Uploaded by Javlonbek Norbayev

1-ma'ruza

advertisement
I-
.
1! "#" $ %%&'&() #" *&(+"#,#"#'
-# "#'( (+ &. , */0'
$# * 1 (.#
,
,
,
.
(,# #"#(:
,
,
,
;
!
;
;
"
)
#
. “
"
”
#
.
x1 , x2 ,..., xn , n ≥ 2
Ω
En $
n−
x
(
1– #2' %. &
α = (α 1 , α 2 , ..., α n )
, !%
"
)
.
n
, "3*
"
n
-
α = α1 + α 2 + ... + α n
($&4
"
,
.
u ( x ) = u ( x1 , x2 ,..., xn )
α = α1 + α 2 + ... + α n
x ∈Ω
# !
∂ ( )u
D u = D1 D2 ...Dn u = α1 α 2
, D 0u = u ( x )
αn
∂ x1 ∂ x2 ...∂ xn
x = αi
. '
"
α
∂ iu
∂u
α
∂ 2u
α
D u = α = D i i u , Di u =
= u xi , D i2 u =
= u x x ,... .
2
i
i i
∂
x
∂
x
∂ xi
i
i
α
α1
α
α2
αn
2– #2' %. (
F ( x, ...,
%
"
Dα u, ...) = 0
u ( x) = u ( x1, x2 , ..., xn )
#
5
# !
"
(1.1)
m − *#'* 5"
.
"
(1.1)
3– #2' %.
.
)
Pα = Dα u, ( α = 0,1, ..., m )
# !
#
4– #2' %.
(
) u xu
. 1. +
2
xy
, F
Pα
. 6 7"
, (1.1)
.
F, α = m
, (1.1)
4-#6 . 6 7"
!". * "
)
1–
u
"
)
#
Pα
"
.
.
+ 2 xuu yy − 3 xyu y − u = 0.
,
.
xy
2
) u y + 3 x u ⋅ u xy + 2 u x − f ( x , y ) u = 0.
3x 2 ⋅ u –
u xy
. 1. &
# !
.
#
%
-
,
u
) sin x ⋅ u xx + xcos y ⋅ u xy − 2 xy ⋅ u x − u = 0.
. 1.
,
u, u x , u xx , u xy
# !. ,
,
.
5– #2' %. Ω
u ( x)
# ! (1.1)
,
" !
(1.1)
'&+ "8' (4"# 4) &. ,
"
, u ( x)
"
[4].
2–
!". a uxxy + b uxxx + c u yyy = 0 . uxxy ∈C ( D) , uxxx ∈C ( D) , uyyy ∈C ( D)
,
"
.
6– #2' %. )"
( –
!
"
(9"&,&(*#') &. ,
"
(
!
, u ( x)
)
"
(1.1)
[4].
1 1
ln , r =
!". E ( x, y ) =
2π r
3–
)
% ($#,&(*#"
2
( x − x0 ) + ( y − y0 )
u xx + u yy = 0
2
# !
.
7– #2' %. (
Lu ≡
α ≤m
aα ( x ) D α u = f ( x )
m − *#'* 5"
"
#
"
(1.2)
.
6
)
x∈Ω
(1.2)
, (1.2)
5 ' : ( " , f ( x)
: ( " 5/",#+#(
"
.
4–
!". 1) 2 x2U xx + 3x yU xy + 5U yy = 0
f ( x)
, 5 '
# !
2) 5 x yU xy + 6U xx = 5 x 2 + y 2
x ≠ 0, y ≠ 0
8 – #2' %. '
;
"
,
.
"
*#'* 5
#
"
-
.
1) a u x − b u y = f ( x ) −
2) U xx −U yy = f ( x, y) −
!".
5–
"
3) U xyy − U xxx = 0 −
;
;
.
9– #2' %. (
∂ 2u
Aij ( x )
+
∂ xi ∂ x j
i , j =1
n
n
i =1
Bi ( x )
∂u
+ C ( x )u = f
∂ xi
n − /6+#' -. "
#
"
"
–
",
"
. ,
(1.1)
!
"
.
"
, !%
. +
(1.3)
"
. (1.3)
Aij u x i x j , A ji u x j x i
(1.3)
i≠ j
# !
#
x∈Ω
Aij = A ji
Aij , Bi , C , f -
,
Aij ( i, j = 1,..., n )
(1.3)
(1.3)
"
Ω
x = ( x1, x2 ,..., xn )
(x)
( Aij + A ji ) u x x
i
j
"
.
m = 2, n = 2
,
"
44 (. *#'* 5" . 6 7" $ %%&'&() #" *&(+"#,#
F ( x, y , u , u x , u y , u xx , u xy , u yy ) = 0
(1.4)
a ( x, y )u xx + 2b ( x, y )u xy + c ( x, y )u yy + a1 ( x, y )u x + b1 ( x, y )u y + c1 ( x, y )u = 0 (1.5)
a( x, y), b( x, y), c( x, y), a1 ( x, y), b1 ( x , y ), c1 ( x , y ) −
,
# ! .
(1.4)
"
#
"
5#'#' " 1 1#'*:
∂F
∂F
∂F
+
+
≠ 0.
∂ u xx
∂ u xy
∂ u yy
7
(1.6)
!". 1) sin2 ( u xx + u xy ) + cos 2 ( u xx + u yy ) − u = 0
6–
#
2)
2
u xx
+ u 2yy
;
"
= f ( x, y ), -
"
#
∂F
∂F
= 2U xx ,
= 2u yy , 2 ( u xx + u yy ) ≠ 0 .
∂u xx
∂u yy
, (1.3)
)
A ij ( x ) = 0
(1.3)
.)
"
.
7–
!". y ( u xx + u yy ) + a u x + bu y + cu = f ( x, y )
!%
,
"
!
.
"#" $ %%&'&() #" *&(+"#,#
.
"
y=0
.,
,
,
1− u = 0 -
y=0
( (+ &. , +# ! $
!""#':
1) ux = 0
2) uxy = 0
3) aux + cu = 0
4) u xxx = 0
u ( x, y ) = ϕ ( y ) ;
u ( x, y ) = ϕ ( y ) + φ ( y ) ;
"
"
u xx = ϕ ( y )
u = f ( y )e
"
ux = ϕ 1( y) + ϕ2 ( y)
−c x
a
;
u ( x , y ) = x 2ϕ 1 ( y ) + x ϕ 2 ( y ) + ϕ 3 ( y ) .
#-6 ( , *#;4#,"#1 . ( #-!""#':
1. m
"
2. .
3.
4. +
"
5. /
6. /
"
.
#
"
#
.
"
.
#
.
#
"
"
"
7. 0
"
8. 0
"
#
#
#
.
! (
)
.
#
(
.
8
)
.
Download