lectures of pde
DESCRIPTION
Partial Differential Equations from Technion - Israel Institute of TechnologyTRANSCRIPT
-
4102 232401 " "
~ 1 ~
01.6131.
= u u u y x Fy x0 , , , ,) (
-
= +x m uy uy x, 2 2) ( .1
m
: =y h y u,0) ( ) ( =u c ux0 .2
=e y k y x ux c ,0 ) ( ) (
= =y h y k y u,0) ( ) ( ) (
=e y h y x ux c ,0 ) ( ) (
=x x u4 0,) (
= =x e k x ux c4 0 0,0 ) ( ) (
:
= xe ux c 7 0,0 ) (
= k7 0) (
. - = k7 0) (
?
: , y x u,) (
y x u,) ( = u y x F0 , ,) (
-
4102 232401 " "
~ 2 ~
:
: =s s0
s t u s t y s t x, , , , , ,0 0 0) () ( ) ( ) (
= +u y x c u u y x b u u y x ay x, , , , , ,) ( ) ( ) (
+ = +u y x c y x c u y x b u y x ay x, , , ,1 0) ( ) ( ) ( ) (
( citsireatcarahC)
. ,
+ = +u y x c y x c u y x b u y x ay x, , , ,1 0) ( ) ( ) ( ) (
= + +c u c ub uay x0 1 0) (
:
y x , ,0 1 , , 1 0) ( ) ( N
+= c u c b au u
. +c u c b a, ,1 0) (
+c u c b a, ,1 0) ( :
:
1 0
t
t
t
a x
b y
c u c u
=
=+ =
.
:
I s s u s y s x; , , :0 0 0) () ( ) ( ) (
-
4102 232401 " "
~ 3 ~
.
I s s u s y s x; , , :0 0 0) () ( ) ( ) (:
= = =s u s u s y s y s x s x,0 ; ,0 ; ,00 0 0) ( ) ( ) ( ) ( ) ( ) ( =s s t,0 ,) ( ) (
2 ) (
2 :melborP
0, :snoitidnoC lanoitiddA
u uy x
x x u
= +
=
.
-
= s s u y x,0 , , , :2) ( ) (
-
) ( ) () ( ) () ( ) (
1
2
3
1,
, 1
2 , 2
t
t
t
xs f t s t x
s f t s t y y
s f t s t u u
=+ =
+ = = + = =
) ( ) () ( ) (2 3) ( ) (
1
2
,0
0 ,0
,0
s s f s x
s f s y
s s f s u
= =
= =
= =
+ + =s t t s t s t u s t y s t x2 , , , , , , ,2) ( ) () ( ) ( ) (
,
22) ( y t s t x
y x y uy x s t y
= + = + =
= =
= cu u b ay x1 , ,0 ,,) ( ) (?
) () () (
, ,
, ,
, ,
t
t
t
u y x a x
u y x b y
u y x c u
=
==
? ,
.1
-
4102
" "
-
232401 " "
~ 4 ~
) () () (
, ,
, ,
, ,
t
t
t
u y x a x
u y x b y
u y x c u
=
==
.
.
s t u s t y s t x, , , , , , () ( ) ( ) ( ""
) () (
) () (
, ,
, ,
y x s s s t x x
y x t t s t y y
= =
= =
) () (
,0
,
t t
s s
y xy xJ
y x s t
= =
( )
,
""
) ( ) (
0 0
0 0
, ,
s s
s s
y x b y x axb ya J
y x = =
?
byx a = =
t y t x, (( ) (
s y s x,0 0) () ( ) (:
s t u s t y s t x, , , , , ,)) .2
J0
:
?= J0
0
0 00
s
s ss
yxb ya
x a= =
t y t x,)) (:
-
4102 232401 " "
~ 5 ~
31.01.02
:
:
= +u y x c u u y x b u u y x ay x, , , , , ,) ( ) ( ) ( .1 : .2
(yhcuaC)
:
s t u s t y s t x, , , , , ,) () ( ) ( ) ( -
s: t:
.3t
t
t
a x
b y
c u
=
==
.4
) () ( ) ( ) () ( ) () ( ) () ( ) (
0 0 0
0
0
0
, ,
,0
,0
0
:
,
s u s y s x
s x s x
s y s y
s u s u
=
=
=
:
.5
) ( ) (
) ( ) (
0 0
0 0
0
0
t t
s s
s s
y xJ
y x
b aJ
y x
=
=
:1
:
) ( ) ( 1
:snoitidnoC lanoitiddA,0
ux
y g y u
==
:
-
4102 232401 " "
~ 6 ~
1
0
1
t
t
t
x
y
u
=
==
( )
s g s ,0 :,) () (
) () () ( ) (
,
,
,
t s t x
s s t y
s g t s t u
=
=+ =
= =y s x t,
+ =y g x y x u,) ( ) ( :
.
:2
: ,
) ( ) ( 1
:snoitidnoC lanoitiddA0,
ux
x h x u
==
:
1
0
1
t
t
t
x
y
u
=
==
(: )
s h s0, :,) () (
!
) () () ( ) (
,
0 ,
,
s t s t x
s t y
s g t s t u
+ =
=+ =
:
0 1
00 1
J
= =
-
4102 232401 " "
~ 7 ~
!
,
+ = =x x h x u3102 0,) ( ) (
:
) () () (
,
0 ,
3102 3102 ,
s t s t x
s t y
x s t s t u
+ =
=+ = + + =
3
:
= + +u u uy x1)1(
2
nis
0 ;
x u
x x x y
= > + =
)1(
= +u u uy x1
+ s s s snis , :, 2) (
:
1 10
0 22 1 1
s
s Js
>
= += =+
) () () (
2
,0
,0
nis ,0
s s x
s s s y
s s u
=
+ =
=
1
1
1
t
t
t
x
y
u u
== =
)2(
) () () ( ) (
2
,
,
t 1 nis 1 ,
s t s t x
s s t s t y
e s s t u
+ =
+ + = + =
)2(
-
4102 232401 " "
~ 8 ~
) (
) () ( ) ( ) (
12
11 2
2
212
1 nis 1 ,x y x
s x y
x y se x y y x u
x y x t
+
+ =
+ = = =
? ?
y x s,) ( "" "
) ( 1 = =C x y s2
+ =C x y2
01.32
= + u ux uyy x)1(
=x x u0,) ( ) ()2(
)3( t
t
t
y x
x y
u u
=
==
,
-
4102 232401 " "
~ 9 ~
s s,0, :) () (
0 0) ( ) (
5 05
0 1
x yJ
s y s x
= = =
= J0 0) ( J0 s0
)3(
)4(
) () () ( ) (
,0
0 ,0
,0
s s x
s y
s s u
=
=
=
yt xt
= =x y xt tt
+x xtt
+ =t s f t s f xnis soc2 1) ( ) (
)4(
) () (
1
0 2
s s f
s f
=
=
)5(
) (
soc
nist
t s x
t s y
e s u
== =
s t,
)6(
2 2
nat1
y x s
yt
x
+ =
=
( , ? )
)5( )6(
-
4102 232401 " "
~ 01 ~
) ( ) ( t1
2na
,2xe xy y x uy
+ =
= + +y x uy u u yy x ) ()1(
+ =x x u1 1,) ()2(
)3( t
t
t
u y x
y y
y x u
+ =
= =
+ s s1,1, :) (
:
1 2
0 10 1 0 1
s y u yJ
+ + = = =
)3(
=e t s yt ,) ()4(
ut xt
t 2 1 ,) ( ) (] [ ) ()5( t
+ = + + = +e s t s u x u x u x
)5( )4(
,
x0
-
4102 232401 " "
~ 11 ~
+ + =x e s e xt tt 2 1) (
+ =e e s t s xtt 1 ,) ( ) ()6(
+ =e es t s ut t ,) ()7(
y x
= + = e es e e es e y xt t tt t t
+ = + + = e y x e e e es ut tt t t2
y x y x u,2 ) ( y
+ =
? , y0
" " . ,
.
= +u y x c u u y x b u u y x ay x, , , , , ,) ( ) ( ) ()1(
.
. !
-
4102 232401 " "
~ 21 ~
u y x, ,) ( u y x c u y x b u y x a, , , , , , , ,) ( ) ( ) ( .
, u y x, ,) ( J0 .1 y x u,) (
.2 .
y x u,) (
)2(
) () () (
,
,
,
t
t
t
a s t x
b s t y
c s t u
=
==
)3(
) ( ) () ( ) () ( ) (
0
0
0
,0
,0
,0
s x s x
s y s y
s u s u
=
==
s t u s t y s t x, , , , ,) () ( ) ( ) (
Ju y x0 , ,) ( ,
( )) () (
) () (
, ,
, ,
y x t t s t x x
y x s s s t y y
= =
= =
u
=y x s y x t u y x u, , , ,) () ( ) ( ) (
:
) () ( ) (
) (
t xx s x , , ,f
s ff t f y x s y x t fx
x h
= =
+
[... , x h) ( ]
y x, u y x u,) ( :
= =y x s y y x t x, , ,) ( ) (
-
4102 232401 " "
~ 31 ~
)1( u :
?
= +c ub uay x
+ + + = + + + = +sb sa u tb ta u s u t u b s u t u a ub uay x s y x t y s y t x s x t y x) ( ) ( ) ( ) ()4(
=t h [ ]
+ = =y t x t tt y t x t1
+ =tb tay x 1)5(
=s h
+ = =y s x s st y t x t0
+ =sb say x 0)6(
, )1( )6(,)5(,)4(
= = += +c u t u ub uat xt y0
...
?
:
:
.
, ,
=u y x lt t t, , 1) ()1(
= u y x l, ,0 0 0 2) ()2( !( ) l l,2 1
-
4102
.
232401 " "
~ 41 ~
=N N N N, ,3 2 1) ()3(
= l Nn0
3 2 1)4(
3 0 2 0 1 0
0
0N u N y N xt t t
N u N y N x
= + += + +
( )
= x xy yt t,,0 0) ( ) (
3 2 0 1 0
3 0 2 0 1 0
0
0N u N y N xt
N u N y N x
= + + = + +
= u ut0
= u y x u y xt t t, , , ,0 0 0) ( ) ( !
...
'
\
N
0
= J0
)4(
!
-
4102 232401 " "
~ 51 ~
s u s y s x, :,0 0 0) () ( ) ( ) (
= +u y x c u u y x b u u y x ay x, , , , , ,) ( ) ( ) (
= +u uy x1)5(
=x x x u,) ()6(
=s s s, ,) (
= = =u y xt t t1 ,1 ,1
...
:1 1
0 11 1
= = J
?
:
= lrahc1,1,1) (
= l1,1,1) (
.
)6(
\
x f) (
-
4102 232401 " "
~ 61 ~
= =f x f x u0 0 , 0,) ( ) ( ) ()7(
=s f s,0,) () (
)5( )7(
) (
t s x
t y
s f t u
+ =
=+ =
= =y x s y t,
+ =y x f y y x u,) ( ) (
x x,) (
= + =x f x x x u0 ,) ( ) (
' , :
-
4102 232401 " "
~ 71 ~
31.01.72
= +up uy uxy x
. p
y x, .1
= +y x1 2 2 = y x u1 ,) ( =p4 .2 . =x x u0,2 ) ( =p2 .3
.1
) () () (
t
t
t
t
p
t
t
e s f x x x
e s g y y y
e s h u up u
= =
= == =
y x s y x t, , ,) ( ) (
=tsnoc y x s. ,) (
) () (
.xs f
tsnocs g y =
=xc y
.2
s s1, nis, soc :) (
nis soc
0 1soc nis
s sJ
s s = =
-
) ( ) () ( ) (4 ) (
soc ,
nis ,
,
t
t
t
e s s t x
e s s t y
e s t u
=
=
=
= +e xyt2 2 2
-
4102 232401 " "
~ 81 ~
+ =y x y x u,2 22) ( ) (
.3
s s,0, :2) (
= +u uy uxy x2)1(
! ) () (
1
2
2
2 2
,
,
x y x u
y x y x u
=
+ =
,
0
00 1
s = J
( ???) )1(
= +u y x b u y x ay x0 , ,) ( ) ()2(
. b a, )2(
) () (
,
,
t
t
y x a x
y x b y
=
=
=C y x F,) (
.)2( =y x F y x u, ,) ( ) (
=t y t x F t u t x u, ,) () ( ) ( ) () ( ) (
+ = + = = =ub ua y F x F F uy x t y t x t t 0: F
= +u uy x0)3(
-
4102 232401 " "
~ 91 ~
:
1
11
t
t
yd x
yxd
== =
= + =tsnoc x y c x y.
) ()4(
,
v
x y F y x u
)2( )4(
0
v
x
v
y
x
v
v
y
vF
xu u
v
u
Fy
F
F u
= = + =
=
=
!)3(
= + + + + + =g uf ue ud uc ub ua u Ly x yy yx xx2 ] [)1(
y x g b a, , , ,) (
=y x g u L,) ( ] [ :
L
+=+ L Lu L u u u2 1 2 1] [ ] [ ] [
: 3 (1) "
L
+ + =uc ub ua u Lyy yx xx2 0] [)2(
bc La 2 ) ( :
-
4102 232401 " "
~ 02 ~
y x,
> 0 = 0 < 0
:
= u uyy xx0 2
:
=uD uyy x (>D0 )
:
= +u uyy xx0 ecalpaL
, " " : =[
!
) () ( ) ( ) (
) ( ) () ( ) ( ) (
, , ,
, , , ,
,
,
y x
y x u
y x y x
yw ux y x
= =
31.01.03
?
y x y x, , ,) ( ) (
0 y x
y x
J
=
:
-
" " 104232 2014
~ 21 ~
( ) ( ), , ,x y x y . ( ) ( ),, ,x y )
( ) ( ),,u x y w (
( ) ( ) ( )( ), , ,,u x y w x y x y =
u
xx xu w w +=
( ) ( )x x x xx x x x x xxxu w w w w ww + + += + +
(3)( ) ( )
( )
2 22
U
xx
xy
x x x x xx xx
U
x y x y y x x y
u w w w
u w w
w w
w
+ + +
+ +
= +
= + +
upcurlybracketleftupcurlybracketmidupcurlybracketright
] [
U " " ) (2)(
(3) (1) ( ),w
[ ] 2l w Aw Bw Cw = + +
(4)
( )( ) ( )( )
2 2
2 2
2
2
,
,
,
x x y y
x x x y y x y y
x x y y
A a b
B a b c
C b
c
ca
+
+ +
=
+ +
+
= +
=
(4)
(5)x y x y
x y x y
A B a b
B C b c
=
( )
( )
2
2
L b
l B
a bac
b c
A BAC
B C
= =
=
=
-
4102 232401 " "
~ 22 ~
= J L l2 ) ( ) (
= ngis ngisL l) ( ) (
!!
= w w w H w, , , ,) ( = w w w H w, , , ,) ( = w w w H w, , , ,) (
.w , > 0 )1(
= =C A0 y x y x, , ,) () ( ) (
)6( = + =+ b a Acy y x x0 22 2
= + =+ b a Ccy y x x0 22 2
= Jx y y x0 :
. " )6(
=A0
= + + + ca b b a ca b b ay x y x0 2 2) () ( ) () ()7( a0
()
=C0 =A0 )7( y x,) ( y x,) (
)8( ) () (
2
2
0
0
y x
y x
ca b b a
ca b b a
+ +
=
= +
( - + )
= + + ca b b ay x0 2) (
-
4102 232401 " "
~ 32 ~
2
2
t
t
ca b b yd a xca b b ya xd
= + = + =
=k y x F,) (()
""
)9( () c a,
) (0
0
,
; 0yy xx
u
uc ua
a
y x
c
= +
< >
> = ca0
)9(
:()8( y x y x, , ,) ( ) ( ) )9(
2 ) a )9( ca
(
)01( = u uyy xx0 2
)8( ,
= + xy0)11(
= xy0)21(
: )11(
1
t
t
yd x
yxd
=
= =
= no yts c x.
= x y y x,) ()31(
: )21(
1
t
t
yd x
yxd
=
= =
+= no yts c x.
+ = x y y x,) ()41(
-
4102 232401 " "
~ 42 ~
a) ( )31( )41(
= x wy y y x ux, ,, ,) () ( ) ( ) (
=+ = + w uw w wx xx
=+ = + w w uw wy yy
) ( ) (
) ( ) (w w xx xx x x
w w w w
w w u
+ + +
+ +
=
+
=
=
[YY ]
, :
,
x t x u,) (
)51( = u c uxx tt0 2
: c
:
)51(
G F, + + =tc x G tc x F t x u,) ( ) ( ) (
!!!)01( )51(
()41(,)31( )
tc x
tc x
=+ =
w t x u,,) ( ) (
= w0 ) ()61(
= U w) ( :
[ ]
-
4102 232401 " "
~ 52 ~
...
( )31.11.3
: . t x u) , (: ,
. c , ( ) = u c uxx tt0 2
G F, , + + =tc x G tc x F t x u) ( ) ( ) , ( :
.
, . : tc x
tc x
=
+ = . = w0 ,
+ = + = G F G du w) ( ) ( ) ( ) , ( . - w
)*( + + =tc x G tc x F t x u) ( ) ( ) , (
G F,- . s F s G) ( ,) (
G F, )*( . ,
"(. " )"" ,
, F- . +tc x G) (- tc x F) (: 2 )*(
. +tc x G) ( , = +x tc x0 . tc x F) (, . = x tc x0
.
( )
(1) = t x u c uxx tt0 , ,0 2
(2)) ( )0, (
t) ( )0, (
x f x u
x g x u
=
= -
( 2(+)1) ". " g- f ( 2(+)1)-
(.2)-)*( )*(. u- .
) ( ) ( ) (
) ( ) ( ) (
x f x G x F
x g x Gc x Fc
= += +
(3)
: )b3(
0
1) ( ) ( ) (
1) ( ) ( ) (
x
x g x F x Gc
C sd s g x F x Gc
=
+ = )c3(
-
4102 232401 " "
~ 62 ~
:)c3(+)a3(0
1 1) ( ) ( ) (
2 2 2
C xx f sd s g x G
c + + =
:)c3(-)a3(0
1 1) ( ) ( ) (
2 2 2
C xx f sd s g x F
c + + =
)*(.- .
) ( ) ( ) ( ) , (1 1] [2 2
tc x
tc x
sd s g tc x f tc x f t x uc
+
)**( + + + =
.
:
.0 4
1 )0, ( ,0 )0, (
xx tt
t
u u
x u x u
=
= = =
)**(:- . = c2,
] [2
2
1 1) 2 ( ) 2 ( 1 ) , (
4 4
t x
t x
t t x t x sd t x u+
= + = =
. ,
.
0 9
2 ,1)0, (
2 ,0
2 ,1)0, (
2 ,0
xx tt
t
u u
xx u
x
xx u
x
=
=
>
> =
? ) , ( mil0 , u)6 /1,0( - t
t x u
=
. . = c3
) (1 1
2 2
1 12 2
2
0
2
7 1 1 1 1 1 1 11 1 1 ) ( ) ( ) ( ) ,0(
6 6 2 6 2 2 2 6
2 1 11 ) ( ) , ( mil
t3 6 2
sd sd s g f f u
sd sd s g t x uc
= + + = + + =
= = =
31.11.6
[ ]
-
4102 232401 " "
~ 72 ~
= u uxx tt0 4
t0, <
= =x g x ut0 0 ,) ( ) (
) (
) (3,
,14
x u
x u
1 2
1 2
t x
t x
= = +
[ , ]
-
4102 232401 " "
~ 82 ~
31.11.01
)1( = u c uxx tt0 2
)2( ) ( ) () ( ) (
0,
0,t
x f x u
x g x u
=
=
"
,1 1) ( ) ( ) ( ) ()3( 2 2
tc x
tc xsd s g tc x f tc x f t x u
c
+
+ + + =
= s g s h) ( ) (
)3(
,1 1) ( ) ( ) ( ) ( ) ()4( 2 2
tc x h tc x h tc x f tc x f t x uc
+ + + + =
[ ]
...
= u uxx tt0
) (
) (
0 0,
1 00,
1 1t
x u
xx u
x
=
> =
tc x
,12 2 2 2 2) ( ) ( ) ( 2
= ++ + = t c x tc x tc x t x u
-
4102 232401 " "
~ 43 ~
+ + =tc x G tc x F t x u,) ( ) ( ) (
t x B t x AB B A A, : , , :) ( ) (
+ + =tc x G tc x F A uA A A A) ( ) ( ) (
+ + = + + =tc x G tc x F tc x G tc x F B uB B A A B B B B) ( ) ( ) ( ) ( ) (
((11) ) = tc x tc xB B A A
[... , ]
-
4102 232401 " "
~ 53 ~
+ =A u E u B u D u) ( ) ( ) ( ) (
! B u) ( E u) (, A u) (
-
4102 232401 " "
~ 63 ~
( mrutS-ellivuoiL) S-L ( reiruoF)
) ( ) ( ) ( - 0
n nnis socn
xn b xn a x f
=
+ =
. ? 12
'1
v u, , V v u, V
=u v v u, ,
v u0 ,
= =u u u0 , 0
+ = +w v b w u a w vb ua, , ,
1 2) ( 1
2
i n; , , , ,2i
x x x x x l
=
< = =
? +l y x2 l y x,2
V
+ V
V u u0
= =u u0 0
=u a ua a
+ +v u v u -
. V
:1 . =u u u,2
j xam l2 j
= x x
-
4102 232401 " "
~ 73 ~
l2 1
2
n
n
x x
=
=
v u v u,
l2
"2
1j j
j
y x
=
. +
2
1
m
j j m
j
y x s=
( =m,2,1 ) + =
2 2
1 1
2
1 1
2m m
j j j j m
j j j j
B y x y x s
= = = =
+ +
:sm
m m
S s
l y x,2 " l2 1
j j,j
y x y x
=
() =
1
m
j j m
j
y x=
=
2 2
1 1 1 12 2 2 2
1 1 1
2
1
2
, ,m m
j j j j mj j j j S C
T y x y x y x y x y x
= = = =
< = =
! < T m
, "
= v u0 , ( " )" v u,
( ") e e ek , , ,2 1) (
0
,i j1
j ie e
j i
= =
1 j =i e
"
-
4102 232401 " "
~ 83 ~
, " V
. " e e ek , , ,2 1) (
1
k k
k
e a u
=
ak =
=e u ak k,
) ( 1 1
2
j j j j k k j k k j, , ,k k
a e a e e a e e a e u
= =
= = = =
!!()
b aj j,
1 1 , 1 1
k k i j j i i i j j, ,k i j i j
a b e e a b e b e a
= = = =
= =
u um , ,1} {)1(
2
1 1
2m m
i i
j j
u u= =
=
11.71
(1,0] [ ) =C V1,0 0) (] [
) ( ) ( 1
0 xd f x g x fg ,
> x r0 ) ( ,
) ( ) ( ) ( ) ( ) ()2( x r) (1
0 xd x f x g x fx r x g ,
V )2(
-
4102 232401 " "
~ 93 ~
2) ( 1
0 =xd x r f f f0 ,
) ( = f f0 , 0
12
x f0 ) ( =xd x r f0
= V3
3 2
" V W V
=e e naps Wn , ,1} {
W u
-
4102 232401 " "
~ 04 ~
) ( W u1
,n
k k
k
e e u u=
=
= w u u W w0 , I
II2 2 2
+ = w u u u w u W w
I
= j e u uj0 ,
=e u e uj j, ,
1
, , ,n
j j k k
k
e u e e e u=
= =
II
2 2
+ = w u u u w u
2 2
)1( + =w u u u
:
2 2 2
+ =u u u u
:
u " " W u II II
2 2
u u w u W w
) ( ) ( =x naps W,1} { , =C V1,1 0) (] [1
1xd x g x f g f,
=
W "
=x x h3 ) (
-
4102 232401 " "
~ 14 ~
= =x x v x v12 1) ( ) (
1
2 11
v vxdx0 1 ,
= =
1 1
1 2
xd v2 1
= =
1 1) ( 2
= x w
= =w w1 2 1
2 21
1
20
2
1
22
3xd x xd x v
= = =
=x x w23 2) (
=x w x w naps W,2 1} {) ( ) ( "
-
4102 232401 " "
~ 24 ~
31.11.02
. e e,2 1} {
V , e e,2 1} { " V
(lesseB)
: V u
: (1) 2
1
e uj ,
: (2) 1
2 2u e uj ,
1
,m
e e u sj j m
. e e napsn , 1} { u sm 2 2 2
+ =u u u u
2 22
1
,m
= e u u uj
.sm
1
2
,m
. e uj
.(2) (1)
-
(nnameiR-eugsebseL) -
n0 , mil V u n
e u
.=
2
1
e uj ,
.
lavecraP
V u
22
1
e u uj ,
=
-
4102 232401 " "
~ 34 ~
.
V w n V wn} {
n0 n
w w
1
,m
=e e u sj j m
? sm
?u
) V e e,2 1} {
1
0 , milm
j jm
e e u u
V u V( =
. V x g x f,) ( ) ( =C V1,0 0] [
) ( ) ( 1
0 =xd x g x f g f,
0
2 1 2 =xd f f
. 1,0] [ " x x,2 1) ( ) (
1
0 = xd f fjj ,
) ( : 1
?, mil
m
j jm
x f f
=
. V u " e e,2 1} { "
-
4102 232401 " "
~ 44 ~
2 2 2
. = u u u u
1
,m
=e e u uj j
n
22
1 1
2
, ,
)?!?(
m m
e e u u e e u uj j j j
erusolC lavecraP
=
. "
(Freiruo)
,] [ x f) ( E:
E
1
, 2 soc, 2 nis, soc, nis,2
x x x x
xd x g x f g f,1) ( ) (
0 soc soc,1 1 1) ( ) ( 2 2
xd xn xn
= =
1 1 1 1 1 1
,2 22
12
22
xd
= = =
xn xn xn xnsoc, soc , nis, nis) ( ) ( ) ( ) (
? " E x f) ( :
xd xm x f xm fsoc soc,1) ( ) ( ) (
... =
?x f) (
) ( ) ( ) ( ) ( ) ()3( 1 1
1 1?nis nis, soc soc, ,
2 2x f xm xm f xm xm f f
= + +
.( , )
-
4102 232401 " "
~ 54 ~
)3(
) ( ) (
) ( ) (
1,3,2,1 soc
1,3,2,1 nis
n
n
n xd xn x f a
n xd xn x f b
= =
= =
)3(
0) ( ) ( 1 1
nis socn n 2a
xm b xm a
+ +
2 , x
=x x f) (
" :
. =x x f) (
b an n,
n xd xn x an0 soc 1) (
= =
) ( ) ( ) (
) ( ) ( ) (
nis1 1soc soc
soc 1 1soc soc
n
x
xn xn xxd xd xn x b
n n
n nxd xn
n n
+ = =
+ =
1 soc2 2) ( ) ( n
nn n
= =
-
4102 232401 " "
~ 64 ~
)4( ) ( ) (
1
1
1 2?nis
n
x xnn
+
:
n2) ( ! 0 ?= x )4(
f f
!
-
4102 232401 " "
~ 74 ~
31.11.42
,] [ x f) ( E:
0) ( ) ( ) ( 1 1
nis socn n 2a
xm b xm a x f
+ +
) ( ) (
) ( ) (
soc 1
nis 1
n
n
xd xn x f a
xd xn x f b
=
=
12 2
xd f f
=
=x x f2 ) (
f
=n bn0
xd xn x ansoc 12) ( an
=
- 2) (
2 0
1 21 43
n
nn a a
n
= =
22) (
21
1soc 4 ~
3n
n
xn xn
=
+
2
1
n
1n
)4(
x f1 ) (
0
0 0
2n nb a
a
= ==
-
4102 232401 " "
~ 84 ~
x f1~) (
0 1 ) ( 0 1
xx f
x
< = n k
2 12) ( ) ( 1
1 1nkk n
k n
d dxd x x
xd xd
1 1) ( ) ( ) ( ) ( 1
1 1 12 2 2 2
1 11 11 1 1 1
n nk n k nk k
k n
x
n
x
k
AB
d d d dx x x x
xd xd xd xd
+
+
, A
x x x x xd d d1 1 1 1 12 2 2 2 22) ( ) ( ) ( ) ( ) ( = k2 xd xd xd
= =
) ( ) () ( ) ( = k3 3 2
2 2 2 2
2x x x xd d1 1 1 1
xdxd = =
n0 2) ( B n
n
dx P
xd
. +n1 "" "
'
x PnsmonyloP erdnegeL:) (
" ,] [ E
=n exni, ,1,0,1 , , } {
xd x g x f g f,) ( ) (
,) (xd e xd e e xd e e e exki xni xki xni xki xnix k n i
= = =
=k n e exki xni0 ,
-
4102 232401 " "
~ 05 ~
31.11.72
?x f) (
x f x smm?) ( ) (
0) ( ) ( 1 1
nis soc2
m m
n n m
a + + =xm b xm a s
x, 0] [
( ) x f x smm0 0) ( ) (
) ( ) ( 2?
n0 miln
x f x s
=
?
1,0] [: I
) ( =n,3,2
10 0
2 1
20
n
xn
x n x fn n
xn
< < =
0
2 1 2 =xd f f
-
4102 232401 " "
~ 15 ~
=n fn1
12
01
2 2nn1
n
= = =xdn xd f f
x fn0 0) ( x1,0 0] [
: E E
) ( ) (
) ( ) (0
0
mil
mil
h
h
x f h x f
h
h x f x f
h
+
+=
=
x0 E x f) (
x f x f x smm2egarevA ehT 1 0} { ) ( ) ( ) (
+ +
td t *) ( E f
,] [ =
td t *) (
=
) (
) ( ) () ( ) ( ) ( ) () ( ) ( ) (1
n 1nis soc soc1 1 12
is
m
n
x s
xn sd sn s f xn sd sdsn s f s f
=
=
+ + =
,
, "
, . ,
.
!"
-
4102 232401 " "
~ 25 ~
) ( ) ( ) ( ) ( ) ( ) (
} { ) () ( ) ( ) (1
1
1 1is soc soc
1n
2nis
. i T nmaD soc 1 12
. . 1g r
m
n
m
n
xn sn xn sn s f x s
f x s
sd
x s n ssd
=
=
+ + =
+ =
x s t
) ( ) ( ) ( 1
1 1soc
12
x
mx
n
t td n x t f x s
=
+ + =
) ( ) ( ) ( 1
1 1c
1so
m2n
td tn x t f x s
=
+ + =
) ( 1
s2
11oc
n
tn
=
+
" "
) ( ) ( 1
1soc
1m2
n
tn t D
=
+
t Dtd x t f x sm m1) ( ) ( ) ()5(
+ =
x f x f x sm m21 0) ( ) ( ) ( + +
t Dm) (
.1
) ( m2 0td t D
=
.2
) ( 2 nis1) (
nis22
m
t m
t Dt
= +
.2
nis nis nis soc1) ( ) ( 2
+ =
-
4102 232401 " "
~ 35 ~
) (1) (
nis1soc 12
2nis
2
m
n
t m
tn=t
+ = +
?) ( ) ( - 1
1 1 1nis soc nis
2 2 2 2
m
n
tt m tn
=
+ = +
) ( ) ( ) ( ) ( 1 1
1 1 1 1 1 1nis nis nis nis soc
2 2 2 2 2 2 2
m m
n n
tt n t n t tn
= =
+ + = +
[ ] + + t t tt t2 2 2 2 2nis nis nis nis nis3 5 3) ( ) (
) ( ) ( ) ( 1
1 1 1 1 1 1nis nis nis nis
2 2 2 2 2 2 2
m
n
tt m t n t n
=
+ = + +
1
nis mil soc mil1 1) ( ) ( ) ( ) ( n n
xd xn x xd xn x
=
2
,0] [ x) (
) ( ) ( ) ( ) ( 0 0
0 nis mil soc miln n
xd xn x xd xn x
= =
) ( ) ( 0 0
0
xx
x x
=
.1
3
,0] [ x v) (
xd x n x v n20 nis mil1 0) ( ) (
= +
-
4102 232401 " "
~ 45 ~
) ( ) ( ) (
) ( ) ( ) () (
) (
1 2
00 soc nis nis soc mil
n2 2
x x
x xxd xn x v xn x v
= +
...
) ( ) ( ) (
,0] [nis2
2
x f t x ft t g
t
++
g
-
4102 232401 " "
~ 55 ~
) ( ) ( ) ( 0
1 1mil
nm2x f td t D t x f
= +
) ( ) ( ) ( 0
1 1mil
nm2x f td t D t x f
= +
)5(
mil1) ( ) ( ) ( nm2
+x f x f x s
+ =
) ( ) (
22
12
1soc 4
3
n
n
xn xn
=
+ =
= x
) ( ) (
) ( ) (
22
2
22
2
1
1
22
1soc 4
3
11 4
3
34 6
n
n
n
n
n
nn
n
=
=
+ =
+ =
=
-
4102 232401 " "
~ 65 ~
31.21.8
:
. f E f E:
- sm
x x f x f x sm m2, 10 0 0 0] [ ) ( ) ( ) ( +
+
x f x sm0 0) ( ) ( x0 f
11 ) (
nis 2 ~
n
xn xn
+
2
x =
) (
) () (
1
11
nis 22 2
1 21 1 17 5 3 2
1 41 1 17 5 3
n
n
nn
=
+
=
+ =
+ =
)ecnegrevnoC mrofinU(
x1,0] [ x hn) (
? x h) ( x hn) (
x1,0 0] [
n mil0 0) ( ) ( n
x h x h
=
-
?yllaeR hO
x N, 00 >
< x h x hn0 0) ( ) ( >N n
n N
-
4102 232401 " "
~ 75 ~
x h) ( x hn) (
>N n N) ( > 0 x1,0] [
< x h x hn0 0) ( ) (
- N
=x x x gnn1,0] [ ) (
n mil) ( n
x g
1 0 0) ( ) ( n n1 1
xx g x g
x= = <
! ?
x gn) (
1
n xn1 =
n n n11 1) ( ) ( n
x ge n
=
xn ,
1 0 1
b a,] [ x hn) ( x h) (
! x h) (
?f x f) (
:
f E x f) ( = f f) ( ) ( , ,] [ x f) ( f f
-
4102 232401 " "
~ 85 ~
0) ( ) ( ) ( 11
nis soc2
m
n n m
n
axn b xn a x s
=
+ + =
x f x sm m ) ( ) (
] [ ) ( 1
n,n
b a x x h
=
x hn n) ( 1
n
n
=
) ( ( M ) 1
n
n
x h
= x f) (
!x f) ( E f
) ( ) ( ) (
) ( ) (
) ( ) (
0
1
nis soc2
1c
1n
so
is
n n
n
n
xn xn x f
xd xn x f
xd xn x f
+ + =
=
=
) ( ) ( ) (
) ( ) (
0
0 1
o
1
ns c 1
f f xd x f
xn x f
= =
=
=
nis) ( ) (
n n
nxd xn x f
bn
=
+
= ann n
". " bn} { an} {
-
4102 232401 " "
~ 95 ~
) ( ) ( 1
2 2 2 212
1
2
1
1n n n n
n n
b an
= =
+ = +
2 2
1 12 2
v u v ui i i i
) ( ) (! 1 1
2 2 12
1 1
2 2
21
1 12 2n n n n
n n nn n
= = =
< + +
1
1
n
n
n
n
a
b
=
=
f bn0 .a
) ( ) (
) ( ) ( ) ( ) (
0
1
0
soc ~2
2 1soc soc
n
n
axn a x f
xd xn x f xd xn x f a
+
= =
) () ( ) ( 0
0
x x fx f
x x f
=
>
f an0 .a
) ( ) (
) ( ) (1
0
nis ~
2soc
n
n
xn b x f
xd xn x f b
= =
-
4102 232401 " "
~ 46 ~
(5 )
V u " 2
: =v v v,
+ += +v u v u v u22 2 2 2) ((1)
(1)
mrutS-ellivuoiL ..."
.b a,] [ x p x q x r, ,) ( ) ( ) (
b a C r q p p, , , ,) (] [(1)
>b a x x r x p, 0 ,] [ ) ( ) ((2)
, , ,
(3)
2 2
2 2
0
0
>
> ++
(6) (5)
u "
u = + + u x r u x q u x p0 ) ( ) ( ) () ()4(
= + b u a u0 ) ( ) ((5)
= + b u a u0 ) ( ) ((6)
x u, ) (
31.21.51
+ u x q up u L) ( ) ( ] [
)4(
=u x r u L) ( ] [
-
4102 232401 " "
~ 56 ~
(6)(5))4((1) x u) (
u0
.c x uc,) () ( x u,) () ( x u) ( x r) ( x q) ( " " " " x p) (
= +u u0
= =L u u0 0) ( ) (
<
-
4102 232401 " "
~ 66 ~
) ( ) (
1 1
1
0
0 hsoc 0 0
hsoc 0
d d u
L d L u=
= = =
0 hnis2 2 = =
0
d L d
u
= +
> 0
) () () (
2 1
1
2
nis soc
0 0
nis 0
x d x d x u
d u
L d L u
+ =
= =
= =
d0 2
= L0 nis
= = n n L3,2,1
=n3,2 ,1 ) (
) (
2
nis
n
n
nL
nx x u
L
=
=
< < <
-
4102 232401 " "
~ 76 ~
=d u1
) 1
1d
0 (
0
0
u1
=
" d1
0
) () () (
2 1
2 2
1
nis soc
0 0 0
nis 0
K
x d x d x u
d d u
L d L u
+ =
= = =
= =
K , ,
= L0 nis
!
= = n n L3,2,1
=n3,2 ,1 ) (
) (
2
soc
n
n
nL
nx x u
L
=
=
) ( 0
0
0
x u1
=
< < < < = 03 2 1 0
= L
(")
xn nis} {
xnsoc} {
-
4102 232401 " "
~ 86 ~
LS
: LS
) ( ) () ( ) (0
0
L u u
L u u
==
r q p p, , ,
LS
... . > =x r r0 0 0) ( ) (
-
4102 232401 " "
~ 96 ~
31.21.81
= + + u x r u x q u x p0 ) ( ) ( ) () ()1(
+ =ur u L] [)1(
= + = a u a u u Ba0 ) ( ) ( ] [)2(
= + = b u b u u Bb0 ) ( ) ( ] [)3(
:"
.2 . )1(
"
x v x u,) ( ) (
: "
(4) ] [ ] [] [ ] [
0a a
b b
v B u B
v B u B
=
x w) ( - )1( . )3( )2(, )1( ", LS "
+ = x vD x uA x w) ( ) ( ) (
)3( )2( w
] [ ] [ ] [] [ ] [ ] [
a a a
b b b
v BD u BA w B
v BD u BA w B
+ =
+ =
. (4) D A,
) ( ) ( 0
0 0 0
u u
L u u
= += =
L x 0
.(4) " ( ) > 0
:
= x x unis ) (
-
4102 232401 " "
~ 07 ~
= x x vsoc ) (
(4)
) ( ) () ( ) (0
0L v u
L v L u
=
1 0
0 L Lnis soc
=
'
0 soc
soc
L
L
=
) (
212
n
n
L
+ =
) ( ) ( ) ( ) (
0
2 0
2 0
u u
u u
u u
= +=
=
x2 0
? "
> 0
soc nis
nis soc
D x D x A u
A x D x A u
= + =
= =
) (
) ( ) (0 1 2 soc 2 soc
0 2 nis 1 2 soc
D A
D A
=
=
2 soc 2 nis
02 nis 1 2 soc
=
-
4102 232401 " "
~ 17 ~
22) (2 2
2
0 1 2 soc 2 nis
0 2 soc2 1 2 soc 2 nis
1 2 soc
n nn 2 2
=
= + +
=
= =
2
? = n n
+ =xn B xn A unsoc nis
xn xnnis, soc) ( = n n2 " n1 !
.2 "
. ba.] [ x v x u,) ( ) ( .1
+ vq vp u) ( + = uq up v v Lu u Lv) ( ] [ ] [) (
) ( ) (
) () ( ) (
) (
vp u vp u vpu up v up v upv
vp u up v
up v upv
+ = + =
=
=
vp u vpu) (
) ( ) (
) ( ] [ ] [
vpu upv
uv uv p v Lu u Lv
=
=
'
b a,] [
) ( ) (] [ ] [ b b
aa = vu uv p xd v Lu u Lv
( ) )3( )2( v u,
] [ ro
0b a
(b a ) = vu vu
) () (
0
0
a u
b u
=
=
:)3()2( b a,] [ x v) ( x u) ( "
-
4102 232401 " "
~ 27 ~
0 ] [ ] [ b b
a a = xd u Lu xd u Lv
b a,] [
,) ( ) ( b
a xd x g x f g f
} {] [ ] [ ] [(5)
== u L v v L uv u L , ,,
) ( ) ( ) (
,b
a r rxd x x g x f g f
.2, " "
r g f
n k k " x vk) ( n " x un) (
= u ur k n0 ,
] [] [
n n n
k k k
ur u L
ur u L
=
=
] [ ] [ b b
n k n n k k k n n ka a = u ur u ur u L u u L u
(5)
0) ( ) ( ) ( ) ( b
n k n ka = xd x r x u x u
LS " .3 (.) "
] [
) (
ur u L
ur uq up
=
= +
= + ur uq up) (
-
4102 232401 " "
~ 37 ~
u u,
. " " x u) ( x u) (: 2
) ( ) ( ) ( ) ( ) ( 2
0b b
a a = =xd x r x u xd x r x u x u
! x u) ( x u0 ) ( " .4
"
( ) 1 "
.x u) (
+ =x wi x w x u2 1) ( ) ( ) (
[: ]
= + + + + + wr wq wp i wr wq wp0 2 2 2 2 1 1) () ( ) (
] [ ] [] [ ] [
2 1
2 1
0
0
a a
b b
w Bi w B
w Bi w B
= +
= +
" w2 " w1
=C x wC x w? 1 2) ( ) ( ) (
1) ( ) ( ) (
+ =x w Ci x w1
2 2 :
-
4102 232401 " "
~ 47 ~
31.21.22
?
= + + u x r u x q u x p0 ) ( ) ( ) () ( )1(
+ =ur u L] [)1(
= + = b u a u u Ba0 ) ( ) ( ] [ )2(
= + = b u a u u Bb0 ) ( ) ( ] [ )3(
= vu uv p v Lu u Lv) ( ] [ ] [ - ' (4)
-
=u Lv v Lu] [ ] [
-
k n u uk k n n, , ,) ( ) (
0 ) ( ) ( ) ( b
k na =xd x r x u x u
,) ( ) ( ) ( b
a rk n xd x r x u x u v u
" - " -
:
LS " -
x v) ( x u) ( 2 LS " "
) (
) (
0
0
ur uq up
vr vq vp
= + +
= + +
-
4102 232401 " "
~ 57 ~
] [] [
ur u L
vr v L
=
=
] [] [0 ] [ ] [
vur u Lv
uvr v Lu
v Lu u Lv
=
=
=
'
= = tsnoc vu uv p vu uv p. 0) ( ) (
0 ) () ( ) ( ) ( ) ( ) ( a x
x v x u x u x v x p=
=
)2(
>p0
vu uv0
!
"
) ( ) (
=x u k x v
!
" LS -
<
-
4102 232401 " "
~ 67 ~
,) ( ) ( ) ( b
a r xd x r x g x f g f
.Er ( )LS "
: an
) ( ) (
) ( ) (0
,
0 mil
n n
n
n nn
n
x u x v a
x u a x v
=
=
=
b a,] [ x f) ( ( )
x f x f 21) () ( ) ( b a x,) ( ++
f (" )
b a,] [ f "
LS ": x un) (
) ( ) ( 0
n n ~n
x u a x f
=
b a,) ( ()x r) ( x uk) (
) ( ) ( ) ( ) ( ) ( ) ( 0
b b
n k n ka a
n
xd x r x u x u a xd x r x f x u
=
=
u un k
2) ( ) ( ) ( ) ( ) (
,
,
b b
k k ka a
r kk
r k k
xd x r x u a xd x r x f x u
u fa
u u
=
=
hgielyaR " 0 -
V:
o] [
2fni 0
b
a
bV u
a
xd u Lu
xdr u
=
x u0) ( o
-
4102 232401 " "
~ 77 ~
31.21.52
hgielyaR
) (:] [
2
b
a
b
a
xd u Luu Q
xdr u
<
-
4102 232401 " "
~ 87 ~
) (
1
2
01
68.9
v Q
=
=
.
hgielyaR
. x u) (
x un n,) () ( LS " "
) ( ) ( 1
n n
n
x u a x u
=
=
x u) (
) () ( ) (
) ( ) ( ) (
) ( ) (
) () ( ) ( ) () ( ) ( ) (
1 1
1 1
1 1
1 ,
n - 1 1 2
b
k k n na
k n
b
k k n na
k n
b
k k n na
k n
b
n k k na
k n
k k k n n
k n
xd x u a L x u a
u Q
xd x r x u a x u a
xd x u a L x u a
xd x r x u x u a a
x u x r a x u a
= =
= =
= =
=
= =
=
=
=
) () ( ) (1
2 2 2
11 1 1
1 12 2
2 2
2 2 2
2 2 2 2
1 1 1
b
a
b
n na
n
r r rk n k n k n nn n n
r r rk n k n k nn n n
xd
xd x r x u a
u a u a u a
u a u a u a
=
= = =
= = =
= = =
x u1) ( . x u Q1) () (
=x u x un1) ( ) (
1
0 1
1a nn
a
= > =
) ( 2
1
2
21
1 1
2 1
1 1
r
r
u au Q
u a
= =
-
4102 232401 " "
~ 97 ~
= =b u a u0 ) ( ) ( - = = b u a u0 ) ( ) ( -
:
) ( 2 2) () (
2
b
a
b
a
xd uq u pu Q
xdr u
=
) ( 2 2 2 ) () ( ) () (
2 2
b b
a a
b b
a a
xd uq u p xd uq up uu Q
xdr u xdr u
+ = =
= + u u u ux0
= =u u0 1 0) ( ) (
2 2) (
2
1
01 1
0
xd v e vx
xd v
+
= =v v0 1 0) ( ) (
LS
< < = +x u u1 0 0 (5)
= + =u uh u0 1 1 0 0) ( ) ( ) ( (6)
>h0
?
-
4102 232401 " "
~ 08 ~
= =B u0 0 0) (
+ = + = A Ah u uhhsoc hnis 1 1 0) ( ) (
hnis
hnath hsoc
= =
>h0 ,
?= 0
+ = =xB A u u0
) () ( ) (
0 0
1 1 0
A u
B Bh u uh
= =+ = + =
>h0 = +h0 1
?> 0
+ = x B x A unis soc
= =A u0 0) (
+ = + = B Bh u uhsoc nis 1 1 0) ( ) (
0 soc nis
nat
h
h
= +
=
."
-
4102 232401 " "
~ 18 ~
t x
t x u,) (
> k0 =uk uxx t (1)
=u x k ux x t) () (
, <
-
4102 232401 " "
~ 28 ~
= = t T0 0 0 0) ( ) ( ) ( (8)
= = L t T L0 0) ( ) ( ) ( (9)
!LS (9) (8), (6)
) ( ) () ( ) (
0
0 0
x x
L
= +
= =
!
) (
) (
2
,2,1
nis
n
n
nn
L
nx x u
L
= =
=
(7) n
= + t T t Tn n n0 ) ( ) (
) ( ) ( 2
n
nt
L te e t Tn
= =
:"
(5)
) ( ) ( (01)2
nis ,n
tL
n
nex t x u
L
=
(3) (2) (01)
(4)
-
4102 232401 " "
~ 38 ~
31.21.92
> < < =t L x u uxx t0 0 (1)
= =t L u t u0 , ,0) ( ) ( (2)
=x f x u0,) ( ) ( (3)
=t T x X t x u,) ( ) ( ) ( (4)
(2) (1)
) ( ) ( =n,3,2 ,1(5)2 2
2
nis ,n
tL
n
ne x t x u
L
=
(. ) (3)
.(3) . (2) (1) (5)
) ( ) ( (6)2 2
2
1
nis ,n
tL
n
n
ne x A t x u
L
=
=
.(2) (1)" " (6)
(3) An} {
) ( (7)) (
) (6 3) ( ) (
1
nnis 0 ,n
nx A x u x f
L
=
= =
x f) ( . LS (7)
.An} { , LS "
) ( ) () (
0
0
2
nis
nis
L
L n
nxd x x f
LA
nxd x
L
=
2 2^ nis 0) ( L n L
xd xL : =
-
4102 232401 " "
~ 48 ~
=L Lxd x x f An nLnis 20) ( ) ( (8) ((3),(2),(1)) " " (8) (6)
> < < =t x u uxx t0 0 (9)
= =t u t u0 , ,0) ( ) ( (01)
) ( ) ( (11)0
0,2
2
x xx f x u
x x
= =
-
) ( ) ( (21)2
1
nt n nis ,n
e xn A t x u
=
=
) ( ) ( (31)1
nnisn
xn A x f
=
=
(8)
) ( ) () ( ) ( ) (
0
0 0
2nis
2nis nis
n
nxd x x f A
L
n nxd x x xd x x
L L
=
+ =
.\
> d0 =u d uxx t
] [] [ ] [
2L
dt
=
=J ux t- : J
. =ud Jx
-
4102 232401 " "
~ 58 ~
) () (
2
2
!
4nis
2
1 41 2
n
k
k
nA
n
Ak
+
=
=
) ( (41)) (
) ( ) () ( ) (1 2 12
12
1 2 nis 1 ,41 2
k kt
k
e x k t x uk
+
=
=
(41) : ? "" (41)
?t x
:
." "
.x 2 . (41) "
) (
) ( ) () ( ) ( ) ( ) (
) ( ) ( ) () ( ) (
2
2
1 2 12
12
1 2
1 1
1 2 nis 1 1 2 1 1 41 2
, 1 2 nis 1 4
t
t
k k
k
k k
n
n k
e x k kk
t x g e x k
+
=
= =
=
"
'2 "
) ( 1
n
n
x h
=
c x hn n) ( 1
n
n
c
=) (
1n
n
x h
=
" M "
) ( 1
n
n
x h
=) ( ) ( hn "
1n
n
x h x h
=
=
) ( ) ( 1
n
n
x h x h
=
. " =
) ( 1
n
n
x h
=
x h) ( "
-
4102 232401 " "
~ 68 ~
!> t0 !
"
) ( ) ( 2
41 2,
kt
e t x gn
> 0, = t
k 41 22) (
e gn
1 22) (
1
k 4
n
e
= !
) ( , M 1
n,n
t x g
= (41) u "
.x
1 "
1
pna
n
e n
= " . >a0
-
4102 232401 " "
~ 78 ~
( 03 )41.1.1
:
[ , ]?(11),(01), (9) (41) 1
? 2
1
.
) (
) ( ) () ( ) () (
2 12
2 2
1 2 nis 11 4 411 2 1 2
t k ke x k
n k
+
12) (
1
n n1 2
= .
, M
(." )
? .
.x 2
) () ( ) () ( ) ( ) ( ) (
) ( ) ( ) () ( ) (
2
2
1 2 12
12
1 2
1 1
1 2 nis 1 1 2 1 1 41 2
, 1 2 nis 1 4
t
t
k k
k
k k
n
n k
e x k kk
t x g e x k
+
=
= =
=
"
) ( ) ( 2
41 2,
kt
e t x gn
1 22) (
1
k 4
n
e
= M
.uxx
t - t .
) (
) ( ) ( ) () () ( ) (
) ( ) () ( ) (
2
2
2
2
1 2 1
1
1 2
1
1 41 2 1 1 2 nis 1
1 2
41 2 nis 1
t
t
k k
k
k k
k
e k x kk
e x k
+
=
=
=
.
. (41) u
-
4102 232401 " "
~ 88 ~
t x x f) (" " .>t0
. f ,
+ t , =t0 .
.
(2) 1
(2) 2
= =B t L u A t u, ,0) ( ) ( (51)
t L u,) ( t u,0) (
t x v x A t x uA B, ,) ( ) ( L + + =
v
t t
xx xx
v u
v u
==
= v vxx t0
= =t L v t u, 0 ,0) ( ) (
) ( ) ( ) (
) ( ) ( ) (0, 0,
0,
A Bx v x A x u x f
LA B
x A x f x vL
+ + = =
+ = =
! "
3
0 , mil) ( t
t x u
=
)...(
...!
!
)1( < < = uL x c uxx tt0 0 2
-
4102 232401 " "
~ 98 ~
= =t L u t ux x0 , ,0) ( ) ( (2)
=x f x u0,) ( ) ( (3)
=x g x ut0,) ( ) ( (4)
)1(
=t T x X u) ( ) ( (5)
= t T x X c t T x X2) ( ) ( ) ( ) (
)6() () (
) (2) (
x X t T
t T cx X
= =
= +X X0 )7(
)8( = +T c T0 2
(2)
= = L X t T X t T0 0) ( ) ( ) ( ) (
= = L X X0 0) ( ) ( (9)
.)7(+(9) LS x X, ) (
:LS )7(+(9)
) ( (01)2
n,2 ,1,0n
nL = =
Lx x un nsoc ) ( (11) =
)8(
) ( 2
n n0 2n
T c TL = +
) (
) ( ) ( ) (00 0 0
n n nnis soc 1
t t T n
n ntc tc t T n
L L
+ = =
+ =
-
4102 232401 " "
~ 09 ~
=t T x X t x un n n,) ( ) ( ) (
.(2) )1(
.(4) (3)
0 0) ( ) ( ) ( ) ( ) ( (21)1
nis soc soc soc ,n n 2
n
n n n n t B Atc x B x tc A t x u
L L L L
=
+ + + =
.(2) )1(
.(4) (3) (21) Bn An
) ( (31)) (
0 3) ( ) (1
soc 0,n 2
n
n Ax A x u x f
L
=
+ = =
) ( (41)) (
0 4) ( ) (1
soc 0,n t2
n
n n Bx c B x u x g
L L
=
+ = =
:
0) ( ) ( (51)1
socn 2
n
n ax a x f
L
=
+ =
LS
=L Lxd x x f an nLsoc 20) ( ) ( (61) x g) (
0) ( ) ( (71)1
socn 2
n
n bx b x g
L
=
+ =
=L Lxd x x g bn nLsoc 20) ( ) ( (81) (31) (51)
=a An n
(41) (71)
=b B0 0
n n n
b c BL =
-
4102 232401 " "
~ 19 ~
,
. ,
.(4) )1(
.(4) )1( t x u, 2) ( t x u, 1) (
[ ]
t x u t x u t x w, , ,2 1) ( ) ( ) (
(91)2
=w c wxx t
= =t L w t wx x0 , ,0) ( ) ( (02)
= x w0 0,) ( (12)
= x wt0 0,) ( (22)
( " " , )"
w0
" "
) ( ) ( 0
12 2 22
L
+ =xd w c w t Ex t w
w0 Ed
td =
) (
) (0
0
2 2 2
2
12
L
x t
L
tx x tt t
d Edxd w c w
td td
xd w w c w w
+ =
+ =
=w w w w w wt xx t x x tx x) (
-
4102 232401 " "
~ 29 ~
) (
2) (0
091
2L
t x x t xx tt t
Edxd w w c w w c w w
td=
+ =
) (
) (
) (
) ( ) (
) (
) (
0
0 002 02
2
0 ,0 ,0 , , 2
L
t x x
t x t x
Edxd w w c
td
t w t w t L w t L w c
= =
=
= =
) ( ) ( ) (
) (
2 2 2
00 022 22
0
10 0
2
L
x t t t
t
dxd w c w E t E
td= =
=
= + = =
t t Ew0 ) (
!
-
4102 232401 " "
~ 39 ~
41.1.5
(1) < < = uL x c uxx tt0 0 2
= =t L u t ux x0 , ,0) ( ) ( (2)
=x f x u0,) ( ) ( (3)
=x g x ut0,) ( ) ( (4)
. LS "
( )
= =t L u t u0 , ,0) ( ) ( (5)
= x F c uut xx tt, 2) ( (6)
(.(2) )(5), (4), (3) (6)
.(5), (4), (3) (6) t x u t x u, , ,2 1) ( ) (
=u u t x w,2 1 ) (
= c wwxx tt0 2
= =t L w t w0 , ,0) ( ) (
= =x w x wt0 0, 0,) ( ) (
""
) ( ) ( 0
12 2 22
L
+ =xd w c w t Ex t w
0 Edtd
=tsnoc t E. ) ( =
= = E t E t0 0) ( ) (
-
4102 232401 " "
~ 49 ~
2 2) ( ) ( 0
0 122
L
+ =xd w c w t Ex t w
=w wx t0
=tsnoc t x w. ,) (
=x w t x w0 0, ,) ( ) (
t x w t x0 , ,) (
=u u2 1
> < < = t L x t x G u uxx t0 , 0 ,) ( (7)
= =B t L u A t u, ,0) ( ) ( (8)
=x f x u0,) ( ) ( (9)
.(9), (8), (7)
. u u,2 1
t x u t x u t x w, , ,2 1) ( ) ( ) (
t x w,) (
= w wxx t0 (01)
= =t L w t w0 , ,0) ( ) ( (11)
= x w0 0,) ( (21)
.w0
" "
) ( ) ( 0
, 122
L
xd t x w t E
-
4102 232401 " "
~ 59 ~
) ( 0) (
0
21
0 0
, 12
x t L
x
d EdL Lxd w xd w xd t x w
td td = = =w w
) (0
2
0
0 0
2
0
0
L
Lx
L
x
xd w w wx x
w t E
w w
xd
=
=
(21)
2) ( ) ( 0
10 0, 0
2
L
= =xd x w E
. t E0 ) ( = E0 0) (, t E) (
= t x w t E0 , 0) ( ) (
=u u2 1
.(2), (4), (3) (6)
"
) ( ) ( (31)0
n n
n
x X t T u
=
=
. LS " Xn
.(31)
=Lx x Xn nsoc) ( ) (
X Lx n soc) ( . L 0 , - LS
0) ( ) ( ) ( ) ( (41)1
1soc ,
n 2n
nx t T t T t x u
L
=
+ =
" t x F,) ( . (41) (2), (4), (3) (6)
-
4102 232401 " "
~ 69 ~
0) ( ) ( ) ( ) ( (51)1
1soc ,
n 2n
nx t h t h t x F
L
=
+ =
."" t hn) (
.(6) (51) (41)
) ( ) ( ) ( ) ( ) ( ) () ( ) ( ) (
01
22
1
01
1soc soc
2
1soc
2
n n
n n
n
n
n n nx t T c x t T t T
L L L
nx t h t h
L
= =
=
= + +
+ =
0) ( ) ( ) ( ) ( ) ( ) () ( ) ( 2
01
0 soc 1 1n n n 2 2
n
n nx t h c t Tc t T t h t T
L L
=
= + +
(61)
) ( ) (
) ( ) ( ) (0 0
2
,2 ,1
n n n
n t h t T
nt h t T c T
L
= =
= +
Lx n soc) ( (2) (4) (3)
) ( ) () ( ) (
0
1
0
1
soc2
soc2
n
n
n
n
n ax a x f
L
n bx b x g
L
=
=
+ =
+ =
. Lx n soc) ( =t0
0) ( ) ( ) ( ) ( ) ( 1
1soc 0 0 0,
n 2n
nx f x T T x u
L
=
= + =
0) ( ) ( ) ( ) ( ) ( 1
1soc 0 0 0,
n t2n
nx g x T T x u
L
=
= + =
-
-
4102 232401 " "
~ 79 ~
) () (0
0
n n
n n
n a T
n b T
=
=
41.1.8
= x F c uut xx tt, 2) (
= =t L u t ux x0 , ,0) ( ) (
= =x g x u x f x ut0, , 0,) ( ) ( ) ( ) (
" LS
x Xn) ( "
) ( ) ( 1
n n
n
x X t T u
=
=
( ) Xn F f g, ,
t Tn) ( "
= t x u uxx tt2 soc 2 soc) ( ) (
= =t u t ux x0 ,1 ,0) ( ) (
=x x usoc 0,2) ( ) (
=x x ut2 soc2 0,) ( ) (
[]
) ( ) ( ) ( ) (
) ( ) (7
01
n t2 soc2 soc 0 0,n
x x g x n T B x u
=
= = + =
= B0 0
= n Tn2,0 0 0) (
= T2 0 2) (
-
4102 232401 " "
~ 89 ~
= = n B n Tn n2,0 0 0) (
= = = B B T2 2 01 2 2 2) (
=n Bn2 0
) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 0
1 1 12 nissoc soc 2 nis soc 2 soc ,
4 2 2A A
tx t x t x t t x u
2 + 2 + 2 + =
. , ! ?
! ?
" "
D
= D y x u, 0) ( (1)
+ = u u uyy xx -
= + = u u u y x uyy xx ,) () (
- u u2
(1) .1
(1) .a
= D y x y x F u, ,) ( ) ( (2)
u .2
(D ) D .3 .a
=D y x y x f y x u, , ,) ( ) ( ) (
.b
= D y x y x g y x un, , ,) ( ) ( ) (
-
4102 232401 " "
~ 99 ~
n - n
.c
D y x y x h u y x u, , ,) ( ) ( ) ( n
= +
= D F u (3)
(4)u
D gn =
,) ( D D
sd g ydxd y x F
=
) (
) ( 4 3) (
n D D D D D
sd g sd u ydxd u ydxd u ydxdF
= == =
)D C) ( ( )D C2) ( , D y x u,) ( .D u (
+w wyy xx0 w 2
> 0
) ( ) ( ) (
) (
, ,2 2
0 4 2 2
x y x u y x v
u v
y
+ =
> = + + =
+
D v
xam D
u M
( ) =
x my xam2 2
= +
( ) D v .
+ D y x m M y x v, ,) ( ) (
) ( ) ( ) () (
, ,2 2
0 ,
m M y x y x v y x u
D y x
+ =
>
+ ( y x v,) ( )
-
4102 232401 " "
~ 001 ~
,
, xam ,) ( ) ( D
D y x u M y x u
=
..
. - D u
.D y x u,) (
.D y x,0 0) ( R y x BR,0 0) ( D y x,0 0) (
( R r 0) Br
2 ,1 0 0) ( (5)Br
sdu y x ur
=
) () ( ) ( ) ( 2
0 00
1 1nis , soc
2 2Br
dr r y r x u sdu r wr r
+ + =
) (
) () ( ) ( ) () ( ) (
0
2
0 00
0 00
? mil
1nis , soc
2, mil
r
r
r w
d r y r x u r w
y x u r w
=
+ + =
=
= nnis, soc ) (
) () ( ) ( ) (
) ( ) (
2
0 00
2
0
2
0
2
0
1nis , soc
2
12
1nis soc
2
12
1 10
2 2
B Br r
n
ud r y r x r w
r
u x uyd
r y r x
u ud
x
n u
u
y
dr sd d
u sdr r
+ + =
+ = + =
= =
= =
=
u
=r w w0) ( ) ( : w = r w0 ) (
:
-
4102 232401 " "
~ 101 ~
.D u . (5) D y x u,) (
[ , ]
,D u . D D c) ( D c2) ( u .D ( )
u , . p0 u
D q . D
=p u q u0) ( ) (
=c p p u p u0 0) ( ) (
= =p u p u p u0 1 2) ( ) ( ) ( :
=p u q u0) ( ) (
...
41.1.21
- D
= D f u
-
4102 232401 " "
~ 201 ~
=D g u
D c u) ( ( ) =D D D D c u2) (
"
D u0 =u u u2 1 ", u u,2 1
0
0
D u
D u
= =
2 1 2 1) (0
u u u u u
f f
= = =
= = = g g u u uD D D0 | | |2 1
) ( ) (
) () (
) (, ,
0 , xam , xam ,D y x D y x
y x u y x u y x u
= =
( )
) ( ) (
) () (
) (, ,
0 , nim , nim ,D y x D y x
y x u y x u y x u
= =
y x u0 , 0) (
= =u u u02 1
> =+y x y x D1 | ,2 2} {) (
0
0
D u
D u
= =
= = + xy y x D1 | ,2 2 2} {) (
= u0 1
-
4102 232401 " "
~ 301 ~
+ =x y x uy nl ,2 22) ( ) (
[ ]
u
+ = r r unl ) (
:
1
u u0r
= +
+ = r r unl ) ( ( ) "
. !
) u
nl (
u2 u1 D
j
= D f u
=j2,1j j
=D g u
) (
) ( ) () (
2 1 2 1) ( ) (, ,
, , xam , , xamD y x D y x
M y x u y x u y x g y x g M
= = =
: =u u u2 1
= D u0
= = g g u u uD D| |2 1 2 1) (
D y x M y x u M, ,) (
M y x u,) (
D n1) ( n
-
4102 232401 " "
~ 401 ~
= u k ut0
z y x, ,3 ) ( t z y x u, , ,) (
zz yy xx
+ + = u u u u
. > k0
[= uk uxx t0 =n1 ]
< < =T t D z y x t z y x QT0 , , | , , ,} {) ( ) (
D z y x T z y x, , | , , ,} {) ( ) ( D z y x z y x, , | 0, , ,} {) ( ) (
( , )
= D z y x z y x T t D z y x t z y x Qp T, , | 0, , , 0, , , | , , ,} {) ( ) ( } {) ( ) ( ) (
:
. D
Q c Q c uT T t2) ( ) (
. u ( )
, , ]
[
v :
< v k ut0 (1)
. QT v
(1) = t z y x vt0 , , ,0 0 0 0) ( ' Q t z y xT , , ,0 0 0 0) (
> t z y x v0 , , ,0 0 0 0) (
!
-
4102 232401 " "
~ 501 ~
T z y x vt0 , , ,) ( T z y x, , ,) (
> T z y x v0 , , ,) (
[ ]
-
4102 232401 " "
~ 601 ~
41.1.51
.
u
) ( ) (
2 ) (, ,
tT 0
t t x u t x v
Q v k v
=
<
) (
, xam) (p T
Qt x u M
=
) ( ) ( ) (
) (,) (
xam , xam ,p pT T
u v
Q Q t xM u t x v t x v
=
) ( 0
+ + = Q t x M u t M t v uT ,
decnuoned eb osla nac
0 sa
2
T t
k
Q F u k u
>
=
=D x x f x u0,) ( ) (
=D x t x g t x u, ,) ( ) (
v u u v2 1 u u,2 1
= v k vt0
= x v0 0,) (
= t x v0 ,) (
) (
0 , xam) (p T
Qt x v
=
0 , xam) ( QT
= t x v
. "
-
4102 232401 " "
~ 701 ~
()
F ( ) 2
( )
sd x n x F xd x Fvid) ( ) ( ) (
=
D
D F u
ug
n
= =
) ( (1)) (
, ) (D uD D D Dvid
ug ydxd y x Fd sd x n u
ndsd us x
= = = =
! -
=v Fu ' v u,
= + v du iv vv u u) (
) () ( ) ( + u v v u u ssuaG
ydxd ydxsd v du v vidn
=
=
D u =v u
u 2
s xdu un
d yd
=
} {
} {
D F u
uD g u
n
= = = = +
( )
.
=u u u2 1 u u,2 1
} {
} {
0
0
D u
uD u
n
= = = = +
) ( 2
200
u uysd xdsd u u
n n d
= =
-
4102 232401 " "
~ 801 ~
2
uydxd0
=
tsnoc u. 0
0 0 u
u n
= =
() u0 =u0
} {
} {
D F u
uD g
n
= = = =
(1)+
=u u u2 1
[ ]
} {
} {
0
0
D u
uD
n
= = = =
+ = =tsnoc u u tsnoc u. .2 1
!
< < < < =b y a x y x D0 , 0| ,} {) (
= D u0
) ( ) () ( ) () ( ) () ( ) (
1
3
2
4
,
,0|
,
0,
D
y g y a g
y g y gg u
x g b x g
x g x g
= =
= = ==
-
4102 232401 " "
~ 901 ~
y Y x X) ( ) (
{) ( ) () ( ) () ( ) (
} ) ( ) () ( ) () ( ) (
1 2
1 1 1 1
3 1 4 2
1 1 2 2
0 0
rof 0 , 0, 0 , ,0 y rof
0 0,
0 0
0,
, ,0
u u
x LS b x u x u y a u y u LS
y g y u x g x u
x u
D D
a x b y
b y a xy g y a u x g b
= =
= = = =
= = =
=
u1
) ( ) ( ) (
=y Y x X y x u,
= + YX Y X0
Y XY X
= =
0 0
0 0
a x X X
b y Y Y
< < = +< < =
= =a X X0 0) ( ) ( x X) (
a x Xx n nnis ) ( =
) ( 2
n
na =
= =b Y Y0 0) ( ) ( (= Y Yn0)y Y) (
) (
) () (hnis
hnis
y n
ny
ae
ny b
a
-
4102 232401 " "
~ 011 ~
u
) () ( ) ( ) ( ) ( 1 1
hnis hnis nisn n n n
n n
n n x ny b y y Y x X u
a a a
= =
+ = =
=y0
4) ( ) ( ) ( 1 1
hnis nis nis 0,n n
n n
n x n nb x A x g x u
a a a
= =
= = =
hnis) (n
n
A
nb
a
=
2) ( ) ( ) ( 1 1
n nhnis nis nis ,n n
n x n x nb B x g b x u
a a a
= =
= = =
hnis) (n
n
B
nb
a
=
2 ,
. " " 2 g
? '
+ =p v u
=g p p
0
D D| |
D v
p g v
= =
) (( y x2 2) ( yx ) 2 2) (
,
y x
yx y x y x p
+ + + =
) ( ) () ( ) () ( ) () ( ) (
0,0
,
,
,0
p D g
a b a p A g
ba b a b a p B g
b b p C g
= =
+ = =
+ + + = =
+ = =
'
-
4102 232401 " "
~ 111 ~
2
r ru u ur rr0 1 1 = + +
< + =R y x y x D| ,2 2 2} {) (
0
D |
D u
f u
= =
= r u) ( ) ( -
< < R r 0 2 0
2
0 1r
= + +
LS
= + r r0 2 -
= + 0 -
) ( ) () ( ) (2 0
2 0
=
=
) () (
0
n nnis soc
tsnoc
n n
=
+ =
= = n n n,2 ,1,0 2
< < = + R r n r r0 0 2 2
) () (
nl 00 0 0
n n nn n 1
r r n
r r r n
+ = =
+ =
+ = = r r nnl 00 0 0) (+ = r r r nn n n n n 1) (
'
0) ( ) ( 1
nis soc ,n n 2
n
n n r r un
=
+ + =
-
4102 232401 " "
~ 211 ~
0) ( ) ( 1
nis soc ,n n 2
n
n n R R un
=
+ + =
"
0 ) ( - LS 1
nis socn n 2
n
a n b n a f
=
+ + =
b an n,
-
4102 232401 " "
~ 311 ~
41.1.91
~~~~~~~~~~~~[ ]~~~~~~~~~~~~
0
D |
D u
f u
= =
= = y x u y x y x r u r u, , , , ,) ( ) () ( ) ( ) (
< + =R y x y x D| ,2 2 2} {) ( R D
2
r ru u ur rr0 1 1 = + +
: u
= r0 r u,) ( ( ) <
-
4102 232401 " "
~ 411 ~
+ = e e ) (
= 0
) (0 0 0) (
+ =
=
= 0 0 "
> 0
+ = nis soc ) (
2
n,3,2,1
n
n n
n
= =
=
+ = n n n nnis soc ) (
< < = + R r n r r0 0 2 2
[ ]
) () (
nl 00 0 0
n n nn n 1
r r n
r r r n
+ = =
+ =
r nl , 1rn
,
""
+ = = r r nnl 00 0 0) (+ = r r r nn n n n n 1) (
'
) ( ) ( ) (
0
1,
nis soc ,2
n
n n
nr u
n n r r un
=
+ + =
-
4102 232401 " "
~ 511 ~
-
) ( ) ( ) (
, 0) (
0 0
1 1
nis soc nis soc ,2 2
r u
n n n
n
n
n
n
an b n a h n n R R u
= =
+ + = = + + =upsquarebracketrightupsquarebracketleft
h) (
h) (
) ( ) (
) ( ) (
2
2
0
0
soc 1
nis 1
n
n
d n h a
d n h b
=
=
0 0 n
n n n
nn
b aa
R R = = =
0) ( ) ( ) ( 1
nis soc ,n n 2
n
r ann b n a r u
R
=
+ + =
~~~~~~~~~~~~~~~[ ]~~~~~~~~~~~~~~~
) r h h r h, ,3 2 1) ( ) ( ) ( ( + + =u u u u3 2 1
) ( ) ( ) ( 1
n, , ,n
D y x y x u y x u
=
=
( . )D
D " " D " D y x un,) ( . . u
h " h ( )
u "
" u Ri
<
-
4102 232401 " "
~ 611 ~
n r un,) ( r
] [
ni2
n
M uRkrR
n
.
D
Ri
. u u B
! D '