º KGKPOMOML_ PK_[KIZL_ JK OEIO[IL JK QEPMEOMLBK_ Kiaknl Iº lpkz lpkz Zlrrks º JMBEFMOE º LXZMFMWEOM LB kipdknl5:Hnfemi.olf
50 jk kbkrl jk 0854
ºMbjmok
5
º JK K[IKP IE KO[EOMLB
0
º JK K[IKP7 oesls kspkomeiks IE KO[EOMLB
?
º JK K[IKP7 Ki oesl jk vermes verme`iks IE KO[EOMLB
:
_uſtomkbome
; 1
Olbjmomlbks jk Zrebsvkrseimjej Fexmfmzeomºlb lb jki `kbkſtoml flblplimste
º JK K[IKP IE KO[EOMLB
Kbolbtrer ie treykotlrme ºlptmfe(kxtrkfl) lptmfe(kxtrkfl) jk ils smnumkbtks aubomlbeiks
5
G (x) 6
(tx + tx + 0 x˝ 0 ) jt
(5.5)
8
s.e
x(8) 6 5 x(5) 6 0 A ( A (t,x,x ) 6 tx + tx + 0 x˝ 0 A x 6 t A x 6 :x ∀ A 6 : x x ∀t
Kuikr Kquetmlbs t :x 6 t ← x 6
:
º JK K[IKP IE KO[EOMLB
mbtknrebjl, mbtknrebjl, l`tkbkfls l`tkbkfls x(t), quk setmsaeok ie treykotlrme nkbkrei7 0 t x 6 + c5 4
⇐
t? x(t) 6 + c5 t + c + c0 0:
epimoebjl ies olbjmomlbks7 x(8) 6 5
x(5) 6 0
← c0 6 5
0? ← 0:5 + c5 + 5 6 0 ⇐ c5 6 0:
plr il tebtl ie treykotlrme lptmfe(koueomº lºptmfe(koueomº lb) lb) pertmouier ks t? 0? ∞ x (t) 6 + t + 5 0:
0:
º JK K[IKP IE KO[EOMLB
0
G ( G (x) 6
(x + t + tx˝
5
s.e
∐ x˝ 0) jt
(5.0)
x(5) 6 ? x(0) 6 : Kuikr Kquetmlb A ( A (t,x,x ) 6 x + t + tx˝ A x 6 5
∐ x˝ 0
A x 6 t ∐ 0x
∀ A x 6 5 ∀t
∐ 0x
5
∐ 0x 6 5 ← x 6 8
mbtknrebjl kbolbtrefls ie sliuomº lb lb nkbkrei x(t) 6 c5 t + c + c0
º JK K[IKP IE KO[EOMLB
epimoebjl ies olbjmomlbks x(5) 6 ?
← c5 + c + c0 6 ? x(0) 6 : ← 0c5 + c + c0 6 : sk tmkbk ub smstkfe jk koueomº lb lb ouye sliuomºlb lb ks7 c5 6 5 y c0 6 0 plr il tebtl ie sliuomºlb lb pertmouier ks7 x∞ (t) 6 t + t + 0
º JK K[IKP IE KO[EOMLB
∐ :8
G (x) 6
5 0 x˝ 0
8
s.e
x(8) 6 08 x(:8) 6 8 sliuomº lb lb
+ c0 x(t) 6 c 5 t + c 5 ∞ x (t) 6 t + 08 0
jt
(5.?)
º JK K[IKP IE KO[EOMLB
∐ 58
G ( G (x) 6
0xx˝ + x˝
8
s.e
0
x(8) 6 58 x(58) 6 588 sliuomº lb lb
x(t) 6 c 5 t + c + c0 x∞ (t) 6 =t + 58
jt
(5.:)
º JK K[IKP IE KO[EOMLB
∐ 0
G (x) 6
50tx 50 tx + + x˝
0
8
s.e
x(8) 6 5 x(0) 6 5> sliuomº lb lb
x(t) 6 t ? + c5 t + c + c0 x∞ (t) 6 t ? + :t :t + 5
jt
(5.;)
º JK K[IKP IE KO[EOMLB
Ils mbnrksls (M ) y ils olstls (O ) jk ube ſtrfe jkpkbjkb jk ie prljuoomºlb lb y jk ie tese jk vermeomºlb lb (x ) jk ie prljuoomºlb lb jk ie smnumkbtk alrfe 0 x M (x, x ) 6 x 0 x0 0 O (x, x ) 6 0x + 0
∐
5
0
Xiebtker ki prl`ikfe jk lptmfmzeomºlb lb jmbº mbºefmo ef moee e trev tr evººks jki jk i oºeio ei ouil ui l jk vermeomlbks. Jktkrfmber ie treykotlrme jk prljuoomº lb e il iernl jki prkskbtk e˙ lb bl bl quk fexmfmok ki veilr jk ils `kbkſtoml (ψ )jk ie ſtrfe. Esufe ube tese jk jksoukbtl (υ) ks mnuei υ 6 5/0 y quk ie prljuoomºlb lb ei mbmoml ks x(8) 6 8 y 8 y ei ſtbei jki e˙ bl bl ks x(5) 6 5/ 5/0
º JK K[IKP IE KO[EOMLB
5). Ki `kbkſtoml jk ie kfprkse kstº k stºe jeje plr7 ψ 6 M Z
∐ O Z ∐
Edlre Edlre pertmfls pertmfls jki supukstl supukstl kb quk ki aubomlbei aubomlbei kstº e jeje kb ki aeotlr aeotlr jk jksoukbtl, jksou kbtl, kstruotureb kstruo turebjl jl ki prl`ikfe e rkslivkr rksliv kr kb tºkrfmbls krfmb ls jki oºeiouil eioui l jk vermeomlbks vermeomlb ks ksterº ksterºĵe piebtkejl piebtk ejl olfl7
t5
fº ex ψ (x) 6 ex
t8
x(t8 ) 6 x8
M (x, x ) ∐ O (x, x ) k∐υt jt
(5.1)
x(t5 ) 6 x5 0).Xere oeiouier ie treykotlrme jk ie prljuoomºlb lb dey quk quk fºexmfer exm fer ψ kstruoturejl `egl ils mbnrksls y ils olstls olfl ie vermeomº lb lb quk quk pljrº jrºĵe tkbkr e il iernl jki e˙ bl. bl. Pksli Pks livm vmkb kbjl jl kb tºkrfm krfmbl blss fetk fetkfº fºetmo etmols ls,,
º JK K[IKP IE KO[EOMLB
∐ 5
fº ex ψ (x) 6 ex
x
8
0
A 6 x x ∐ 0x˝ k∐ A 6 (5 ∐ 0x) k∐ t x
5 0
5 0
t
:x ¨
5
A x˝ 6 ∐:xk ˝ ∐ t A 6 0xk ˝ ∐ ∐ :xk x ¨ k∐ t
t
0
0
:r 0
Kuikr Kquetmlb3 (5 ∐ 0x) k∐
5 0
t
6 0xk ˝ ∐
t
0
∐ 0x˝ ∐ 0x 6 ∐5,
ki plimblfm pl imblfml l oereotkrº oereotkrºĵstmol ĵstmo l ks,
0
x,t ˝
(5.>)
smfpimſtoebjl y rklrjkbebjl ie koueomºlb lb jmakrkbomei3
sliuomº lb lb 0
5 0
x ∐ 0x˝ k∐ t jt
x(8) 6 8 5 x(5) 6 0
∐
t
∐ :xk x ¨k∐ , 0
∐ 0r ∐ 0 6 8, ouyes remoks slb7 r5 6 ∐5/0 y r 0 6 5.
º JK K[IKP IE KO[EOMLB
Ie sliuomº lb lb dlflnºkbke kbke ks jk ie alrfe t
x(t) 6 c5 k∐ + c0 kt . 0
Xlr il tebtl ie koueomºlb lb nkbkrei ks3 5 ∐ t x(t) 6 c 5 k + c0 k + . t
0
0 Epimoebjl ies olbjmomlbks mbmomeiks l`tkbkfls quk3 c5 6
k
c0 6
∐
0 k
5
k
0
k∐
5 0
∐
0 k
k
5 0
Ambeifkbtk ie koueomº lb quk setmsaeok ie treykotlrme quk fexmfmze ki veilr lb jk ils `kbkſtomls esufmkbjl ie tese jk jksoukbtl rkstrmbnmjl ks x∞ (t) 6
5∐ 0t
k
+
t∐ 50
k
∐ ∐
0 k
5 0
k
0 k
k∐
5 0
5 + 0
º JK K[IKP IE KO[EOMLB
5
G (x) 6
0
0
x˝ + x + :xk :xk
8
s.e
t
jt
x(8) 6 8 x(5) 6 5 sliuomº lb lb
Kuikr Kquetmlbs A ( A (t,x,x ) 6 x˝ 0 + x0 + :xk :xkt A x 6 0x + :k : kt A x 6 0x
0x 6 0x + :k : kt smfpimſtoebjl
∀ A x 6 0x ∀t
x
rkslivmkbjl
∐ x 6 0kt
(5.4)
º JK K[IKP IE KO[EOMLB
xd 6 c5 kt + c0 k∐t pere ie sli. pertmouier ks jk ie alrfe x p 6 t( t (Ekt ) E(0k (0kt + tkt ) 0Ekt 6 0kt
∺
∐ Etkt 6 0kt E 6 5 ⇐ x p 6 tkt
x(t) 6 c 5 kt + c0 k∐t + tkt
º JK K[IKP IE KO[EOMLB
Epimoebjl ies olbjmomlbks x(8) 6 8 x(5) 6 5 fuitmpimoer7 pere c0 7
← c5 + + c c0 6 8
← c5k + + c c0 k∐5 6 8
(e) (`)
Kq.(e) y sufer ksk rksuitejl kb Kq.(` Kq.( `), jkspuks rkslivkr ∐k ie Kq.(e k c0 6 k + 5
pere c5 , tkbkfls kb Kq. (e (e)7 c0 6
∐c5 , kbtlboks7 k c5 6 ∐ k + 5
Xlr il tebtl ie koueomºlb lb pertmouier quk setmsaeok ils kxtrkfls jki aubomlbei ks 5∐t t+5 k k x∞ (t) 6 + tkt k + 5 k + 5
∐
º JK K[IKP7 oesls kspkomeiks IE KO[EOMLB
Jki smn. aubomlbei, olfprl`er quk ie sliuomº lb ks ube imbke sm e 6 8 y e 6 0 lb ?
G (x) 6
0
x (5
5
∐ x˝ )
0
jt
(0.5)
Iiknerkfls e ie sliuomºlb lb nkbkrei epimoebjl ki oesl jlbjk ki aubomlbei slil jkpkbjk jk x y x˝ , ks jkomr3 G (x, x˝ )
∐ ∐ A
x˝
∀ A ∀ x˝
6 c5 .
(0.0)
Olb eyuje jk ie koueomºlb lb (0.0 0.0)) pere ki aubomlbei (0.5 (0.5). ). Ie sliuomº lb lb nkbkrei ks3 x(t) 6
(t + c + c0 )0 + c5
º JK K[IKP7 oesls kspkomeiks IE KO[EOMLB
Kbolbtrer ki kxtrkfl jki smn. aubomlbei 5
G (x) 6
8
x˝ 0 0
+ xx˝ + x˝ + x + x jt
(0.?)
Jk mnuei alrfe pukjk kfpikersk ie koueomº lb lb (0.0 0.0)) pere ki aubomlbei (0.? (0.?)) Jkserrliiebjl sk iikne quk7 (t + c + c0 )0 x(t) 6 + c5 0
º JK K[IKP7 Ki oesl jk vermes verme`iks IE KO[EOMLB
KBOLBZPEP KI KTZPKFL JKI A[BOMLBEI
ψ/0 ψ/ 0
G (x5 , x0 ) 6
8
s.e
x5 sliuomº lb lb
x˝ 05 +
x˝ 00 +
0x5 x0 jt
x5 (8) 6 8 x0 (8) 6 8 ψ ψ 6 5 x0 68 0 0
A ( A (t, x5 , x0 , x˝ 5 , x˝ 0 ) 6 x˝ 05 + x˝ 00 + 0x5 x0 pere x5 A x 6 0x0 A x 6 0x5 ∀ A 6 0 x x 5 ∀t 0x5 6 0x0 x5 6 x 0 5
5
5
←
pere x0 A x 6 0x5 A x 6 0x0 ∀ A 6 0 x x 0 ∀t 0x0 6 0x5 x0 6 x 5 0
0
0
←
(?.5)
º JK K[IKP7 Ki oesl jk vermes verme`iks IE KO[EOMLB
∀ (x 6 x 0 ) ∀t 5
← x5(:) 6 x0
_ustmtuykbjl kb x0 6 x 5 (:)
x5 6 x5
(:)
x5
∺
mfpimoe ube verme`ik jk :l lrjkb, rkslivmkbjl l`tkbkfls ki plimblfml oere ereotkr ot krººĵstmo st moll r: 5 6 8 jki ouei ies reºĵoks slb7
∐
r5,0 6
∐ x5 6 8
µ5
r?,: 6
Xlr il tebtl ie sliuomºlb lb nkbkrei jk ie K.J x5 (t) ks7
L`tkbkfls ube K.J quk slil
∈
x5 (t) 6 c5 kt + c0 k∐t + c? ols(t ols(t) + c + c: smb(t smb(t) Ie sliuomº lb lb nkbkrei jk ie K.J
∈ x0(t) ks7
← x0(t) 6 c5kt + c0k∐t ∐ c? ols(t ols(t) ∐ c: smb(t smb(t)
x0 6 x5
µm
º JK K[IKP7 Ki oesl jk vermes verme`iks IE KO[EOMLB
Epimoebjl ies olbjmomlbks x5 (8) 6 8
← c5 + + c c0 + + c c? 6 8 + c x0 (8) 6 8 ← c5 + c0 ∐ c? 68 ψ/0 0 ψ/0 0 x5 (ψ /0) 6 5 ← c5 kψ/ + c0 k∐ψ/ + c: 6 5 ψ/0 0 ψ/0 0 ∐ c: 6 8 x0 (ψ /0) 6 8 ← c5 kψ/ + c0 k∐ψ/
º JK K[IKP7 Ki oesl jk vermes verme`iks IE KO[EOMLB
Ouejrl7 Kstruoture Ouejrl7 Kstruoture e rkslivkr
c5 5 k 5 k
rlw 5 rlw 0 rlw ? rlw :
ψ
0
ψ
0
c0 5 k∐ 5 k∐ ψ
0
ψ
0
c? 5 8 -5 8
c: 8 5 8 -5
8 5 8 8
sufefls ies oliufbes3
0 + 0k
ψ
0
c5 + 0 + 0k∐
ψ
0
c0 6 5
edlre tlfefls ki rkbniº lb 5 y ? y sufefls ies oliufbes3 lb rlw 5 rlw ?
c5 5 5
c0 5 5
c? 5 -5
c: 8 8
8 8
(?.0)
º JK K[IKP7 Ki oesl jk vermes verme`iks IE KO[EOMLB
0c5 + 0c0 6 8
(?.?)
jk ies koueomlbks (?.0 (?.0)) y (?.? ?.?)) tkbkfls ub smstkfe quk slil mfpimoe c5 y c0
0 + 0k 0k
ψ
0
c5 + 0 + 0k 0 k∐
ψ
c0 6 5
(?.:)
0c5 + 0c0 6 8
(?.;)
0
ψ
fuitmpimoefls7 (5 k ) ie koueomºlb lb (?.; ?.;), ), jkspuºks ks sufefls ki rksuitejl rksui tejl kb ie koueomº lb lb (?.: ?.:)) y jkspkgefls e c0 3
∐
0 + 0k 0k
∐
ψ
0
0
0 + 0k
+ 0 + 0k 0 k∐
c5
ψ
0
∐ c5
0 + 0k 0k
ψ
0 + 0k 0 k∐
0
ψ
0
∐ ψ
0
c0
65
c0
68 0 + 0k
ψ
0
c0
65
º JK K[IKP7 Ki oesl jk vermes verme`iks IE KO[EOMLB
c0 6
5
0 + 0k 0 k∐
ψ
0
∐
0 + 0k
ψ
0
ψ
6
5
0 k∐
ψ
0
∐ k
ψ
0
(?.1)
Edlre fuitmpimoefls7 (5 ∐ k∐ ) ie koueomºlb lb (?.; ?.;), ), jksp jkspuº uºks ks sufe sufefls fls 0
(olfl il dmomfls ebtkrmlrfkbtk) ki rksuitejl kb ie koueomº lb lb (?.: ?.:)) y jkspkgefls e c5 3 c5 6
5
0 + 0k 0k
ψ
0
∐
0 + 0k 0 k∐
ψ
0
5
6
∐ 0 k
ψ
0
k∐
ψ
(?.>)
0
pere c? utmimzefls slil ki rkniºlb lb 5 y ? jki ouejrl (5 (5), fuitmpimoefls plr3 (-5) ki rkbniº lb 5 y sufefls olfl il dkfls vkbmjl deomkbjl y tkbkfls lb quk7
∐ 0c ? 6 8 ← c ? 6 8 .
(?.4)
º JK K[IKP7 Ki oesl jk vermes verme`iks IE KO[EOMLB
Xere c: utimzefls slil ki rkbniºlb lb 0 y : jki ouejrl (5 (5), fuitmpimoefls plr3 (-5) ki rkbniº lb 5, plstkrmlrfkbtk sufefls y tkbkfls quk7 lb
∐ 0c: 6 ∐5 ←
5 c: 6 0
(?.=)
Ambeifkbtk ie sliuomº lb lb pertmouier pere x5 y x0 ks7 x∞ (t) 6 ∐ 5
x∞ (t) 6 0
kt
∐
0 k∐
0 k∐
ψ
0
∐ k
ψ
0
kt ψ
0
∐ k
ψ
0
+
+
k∐t
0 k∐
ψ
0
∐ k
ψ
0
k∐t
0 k∐
ψ
0
∐ k
ψ
0
smb(t smb(t) + 0
∐
smb(t smb(t) 0
Ks l`vml quk ils pesls aukrlb kxtkbsls, kxtkbsls, pkrl auº k slil pere pere ube feyl feylrr olfprkbsmº lb. lb. Jksjk ki ouejrl (5 (5) ki ikotlr pujl pkroetersk olfl sk pljrº jrºĵeb febmpu febmpuier ier ils rkbnil rkb nilbks bks,, ejkfº ejkfºes es kxmstk kxmstkb b bufkrl bufkrlsls sls fºktljls ktl jls pere rkslivkr smstkfes jk koueomlbks.
º JK K[IKP7 Ki oesl jk vermes verme`iks IE KO[EOMLB
KBOLBZPEP KI KTZPKFL JKI A[BOMLBEI
t5
G (x5 , x0 ) 6
t8
x05 +
x˝ 5 ˝x0 + x + x00
jt
sliuomº lb lb
A ( A (t, x5 , x0 , x˝ 5 , x˝ 0 ) 6 x05 + x˝ 5 ˝x0 + x + x00 pere x5 A x 6 0x5 A x 6 x 0 ∀ A 6 x x 0 ∀t
pere x0 A x 6 0x0 A x 6 x 5 ∀ A 6 x x 5 ∀t
5
0
5
0
5
0
x0 6 0x5 ∀ x5 6 0x0 ∀t
←
x5 6 0x0 (:) x5
∗ ∗ 6 0x0 ⇐ ∈ r5,0 6 µ 0 y r?,: 6 µ 0m
(?.58)
º JK K[IKP7 Ki oesl jk vermes verme`iks IE KO[EOMLB
Xlr il tebtl ie sliuomºlb lb nkbkrei ks7
∗
x5 (t) 6 c 5 k
0t
+ c0 k∐
∗
0t
∗
∗
+ c? ols( 0t) + c + c: smb( 0t)
Ie sliuomº lb lb pere x0 ks7 5 x0 6 x5 0
∗
← x0(t) 6 c5k
0t
+ c0 k∐
∗
0t
∗
∐ c? ols(
0t)
∐ c: smb(
∗
0t)
_uſtomkbome
∐
5
G (x) 6
(x0 + x˝ 0 ) jt
(:.5)
8
s.e
x(8) 6 8 x(5) 6 5 sliuomº lb lb
A ( A (t,x,x ) 6 x0 A x 6 0x A x 6 0x ∀ A 6 0 x x ∀t
∐ ∐
∐ ∐ x˝ 0
∐
Kuikr Kquetmlbs
∐0x 6 ∐0x ← x ∐ x 6 8 Nkbkrei _liutmlb7 x(t) 6 c 5 kt + c0 k∐t
_uſtomkbome
olbjmomlbks7 x(8) 6 8 x(5) 6 5
+ c c0 6 8 ← c5 +
← c5k + + c c0 k∐ 5 6 8
(e) (`)
fuitmpimoer plr7
∐k ie Kq. (e (e), suferil olb ie Kq. (` (`) y rkslivkr pere c0 5 c0 6 ∐5 k k
∐
∐k∐5 ie Kq. (e (e), suferil olb ie Kq. (` ( `) y rkslivkr
pere c5 3 fuitmpimoer plr7 pere c5
5 c5 6 k k∐5
∐
_uſtomkbome
x∞ (t) 6 Fetrmz Dkssmebe
kt k∐t + ∐5 ∐ 5 k k k k
∐
D 6 D 5 6
∐
∐ 0 8
{∐0, ∐0} ≪ 8
Ie sliuomº lb lb ks ub Fº exmf ex mfl l rkie rk ietm tmvl vl
8 0
∐
D 0 6 :
≫8
_uſtomkbome
Kgkfpils olb ki tklrkfe jk suſtomkbome
t5
G (x) 6
:tx
t8
_liuomº lb lb Nkbkrei7
x(t) 6
∐
0
∐ x˝
jt
t? + c5 t + c + c0 ?
Fetrmz dkssmebe7
8 D 6 8 D 5 6 8, 0
{ ∐ }≪8
exmf ex mfl l rkie rk ietm tmvl vl Ie sliuomº lb lb ks ub Fº
8 0
∐
D 0 6 8
≫8
(:.0)
_uſtomkbome
t5
G (x) 6
:tx˝
t8
_liuomºlb7 lb7
0
∐ 0x˝
jt
t0 x(t) 6 + c5 t + c + c0 0 Fetrmz dkssmebe7
8 D 6 8 D 5 6 8, :
{ ∐ }≪8
exmf ex mfl l rkie rk ietm tmvl vl Ie sliuomº lb lb ks ub Fº
8 :
∐
D 0 6 8
≫8
(:.?)
_uſtomkbome
∐ t5
G (x) 6
0
0
x + ?x ?xx˝ + 0 x˝
t8
_liuomºlb7 lb7
x(t) 6 c 5 ols
∗ 0 t 0
+ c0 smb
jt
∗ 0 t 0
Fetrmz dkssmebe7 D 6 D 5 6
∐
{∐0, :} ≪ 8
Ie sliuomº lb lb prkskbte pubtl jk smiie
0 ? ? :
D 0 6
∐5> ≪ 8
(:.:)
_uſtomkbome
∐ t5
G (x) 6
x0
x˝ 0 jt
t8
_liuomºlb7 lb7
x(t) 6 c5 ols(t ols(t) + c + c0 smb(t smb(t) Fetrmz dkssmebe7
0 D 6 8 D 5 6 0, 0
{ ∐}
8 0
∐
D 0 6
∐:
(:.;)
_uſtomkbome
t5
G ( G (x) 6
t8
x˝ 05
∐
x˝ 00 +
0x5 x0
∐
0x00
jt
(:.1)
sliuomº lb lb
A ( A (t,x,x ) 6 x˝ 05
∐ x˝ 00 + 0x5x0 ∐ 0x00
Xere x5 A x 6 0x0 A x 6 0x5 ∀ A x 6 0x5 ∀ t 0x5 6 0x0 x5 6 x 0
Xere x0 A x 6 0x5 :x0 A x 6 0x0 ∀ A x 6 0x0 ∀ t 0x0 6 0x5 :x0
5
0
0
5
0
←
sm x5 6 x 0
∺
x 0 6 0x5
∐
5
∐
∐
∐
∐ x5 ∀ x5 6 x 0 ∀t
←
(:)
x5 6 x0
∐ ← x0 6 0x0 ∐ x5
_uſtomkbome
x(:) 6 0x5
∐ x5 ← x(:) ∐ 0x5 + x5 6 8 | ∈ r5 6 r0 6 5 y ∈ r? 6 r: 6 ∐5 x5 (t) 6 c 5 kt + c0 tkt + c? k∐t + c: tk∐t
pere x0 (t) dey quk jkrmver jls vkoks ie sliuomlb jk x5 (t), ks jkomr7 x5 6 x0
0c0 kt + c? k∐t ∐ 0c: k∐t + c: tk∐t ← x0(t) 6 c5kt + c0tkt + 0c
Xere ki oesl jk suſtomkbome jk jls verm`iks olfl sk prkskbtº l kb ki kgkfpil ebtkrmlr vefls e jkſtbmr ies fetrmoks jk sknubjes jkrmvejes.
_uſtomkbome
Ies olbjmomlbks jk suſtomkbome sk kstruoture kb ie fetrmz jk sknubjes jkrmvejes olfl7
∎
A x A x 0 A ( A (t,x, x˝ ) 6 A x˝ A x˝
5
,x5
0
,x5
5
,x5
0
,x5
A x A x A x˝ A x˝
5
,x0
0
,x0
5
,x0
0
,x0
A x A x A x˝ A x˝
5
,x˝ 5
0
,x˝ 5
5
,x˝ 5
0
,x˝ 5
A x A x A x˝ A x˝
5
,x˝ 0
0
,x˝ 0
5
,x˝ 0
0
,x˝ 0
Ils fkblrks jk lrjkb 5 slb7
F 5 6 A x
{
5
,x5 , A x0 ,x0 , A x˝ 5 ,x˝ 5 , A x˝ 0 ,x˝ 0
}
Ils fkblrks jk lrjkb 0 slb7
F 0 6 Jkt
A x A x
5
,x5
0
,x5
A x A x˝
0
,x0
5
,x0
A x A x
5
,x0
0
,x0
A x A x˝
0
,x˝ 5
5
,x˝ 5
A x , A x˝
A x , A x˝
5
,x5
5
,x5
0
,x0
0
,x0
A x A x˝
5
,x˝ 5
5
,x˝ 5
A x A x˝
0
,x˝ 0
0
,x˝ 0
A x , A x˝
5
,x5
0
,x5
A x˝ , A x˝
5
,x˝ 5
0
,x˝ 5
A x A x˝
5
,x˝ 0
0
,x˝ 0
A x˝ A x˝
5
,x˝ 0
0
,x˝ 0
,
_uſtomkbome
Ils fkblrks jk lrjkb ? slb7
F ? 6 J Jkt kt
A x A x A x˝
5
,x5
0
,x5
5
,x5
A x A x˝ A x˝
5
,x5
5
,x5
0
,x5
A x A x A x˝ A x A x˝ A x˝
5
,x0
0
,x0
5
,x0
5
˝5 ,x
5
,x ˝5
5
,x0
[be aubomºlb lb ks olbvkxe sm ks skfmjkſtbmje plsmtmve. Kb kstk oesl tljls tlj ls ils fkblrks jk`kr jk` krººĵeb skr plsmtmvls7 F 5 F 0 F ?
≫ 8 ≫ 8 ≫ 8
A x A x A x˝ A x A x˝ A x˝
5
,x˝ 5
0
,x˝ 5
5
,x˝ 5
5
˝0 ,x
5
,x ˝0
0
˝0 ,x
A x , A x A x˝
A x , A x˝ A x˝
5
,x5
0
,x5
0
,x5
0
,x0
5
,x0
0
,x5
A x A x A x˝ A x A x˝ A x˝
5
,x0
0
,x0
0
,x0
0
˝5 ,x
5
,x ˝5
0
˝5 ,x
A x A x A x˝ A x A x˝ A x˝
5
,x˝ 0
0
,x˝ 0
0
,x˝ 0
0
˝0 ,x
5
,x ˝0
0
˝0 ,x
Kb oesl jk olboevmjej sk skrºĵe7 F 5 F 0 F ?
≪ 8 ≫ 8 ≪ 8
Jk bl oufpimskr bmbnubl jk ils ksqukfes ebtkrmlrks, ki kxtrkfl olbstmtuyk ub pubtl jk smiie
,
_uſtomkbome
Pkslivkr
t5
G (x5 , x0 ) 6
t8
(x˝ 05 + x + x0 + x˝ 5 ˝x0 ) jt
(:.>)
sliuomº lb lb
Ie koueomº lb lb jk kuikr pere x5 ks7
Ies sliuomlbks slb7
0x˝ 5 + x ¨ 0 6 8. Ie koueomº lb lb jk kuikr pere x0 ks7 x ¨ 5 6 5. _k tmkbk ki smstkfe3 0x˝ 5 + x ¨0 6 8 x ¨5 6 5
t0 x5 (t) 6 + c5 t + c + c0 0 t? x0 (t) 6 c5 t0 + c0 t + c + c? ?
∐ ∐
Ies olbjmomlbks jk suſtomkbome sk pukjkb vkrmſtoer olb ies fetrmoks ebtkrmlrfkbtk piebtkejes.
_uſtomkbome
Pkslivkr
t5
G (x, y ) 6
(x0 + x˝ y˝ + y + y 0 ) jt
(:.4)
t8
sliuomº lb lb
Ie koueomº lb lb jk kuikr jk x ks7
Jkrmvebjl7 x7
∀ 0 ie ∀t 0
koueomºlb lb jk kuikr jk
y¨ 6 0x. Ie koueomº lb lb jk kuikr jk y ks7 x ¨ 6 0y. _k tmkbk ki smstkfe3 y¨
∐ 0x 6 8 x ¨ ∐ 0y 6 8
y (:) 6 0x, x ¨, sustmtuykbjl kb ie koueomº lb lb jk kuikr jk y7 y (:) 0y 6 0 ouye sliuomº lb lb ks7
← y(:) ∐ :y 6 8
_uſtomkbome
y (t) 6 c 5 k
∗
0t
+ c0 k∐
∗
0t
+ c? ols
Jkrmvebjl7 y˝ (t) 6
∗
0c 5 k
y¨(t) 6 0c5 k
∗
∗
0t
0t
∐
∗
0c0 k∐
+ 0c 0c0 k∐
∗
0t
∗
0t
∐
∗
∗
0t + c: smb
0t .
∗ ∗ ∗ ∗ ∐ ∗
0c? smb
∐ 0c? ols
∗
0t +
0t
0c: ols
0c: smb
0t
Jk ie koueomº lb lb jk kuikr pere x, rksuitºl quk7 y¨ 6 0x
←
y¨(t) x(t) 6 . 0
Ambeifkbtk ie sliuomº lb lb pere x(t), ks7 x(t) 6 c 5 k
∗
0t
+ c0 k∐
∗
0t
∐ c? ols
∗ ∐ ∗ 0t
c: smb
0t .
Ies olbjmomlbks jk suſtomkbome sk pukjkb vkrmſtoer olb ies fetrmoks ebtkrmlrfkbtk piebtkejes.
0t
Olbjmomlbks jk Zrebsvkrseimjej
Pkslivkr
Z
G (x) 6
(t0 + x˝ 0 ) jt
(;.5)
8
Ouebjl7 5
x(8) 6 :, :, Z 6 0 y x(0) 6 Im`rk 6 Im`rk
0
x(8) 6 :, :, Z 6 Im`rk y x(Z ) Z ) 6 ;
5. Ie 5. Ie sliuomº lb lb nkbkrei ks7 x(t) 6 c 5 t + c + c0 sugktl e x(8) 6 :, :, sk tmkbk quk7 x(t) 6 c5 t + :
(;.0)
Olbjmomlbks jk Zrebsvkrseimjej
Kb kstk oesl sk trete jk ub aubomlbei olb dlrmzlbtk tkfplrei, jki ouei sk epimoe ie smnumkbtk olbjmomº lb7 lb7
∀ A ∀ x˝
plr il tebtl, 0x˝ t60 6 8
|
68
(;.?)
t6Z
← 0(c 0(c5 )|t60 6 8 ⇐ c5 6 8
Ambeifkbtk x∞ (t) 6 : 0. Kb 0. Kb kstk oesl vefls kfpiker ie olbjmomºlb7 lb7
∐ ∐ A
x˝
∀ A ∀ x˝
t6Z
68
(;.:)
Olbjmomlbks jk Zrebsvkrseimjej
sk de l`tkbmjl ebtkrmlrfkbtk quk7 x(t) 6 c 5 t + : kfpikebjl ie kxprksmºlb lb (;.: ;.:), ), sk tmkbk quk7 0
0
t + x˝
smfpimſtoebjl, t Z 0
0
∐ x˝ (0x˝ ) t6Z 6 8 0
∐ x˝
t6Z
68
∐ c50 6 8 ← Z 0 6 c50 ⇐ Z 6 c5
sustmtuykbjl kb (;.0 (;.0)) x(Z ) Z ) 6 Z ( Z (Z ) Z ) + : epimoebjl ie olbjmomº lb lb x(Z ) Z ) 6 ; Z 0 + : 6 ;
← Z 0 6 5 ⇐ Z 6 µ5
Jki ouei slil tlferkfls e Z 6 5. Ambeifkbtk x∞ (t) 6 t + :
Olbjmomlbks jk Zrebsvkrseimjej
Pkslivkr
Z
G (x) 6
(x + x˝ 0 + t) t) jt
(;.;)
8
Ouebjl7 5
x(8) 6 5, 5, Z 6 0 y x(0) 6 Im`rk 6 Im`rk
0
x(8) 6 8, 8, Z 6 Im`rk y x(Z ) Z ) 6 8
5. Ie 5. Ie sliuomº lb lb nkbkrei ks7 t0 x(t) 6 + c5 t + c + c0 : epimoebjl ie olbjmomº lb lb x(8) 6 53 53 t0 x(t) 6 + c5 t + 5, 5, :
(;.1)
Olbjmomlbks jk Zrebsvkrseimjej
Kfpikebjl ie koueomº lb lb (;.? ;.?), ), 0x˝ t6Z 6 8
|
_mfpimſtoebjl3
←0
c5 6
t + c5 0
68
t60
∐5.
Ie sliuomº lb lb ks7 t0 x(t) 6 t + 5 : 0. Epimoebjl 0. Epimoebjl olbjmomº lb lb kb ie koueomºlb lb (;.1 ;.1), ), sk iikne quk
∐
t0 x(t) 6 + c5 t. :
Olbjmomlbks jk Zrebsvkrseimjej
E olbtmbueomº lb sk deok usl jk ie koueomº lb lb lb (;.: ;.:)) 0
∐ ∐ ∐ ∐ ∐ ∐ ∐ ∐
x(t) + x˝ (t) + t + t
x˝ (t)S0˝x(t)V
t6Z
68
x˝ 0 (t)
t6Z
68
x(t) + t + t
x(t) + t + t
t + c5 0
0
t6Z
t0 x(t) + t + t + tc5 + c + c50 : t0 t0 + c5 t + t + t c5 t c50 : : t
68
c50
68
t6Z
68
t6Z t6Z
68
Z 6 c 50
(;.>)
Olbjmomlbks jk Zrebsvkrseimjej
epimoebjl ie olbjmomº lb lb x(Z ) Z ) 6 8,
0 c50
: sk iikne quk
0 + c5 c50
68
←
c5: :
+ c5?
68
c5 6 8 c5 6
∐:,
sustmtuykbjl kb (;.> (;.>)) Z 6 8 Z 6 51 51,, sk tlfe slil Z 6 51 51.. Ambeifkbtk7 t0 ∞ x (t) 6 :
∐ :t
⇐
c5?
c5 +5 :
68
Fexmfmzeomº lb jki `kbkſtoml flblplimste lb
[b flblplimste tmkbk olfl l`gktmvl fexmfmzer sus `kbkſtomls, pere kiil jmsplbk jk ub pljkr jk fkroejl quk ik pkrfmtk ſtger ki prkoml e ie oebtmjej jk prljuotls, bl ef`es. _m jkomjk plr ie ſtgeomºlb lb jk prkomls, kbtlboks pukjk kiknmr kbtrk jmvkrses treykotlrmes treykotlrmes jk prkomls. Einubes E inubes jk kiies eufkbterº eb eb sus mbnrksls prkskbtks pkrl p krl kb ki auturl oekrºeb eb il suſtomkbtk pere olbtrerrksteril, l `mkb pukjk kiknmr prkomls `egls y eufkbter ils mbnrksls prkskbtks pkrl ks plsm`ik quk ils olstls skeb teiks quk l`tkbneb `kbkſtomls buils. Ie jkfebje bl slil jkpkbjk jk ils prkomls smbl jki pkrmljl pkrml jl olrrmkbtk ejkfºes es jki oef`ml kb ils prkomls.
_keb q 6 e
∐ `p( `p(t) + d + d p( p˝ (t)
e, ` 2 8, 8 , d 6 8.
(1.5)
Ils olstls slil jkpkbjkb jk ie oebtmjej prljuomje y plr smfpimſtoer ks ube aubomºlb lb ouej ou ejrº rºetmo et moee O 6 εq 0 + ΰq ΰ q + + ο
ε, ΰ , ο 2 8. 8 .
(1.0)
Fexmfmzeomº lb jki `kbkſtoml flblplimste lb
Ki prl`ikfe quk kbarkbte ki flblplimste ks7
t5
fº ex 6 ex ψ
t8
S pq pq
∐ o(q )V)V jt
p( p(t8 ) 6 p 8
(1.?)
p( p(t5 ) 6 p 5 Kbtlboks, sm7 ψ 6 pq ψ 6 p( p (e
∐o
∐ `p + `p + d d p) p˝ ) ∐ ε(e ∐ `p + `p + d d p) p˝)0 + ΰ (e ∐ `p + `p + d d p) p˝ ) + ο + ο
0εe`p + + 0εed 0 εed p˝ ∐ εd0 p˝ 0 ∐ `p0 + dp p˝ ∐ εe0 + 0εe`p + 0ε`pd 0ε`pd p˝ ∐ εd0 p˝ 0 ∐ eΰ + ΰ + ΰ`p `p ∐ ΰ d p˝ ∐ ο ψ ( p, p) p˝ ) 6 ∐ (εe0 + ΰe ΰ e + ο + ο ) + (e + 0εe` 0 εe` + + ΰ ΰ``) p ∐ `(5 + ε` + ε`)) p0 0 0 (0εe + + ΰ ΰ ) p + p˝ + d d(5 (5 + 0ε` 0ε`)) p p˝ ∐ εd p˝ ∐ d(0εe ψ ( p, p) p˝) 6ep
(1.:)
(1.;)
(1.1)
Fexmfmzeomº lb jki `kbkſtoml flblplimste lb
Ki oefmbl jki prkoml ºlptmfl3 lptmfl3 ∀ψ 6 (e + 0εe` 0 εe` + + ΰ ΰ``) 0` (5 + ε` + ε`)) p + p + d d (5 + 0ε` 0ε`)) p˝ ∀p
∐
κ5
∀t ∀t
κ0
∀ψ 6 d(0εe (0εe + + ΰ ΰ ) + d + d(5 (5 + 0ε` 0ε`)) p ∀ p ˝ ˝ ∀ψ 6 d 6 d (5 + 0ε` 0ε`)) p˝ 0εd0 p¨ ∀ p ˝ ˝
∐
κ?
∐ 0εd0 p˝
∐
κ?
epimoebjl ie koueomº lb lb jk kuikr dκ ? p ˝ jmvmjmkbjl7
5 0εd0
∐ 0εd0 p¨ 6 κ 5 ∐ 0`pκ 0 + dκ + dκ ? p ˝
y rklrjkbebjl ie koueomºlb lb (1.> 1.>), ),
(1.>)
Fexmfmzeomº lb jki `kbkſtoml flblplimste lb
p¨
∐ ∐ `κ ` κ 0 p 6 p 6 0 εd
κ 5 0εd0
κ:
p¨
κ;
∐ κ : p 6 p 6 ∐κ ;
ie sliuomº lb lb dlfl dlflnº nºkbke kbke ks3 ks3
∗
pd (t) 6 c 5 k
κ: t
+ c0 k∐
∗
κ: t
ie sliuomº lb lb bl dlflnºkbke kbke vmkbk jejl plr7 κ κ ; κ ; ∐ εd p 6 p¯ 6 6 6 0`κ ∐κ : κ : εd 5
0
0 0
6
κ 5 . 0`κ 0
Xlr il tebtl ie sliuomºlb lb nkbkrei ks7
∗
p( p(t) 6 c5 k
κ: t
+ c0 k∐
∗
κ: t
+ p¯
Fexmfmzeomº lb jki `kbkſtoml flblplimste lb
∗
p( p(t) 6 c5 k
κ: t
+ c0 k∐
∗
κ: t
+ p¯
(1.4)
epimoebjl ies olbjmomlbks kb ie sliuomº lb lb (1.4 1.4), ), sk iikne quk7
c5 6
p8
∐ p¯ ∐ ( p5∗ ∐ p) p¯)k 5 ∐ k0 κ t : 5
∗
κ: t5
c0 6
p8
∐ p¯ ∐ ( p5 ∐∗ p) p¯)k∐ 5 ∐ k∐0 κ t : 5
∗
κ: t5
.
Kstk kgkfpil auºk kstruoturejl fes smfpimste kb ki im`rl jk Odmebn, Eipde O , pkrl auºk N. O. Kvebs qumkb egustº l kstk fljkil flj kil jmbºefmol. efmol . Xere ube feylr feylr Kvebs, ‖Zdk Jybefmos la mbtkrprkteomº lb lb kolbº lfmoe, lfmoe, rkvmser7 N. O. Kvebs Flblpliy‖ , Efkrmoeb Fetdkfetmoei Flbtdiy, Ak`ruery 5=0:, pp. >>-4?.
