! "$#&%(')*$+,-
. 0/ 1
2 34156-67
8:9 =; <?>A@B<DC EF9AG JH IK LM<N<OQP 9R)LTS HVUWLYX Z:XQ[ \$9] >^S _-X3X&>3O
` a&> b Z<Aa&>K Z<cH deU Lf g+HJXMZ3XQdhH,Z:IQ] >jikHmlMZ<cHnl&>U ;po a$9
q OsrOst*CAOuo Z3K vhi=wH [:x> y XjHnl&>jizU{Hn<4_ E i|>U}U ;~8t3t q C^9
€9 o 9u‚ I I,HwLM< Z:XQ[ $\ 9Fƒ d„(>jX&[3LYX OG >&Z:„ 3Z XQ[ … Z†ikHˆ‡
Z<cH‰LXMZ3I b >A<Aa&L[U =f L*i b LljHJXQx Š)L‹:X&[ ZhŒi _ E i Lve‡
Im>K UO)E HnN< <AK Z:X E ‹::v I,HVUeaHŽX&x ;~8t:€ C^9
‘9 ’ 9-G ja H“<?>3O1” X =• X<Ai|L[Y‹&dA<cH‰LYX <?L <Aa> g+HJXjH“<?> ‚ I–‡
>K >jX< b ^> <AaL[jU S “H <Aa ” —:—:I,HwdZ<cH‰LXTU <?L ˜uLX:I,HJX&>&Zhi
E iLv:Im>K U
O \&La:X G ŽH Im>^_ Z:X&[ PLYXU ;~8t q C^9
™ 9 ` 9 PuaL i|>UBO ˜ ‹3K >i=Hwd&Z3I EšZhi›<cH,Z:I ] Hnœ>i|>X<cH,Z:I
‚+žY‹MZ<cH‰LXUBŸ ” X •=X<AiL[Y‹&dA<cH‰LX*O Im>TdA<A‹3i|> XL6<?>U ZBlMZjHJI–‡
Z3v:Im> HJX E ‹3v:I,Hwd [5HJi|>TdA<?L*iŒ_ ;€* 3 ‘$C^9
¡ ¢ £ ¤¦¥ § ¡ ¨ £ © § ' § ¤ § ª £ ¢ 2 § ' 7
8-9 «Y¬®­?¯(°²±´³jµ·¶T¬–¸N¹ º ¶B»{¸D¬{¼½¶A¾1¿sÀ ¸(¾ÂÁ®¯}­cÃ Ä ÅÇƲÁȶTÃp¹
;ÉZMC Ä ÊW¸D¾
ÊcÁȸÉË
;?vMC ÌF¾Í­cÃp¹Î¼½¶A¾ŒÀ ­ÏƲÁȶTÃp¹
€F9ÐY­·¾½Áˆ­NÆÁѶÃ~­c¬^ÒÍÃW¸ÇÓBÔ~­{¬“ÁÕƲÁ®¸É¹
;ÉZMC Ö*¬V¬“Á‰ÅÏƲÁ®¯ ×$¾Ø¶TÙ{¬Ñ¸DÀ ¹
;?vMC ×h­·¾Í­cÙÚ¶T¬nÁ®¯ × ¾Û¶TÙ{¬Ñ¸DÀ ¹
‘F9 ÜÔ{À ¸(¾ÂÁ®¯}­c¬³¶¬mÔÏÆÁѶÃp¹uÆζ ÐY­·¾½Áˆ­NÆÁѶÃ~­c¬^ÒÍÃW¸ÇÓBÔ~­{¬“ÁÕƲÁ®¸É¹
;ÉZMC Ö*¬V¬“Á‰ÅÏƲÁ®¯ ×$¾Ø¶TÙ{¬Ñ¸DÀ ¹
;?vMC ÜÔ{À ¸D¾½Áȯ?­c¬Ä ÝÞÖh¹
;ßd&C ×h­·¾Í­cÙÚ¶T¬nÁ®¯ × ¾Û¶TÙ{¬Ñ¸DÀ ¹
™ 9 ÜÔ{À ¸(¾ÂÁ®¯}­c¬hÖQàáÅc¸(¾ÂÁ‰À ¸(ÃÏÆâ¹
ã
ä 7 ! "5$#%(') ,: . 0/1 ¡ å %
A æ ç ¡ è è é ê
G H“<Aa X&LM<DZ<cH‰LX
•
Ÿ
T
Z
?
U
W
U
A
>
<
h
—
=
i
w
H
B
d
3
>
ë
S
• t
Ÿ <cHJK >3ë
Ÿ lMZ:I‹&> L6f 3Z X LY—<cH‰LX
f=L*i Z :— ‹<ë
• V (S, t)
Z3XQ[ V = P
•
V =C
f=L*iìZ dZ:II
Ÿ
&
l
L
w
I
Z
c
<
Ž
H
,
I
“
H
²
<
_
6
L
f
3
‹
Q
X
3
[
>
4
i
®
I
j
_
Ž
H
&
X
x
T
Z
?
U
W
U
A
>
<
ë
σ
• E
Ÿ >A@h>iídhHîU·> —†i=Hwd>:ë
• T
Ÿ >A@-—jHJiŒ_ <cHJK >3ë
ï
Š IwZ&d„(‡ÏPFdaLIm>U fkL i ” K >i=Hwd&Z3X L—<cH‰LX i|>T[Y‹Qd>U <?L
>A@HîUÏ<?>jX&dB> L6fuZ:X ZU?U·>^< —†ikHd> S (t) fkL i>&ZQd6a <cHJK > t
;ÉZ ð4fÉi|>B> vjL‹3XQ[ ZhiŒ_ñÇC)Ue‹&da <AaMZf<ì<Aa> L—<cH‰LYX UeaL‹3Iî[
XL6< vj> >^@Q>ji|d†HVUW>T[ LYX LX&> UáHw[3> Lf
Z:X&[ Uea&LY‹:IV[
Sf (t)
LYX <Aa&> L6<Aa&>i UáHw[3>3OìUWL <Aa†Z< S (t) HîU Z3X L—<cHŽK Z:I
—hi=Hwd>-9 Pu—j>Td†HnòdZ:II®_eOUW>A< F = Sf− E Hmf <Aa&> L—<cH‰LX
HVU Z d&Z3IóIZ:XQ[ F = E − S Hnf <Aa&> LY—<cH‰LX HVU Z —:‹<9
` a&>X V (S, t) K ‹UÏ< Z:IÈS Z{_U U(Z<cHVUpfâ_ <Aa> dLXTUÇ<AiôZHŽX<
V (S, t) ≥ max{F, 0}.
G a >ji|> <Aa> dLXTUÇ<AiôZHŽX< HVU Z3X >TžY‹†Z:I,H“<²_eOS > aMZlQ> Z
Š IwZ&d„(‡ÏPFdaLIm>U õö3÷Tø ù†újûõØüŽý
∂V
1 2 ∂ 2V
∂V
+ S
+
rS
− rV < 0
2
∂t
2 ∂S
∂S
Z3XQ[ S a&>i|> <Aa> d LXTUÇ<AiôZHŽX< HVU UÇ<Ai=HwdA<O:S > aMZBl&> <Aa&>
Š IwZ&d„(‡ÏPFdaLIm>U ÷Tø ù†újûõØüŽý
1 2 ∂ 2V
∂V
∂V
+ S
− rV = 0.
+
rS
2
∂t
2 ∂S
∂S
G > Z:I“UWL aMZBl&> <Aa&> ‹TUe‹MZ3I vL‹:X&[ ZhiŒ_ Z3XQ[ HJXHn<cH,Z3I
dLX&[5H“<cH‰LXTU Lf <Aa&> dZ:II“U Z3XQ[ —:‹<pUB9
þ
ÿ+- Ž,6êâê´ê
] L1>U <Aa&> fNi|>B> vL‹:X&[ ZhiŒ_ >^l&>jX >A@HîUÏ< •Âf XL6<O
<Aa> LY—<cH‰LX U^a&L‹3IV[ X&L6< v> >A@h>iídhHîU·>[ HŽK K >[H"Z<?>jI®_
;ÉZhi4vHn<AiâZQx3>&CAO UWL H“< HVU L—<cHJK Z:I+<?L S ZjH“< ‹:X<cHJI<Aa&>
>A@-—jHJiŒ_ [*Z<?> <?L >A@Q>jiídhHVUW> H“< Z:X&[ <Aa&> ” K >i=Hwd&Z3X
LY—<cH‰LX i|>T[Y‹Qd>U <?L Z:X ‚ ‹:i|LY—j>&Z:X L—<cH‰LYX$9 G >
a†ZBl&>-Ÿ
•
¿sÀ ¸D¾½Áȯ?­cà ÅáÔÏÆϸɹÆßµW¸D¾Û¸ Á–¹ ­ ¼´¾Û¸p¸ Ùc¶TÔ{÷»A­W¾
¿sÀ ¸D¾½Áȯ?­cà ¯?­c¬m¬ËuÁÕÆßµ ¯Ï¶Ãp¹ôÆ ­{ÃÏÆ »Á ÊÚÁ®»{¸D÷»
• N¸N¹
•
0 < D0 < r ¿sÀ ¸D¾½Áȯ?­cà ¯?­c¬V¬áËuÁ Ƶ?¶TÔÏÆ)»ÁÕÊcÁÈ»{¸(÷»(¹Ã?¶
•kX <Aa&> dZUW> <Aa†Z< <Aa&>i|> HVU Z fÉi|>B> vjL‹3XQ[ ZhiŒ_eO S >
X>T>T[ >^@B<AiôZ dBLYXQ[Hn<cH‰LYXU Z<ì<Aa&> fÉi|>T> vL‹:X&[ ZhiŒ_ <AaMZ<
a>jI— <?L U^—j>dhHnfâ_ Hn<O*XMZ3K >I®_ <Aa†Z< V (S, t) HVU UeK L1L6<Aa
<Aa>ji> S H“<Aa
‹TUáHJXQx
+
f=L*i Z
∂V
(Sf (t), t) = ±1
d&∂S
Z3II1Z:X&[
fkL i Z —3‹<9
−
æ ÿç £ ² cN -A 6ê
` L „(>T>— <AajHJX&xjU UáHJK —3IV>3OQS > IóIáf=Ld6‹U LX ” K >i=Hwd&Z3X
—3‹<pUBOQUáHJXQd> ” K >jikHdZ:X d&Z3IInUìS Hn<Aa [5HnlHw[>jX&[ i>ž‹jHJi|>
U^I"Hwxa<AI®_ [Hmœ)>i|>X< Ue‹3vUÇ<cH“<A‹<cH‰LYXU ;4UW>B> <?>A@T<O P>Td{‡
<cH‰LYX 9€9€ C^9 ’>Td&Z3II;ßP>dá<cH‰LX q 9 ™ L6f <?>A@T<DC <AaMZ<
S > K Z3„Ú> <Aa&> Ue‹3vUÇ<cH“<A‹<cH‰LYXU
1 2
x
S = Ee , t = T − τ / σ , P = Eeαx+βτ u(x, τ ),
S a&>i|>
2
1
1
1
k = r/ σ 2 , α = − (k − 1) , β = − (k + 1)2
2
2
4
Z3XQ[
g(x, τ ) =
1 (k+1)2τ
4
e
1
1
(k−1)x
2
− e 2 (k+1)x , 0}.
max{e
ƒ X > Lv<DZjHJXU <Aa&> Š IwZQd6„Ú‡ÏPFd6a&LYIm>U >ž‹MZ<cHwLYX 3Z XQ[
vL‹:X&[ ZhiŒ_ HJXjH“<cH,Z:I dL X&[5H“<cH‰LXU HŽX <Aa&> K ‹Qd6a UáJH K ‡
—3Im>ji f=L*i4K
∂u
∂τ
u(x, 0)
u(−∞, τ )
u(∞, τ )
=
=
=
=
∂ 2u
, −∞ < x < ∞, τ > 0,
2
∂x
g(x, 0), −∞ < x < ∞,
g(∞, τ ), τ > 0,
g(∞, τ ), τ > 0.
•kX h— iôZ&dA<cHwd>:OS > i|>—:IwZQd>
vc_
OYS a>ji|>
±∞
±x
x
HVU Iw†Z iíx>-9 ” InU·LO)<Aa> dBLYXUÇ<AiâZjHJX< Bd LYXQN[5H“<cH‰LX <AiôZ3XUNN‡
fkL i K U <?L
u(x, τ ) ≥ g(x, τ )
3Z XQ[ A< a&> fÉi|>T> vL‹:X&[ ZhiŒ_ <AiôZ:XTUpf=L*i4K U <?L
:Z XQ[
x (τ )
<Aa> UeK L1LM<Aa:X&>U?U dLX&[5H“<cH‰LX LYX P (S, t) dZ†fi i=H‰>U 6L lQ>i
9
<?L
u(x, τ )
g HJXMZ:II®_eO$S &a >i|> <Aa&> dLXUÏ<AiôZjHJX< HVU Z3X >TžY‹†Z:I,H“<²_eOS >
a†ZBl&> Z Š IwZ&d „(‡ÏPFdaLIm>U õö3÷Tø ù†újûõØüŽý
L i ∂u − ∂ 2u > 0
∂τ
∂x2
Z3XQ[ S a&>i|> <Aa> dLXTUÇ<AiôZHŽX< VH U UÇ<Ai=HwdA<O:S > aMZBl&> A< a&>
>TžY‹†Z:I,H“<²_
∂ 2u
∂u
+
<0
−
∂τ
∂x2
∂u ∂ 2u
−
= 0.
2
∂τ
∂x
˜ L6<cHwdB> <AaMZ< <AajHVU ð d6IwZU?U{HdZ:I—hi|LYv:Im>jK ñÚOdBLYXUáHVUÇ<cHJXQx
u
L6fZ3X HJX&>TžY‹†Z:I,H“<²_ dBLYXUÇ<AiâZjHJX< LX <Aa> UeK L5L6<Aa UWLI‹^‡
<cH‰LYX O1Z E ] ‚ S aHwda <Aa&> UWLI‹<cH‰LYX UÚZ<cHVUpò>U LYX <Aa&>
i>xHwLYX Lf UÇ<AikHdá< HJX&>TžY‹MZ3I,Hn<4_eOÞvjL‹3XQ[*Z†iŒ_ Z:XQ[ HJXHn<cH,Z3I
dLX&[5H“<cH‰LXTUBO)HVU >A@Z&dA<AI®_ >TžY‹jHnlMZ:Im>X< <?L Z UWLh‡ÏdZ:IIm>[
ûõö3÷ú 5û ÷ ÷Töü ú 4õØüŽý "!:û ÷ ∂ 2u
∂u
−
∂τ
∂x2
!
· (u − g ) = 0,
∂u ∂ 2u
−
≥ 0,
∂τ
∂x2
u − g ≥ 0.
` a&> vjHwx ZQ[Ml6Z3X<DZ&x3> Lfì<AaHîU IwZ<N<?>ji¦f=L*i4K ‹3IwZ<cH‰LX HîU
A< a†Z< <Aa&>i|> HîU XL K >X<cH‰LYX Lf <Aa> fNi>T> vjL‹3XQ[ ZhiŒ_9
•Í< HîU dLK —:‹<?>[ Z —jLjUÇ<?>jikHwL*i=HMfNi|LYK <Aa> dLK —:‹<?>T[
U·LI‹<cH‰LX <?L <Aa&> I"HJX>QZhi dLK —:Im>jK >X<DZhi=H“<²_ —hi|Lv^‡
Im>K 9 ˜ìLS HJK ZQxHŽX> fkL i0Z K LK >X< <Aa†Z< f=L*i¦UWLK >
UÏ<AiôZ:X&x3> i|>&ZUWLYX S > ZQdá<A‹MZ3IóI®_ #†ö3÷ $ S aMZ< ∂u/∂τ
S >i|>:O UÚZc_
9 ;Ƀ [:[O v:‹< S > a†ZBl&> ‹:I–‡
<?>i=H‰L*i K LM<c∂u/∂τ
Hml&>UBOÞUWL =—3fIZc_ Z3ImLXQx S H“<Aa H“<9ŽC ` ajHVU HîU
<Aa> UWL*i›< Lf —†iLv:Im>K S > S Z:X< <?L <DZQd6„MIm> HJX <Aa&>
X>^@B< UW>Tdá<cHwLYX$9
%
&
7
'
$-A$ ¥ )(só,A-
æ ç § Jó å ^
*
4ÿ, ê
` &a > —hi|LYv:Im>jK S > Zhi|> xL1HJXQx <?L dLXTUáHw[3>i HVU Z UWL†‡
dZ:IIm>T[ !+Tü ú,Bû ÷ "!:û ÷- O<AaL‹Qxa <AajHVU >X<cHJi|> [5HVUN‡
d6‹U}UáH‰LX dZ†i4ikHw>U L6lQ>i <?L K L i|> dLK —:I,Hwd&Z<?>[ UW>Td{‡
LYXQ[ L ií[>ji >jII,HJ—<cHwd vjLY‹:XQ[*Z†i›_ lMZ:I‹&> —hi|LYv:Im>jK U HJX
Z iôZ<Aa>jiZ3vUÇ<AiôZ&dA< U·>^<N<cHJXQx ;4UW>T> G aH“<?>""U <?>^@B< i|>^f}9
‘ C fkL i Z i|>&Z:II®_ XjHwd> v:‹< x>jX<AIm> —†i>UW>X<DZ<cH‰LX Lf
<Aa> K L*i|> Z3vUÇ<AiâZQdá< UW>A<N<cHŽX&x9 Rì>ji|> "U Z —jHwdá<A‹:i|> Lf
Z —jLjU?UáHJv:Im> U~d>jX†Z†i=H‰LŸ
.
•kK Z&x5HJX&> Z:X Hw[3>&Z:I UÇ<Ai=HJXQx HJX UÇ<?>&ZQ[6_ UÇ<DZ<?> Iˆ_jHJXQx
L6lQ>i)Z3X LYvUÇ<DZ&dIm> S H“<Aa <?>XU{HwLYX T Z:X&[ Z —†i>U?U^‹:i|>
fɋ:X&dA<cH‰LX
O
9 /$>A<
9
p(x) 0 ≤ x ≤ 1
f (x) = p(x)/T
/$>A<
vj> <Aa&> lQ>i›<cHwd&Z3I [5HVUe—:IwZ&dB>K >X< L6f <Aa&>
u(x)
UÏ<Ai=HJXQx Z3XQ[
<Aa&> xYiâZ:—:a Lf <Aa&> UeK L5L6<Aa Lv^‡
g(x)
UÏ<DZQd6IV>:9 Pu‹:—:—LUW> <Aa&> UÇ<Ai=HJX&x HVU f4ZUÇ<?>X&>T[ Z< a&>†HwxYa<
0 >ji|L LYX vL6<Aa UáHw[3>U ZÚS Z{_ fÉi|LK <Aa&> LYvUÇ<DZ&dIm>3O*UÚZc_
Z< —jL1HJX<pU (0, 0) Z3XQ[ (1, 0) 9 /$>^< D vj> <Aa&> UW>^< L6f
—L1HJX<pU HJX (0, 1) Z< S ajHwd6a <Aa&> UÏ<Ai=HJXQx K Z:„(>U dLX^‡
<DZ&dA< S H“<Aa <Aa> LvUÏ<DZQd6IV>:9 G > Z:I“UWL U^‹:—:—LUW> <AaMZ<
<Aa> LYvUÇ<DZ&dIm> HVU Ue‹ 1 dhH‰>jX<AI®_ dLXlQ>A@ Z< —jL5HŽX<pU Lf
dLX<DZQdá<O <AaMZ< HVUBO −g ≥ f (x) f=L*i x ∈ D 9 g$L i
xx
ß
;
[

L
S
X
S
†
Z
|
i
[
—hi|>U?Ue‹:i>QC <AaHVU HVU Z l6Z&d‹^‡
f (x) ≤ 0
LY‹U aB_:—jL6<Aa>U{HîUB9 G > Z3InU·L S Z3X< <?L Z:IImLTS f=L*iì<Aa&>
—LU?UáHJvHŽI,H“<²_ <Aa†Z< <Aa&>i|> HVU Z f=L*iídB> —†i|LY—jL*i›<cH‰LXMZ3Ij<?L
<Aa> [5HVUe—:IwZ&dB>K >X< ZQdá<cHJXQx LX <Aa&> UÇ<AikHŽX&x ZTU S >jIIJO
U(Z{_ −cu OS a&>i|> c ≥ 0 9
2
ª , $áój ¨ N ":A 7 g HJXQ[ Z fN‹:X&dA<cH‰LYX
S H“<Aa dBLYX<cHJX:‹L‹U WU >TdBLYXQ[ [3>i=Hnl6Z<cHml&> <Aa†Z<ìUÚZ<cHîu(x)
UÇò>U
LYX [0, 1]\D,
−uxx +cu = f
L
X
u
=
g
D,
LYX
−uxx +cu ≥ f
[0, 1],
LX
u
u(0)
u(1)
≥
=
=
g
0,
0.
[0, 1],
˜uL6<?> <Aa> a&LYK L:x3>X&>BL‹U vjLY‹:X&[ ZhiŒ_ dBLYXQ[Hn<cH‰LYXUBO
S ajHwd6a S > S LY‹:IV[ XL6< a†ZBl&> HJX <Aa&> Š IwZQd6„Ú‡ÏPFd6a&LYIm>U
dLX<?>^@B<9 ˜ìLYXe‡(aLK L-x3>X&>BL‹TU vjL‹3XQ[*Z†iŒ_ dLXQ[H®‡
<cH‰LYXU Zhi|> f²ZHŽi Iˆ_ >&ZUp_ <?L aMZ3XQ[YIm>3Ov3‹< S > >^@hdI‹&[3>
<Aa>jK f=L*i dBLYXBl&>jXH‰>jX&dB>:9 ` a&> <Aa>TL*i|>K S > Zhi|>
xL1HJXQx <?L [>el&>jImLY— aLIV[jU f=L*i X&LX^‡(a&LYK L:x3>X&>BL‹TU
dLX&[5H“<cH‰LXTU ZU S >IIØ9
G > S LX) < xHnlQ> Z f=L*i4K Z:I [>eòXjH“<cH‰LX Lf >jII,HJ—<cHwd
—hi|LYv:Im>jK UBO*v:‹<0<Aa>UW> Zhi|> <cHJK > HJX&[3>—j>jX&[3>X< —hi|Lv^‡
Im>K U UÇ<?>&ZQ[M_ UÇ<DZ<?> vjL‹3XQ[*Z†iŒ_ lMZ:I‹> S ajHwd6a HŽX&dI‹&[3>
>TžY‹†Z<cH‰LXTU I,HJ„Ú> −u = f O HŽX <Aa&> d&ZTUW> Lf LYX&>
U^—MZQd> l6Zhi=H,Z:v3Im>-9 xx
ã43
§ 651 ¨ N ":^++7 ] >eò0X> Z ;ßd6IVLjUW>T[OdLXe‡
l&>A@C UW>A< K dLXTUáHVUÇ<cHJXQx Lfu<?>UÇ< fɋ:X&dA<cH‰LXTU UeK L1L6<Aa
Z3XQ[ UÚZ<cHVUpf´_HJX&x
LYX
O Z3XQ[ UÚZ<cHVUN‡
v(x) ≥ g(x)
[0, 1]
f´_HJX&x <Aa&> vjLY‹:XQ[*Z†i›_ dLXQ[H“<cHwLYXU v(0) = 0 = v(1) 9
` a&> —hi|Lv3IV>K HVU <?L ò0X&[ Z
<AaMZ< K HŽXH®‡
u(x) ∈ K
K H 0 >U <Aa> >jX>jiíxM_ fN‹:X&dA<cH‰LYXMZ:I
Z
1 1 2
2
vx + cv − 2vf dx
E(v) =
L6lQ>i Z:II
9 2 Pu0‹Qd6a Z fN‹:X&dA<cH‰LYX HVU Z:X
v ∈ K
+9-ûJù*üõ: ö*9
'
÷Tö3÷-87&ý
ó:^ ¨ N 7 g+HJXQ[ Z
Ue‹&da <AaMZ< <Aa&>
u∈K
÷Tûûõ;üõ< =cú4õ,ú*üõ:-öújûõö÷ø*ùMújûõØüŽý ;W‚¦… •íC
Z 1
(ux (v − u)x + u (v − u)) dx ≥
a 0LIV[jU fkL i Z3IóI
9 Pu‹&da Z
v∈K
ü:õ :öjújûFõö÷ø*ùMújûõØüŽý +@:ûJù*üõ:-ö*9
u
HVU
Z 1
f ·(v − u) dx
d0Z:IIm>[ Z Ú= ú>4 õ,ú?
G †a Z< †Z i|> <Aa&> dLX3X&>TdA<cH‰LXTU v>^<4S >B>jX <Aa>UW> f=L*i4K U
L6f <Aa> †— i|LYv:Im>K O0Hnf Z3XB_
ãã
¸ Æ Ùc¸ »{¸MLj÷¸Ç» ­Ç¹ ­cÙÚ¶(ÊW¸9
K
­ Ö Ã·¸D¾ON ¹ ¶T¬VÔÏƲÁȶTÃp¹ì­·¾ ¸ Ê~­·¾½Áˆ­NÆÁѶÃ~­c¬Á‰Ã·¸pÓBÔ~­c¬nÁ ÆP ¹ ¶T¬VÔÏƲÁȶTÃp¹Q
Ù ÐY­·¾½Áˆ­ÏƲÁȶTÃ~­c¬Áî÷¸pÓBÔ~­c¬nÁÕGÆ ¹ ¶T¬VÔÏÆÁѶTÃ}¹Þ­·¾ ¸ ¸D÷¸(¾RN ¹ ¶T¬VÔÏƲÁȶTÃp¹Q
¯ S ¬È­p¹k¹Á®¯}­c¬0¹ ¶T¬VÔÏÆÁѶTÃ}¹ ­·¾ ¸ ÊD­·¾ÂÁ“­NƲÁȶTÃ~­{¬ Áî÷¸pÓBÔ~­c¬nÁÕÆG ¹ ¶T¬VÔN±
Ʋ ÁȶTÃp¹ » ÐY­W¾ÂÁ“­NƲÁȶTÃ~­c¬hÁ‰Ã·¸ÇÓBÔD­c¬“ÁÕGÆ ¹ ¶T¬mÔÇƲÁȶTÃp¹ ­·¾Û¸ ÔáÃÚÁÈÓTÔ·9¸ ¸ ÐY­W¾ÂÁ“­NƲÁȶTÃ~­c¬hÁ‰Ã·¸ÇÓBÔD­c¬“ÁÕGÆ ¹ ¶T¬mÔÇƲÁȶTÃp¹ ¸à{Á ¹ÎÆ A BDC9E>FGCH
IJ K
TUCVH W@FYX-J
¾Û¸}­p¹ ¶TÃp¹
Ì µÚÁ–¹ Æßµ·¸Ç¶A¾Û¸(À Á–¹ Ê·¸(¾ Ôp¹ ¸É¼íÔá¬B¼Â¶^¾s­ Ã{Ô{À Ùc¸(¾¶(¼
"
Ò
)
Æ
É
¹
?
µ
}
¶
¦
Ë
ì
¹
Æ
~
µ
N
­
Æ
Á
¼
ß
Æ
·
µ
(
¸
Û
¾
¸
–
Á
¹
­
D
¯
®
¬
Ç
­
k
¹
¹
È
Á
?
¯
c
­
^
¬
¹

¶
m
¬
Ï
Ô
Æ
Ñ
Á

¶
Ã
ß
Æ
W
µ
D
¸
Û
¾
¸
Á
¹
¶T Z Ã{ ¬[ ¶Ã·¸9
Ì
W
µ
¸
Ç
¯
T
¶
·
Ã
p
¯
D
¸
Ï
Å
ô
Æ
¹
(
¶
¼
D
¸
·
Ã
(
¸
RN ­c÷» Ê~­·¾½Áˆ­ÏƲÁȶTÃ~­c¬Á‰Ã·¸pÓBÔ~­c¬nÁ ÆP
¾
¹ ]¶T\ ¬VÔÏƲÁȶTÃp¹ ¸(Ã{¬®­·R¾ N{¸ Æßµ·¸ ¹ ¯Ï¶TÅ{¸ ¶(¼ ÅW¾Û¶TÙ{¬Ñ¸DÀ ¹ Ƶ~­NÆ Ë¸ ¯}­cÃ
µ~­c÷»B¬Ñ9¸ Ü:¶(Æk¸ Æßµ~­NƝ˸ Ã?¶ ¬ ¶TÃ^N{¸D¾+¾Û¸pÓBÔÚÁ‰¾Û¸ ÆßµD­NÆ u(x) µ~­ÏÊW¸
­ì¹ ¸p¯Ï¶TÃW» »{¸(¾ÂÁ Ê~­NÆÁÕÊW9¸ ¿5¹š­ À ­NÆ´Æk¸(¾h¶Ú¼¼Œ­?¯ Æ ÁÕÆ»c¶B¸N¹ÉÃ`_®Æ¸ÉÊ·¸DÃ
µ~­ÏÊ·¸šÆζ µ~­ÇÊ·¸ìa
­ L¾J¹ôÆ»{¸D¾½ÁÕÊD­NƲÁ Ê·¸ ÁîÃ0Ƶ·¸ Ô}¹ÉÔ~­c¬N¹ ¸DÃ}¹ ¸Ub ¾Í­NÆßµ·¸(¾
­ cŒË¸}­ce° d »á¸D¾ÂÁ Ê~­ÏƲÁ Ê·9¸ Ì µ·¸ ¸D÷¸(R¾ N ¼Â¶A¾ŒÀ Á g
¹ Nc¶Á‰hà N Æζ NÁÕÊW¸ Ôp¹ ­cà ¸}­pY¹ ËFi­ Æζ
»{]f ¸ ÊW¸D¬–¶TÅ ­ Ã{Ô{À ¸(¾ÂÁ®¯}­c¬A¹4¯Dµ·¸(À ¸ ¼½¶A¾$¹ ¶T¬ ÊÚÁ‰hà N ÆßµÚÁ–¹ ÅW¾Ø¶TÙᬖ¸(À ãkï
æ ÿç
*
²ÿó
*
4+ÿ,1 G †a Z< S > Z†i|> K ZjHJX3Iˆ_ HJX<?>ji|>UÇ<?>T[ HŽX HVU <Aa&> fkLIImLS ‡
HJX&x —†i|LYv:Im>K Ÿ /$>^< D vj> <Aa&> UW>A< Lf Z3IóIš—jL1HJX<pU HJX
<Aa> ‡(—3IwZ:X&> U^‹Qd6a <Aa†Z<
Z:X&[ Im>A<
xt
u(x, t) = g(x, t)
U·>^<pU K(t) dLXTUáHVUÇ<cHJXQx Lf UeK L1L6<Aa <?>UÇ< fɋ:X&dA<cH‰LXTU
LYX
OÞZ3XQ[
U(Z<cHVUpfâ_jHJXQx
v(x, t) ≥ g(x, t)
[0, 1] × [0, T ]
U(Z<cHVUpfâ_jHJXQx <Aa> vjLY‹:X&[ ZhiŒ_ dLXQ[H“<cHwLYXU v(0, t) =
9
0 = v(1, t)
ª , $áój ¨ N "-A 7 g HJXQ[ Z fɋ:XQdá<cH‰LX
u(x, t)
S H“<Aa UeK L1LM<Aa UW>dLX&[ —†Z†i›<cH,Z3InU <AMa Z< UÚZ<cHVUpò>U
ut − uxx = f
u = g
ut − uxx ≥ f
u ≥ g
L X
LX
LYX
LX
[0, 1] × [0, T ]\D,
D,
[0, 1] × [0, T ],
[0, 1] × [0, T ],
u(0, t) = 0,
u(1, t) = 0,
u(x, 0) = u0(x) ∈ K(0)
ã
ó:^ ¨ N 7 g+HJX&[ Z fN‹3XQdá<cH‰LX
U^‹Qd6a
u(x, t)
<Aa†Z< u(·, t) ∈ K(t) f=L*i t ∈ [0, T ] Z:X&[ u HVU [>eò0X>[
Z3XQ[ HJX<?>TxYiôZ3v:Im> HJX Z d>ji›<DZHJX UW>jXTUW> Ue‹&dta <AaMZ< <Aa&>
—†Z†iâZ:vjLYI"Hwd =Ú>ú õ"ú-ü:õ -öjúûõö3÷Tø ù†újûõØüŽý ;WEs… •íC
'
Z 1
ut (v − u) dx+
Z 1
ux (v − u)x dx ≥
Z 1
f (v − u) dx
a LIV[jU f=L*i :Z II v ∈ K(t) Z:X&[ ðÏZ3IK LjUÇ< 3Z II®ñ
Pu‹&da Z HVU dZ:IIm>[ Z =Úú4õ,ú*üõ:-öújû1õö3÷Tø ù†jú ûõØüŽý
u
ü:õ :ö 9
0
0
0
t
:Z X&[
+@-ûJù?
G †a Z< †Z i|> <Aa&> dLX3X&>TdA<cH‰LXTU v>^<4S >B>jX <Aa>UW> f=L*i4K U
L6f <Aa> †— i|LYv:Im>K O0Hnf Z3XB_
¸ Æ
cÙ ¸ {» ¸kL÷¸Ç» ­Ç¹ ­cÙÚ¶(ÊW¸9
K(t)
¬È­Ç¹=¹Áȯ?­c¬¹ ¶T¬VÔÏƲÁȶTÃp¹ W­ ¾Û¸ DÊ ­·¾ÂÁ“­NÆÁѶTÃD­c¬ ÁîÃW¸ÇÓBÔ~­{¬“ÁÕÆG ¹ ¶T¬VÔN±
A BDC9E>FGCH
j"J K
­ S
Ʋ ÁȶTÃp¹
Ù ÐY­W¾ÂÁ“­NƲÁȶTÃ~­c¬hÁ‰Ã·¸ÇÓBÔD­c¬“ÁÕÆG ¹ ¶T¬mÔÇƲÁȶTÃp¹ ·­ ¾Û¸ ÔáÃÚÁÈÓTÔ·¸9
¯ ÐY­W¾ÂÁ“­NƲÁȶTÃ~­c¬hÁ‰Ã·¸ÇÓBÔD­c¬“ÁÕGÆ ¹ T¶ ¬mÔÇƲÁȶTÃp¹ ¸{à Á ¹ÎÆ
` &a >UW> <Aa>TL*i|>K U Zhi|> iôZ<Aa&>i K L*i|> eU ‹:v<AIV>:9 P>B>
’>^f}9 € f=L*i Z dZ†i|>^fN‹3I —hi|>UW>jX<DZ<cH‰LX L6f <AaHîU <Aa&>BL†‡
i>jK Z3XQ[ L6<Aa>ji i|>U^‹:IÈ<pUT9
4ã l 7
(š$,^ -
æ ç § Jó å ^
'
'uŽ:^+$ c
'
ó:^ ¥ %
ó:A$ ¥ )(só,A-
G >
<?>UÇ<
S >
HJX&x
Z†i> xL1HJXQx <?L dLXTUáHw[3>i <Aa&> —jLU?U{HŽvHJI"H“<²_ <AaMZ<
fN‹3XQdá<cH‰LXU <DZ:„(> l6Z3Ió‹>U LM<Aa&>iì<AaMZ:X 0 >jiLF9 PL
i|>T[3>^ò0X&> Z ;ßdImLjUW>[O dLXBl&>A@C U·>^< K dLXU{HîUÏ< ‡
Lf <?>UÇ< fN‹:X&dA<cH‰LYXU UeK L1L6<Aa Z:XQ[ U(Z<cHVUpfâ_jHJXQx
LX [0, 1] O1Z:X&[ U(Z<cHVUpfâ_jHJXQx <Aa> vjL‹3XQ[Q‡
v(x) ≥ g(x)
ZhiŒ_ dBLYXQ[5H“<cH‰LXTU v(0) = g(0) Z3XQ[ v(1) = g(1) 9
<AaMZ< K HŽXH®‡
` a&> —hi|Lv3IV>K HVU <?L ò0X&[ Z
u(x) ∈ K
K H 0 >U <Aa> >jX>jiíxM_ fN‹:X&dA<cH‰LYXMZ:I
L6lQ>i
Z
1 1 2
2
E(v) =
vx + cv − 2vf dx
Z3II
9 2 0
v∈K
G > Z†i> x3L5HJXQx <?L :Z —3—†iL@HJK Z<?> <AaHîU HJXò0XjH“<?> [Hˆ‡
K >XUáH‰LX†Z:I—†iLv:Im>K cv _ Z ò0XHn<?> [5HJK >jXTUáH‰LXMZ3I-LYX&>
<Aa†Z< S > dZ:X —:I‹Qx HJX<?L Z dLK —:‹<?>jiß9
ã%
{:A 7
N
Ÿ X3‹:K vj>ji Lf ‹:X3„MXLS X X&L1[3>UT9
1 Ÿ
h=
N +1
X3‹:K vj>iL6f+U^‹:vjHJX<?>jiŒlMZ:I“U ;4UÇ<?>— U{H 0 >QC^9
Ÿ <Aa X&L1[3>3O
xj = j · h j
kj = k(xj )
j = 0, . . . , N + 1
Ÿ f=L*i Z x5Hnl&>jX fN‹3XQdá<cH‰LX
9
k(x) ∈ C[0, 1]
Ÿ <Aa&> UW>A< Lf Z3II
Ue‹&da <Aa†Z<
Kh
v(x) ∈ C[0, 1]
Z:XQ[
HVU I,HJX&>&Zhi LX <Aa&> Ue‹:vHŽX<?>ji›l6Z3InU
vj ≥ g j
v(x)
O
9
[xj , xj+1] j = 1, . . . , N
Ÿ <Aa> —jH‰>d>AS VH UW> I,HŽX>QZhiÞfɋ:X&dA<cH‰LX h; mjú-ü n|ùö"o?
n|ùjöhüõ::ö+op eú>sr3õöjújû nj
ù ö"Qüõ:-ö+†C
LYX Ue‹3vjHJX<?>iŒlMZ:I“U [x , x ] Ue‹&da <AaMZ<
Ni(x)
üõ:: öQU9p -múq÷ú:ù
j
Ni (xj ) =
j+1

1,
0
nH f j = i
L6<Aa&>i›S HîU·>
.
ãt.
9
y
N
1 0
x
0
N1
N2
Nn
x
x
x3
xn−1
xn
1
2
Hat Functions on [0,1]: x = 0, x = 1.
1
0
Nn+1
xn+1
3 ¨ †h7 ” XB_ dBLYX<cHJX:‹L‹TU fɋ:X&dA<cH‰LX
A
<
M
a
Z
<
k(x)
VH U I"HJX>QZhiLYX U^:‹ vjHJX<?>jiŒlMZ:I“U
O
O
[x , x
] j = 0, . . . , N
VH U ‹:XjHwž‹&>Iˆ_ i|>†— i>UW>X<?>T[ v{_ j <Aa&j+1
> fkL i K ‹3IZ
u
k(x) =
NX
+1
j=0
k(xj )Nj (x).
` &a > i|>QZTUWLX HVU >QZTUáHJI®_ UW>B>X$Ÿ I"HJX>QZhi dLK vjHJXMZ<cHwLYXU
L6f dLX<cHŽX3‹&LY‹U fɋ:XQdá<cH‰LXTU I,HŽX>QZhiLYX <Aa> U^‹:vjHJX<?>ji‡
lMZ:I“U aMZBl&> <Aa&> UÚZ3K > —†i|LY—j>i›<²_9 PL <Aa&> i=HwxYa< a†Z:XQ[
U{H[> HVU dLX<cHJX3‹&L‹TU Z3XQ[ —H‰>d>^S HVUW> I,HJX&>&Zhi9 b L*i|>á‡
L6lQ>ißOuHn< ZQxYi>T>U S H“<Aa k(x) Z< >&ZQd6a L6f <Aa> XL[3>U
O
9
P
J
H
&
X
B
d
>
,
I
J
H
&
X
&
>
†
Z
)
i
É
f
:
‹
Q
X
á
d
c
<
‰
H
L
T
X
U
h
Z
|
i
>
xj j = 0, . . . , N + 1
‹3XjHwžY‹>jI®_ [3>A<?>ji K HJX&>T[ vc_ < S L l6Z3I‹&>UBO)<Aa&> i>Ue‹3IÈ<
fkLIImLS UT9
ã42
•f
HVU XL6< —H‰>d>^S HVUW> ,I HŽX>QZhißO A< a> Z:vjL6lQ> =f L*i‡
k(x)
K :‹ IwZ xHml&>U <Aa&> U·L†‡ÏdZ:IIm>[ û õö3÷ú> õö ü²÷-vw:û"úöü qú ?
yx <õ ú*ü:õ -ö
<?L
O <Aa†Z< VH UBO
h
k (x)
kh (x) =
k(x)
NX
+1
k(xj )Nj (x).
j=0
ÿ+- /Y è :A è -6êâêâê
” ?< >d6a:XHwd&Z3I [3>A<DZjHJI Z:vjLY‹< a†Z< fɋ:XQdá<cH‰LXTU HîU <Aa&>
K Z<N<?>iL6fuZ [3>i=Hnl6Z<cHnlQ> fN‹:X&dA<cH‰LYX$9 ƒ vlH‰L‹TUeI®_eOLYX&>
[L1>UezX < >A@HîUÏ< Z< vj>jX&[jUBO UWL N (x) [3L1>UeXz< aMZBl&>
Z [3>i=Hnl6Z<cHml&> HŽX <Aa> d6IZTU?UáHwdZ:I¦UW>XUW>:9 G > r÷{:ö3÷
?
<
L
j
v
>
Y
L
X
A
<
&
a
>
Ž
H
X
?
<
j
>
›
i
6
l
3
Z
I
O
0
Nj (x)
1/h
(x
, x ] −1/h
LYX <Aa&> HJX<?>iŒlMZ:I (x , x ] Z3XQ[ 0 L6j−1
<Aa>ji›S j HVUW>-9 •=X
<AaHVU S Z{_ S > i|>TdBL6lQj>i <Aj+1
a&> fN‹3XQ[*Z:K >jX<DZ:I <Aa>TL*i|>K
L6f dZ:IVd6‹:I‹UBŸ
fkL i Z:XB_
Z b
a
Nj0 (x)dx = N (b) − N (a)
0≤a≤b≤1
9
ïM3
G > dZ:X X&LTS f=L*i4K ‹:IwZ<?> <Aa> [HîUWdi|>A<?> K L[>jI$f=L*i
<Aa> ‚ X&>iíx6_ gL*i4K ‹:IwZ<cH‰LYX$9 ” —:—hi|LT@3HJK Z<?> <Aa&> UWL†‡
I‹<cHwLYX v(x) O i=HwxYa< a†Z:XQ[ U{H[> f (x) Z:X&[ vMZhi4i=H‰>i
fɋ:X&dA<cH‰LX g(x) v{_ <Aa&>†HJi I,HŽX>QZhi HJX<?>i4—jLYIwZ:X<pUB9 ƒ ve‡
U·>jiŒl&> <AaMZ< <Aa> ò0izUÏ< Z:X&[ IZTUÇ< l6Z3Ió‹>U Lf v(x) O v
Z3XQ[ v
OZ†i|> „MXLS X OTS ajHJIm> <Aa> lMZ:I‹>U v , v , . .0. , v
1 2
N
Zhi|> ‹3NX:„M+1
XLS X$9
•kX<Ai|L1[Y‹Qd> X&LM<DZ<cH‰LX fkL i<Aa&> „†X&LTS X ;ÇHJX —†i=HJX&d†HJ—:Im>&C
dLXTUÇ<DZ:X<pU
ajk =
Z 1
0
0
0
Nj (x)Nk (x) + cNj (x)Nk (x) dx,
bj =
Z 1
0
Nj (x)f h (x)dx.
ï?ã
˜u>^@B<O¦d&Z3Iîd6‹:IwZ<?> <Aa&> >X&>iíx6_ fɋ:XQdá<cH‰LX†Z:I
Z
2
1 1 h2
h
E(v ) =
vx + c v h − 2v hf h dx
2 0 


Z 1 NX
NX
+1
+1
1

=
vk Nk0 (x) dx
vj Nj0 (x) 
2 0
j=0
k=0



NX
+1
NX
+1
c
vj Nj (x) 
+
vk Nk (x) dx
2 0
j=0
k=0
Z 1 NX
+1
vj Nj (x)f h (x)dx
−
0 j=0
Z
1 1
+1 NX
+1
NX
+1
1 NX
vj ajk vk −
vj b j
=
2 j=0 k=0
j=0
` ajHVU HîU)|D‹UÇ< Z ž ‹MZQ[iôZ<cHwd fN‹:X&dA<cH‰LYX HJX <Aa> l6Zhi=H,Z:v3Im>U
9 ” US > X&LM<?>[ vj>^f=L*i|>:O v Z3XQ[ v
v1 , v 2 , . . . , v N
Zhi|> X&LM< l6Zhi=H,Z:v3IV>U } <Aa&>^_ Zhi|> „MX&LTS X 0l6Z3Ió‹>U N +1
v(0)
Z3XQ[ v(1) 9
ïï
$g L*i <Aa&> K LK >jX<O Im>A<` U HxX&L*i|> <Aa&> dLXUÏ<AiôZjHJX<pU
LYX <Aa&>
UB9
” U HJX d&Z3Iîd6‹:I‹TUBO S > S L‹3IV[ [5Hnf|‡
vj
fk>ji>jX<cH"Z<?> <Aa&> >X&>iíx6_ fɋ:XQdá<cH‰LX†Z:I S Hn<Aa i|>Ue—j>TdA<
<?L
Z:X&[ UW>A< <Aa> i>Ue‹3IÈ<pU >ž‹MZ3I<?L 0 >i|L
v1 , . . . , v N
<?L IVL1d&Z<?> <Aa&> dikHn<cHwdZ:I—jL5HJX<pUT9 yuUW> <Aa> Up_:K K >{‡
<Ai›_ —hi|L—j>i›<²_
Z3XQ[ Z I,H“<N<AIV> d&Z3IVd‹3IZ<cH‰LX
ajk = akj
U^a&LTS U <Aa†Z<
˜ LS
u
<Aa>
NX
+1
amk vk − bm = 0, m = 1, 2, . . . , N.
k=0
IV>A< <Aa> N × N K Z<Ai=H“@ A a†ZBl&> >X<AikHw>U a O
jk
lQ>Tdá<?L i aMZlQ> >jX<Ai=H‰>U
N ×1
b
b1 −a01v0, b2, . . . , bN −
Z3XQ[ <Aa&> N × 1 l&>dá<?L*i v aMZBl&> >jX^‡
aN,N +1vN +1
<AikHw>U
9 ` a>jX <Aa> —hi|Lv3Im>jK i|>[‹Qd>U
v1 , v 2 , . . . , v N
<?L <Aa&> U{HŽK —:Im> I,HJX&>&Z†i Up_UÇ<?>jK
Av = b.
˜uLS H“< d&Z3X vj> —†i|L6lQ>T[ <AaMZ< <Aa&> K Z<Ai=H“@
HVU Up_:K ‡
A
K >A<Ai=Hwd —LUáH“<cHnlQ> [3>^ò0XHn<?>3O a>jX&dB> X&YL Xe‡ UáHJX&xY‹:IwZhi9 ` a&>i|>{‡
fkL i>:O<AajHVU UÇ_UÇ<?>K a†ZU Z ‹3XjHwžY‹> UWYL I‹<cH‰LX 9
ïþ
G > dL‹3Iî[ ò0XQ[ <Aa&> UWLYIó‹<cH‰LX v v{_ ~ Z3‹U?UáH,Z3X >I,HŽK ‡
HJX†Z<cH‰LX*O ZTU HJX b Z<Aa ‘8 ™ 9 R)LS >^lQ>ißO <Aa&>i|> HîU
Z3X&LM<Aa&>i Z3—:—hi|L$Z&da <Aa†Z< S > S Z:X< <?L >A@*—3ImL i|> vj>á‡
dZ:‹TUW> H“< dZ:X vj> >A@B<?>jX&[3>T[ ?< L a†Z:XQ[Im> <Aa&> dLXe‡
UÏ<AiôZjHJX<pU Lf <AajHVU —†i|LYv:Im>K 9

$áD%Ú'u/V%Ú' ¡ 2
. 0/ì7
` a&> H [>QZ HVU <?L H“<?>jiâZ<cHnlQ>I®_ Z:—3—†i|LT@HŽK Z<?> Z UWLI‹^‡
c< H‰LYX <?L Av = b OhUÇ<DZhi›<cHJXQx fNiLK UWLYK > HŽXH“<cH"Z3I{x‹&>U?U
g$L*i>^@6Z:K —:Im>3OS > dL‹:IV[ ‹TUW> <Aa&> ò0izUÇ< >TžY‹†Z<cH‰LX
v0
<?L U·LI®lQ> f=L*i v1 HJX <?>i4K U Lf v0, . . . , v0 O5<Aa&>X ‹TUW>
<Aa> UW>dLX&[ >TžY1‹MZ<cH‰LX <?L UWLYIˆl&> 2 f=L*i v1NHŽX <?>ji K U Lf
2
0 O¦>A<·d39 •=X >ž‹MZ<cHwLYX fkL i K
1 0
v1 v3 , . . . , v N


X
X
n+1
n+1

−
ajk vkn /ajj , j = 1, 2, . . .
= bj −
ajk vk
vj
k>j
k<j
S &a >i|> <Aa&> H“<?>iôZ<cH‰LYX îH U i|>—j>&Z<?>T[ f=L*i n = 0, 1, . . .
‹3X<cHJIdBLYXBl&>ji|x3>XQd>-9 R)TL S >^l&>jißOÞLX> dZ:X <AajHJX:„ Lf
<Aa> <?>ji4K Z:vjL6lQ> ZTU Z Wð i|>IwZ@6Z<cH‰LX <?>i4K ñ HJX <Aa&>
U·>jXTUW> <AaMZ<OYS Hn<Aa
O
ω=1
n+1
n+1
n+1
n
n
n
n
− vj .
− v j = v j + ω vj
= v j + vj
vj
ï
` ja HVU Ue‹Qx3x3>UÇ<pU A< aMZ< S > Bd LY‹:IV[ ð L6lQ>i4i|>IwZ@Bñ A< a&>
Z3—:—hi|LT@3HJK Z<cH‰LX vc_ ‹UáHJXQx wI Z†iíx>ji 9 R)>jX&dB>3O <A&a >
ω
Pšƒ ’ K >A<Aa&L1[Ÿ


X
X
n+1
n+1

=
vbj
−
ajk vkn /ajj
bj −
ajk vk
k>j
k<j
vjn+1 = vjn + ω vbjn+1 − vjn = (1 − ω ) vjn + ω vbjn+1
£ -4 l ê •Âf
HVU Z Up_:K K >^<AikHd —jLjUáH“<cHnlQ> [>ef‡
A
HJXHn<?> K Z<Ai=H“@ Z3XQ[ 0 < ω < 2 O <Aa&>X <Aa&> ~ Z:‹TU?UN‡
P>†Hw[3>I–‡ÏPšƒ ’ K >A<Aa&L1[ dLXBl&>iíx3>U fNi|LYK Z:X_ UÇ<DZhi›< ‡
HJX&x xY‹>U?UB9 b L i>TL6lQ>ißO+<Aa>ji|> HVU Z:X L—<cHŽK Z:I ω =
;?K LjUÇ< iôZ3—jHw[ dLXlQ>iíx>jX&dB> iôZ<?>&CAO)S a&>i|>
∗
ω
1<
9
ω∗ < 2
Š Z&d„ <?L
<?>X<cH‰LYX
<DZ3„Ú> <Aa&>
‹TUW>T[ <?L
vj > g j
LY‹:i —†i|LYv:Im>K O S a&>i|> S > X&LTS —MZc_ Z< ‡
<?L <Aa> dLXUÏ<AiôZjHJX<pU LYX <Aa>
"UT9 •Âf S >
vj
—jL1HJX< Lf ljHw>AS <AaMZ< <Aa&> j <Aa >TžY‹†Z<cH‰LX HVU
U·LI®lQ> fkL i O<Aa&>X S > K ‹UÏ< a†ZBl&> >MH“<Aa>ji
vj
L*i <Aa> j <Aa >TžY‹†Z<cH‰LX a&LYIV[jUT9
ï
ƒ X > dL‹3IV[ ‹U·> ðWI,HJX&>&Z†idBLYK —3Im>jK >jX<DZ†ikHn<4_Bñ <?L >A@‡
—hi|>U?U <AajHVU dBLYXQ[5H“<cH‰LX 9 G > UáHJK —3Iˆ_ LvTUW>iŒlQ> <AaMZ< Z
U^K Z3IóIZQ[h|D‹UÏ<AK >X< HJX Pšƒ ’ >jX†Z:v3IV>U ‹TU <?L i|>Ue—j>TdA<
<Aa>U·> dLXTUÇ<AiôZHŽX<pUT9
£ -4 € ê Pu‹3—:—jLjUW> <AaMZ<
HVU Z Up_:K K >A<Ai=Hwd
A
—LUáH“<cHnlQ> [>eò0XH“<?> N × N K Z<Ai=H“@O 0 < ω < 2 Z3XQ[
Z:XQ[ 0 Zhi|>
l&>TdA<?L*izU Z:XQ[
g
v
N ×1
HVU
Z:X&[
K = {v | v
N ×1
vj ≥ gj , j = 1, . . . , N }.
` a&>X <Aa> ‚A-÷ Qü²÷ r ƒ „ … ÷†iü m†r


X
X
n+1
n+1
vbj
= bj −
ajk vk
−
ajk vkn /ajj
k<j
k>j
n
o
n+1
n+1
n
= max gj , (1 − ω ) vj + ω vbj
vj
{v n } ∞
n=0
x >jX>jiâZ<?>U Z UW>TžY‹>jX&dB> L6f l&>TdA<?L*izU
<AaMZ<
dLXlQ>iíx>U <?L <Aa&> UWLI‹<cHwLYX <?L <Aa&> —†i|LYv:Im>K Lf
K HJXjHJK H 0 HŽX&x
n
T
T
o
E(v) = min v Av − v b .
` ja HVU —†i>^<N<²_ S >IIdLK —:Im>A<?>U A< a&> [ >UWdi=HJ—<cHwLYX Lf
LY‹:i X3‹:K >ji=HwdZ:IZ3IîxL ikHn<Aa3K 9 ” f=A> S K L i|> K >d6aMZ3Xe‡
HwdZ:I$[>^<DZHJInU Zhi|> X&>B>[>[9
Mï v∈K
ÿ+- Û-Ú)5s²,6êâê´ê
` a&>UW> f=L*i4K ‹:IwZU Z†i> >&ZU{HŽI®_ d6a&>Td„(>T[ [HJi|>dá<AI®_Ÿ


0,


Z 1

−1/h,
Nj0 (x)Nk0 (x)dx =

2/h,
0


1/h,


0,


Z 1

h/6,
Nj (x)Nk (x)dx =

2h/3,
0


h/3,
|j − k| > 1
k =j±1
0<k =j <N +1
N +1
k=j=0
¶A¾
|j − k| > 1
k =j±1
0<k =j <N +1
k=j=0
N +1

h

(2f0 + f1 ) ,

6
R1
h
h (x)dx =
N
(x)f
f
+
4f
+
f
j
j−1
j
j+1 ,
0
6

 h (f + 2f
N
N +1 ) ,
6
¶A¾
j=0
0<j <N +1
j =N +1
ï
%
æ ÿç
¡
‡
§ G > IIJOXL6< i|>QZ3IIˆ_9 ” dA<A‹†Z:II®_eO†<Aa> žY‹>UÏ<cHwLYX S > aMZBl&>
<?L [3>&Z:IS H“<Aa HVU UáHJK —:Im>ißO v3‹< <Aa> dBLYX:X&>Tdá<cHwLYX <?L
ƒ ] ‚šU HVU <AaMZ< LX> ‹UW>U <Aa&> ` Zc_:IVL*i UW>i=H‰>U „&HJXQ[jU
L6f Z†iíx‹:K >jX< S >Âi> Z:vjLY‹< <?L UW>T> HJX Z3X >U?UW>X<cH,Z3I
S Zc_ HJX Z:XMZ3I®_ 0 HJX&x <Aa> K >i=H“< L6fšlMZ†ikHwLY‹U X3‹:K >jikHdZ:I
ƒ ] ‚ K >A<Aa&L[UT9
6ù ÷y+Büõ<:öŠ‰ )
R LS [3L S > Z3—:—hi|L@HJK Z <?> Z Nf ‹3XQdc‡
x5Hn&l >jX lMZ:I‹&>U L6f
Z< lM†Z ikHwLY‹U &X L[>
<cH‰LYX
dy/dt
y(t)
—L1HJX<pyU
ˆ
$g L*i —:‹3i4—jLjUW> LfUáHJK —3I,HdhH“<²_eOT•Œ>^<`"U UÚZ{_ <AaMZ<uS > aMZBl&>
>TžY‹†Z:II®_ Ue—†ZQd>[ XL[3>U HJX UÇ<?>—U Lf k 9 /$>^< t = j
Z3XQ[ y = y(jk) 9 ’>Td&Z3II<Aa&> ` Z{_:ImL if=L*i4K ‹:IwZjU
j
k
y(tj+1) = y(tj ) + y 0(tj )k + y 00(tj )
2
2
+ O(k3 )
k2
y(tj−1) = y(tj ) − y (tj )k + y (tj ) + O(k3 )
2
ïe.
0
00
y UW> A< a> ò0izUÏ< ?< L UWLI®l&> fkL i y0
Þ
0(t ) Z3XQ[ LYv<DZjHJX
=
y
<Aa> nGk$úsr r:Rõ ‹F÷ ÷öB÷ nŒs j ùjû,ú j
yj0 =
yj+1 − yj
+ O(k).
PHJK HJIZhi4I®_eO <Aa&> UW>TdBLYXQ[ k =f L*i4K ‹:IwZ xHml&>U <Aa&> ! ú,V#-?
$)
ú sr r3õR‹¦÷- ÷Tö"÷ nGs ùjû,ú
yj − yj−1
0
yj =
+ O(k).
˜ >^@B<O S > ^U ‹:v<AiâZQdá< A< ak> ·U >dLX&[ ` cZ _:IVL*i f=L*i4K ‹:IwZ
u
fÉi|LK <Aa> ò0zi UÇ< Z:XQ[ UWYL Iˆl&> =f L*i 0 <?L YL v<DZHŽX
yj
yj+1 − yj−1
+ O(k2 ).
UWLh‡ÏdZ:IIm>2k
[ B÷Töü²÷- ÷-r r3õO‹F÷- ÷Tö"÷
yj0 =
` ajHVU HVU <Aa&>
nŒ?
ùjû,ú39 ` a&>UW> L*ií[>ji fkL i4K ‹:IwZTU Ue‹&x:x3>UÇ< <AaMZ< ZU
x>^<pU UeK Z:IIm>ißO:<Aa> d>jX<?>ji|>T[ [Hmœ)>i|>XQd> fkL i4K ‹:IwZ
k
U^a&L‹3IV[ >^@hd>T>T[ <Aa> LM<Aa&>izU HJX Z&dTd6‹:iâZQdA_9• fS > Z&[:[
<Aa>U·> <4S L ` Z{_:ImL i f=L*i4K ‹:IwZU <?L:x3>A<Aa&>ißO S > Z:I“UWL
x>^< U·LK >^<AaHŽX&x iâZ<Aa&>iHŽX<?>ji>UÇ<cHJX&x f4ZQdá< Z:vjLY‹< Zl‡
>iôZ&x3>UTŸ
yj =
yj−1 + yj+1
2
+ O(k2 ).
ïM2
ƒ X > K L*i|> —jL5HJX< Z3ImLXQx <Aa&>UW> I,HJX&>UTŸ S L‹3Iî[Xz<
H“< vj> XHwdB> Hmf U·LK >jaLS <Aa> UáHJK —:Im> fkL i½S Zhií[ [5Hnf|‡
fk>ji>jX&dB>UBO S ajHwd6a a†ZBl&> <Aa&> K >i=H“< Lf LYX:I®_ ‹UáHJXQx
lMZ:I‹>U Z< <4S L ZQ[Q|}Z&dB>X< XL[3>UBO1S >i|> U·>dLX&[ L i‡
[>ji ZQdBd‹3iôZ<?>> G >IóIJO Hmf _eL‹ U~žY‹HŽX<O <Aa&>^_ Z†i>zŽ
Pu‹3—:—jLjUW> <Aa†Z< S > a†ZQ[ S ikHn<N<?>X L‹< <Aa> ` Z{_:ImL i
fkL i K ‹3IZTU Z3vjLl&> f=L*i a†Z:I®f UÇ<?>j—TU
9 G a†Z< S >
k/2
S L‹3IV[ aMZlQ> >XQ[>[ ‹:— S H“<Aa HVU
0
yj+1/2
=
yj+1 − yj
k
+O(
2
k
2
)=
yj+1 − yj
k
+O(k2 )
S a&>i|> S > ‹:XQ[>ji UÇ<DZ:X&[ <AaMZ< y
j+1/2 = y(tj+1/2 ) =
`
a
î
H
U
V
H
U
Z
W
U
†
L
Ï
‡

d
:
Z
I
m
I
T
>
[
M
l
:
Z
I
&
‹
>
6
L
f
Z
<
Z
y((j +1/2)k).
y
ð·aMZ:I®f‡(X&L[>^ñÚ9 • f S > Z:I“UWL S Z:X<?>[ Z UW>TdBLYXQ[ L i|[3>i
Z&dTd6‹:iâZ<?> l6Z3I‹&> f=L*i y H“<pUW>jI®f Z< <AajHVU a†Z:I®f|‡DX&L[>:O
<Aa>jX S > S L‹:IV[ ‹U·> <Aa&> ZlQ>iôZQxHJXQx f=L*i4K ‹:IwZ <?L
LYv<DZHŽX <AaMZ<
yj+1/2 =
yj+1 + yj
2
+ O(k2 ).
þM3
æ *ç
*
²ÿóó
'
ó:^ ¥ )(š$ó,A-
G > i|> XLS x3L5HŽX&x <?L dLK vjHJX> —†Z†i½<pU ;ÉZMC Z3XQ[ ;?v†C
<?L <DZQd6„MIm> <Aa> —MZhiôZ:vjLYI,Hd lMZhi=H,Z<cH‰LX†Z:IYHJX>ž‹MZ:I,H“<²_9
þ?ã
€
7
ój §

å 1 1-Ú
þï