3 45:B3]
b=c
a = 0,
OW023 023BO-H"O?Xg g'L´6Ó4SLM?X84,
a}3]1=023 \ 3 O?eO-HZs3 45:B3 35±;OJ H´3P6A< \ 3 g3Ü?B3 O
ã Ëã äà" $ 41 FK3 O-?A 698jO-HTOW0Ü81vJ K1v?j3K?P4 1; 45:B3
6 !
1Gc45c4 @X3d8541=0 1=0}45:j3 023P6RK1G?Z6D3 @B3d8541=0 O4 O O0Q
4SLT8c-HTOW0 4SLMJ 3), f 3 \ LMHæH O-H \ OËZ6 8A1Gc-?4 45:B3 Ü0C64
4SLMJ 3 1; J 3XOd6Rc-023;J 3K?P4 OZ6
:-cd6A< 1h0 35±PQ
t = 0.
OJ HI3-<VL| P1Gc :OA@B3 LM?KL¡4SL¶O-Hl81v?Bg'L¡4SL[1v?Z6 1=0Ü45:j3 023Q
6]K1v?d6D3 °c-?B854SL[1v? 1; 1vc0 35±;OJ H´3 67j643KJ < 45:j3K?
R1Gc :eOA@B3 O-Hæ023BOXg 4#\ 1 ghOP4ÕO K1'L?P476d, a f < \ 3
J O-pÎ3 ?j1 ïlê#é ðê#é%OZ66]c-J 4SL[1v?d6 OFK1vcP4 pe?j1Z\ HI3Zg^3
1 LM?;L|4SL¶OHÜ81v?Xg÷L¡4SLT1G?Z6 1h0 45:B3 0u36R;1v?Z63 °c-?B854SL[1v?
1 1vc-0 J 1%g3;H , f :eOP4 \ 3 \ O?4 41 pe?B1d\ L´6 \ :eO4
g=O4ÕO K1'LM?476A<%L|O-?R<hOW023 6RMc : 8L[3;?4 41 LTg3;?4SL| 45:B3
0bO43 eOW0OJ 3543K0t6 1; 45:KL´6 W021vFHI3KJ ,
® ã ( Ù * D³3;K3XOP43Zg 021%1e476q >? 45:;L´6 8BOd6D3 \ 3 OZ6²Q
6]c-J 3 45:eO4 \ 3 p?B1d\ O W0L[1h0L=45:O4 45:B3 021%1e476 OW023
0u3KK3BO43dg`<L , 3-,<
O-?Bg
>? 45:;L´6 8jOZ63 \ 3
a=0
b = c.
p?B1d\ 45:eOP4 45:B3 1v?-H 3eLT^3;?A@eO-Hæcj3 41 45:j3 67Z6Ó43;J
L´6 ^'L|@B3K? FS 2
6D1
:j3
λ + 2cλ + c2 = 0,
λ = −c.
^l3K?j3K0bO-H=6D1GHëcP4SL[1v? 41 45:j3 6Ój6Ó43;J L´6
x1(t) = d1 cte−ct + d2e−ct
x2(t) = d1 e−ct
r c0!Ü0C6Ó4g=O4ÕO K1%LM?41"8A8c-0C6 OP4
6qOS
t = 0,
x1(0) =
(0) 6D1 \ 3 ^l3]4
(0) > \ 3 4ÕO 45:j3 1vc4©Q
x1 ,
c476 OP4 4#\ 1 HTO43;d0 24SLM=
J 3x6A1< 6q.OS
O-?Xg
\ 3 ^354
t1
t2 ,
4\ 1 ?B1v?H¶L?j3XOW0 3dýGcO4SL[1v?d6 LM?
O-?Bg
45:O4 8jO-?
d1
c
F;3 61vH@X3dg FS O y3]\ 41v?hþ~6 J 3]45:j1"gn, > 1G?B3 1;
1Gc-0}ghOP4ÕO K1%LM?P476 L´6 ?B1e4 O-? 1Gc45c4P<F-cP4 O-? LM?;L|4SL¶OH
81v?Bg'L¡4SL[1v?=< 6qOS
(0) 45:B3;? \ 3 LMJ J 3dg'LQ
OP43KH 1GF4ÕOKLM? 45:eOPx4 2(0) = x2O-?B,g 1G?B3 J 1=023 1Gc45-cP4
d1 = y0
\ LMHæHÜ
^÷LI@B3 cd6 3;?B1GcX^": LM?1=0#J OP4SL[1v?
(1)
x1 = x1(t1 )
41 6D1vH@B3 1=0 8-, :-cd6 \ 3 ?j3d3dg 45:023d3 °O4 HI3BOZ6ÓH4
g=O4ÕO ;1%LM?476 41 LTg3;?4SL|b 45:B3 O0bO-J 3]43;0C6 c?KLTýGcj3KHP,
¨ ù Ú ù f :A ú OP4 HI3XOd6Ó4²üÓç :j3 W021GF-HI3KJ L´6 O H¶L¡4²45HI3
54 0L8pXL[3;0Ü45:O-? L¡4 O-K3BO0C6Ax 1G?B3 8jO-? ^354 45:B3 6ÎO-J 3
@eO-Hæcj3
OP4 4\ 1 63KO0OP43 4SLMJ 36<%KLT3;H´g'LM?X^ 45:B3
x1(t)
6ÎO-J 3 3dýGcO4SL[1v?d6 L? 45:B3 @eOW0L¶O-F-HI3
81vJ Q
x = ct.
J 1v? 023;J 3dg L´6 41 6qO-J -HI3 J O?A ghOP4ÕO K1'LM?476A<
45:j3K? 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).
© Copyright 2026 Paperzz