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InverseIntro.pdf

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Kx = y,
9:C# Du
D & T !@ @g`M #cR` } GM@ 125:/p\:U?5 & 1 GM>CDE@ D˜1y
K −1
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D & * N‹ŠC]$D X$cG+@ @zyJ6PT 5G(9:
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9CD G 125: W\+@A
D & G /p#: In1…/PI /p#c
Kf (x) = 0x f (y) dy
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!
kf k = sup |f (x)|
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† 5:!# /p#c
kf − gk .
&5:/]T 125^G1 125c /?ŠCC[G1…/7
D & $/p#H1KDE#+\:/?\R& DE#
K
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GM#%`
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f (x)
g(x)
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GM#%`
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8
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M
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† :5 !#
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x
gε(x) = ε sin 2 ,
ε
T 5:!"
_ 5G(9:
* Ng/
ε > 0.
kgεk = kgε − 0k ≤ ε
ˆ/a J& B G+@ @
D & $@A/e&< 1…/ 125c l!"/ \M#%$•1KDL/?# *
ε, gε(x)
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x
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> ]$/pB &
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K gε
gε = ε
K
WG "/?B l!"/ G]&
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h C#:$(
ε → 0.
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#!DL$(!@zy DE@ @ \R& 12[G1…H& 125c Jtv]$213& /! #:/ZD & y `7G1 G *
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#c/.`MH&
}  GH1 /?\G+@ @zy &JŠG%$R`
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xk = kh, k = 0, 1, . . . , n
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8
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T D˜125 :G%$5
F = F (xk ),
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DEB G1… 125: `M!Du9GH1KDA9: 9GM@ \c&
Fk0 = F 0(xk ), k =
_ i#:/]T "/pB $cG+@ $\+@ \R& 125^G1
1, . . . , n − 1.
Fk+1 − Fk−1
= Fk0 + O(h2 )
2h
- B :G+#R& o/p\%U.5+@zy , G /?\^GM#1KD˜1y :
# / >CDLUPI
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t
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t
c0
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k =
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=
+
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2h
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2
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h
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n
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8
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&5:/]T & $!@A:GW@zy 125^GH1 D˜1 D & #:/^1 T D &‰ 1…/ @A1
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δ.
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k $/p\+m&<+6 125:!" GP" /125cC GMŠ+ŠP"/”G:$!5c& * k #:
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8
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‹
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/ B 1 GM>C/p@ D˜1… >C^DE#:U 12G+#]&JŠC/7‡1…]` G%$"/C&…&‹125: B !B I
>PGM#:H& G+#:` 125^GH1 125c [G1… $(/Z $PDL!#13& Go #c/p#I
#cRU7G1KDu9: $/p#]& 1 G+#H13& * h Co D & G &4$5:!B GH1KDL$ /7
125c G%$•1KDL/?# } D & 1KDEB  '
t
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2
a
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b
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c
8
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"
x1(t)
x2(t)
#0
=
"
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a
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#"
x1(t)
x2(t)
#
+
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i1(t)
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5 D & D & GM# G+B Š+@A /! G , $(/?B ŠG 12B !#1 B /.`eC@ * † C
k #c $(/?\+@ ` DEB G%U DŒ#c B /a" $/pB Š^GP‡12B C#H13&(6 B /7"
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125G1 T /p\M@ ` #:/1 >! "!ŠP"&‰C#H1 G1…]` >Ky GH1… $(/0"I
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8
#!
!
#
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x(t)0 =
Ax(t) + B w(t)
• x(t) = (x1 (t), x2(t))
*
h
i
• y(t) = 1 0
% *
• C
D & 125:
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x2(t)
#
= C x(t)
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8
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G @ @ /!?125CD & :# /1 GH1KDS/p#a6%125c UM/!9:!W#CDE#%U ` y+#^GMB I
LD $J& / 21 5!D & $/pB Š^PG ‡12B C#H1 B /.`eC@:$:GM# >C UZuD 9:C# GR&
x0(t) = Ax(t) + B w(t)
'
% & ' #
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125c DŒ#MŠ+\H1 \M#%$•1KDL/?#
GM#%`
u(t) = B w(t)
\+#:$21KDS/p#
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y(t) = C x(t).
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ŠM\13&(6 $/pB Š+\1… 125: B GH12D '
'
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8 i#:/]T @A]`+UM / 125c DE#MŠ+\13& G+#%` /p\H12Š+\H13& D & &J\ I
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a
−b
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y(t) =
h
1 0
i
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x2 (t)
#
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x1(t)
x2(t)
#
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−(a + c) − s
b
p(s) = a
−b − s
= s2 + (a + b + c)s + bc
T 5CDL$5 5GR& ` D &4$!ˆDŒB DE#^GM#1
(a + b + c)2 − 4cb = (b − c)2 + a2 + 2a(b + c).
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Go " I
† C
A
Š!%GH1…R` Du
G+#:`
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b=c
a = 0,
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† 5+\]&(6 ˆ/a HI
t = 0.
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λ = −c.
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x1(t) = d1 cte−ct + d2e−ct
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x1 ,
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c
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(0) 125:!# T DEB B ]`ZDzI
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x $/pB I
x = ct.
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2
B D l 125: &J\+B /!?& /?\^GP"H& /! "H&•DL`?\G+@˜& (k)
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T :5 !"
GP" 125: ^DLUM!#(9^G+@ \c& /
_ G+@˜&</
λ1 , λ 2
A.
5G(9: 125^GH1
λ1 λ2 = bc
λ1 + λ2 = −(a + b + c)
‰& / 125^GH1 >C/125
GP" #:/p#I ŠC/C&•D˜1KDA9: * † 5M\R&(6
λ ,λ
ˆ/p\MZ/?\12ŠM\13&&J5c/p1\+@ ` 2&J\ $( 1…/ `M21…CWB DE#:
d , λ1 , d12
GM#%`
D & `M21…!WB DE#cR` DE# ŠP11
DE#%$PD I
† 5+\R&
λ .
x (t)
ŠM@A GM#%2` G+@ @M \+‡125cC1 /p\H12Š+\13& G%`M` #c/125CDE#:U 1…/ /p\M
i#:/]T @A]`+UM >+\H1 $/p# WB G1KDS/p# /!Z125c ˆ/aWB /!
x (t).
† 5CD & DE#/7WB GH1KDS/p# $(/?B H& $!@A/e&‰ 1…/ `e1…!WB DŒ#!1DE#%U
125c $(/ $DS!#13& /
DŒ# 9eDS2T /! 125: GM>C/!9: 1WT /
A
/?\G1KDS/p#]& * h /]T 9:CV6!125c&‰ /?\G1KDS/p#]& Go DE#]&J\(oI
~$DS!#1 1…/ `M21…CWB DE#: GM@ @~125M"( /
† 5cCo•I
a, b, c.
ˆ/ao+6P125CD & &3yR& 1…!B D & GR& D˜1 & 1 G+#:`C& 4
' #
' '
4} ): 8 ŒD #!D˜1KDwGM@7$/p#:`ZD˜1KDS/p#]& GP" G 9G!DE@GM>+@AM6 \R&‰ 21 5:!B
G]&‹125: ;& 1 1WT / `aG1 G ŠC/0DE#H13& * † 5:!# &jG+B Š+@A 21 5+"(
B /7" /p\H21 Š+\13& * x & GM>C/!9:M6pT c$ G+# `M21…CWB DE:# x1 (t) = d11 eλ1 t + d12 eλ2 t
ƒ
Š+@ DL$D˜12@zy * s uD tvCoC#H1KDwGH1… 21 5!D & <& /?@ \1KDS/p# /7 x (t)
1
GM#%` J9^G+@ \G1… HG 1
1…/ p
/ >H1 GCED #
t=0
d11 λ1 +d12λ2 = x01(0) = −(a+c)x1(0)+bx2(0)
/ \^G1KDS/?# T /p\+@ ` #c/1 G%`M` #cT
† 5: /p#M@ y 1KDEB 125!D & .
ED #/7WB GH1KDS/p# D & T
lHC"/ /a / &jGMB r‹^D˜125:!‹/!v125c&<
$/p#:`ZD˜1KDS/p#]& * † 5CD &
125!D & $/p#1… (1H6.125:
5cC# >C/^125
G+#:`
GP"
x1(0)
x2(0)
B G%U.#CD˜12\%`e GM#%` /pŠ+Š!/e&•D˜1… & DU.# *
$(/?#%`ZD˜1KDS/p#]& UZDu9% Š+5(yR&•DL$cG+@aDE#!Du1KD GM@
T DE@ @a&</p@z9: 125c ŠP"/p>M@ !B GM#%` DE#
&3yR& 1…CB D & *
/ \12ŠM\13& T D˜125 DE#+ŠM\1 \+#:$21KDS/p#
}nfa NG+B Š+@A 125c ?
x # +
G Š+ŠP"/?ŠD GH1… DŒ#MŠ+\H1H6 * U * 6
u(t) = [u2(t)].
/7
0,
/7 0 ≤ t < 1
u2(t) =
1,
1≤t<∞
} /7 B /e&‚1 DE#MŠ+\13&6 ":G+@ @zy  T DE@ @”yCDSC@ ` &J\ P$ DSC#H1 ED #JI
ˆ/aŽB HG 1KDL/?# 1…/ DL`M!#1KDuy 125: &3yR& 1…CB *
(
4
x # –7ŠC/?#:!#1KD G+@ - "/]T 125 F /.`eC@
N \MŠ+ŠC/C&< G C
‹
Š /pŠM\+@LG1KDS/p# \+#:$21KDS/p#
$cG+# >C B /.`PI
u(t)
!@AR` >Ky 125c ƒŠC/p#cC#H1KD G+@”U."/RT 125 @LGjT
du
= ru
dt
T 5:!"
'
r = r(t)
%
D & G #c/p#I‚$(/?#R& 1 GM#1 U."/RT 125 [G1… *
&(' #
- Du9:C# 125c B /.`eC@E6eD * * 6U?"/]T 125
[G1… \+#:$21KDS/?#
6!$(/?B ŠM\1… 125: ŠC/pŠM\+@LG1KDS/p# U?"/]T 125
r(t)
\+#:$21KDS/p#
u(t).
'
1KDS/?#
'
$&(' #
- Du9%!# 125: ŠC/pŠM\+@LG1KDS/p# \M#%$KI
$/pB Š+\H1… 125: U."/]T 125
G1… \+#:$21KDS/?#
8ˆ# 125!D & $cGR&<M6?125: DE#(9:Cm&< Š"/?>+@A!B
&<]!B & :GR&3y”'
u(t),
r(t).
r(t) =
du/dt
d
=
ln u(t).
u
dt
4
_
N / G+@ @‹T 5G(9: …1 / `M/ D & 1 GMij 125c #G12\MG+@ A@ /ƒU
g
/ 125c `7G1 G G+#:` ` DAtv!"!#1KD G1… 125G1 \+#:$21KDS/?# } /7
` Dut!"!#1KD G1… G+#%` Z
` Du9CDL`M > y 125c \+#:$21KDS/p# *Œ
5G1 $/p\M@ ` >C &•DEB Š+@A!V6‹DLU?5H1
_
o/p#%U  /?\ i^#:/]T T 5G1 125: $:GH1‰$!5 D & "/?B
q”]$212\M" ) ' #+\+B !DL$:GM@ `ZDutv!"C#H1KD G1KDS/p# / `7G1 G
T D˜125 CŽ"/am& DE# D˜1 D & B \%$5 B /ao `ZD $\+@ 1 125G+#
D˜1 &</p\M#%`!&(6 GM#%` 1 G+i:DŒ#:U 125: @A/+U / 125: `aGH1 G ;&‚1
$/p\M@ ` B GMij 125CDE#%U!& J9:C# T /a;&‰ * 8 pT 5^G(9: 1KDEB M6
T 0@ @?5^G9% &</?B \+# T D˜125 125!D & G+B Š+@A @LG1…! > y
T GKy / 125c F x † q”x Y - — 8 r \+B `M!Du9 *
4
'
' % ' $
'
%
&(! ' &
† :5 ` DE"R$•1 Šo/p>+@A!B 5:!" D & G B /.`M!@/7 G ŠC/pŠI
\M@LG1KDS/p# `eC#]&•D˜1y \M#%$•1KDS/p#
T 5:!"
v = v(x, t)
t≥0
D & 1KDEB G+#:`
D & G &4$cG+@AR` &•D l $@G]&…&
x, 0 ≤ x ≤ 1
9^GPD G+>+@A * † 5!DŒ#Mi /!
G]& G i:DŒ#:` /! ŠG[G+B 1…!
x
125G1 /.\^G+#H1KD ~& &</pB $!5GG:$21…!D & 1KDL$ /! 125: ŠC/pŠI
\M@LG1KDS/p# * /7 G+B Š+@AM6 $(/?\+@ ` >C G%Ue "!B CB I
x
>!C 125G1 T 5^G9% &4$cG+@AR`
&</ 125^GH1 DŒ# &4$:GM@AR`
x
#M\+B >C!;&
D & 125: G%UM / G #cT >C/7W# G+#:`
D&
0
1
125c @LG"UMH& 1 G:UM G:$!5!DSJ9:R` > y GM#(y B !B >C! / 125:
Š!/pŠ+\M@LG%$ * x #:/^125:!0 G+B Š+@A D & 125G1 $/p\M@ ` "!ŠJI
x
o&<!#1 &4$cG+@A]` &•D l * _ 5^G1…9:Cp125c $5^GPG%$•1…!D & 1KDL$
/ 125!D & Š^GPGMB 21…Cn6 T GR& &J\+B 125^GH1 B CB >C!;&
/ 125: ŠC/?Š+\M@G:$( Š"/+U?"H&…& & 1…%G:`ZDE@zy DE# 1KDEB "/?B
$@LGR&…& &•D l
1…/ $!@LG]&…& &•D l
GM#%` G+@A/?#%U 125: T GKy
0
1
U Du9% >!DŒ 125 1…/ #cT >C/aŽ#R& /! $!@LGR& & & D lH
G:$]$/7 `ZDE#%U
0
1…/ &</?B $@LGR&…& &•D l >CDE‡125 G1… \+#:$21KDS/p# *
4 † 5: >^G]&•DL$ Š^GPG+B 1…! \M#%$•1KDS/p#R& /7 125CD & B /.`eC@Œ'
D &v125: Š!/pŠ+\M@LG1KDS/p# `M!#R&•D˜1y / vB !
B >C!;&
/ 125c &JŠC]$DSH& T D˜125 $!@LG]&…& & D lH
G 1 1KŒD B • v(x, t)
x
• g(x)
• µ(x)
• p(x)
• ψ(x)
x
1KDEB t.
D & G &•D l `M!ŠC!#%`M!#1 U?"/]T 125 G1… *
D & G &•D l `M!ŠCC#:`M!#1 B /7‡1 G+@ D˜1y GH1… *
D & G &•D l `M!ŠC!#%`eC#H1 >CDE‡125 GH1… *
D & 125c DE#!Du1KD GM@!ŠC/pŠM\+@LG1KDS/p# `ZD & 12DE>+\H1KDS/p# GH1
t = 0.
>^GM@LG+#%$ @LGKT
G+#G+@zyc& D & yCDL!@ `C& B /.`eC@!
/.\^G1KDS/?#R&]'
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
4
'
- Du9:C#
G #:`
+
$/pB Š+\H1… 125: Š!/pŠ+\M@LG1KDS/p# `M!#R& Du1Wy \+#:$21KDS/?#
&' #
g(x), µ(x), p(x)
Ψ(x),
v(x, t).
'
'
$&(' #
- Du9:C# GM@ @ >+\H1 /?#: / 125:
+
\ #:$21KDS/p#]&
GM#%`
GM#%` &J\CD˜1 I
g(x), µ(x), p(x)
Ψ(x),
GM>+@A `7G1 G &jGMB ŠM@wDE#:U /!
$(/?B ŠM\…1 125c " I
v(x, t),
B GCDE#!DŒ#:U Š^GPG+B 1…! \M#%$•1KDS/p# *
5 D & D & /p#:UM/0DE#:U "&‰%GP $5 /0DE#12@zy T D˜125 x @ G+#%`7G
† C
s !#R&•DS/!9 * h !" D & G &jGMB ŠM@wDE#:U /! T ^
5 GH1 T 0 9: `ZD & I
$/!9:!"R` &</ GP * ]UaGP `ZDE#:U 125c ` DE"R$•1 ŠP"/?>+@ACB '
)ƒ* – 9:!# 125: ` DE"R$•1 Š "/?>+@A!B D & #c/p#H12Du9eD G+@ * k #:
5GR& 1…/ `M/ G @wD˜1 12@A T /7Wi 1…/ J& 5c/RT D˜1 D & G T !
@ @I
ŠC/C&<]` ŠP"/p>M@ACB } D˜1 D & *E
fO* r \MB !DL$cG+@C$/pB Š+\H1 G1KDS/p# / 21 5: ` DŒoR$•1 ŠP"/p>I
@A!B D & GM@u&‰/ #c/p#12ˆDA9CD G+@E6 >M\1 T 5^G 9% GM# G+ŠJI
ŠG"!#12@zy `eR$C#H1 B 125c/.` *
]UaGP ` DŒ#:U DE#9%!;&< ŠP"/p>M@ !B &]'
D & G $(/?#R& 1 GM#1H6
)ƒ* N‹\MŠ+ŠC/C&<
g(x) = g
T = 1/g
GM#%` GM@ @aŠGGMB 21…!;& P$(!Š1
GP" U Du9%!# * — #JI
µ
`eC &</?B o%G]&</p#G+>+@A o& 12ˆD$•1KDS/p#]&(6
D&
µ(x)
\M#CD /?\cC@zy `M21…!WB DE#cR` >Ky 125: `7G1 G
v(1, t), 0 ≤
G+#:` Q /p@ 1…!WG DE#1…]U?GM@a
/.\^GH1KDL/?#
t≤T
Z x
1
µ(s) ds = g ln

1−x
u(1, g )



φ(x)
, 0 ≤ x ≤ 1.
D & G $(/?#R& 1 GM#1 +
G #:` G+@ @?Š^GI
fO* N‹\MŠ+ŠC/C&<
g(x) = g
GMB 21…!;& P$(!Š1
GP" UZDu9:C# * ˆ8 # UM!#:!G+@E6
q(x)
$cG+#+#c/1 >! \M#CD /?\cC@zy `M21…CŽB DE#:] ` >Ky q(x)
B %G]&J\+oCB C#H13& /!
v(x, t).
78

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