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Международный университет информационных технологий

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1 / 31 2 3 > Следующая >>>

“ ” .

, U V –

, a U + b V a b

. ,

, , .

, . ,

, ,

. ,

, .

( ),

( )

u= u(x;y;z; t ,

t – , ,

, .

« u »

x, y, z, t .

1.	∂2 u		= a	2 ∂ 2 u
							.	(1)
	∂ t	2			∂ x	2

, ,

, , , .

2. :	∂2 u	= a 2	∂2 u
			∂ x 2 .	(2)
	∂ t

3. :	∂ 2 u	+	∂2 u	= 0 .	(3)
	2		2
	∂ x		∂ y

, , .

1. ( ).

. [0;l] .

, .

, -

, , : u = u(x; t .


			u
MM1 .			u
MM1 .
					M1		ϕ +	ϕ
.				M	M1		ϕ +	ϕ
				M

		O		ϕ
ϕ ϕ +	ϕ .			ϕ
ϕ ϕ +	ϕ .			x	x + x		l	X
Ou ,				x	x + x		l	X
Ou ,
MM1					1.
T sin(ϕ +	ϕ − T sin ϕ .
ϕ Dx O( x2 + ϕ 2						:

T sin(j + Dj -)T sin j = T ×tg j + Dj( -T ×tgj =) T çæ						¶u(x + Dx;t

	¶2u(x +q Dx;t		¶2 u(x;t		è		¶x
= T		Dx = T		Dx , 0	< θ < 1 .
	¶x		¶x

-	¶u(x;t	ö	=
		÷
	¶x	÷
	¶x	ø

, , -

, . ρ −

, – ρ x , : ρ x

∂ 2u

= T

∂2u

x .

∂ t

∂ x

= a

∂2 u

= a

2 ∂2 u

, :

∂ t

∂ x

u(x; t

( )

( t = 0 ).

	u(0;t )= 0, u l; t (= 0 – .	(4)
	t = 0
, , .
f (x) , u(x;0 )= u \|t =0 = f x .		(5)

, ϕ(x ). ,

:	∂ u	\|t =0 = ϕ (x ).	(5′
	∂ t

(5) (5′ .

2. .

(1)

(5,5′ .


											Ψ1 (ξ ) Ψ2 η (.
						u(x;t		)= Ψ1 x −( at + Ψ2)		x + at				(6)
								(1),
(6) (1).
		Ψ1 (ξ ) Ψ2 η (, -
, (6) (5,5′ .
	t = 0		:	Ψ	(x )+ Ψ x (=)f x , ( −)aΨ ′					x + aΨ(	′ )x = ϕ x	.	( )	(7)	( )
			:	1	2				1	2		.		(7)
(7):
						1	x
		Y1 (x )- Y2 x( =)-				1	xòj (x )dx + C,		C = Ψ2 (x0		)− Ψ1 x0( .			(8)
		Y1 (x )- Y2 x( =)-				a	xòj (x )dx + C,		C = Ψ2 (x0		)− Ψ1 x0( .			(8)
					0

(x )=

ê f (x )-

j (x )dx + Cú,

(7) (8) :

a xò

(x )=

ê f (x )+

j (x )dx - Cú.

a xò

(9)

(10)

(9) t, (10) t (6),

	f (x - at +) f x +( at		1	x+at
: Y2 (x )=		+		òj (x )dx .
: Y2 (x )=	2	+	2a	òj (x )dx .
				x−at

:	∂2 u		= a	2	∂2 u		(1)
	∂ t	2			∂ x	2

(4) (5), (5′ .
							(1),
(4) : u(x; t )= X x(×T) t					. (11)

(11) (1):


X(x )× T¢¢ t( =) a 2 X¢¢ x ×(T t) .		(t )		X	(12)
	T				(x )
(12) a 2 X(x )×T t( :		′′	=		′′
						. (13)
	a 2 T(t )			X(x )

(13) t, – .

′′

(t )

X (x )

′′

= − , > 0 , ( < 0 ).

a 2 T(t )

X(x )

¢¢

( =)0,

t + a( )×T t

= 0

(14)

X (x )+ × X x

(7) :

X(x )= Acos

x + Bsin

T(t )= Ccos

t + Dsin

t ,

(15)

A, B, C, D – . (15) (11),

u(x; t = (Acos

x + Bsin

x (Ccos(a

t + Dsin (a

t )

(16)

(4). (4)

X(0 )= 0, X l

= 0 , A ×1 + B ×0 = 0, Acos

l + Bsin

l = 0 .

A = 0,

B ¹ 0,

sin

l = 0,

= n l , n =1,2,.....

(17)

(17) ,

X(x )= Bsin (np l x – .

(14) < 0 , :

X(x )= Ae -λ x + Be− -λ x ,

(4). ,

n c (16), (17) (11): u(x; t = sin (npl x(Cn cos(anpl t + D n sin (anpl t ), n =1,2,....

(1) ,

∞
: u(x; t )= å(Cn cos(anp l t )+ Dn sin (anp l t )×sin (np l x).	(18)

n=1

(18) (5,5/) :

			∞			∞		x).
			f (x )= åCn sin (np l x),	j(x )= åDn (anp l sin) (np l				x).
			n=1			n=1
Cn Dn :
		2	∞ l			2	∞ l
C n	=	2	åòf (x )sin (np l x),	Dn	=	2	åòj(x )sin (np l x).	(19)
C n	=	l	åòf (x )sin (np l x),	Dn	=	anp	åòj(x )sin (np l x).	(19)
		l	n=1 0			anp	n=1 0

, (1,4,5,5/) (18,19).

2. .

, ,

– .

. ,

, .

, : L,

, R G.

i(x;t −

U (x;t ) − x

t .
x x +	x .
o( x) R x

x .

∂)i(x; t

U (x; t

)− U x (+

x; t = R) xi x; t

+ L( x

(20)

, o( x)

∂t

∂U (x; t

U (x +

x;t − U)

x;t (=

)

(21)

∂x

(20) (21), :

∂U

+ L

∂i

+ Ri(x;t )= 0

(22)

∂x

∂t

[x; x +

i(x; t )− i x(+

x;t = −)∂i(x;t

(23)

∂x

G xU (x; t )

∂U (x; t

∂t

∂)U (x; t

i(x; t )− i x(+

x; t = G) xU x; t

+ ( x

(24)

(23) (24), :

∂t

∂i

∂U

+ GU = 0 .

(25)

∂x

∂t

(22) (25) .

(22) x , (25) –

t ,									∂2i
										,
									∂x∂t
U (x; t )
	∂U			∂U			∂U
			= LC			+ (RC + GL	)	+ GRU .
	∂x	2		∂t	2
							∂t
:
	∂i		= LC	∂i		+ (RC + GL	∂i)+ GRi
	∂x2			∂t 2
							∂t

, :

∂w

= a

∂w

+ 2b

∂w

+ c

(26)

∂x2

∂t 2

∂t

= LC,

b =

RC + GL

c = GR

(R = 0

(G = 0 ,

b = 0

(19)

∂w

= a

2 ∂2 w

, a =

LC .

(27)

∂t

∂x

, ,

3..

D ,

Γ .

T .

, T ,

. u

M (x; y; u

u = u(x; y; t . t = t0 u = u(x; y; t0

(S ′ t ,

(S x

y . (S ′

x2 +

y .

S ′

Γ1

(x; y

, ,

u = u(x; y; t .

OU ,

, Γ1

(S ′ . M

n = i cos α + j cos β + k cos γ ,

Γ1 : r = i cosα1 + j cosβ1 + k cosγ1 .

M [r ; n ,

, dS ′ Γ1

T (cos α1 cos β − cos α cos β1 )dS′ .

Γ1 , :
T ò(cos α1 cos β − cos α cos β1 )dS′ = T òcos β d x1 − cos α dy1 ,			(28)
Γ1		Γ1
	cos α1dS ′ = d x1	cos β1dS′ = d y1 .

u = u(x; y; t

(

OU − )

′

− u′

cos α =

− ux

cos β =

(ux

) + (u y

)

1 + (ux

) +

(uy

)

1 +

′

′ 2

cos γ =

′

2 .

1 + (ux

) + (uy

)

x2 +

y 2

′

cos γ = 1,

cos α = −ux ,

cos β = −u y ,

(28) :

′

(29)

− T òu y d x1 − ux dy1 .

Γ1

dx1 = dx

dy1 = dy ,

dx, dy −

σ .

(29)

σ ,

ax = u′y , ay = −u′x .

′

′′

dS .

− T òu y d x1 − ux dy1

= −T òu y d x − ux dy = T ò(ux

+ u y

Γ1

( , ),

OU ,

ρ g(x; y; t .

−ò ρ g(x; y; t dS .

, ρ −

(S ′

	′′
	ρ ò utt (x; y; t dS ,
	S
t ³ 0
	′′	′′	′′	= 0 .	(30)
	ò [ρutt	− T (uxx	+uyy + ρ g dS	= 0 .	(30)
	S
(S − ,
(30) .
′′	′′	′′
ρutt	− T (uxx +u yy )+ ρ g(x; y; t = 0 ,

′′	− a	2	′′	′′	= −g(x; y; t ,	a =	T	.	(31)

utt			(uxx	+uyy			ρ

(31) .

g(x; y; t ≡ 0 ,

′′	2	′′	′′		(32)
utt = a		(uxx	+uyy .
(31)
.
.
					D
,				Γ − .
:
u \|t =0 = f (x; y ,)				u′ \|t =0 = F x; y ,	(33)
D			f (x; y ,)F x;( y

(x; y .

u |Γ = 0 ,

(34)

u |Γ u Γ .

D r0

( . 3).

r ϕ

D O X

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