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Национальный исследовательский университет «МЭИ»

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Высшая математика

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УрМатФиз / УрМатФиз с теорией

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20 r

1 e

1 1

fr ' t = 2J0

= nX kX fnk t vnk r ' =

0 k

1 1

= nX kX fnk

t J2n

cos n'

12 r

1 1

gr ' = J2 0

1 cos 2' = nX kX gnkvnk

r ' =

=0 =1

1 1

= nX kX gnkJ2n

cos

24 r

1 1

pr ' = 2J4

1 cos 4' = nX kX pnkvnk

r ' =

r 1

1 1

= nX kX pnkJ2n

cos

fnk

t =

ZZ f r ' t vnk

r ' r drd'

jjvnkjj

gnk =

ZZ g r ' vnk

r ' r drd'

jjvnkjj

ZZ p r ' vnk

pnk =

r ' r drd'

n = 0 1 k = 1 1:

jjvnkjj

:2:22

:2:23

:2:24

						fnk t, gnk pnk
! " ,		#$ % & '. ) , v02 r ' =
0 20 r1			0 12 r 1			0 24 r 1
= J0 B	b C v11	r ' = J2	B b	C cos 2' v22 r ' =		J4 B	b C cos 4':
@	A		@	A		@	A
* & # & + $& + %+ &
		#
	f02	t = 2e 2t	fnk	t 0 # n k 6=0 2
		g11 = 1	gnk = 0 #		n k 6=1 1
		p22 = 2	pnk = 0 #		n k 6=2 2:

251

4.2.21 , 4.2.22 4.2.6

		2	t	+ a2 nkunk t fnk t 9 vnk r ' = 0
	8d u 2		t	+ a2 nkunk t fnk t 9 vnk r ' = 0
		nk			=
k1		dt			=
2	:
d unkt		2
d unkt		2			n = 0 1 k = 1 1:
dt2		+ a nkunk t = fnk t			n = 0 1 k = 1 1:	:2:25

X X

r ' = 0

= gnk

:2:26

n=0 k=1funk

gnkgvnk

unk

X X

dunk

pnk9 vnk

r ' = 0

dunk

= pnk

:2:27

n=0 k=1

n =

0 1

k =

1 1

$ % & '

) ! ) *: ) n k =

0 2 ,

k = 1 1 n k =

2 2 ! , ) -. )

, unk t 0 )

k 6=0 2 ,

n k

6=1 1 , n k 6=2 2:

/ ' ! % & '

+ a2 02u02 t = 2e 2t u02

= 0

= 00

d u

dt2

d2u11t

+ a2 11u11 t = 0 u11

= 1

du11

= 00

dt2

+ a2 22u22 t = 0 u22 0 = 0

= 2

dt2

, 1 2

2 "e

cos aq 02t +

sin aq 02t #

u02t =

a 02

:2:28

4 + a2 02

u11t = c os a 11t

:2:29

u22t = 2 sin ap

:2:30

ap 22

252

02 =

4.2.28 | 4.2.30 unk t 0 n k 6= 0 2 , n k 6= 6= 1 1 n k 6= 2 2 4.2.21 4.2.6 | 4.2.10 :

2 "e 2

sinaq 02t # J0 q 02 r

cosaq 02t +

a 02

u r ' t =

4 + a2

2 sinap

t J4 p

r cos 4'

+ cosaq 11t J2 q

11 r cos 2' +

t sin x cos 2y + 4 sinap

1 u x y t = 2 cosap

t sin x cos 4y +

ap17

sin 2x cos 4y

sin 2ap5t cos 2ap5t + e

+3 "

2ap

1 + 20a2

"e 2

sinaq 02t # J0 q

cosaq 02t +

02 r

a 02

2 u r ' t =

4 + a2

2 sinap

t J4 p

r cos 4'

+ cosaq 11t J2 q

11 r cos 2' +

q nk =

| k-& ' ( ) J2n = 0

J2n x |

* '+ ) , ) 2n-

) ' .

4.2.1. - & (-' & ) -

)

@2u = a24u + f @t2

) ( ' ' ' . ) , ' ' 2.8.1, ( . )

u t=0= g 253@t t=0= p:

= u

= uy

y=0

= u

y= 2

= 0

f x y t = 2 sin x cos y e t

g x y = sin 2x cos 3y

p x y = 3 sin x cos 3y:

u '=0= u' '= 4= u r=b= 0

f r ' t = 2J 2 r=b sin 2' t

g r ' = J

6 r=b sin 6'

p r ' = 3J2 22 r=b sin 2':

ux x=0= u x= 2= uy y=0= uy y= = 0

f x y t = 3 cos x cos y e 2t

g x y = 2 cos 3x cos 2y

p x y = cos x cos 2y:

u '=0= u '= 3= ur r=b= 0

f r ' t =

3 r=b sin 3'

g r ' = J

6 r=b sin 6'

p r ' = 2J 3 r=b sin 3':

ux x=0= ux x= = u y=0= uy y= 2= 0

f x y t = cos 2x sin y

t + 1

g x y = 3 cos x sin 3y

p x y = 2 cos 2x sin 3y:

u' '=0= u '= 2= u r=b= 0

f r ' t =

1 r=b cos '

e t

g r ' = J

3 r=b cos 3'

p r ' = 2J1 11 r=b cos ':

u x=0= ux x= 2= u y=0= u y= = 0

f x y t = 2 sin x sin 2y

g x y = sin 3x sin y

p x y = 3 sin x sin y:

u' '=0= u' '= = u r=b= 0

f r ' t = J1 21 r=b cos '

g r ' = J2 12 r=b cos 2'

p r ' = 3J1 11 r=b cos ':

254

u 0 = u x= = uy y=0= uy y= = 0

f x y t = 2 sin 2x cos y e

g x y = 3 sin x cos 2y

p x y = sin 2x cos 2y:

10.

u '=0= u' '=

3= ur r=b= 0

f r ' t = 2J

3=2 r=b sin 3'=2 t

3=2

g r ' = J

5=2

5=2 r=b sin 5'=2

p r ' = J

3=2 r=b sin 3'=2 :

3=2

11.

ux x=0= ux x=

2= u y=0= uy y=

2= 0

f x y t = cos x sin y e 2t

g x y = 2 cos 3x sin 3y

p x y = 3 cos x sin 3y:

12.

u '=0= u '=

2= u r=b= 0

f r ' t =

2 r=b sin 2'

e t

g r ' = 2J

4 r=b sin 4'

p r ' = J2 22 r=b sin 2':

13.

ux x=0= ux x= = u y=0= u y= = 0

f x y t = cos x sin 2y t

g x y = cos 2x sin y

p x y = 2 cos x sin y:

14.

u' '=0= u '= = ur r=b= 0

f r ' t =

1=2 r=b cos '=2

e t

1=2

g r ' = J

3=2

3=2 r=b cos 3'=2

p r ' = 2J

1=2 r=b cos '=2 :

1=2

15.

u x=0= ux x=

2= uy y=0= u y=

2= 0

f x y t = 2 sin 3x cos y t + 1

g x y = 2 sin x cos 3y

p x y = sin 3x cos 3y:

16.

u' '=0= u' '=

4= u r=b= 0

f r ' t =

4 r=b cos 4'

e t

g r ' =

4 r=b cos 4'

p r ' = J2 18 r=b cos 8':

255

17. u	0	= u	x=	= u	y=0	= uy	y= 2	= 0
	0		x=		y=0		y= 2

f x y p x y

18. u '=0=

t = 2 sin 2x sin y e 2t = 3 sin 2x sin 3y:

u' '= 2= u r=b= 0

f r ' t = J	1 r=b sin ' t
1	2

p r ' = 2J1 11 r=b sin ':

g x y = sin x sin 3y

g r ' = J3 13 r=b sin 3'

19.	ux x=0= u x=			2= u y=0= u y= = 0
	f x y t = 2 cos x sin 2y e t					g x y = cos 3x sin y
	p x y = 3 cos x sin y:
20.	u '=0= u '= = ur r=b= 0
	f r ' t = 2J				1 r=b sin '	t	g r ' = J		2 r=b sin 2'
				1	2			2	1
	p r ' = J	1	1 r=b sin ':
		1		1
21.	ux x=0= ux x= = uy y=0= u y=					2= 0
	f x y t = 2 cos 2x cos y t					g x y = cos x cos 3y
	p x y = 3 cos 2x cos 3y:
22.	u' '=0= u '=			4= ur r=b= 0
	f r ' t =		J	2 r=b cos 2'		e	2t	g r ' = 2J			6 r=b cos 6'
			2		1					6	1
	p r ' = 2J 2 r=b cos 2':
			2		2
23.	u x=0= ux x=			2= uy y=0= uy y= = 0
	f x y t = 2 sin x cos y e t					g x y = sin 3x cos 2y
	p x y = 3 sin x cos 2y:
24.	u' '=0= u' '=				3= u r=b= 0
	f r ' t =		J	3 r=b cos 3' e t				g r ' = 2J		6 r=b cos 6'
			3		2				6		1

p r ' = 2J3 13 r=b cos 3':

256

25.

u 0 = u x= = u y=0= u y= = 0

f x y t = 2 sin x sin y t

g x y = sin 2x sin y

p x y = 3 sin x sin 2y:

26.

u '=0= u' '= = ur r=b= 0

f r ' t = 2J

1=2 r=b sin '=2

1=2

g r ' = J

3=2

3=2 r=b sin 3'=2 p r ' = J

1=2 r=b sin '=2 :

1=2

27.

ux x=0= u x=

2= uy y=0= u y=

2= 0

f x y t = 2 cos x cos 3y e

g x y = cos 3x cos y

p x y = 3 cos x cos y:

28.

u '=0= u '=

4= ur r=b= 0

f r ' t = 2J 4 r=b sin 4'

e t

g r ' =

8 r=b sin 8'

p r ' = 3J 4 r=b sin 4':

29.

ux x=0= ux x= = uy y=0= uy y= = 0

f x y t = 2 cos x cos 2y

g x y = cos 2x cos y

p x y = 3 cos x cos y:

30.

u' '=0= u '=

3= u r=b= 0

f r ' t = J

3=2 r=b cos 3'=2 e 2t

3=2

g r ' = J3=2 23=2 r=b cos 3'=2 p r ' = 2J3=2 33=2 r=b cos 3'=2 :

4.2.2.

1 - !

" # ! $ !% #" % $ $ $

D = f x y z : 0 x 0 y =2 0 z g

@2u

@t2 = a24u + f x y z t x y z 2 D t 0

4:2:31

( " )% #( !% @D, $ % 2.8.2 2.8.30 , 2.8.31 :

257

		@u	@u	@u	= u z= = 0
u	0	= u x= = @y y=0= @y y= 2= @z z0			= u z= = 0

u t=0= g x y z = 3 sin 2x cos 2y cos 3z=2

4:2:32

4:2:33

	@u		= p x y z = 2 sin x cos 2y cosz=2 x y z 2 D				4:2:34
	@t	t=0

				f x y z t = sin x cos 4y cos 3z=2 e		2t:	4:2:35

	2 -				-
! "
			D = f r ' z : 0 r b 0 ' 2 0 z hg
		@2u		= a24u + f r ' z t r ' z	2 D t	0	4:2:36
		@t2

@D, ! 2.8.2 2.8.32 , 2.8.33 :

		@u		=	@u			=	@u		= 0
		@z z=0			@z z=h				@r r=b

				0		12 r1
	u t=0= g r ' z = J2 B					b	C cos 2' cos z=h
				@		b	A
@u				0	11 r1
@t		t=0= p r ' z = 2J1 B				b	C cos '			r ' z 2 D
@t				@		b	A
		0	1 r1
		0	1 r1
f r ' z t = J1 B			2	C sin ' cos z=h t + 1
		@	b	A

4:2:37

4:2:38

4:2:39

4:2:40

n		0	Jn x \| -" . n-
k	\| k- Jn = 0
! .
3 -			-
! "
	D = f r ' z : 0 r b 0 ' =2 0 z hg
	@2u	= a24u + f r ' z t r ' z 2 D t 0		4:2:41
	@t2

258

@D, 2.8.2 2.8.34 , 2.8.35 :

= u

= 0

4:2:42

@' '=0

@' '= =2

r=b

z=0

z=h

= g r ' z = 3J2 0 22 r1 cos 2' sin 3 z=h

4:2:43

t=0

= p r ' z =

J4 0 14 r1 cos 4' sin z=h r ' z 2 D 4:2:44

t=0

f r ' z t = 2J2 0 12 r1 cos 2' sin z=h

4:2:45

| k-" J

= 0

x | #$ %

2n- .

1 ' ( ) 4.2.31 | 4.2.35 ) *

. + ,-

) * 4.25

u x y z t =

unkm t vnkm x y z

4:2:46

n=1 k=0 m=0

X X X

vnkm x y z = sinnx cos 2ky cos

2m + 1

4:2:47

) 2- |3 , - -

)- - -

- )

4 - -

4.2.31

4.2.32 . 6- )

|3 (

2.8.2, /

) nkm

2.8.46 , - -

#$ 5-

" 2.8.47 . 7 )

8##$ -

unkm t 5-

(

) 9 ( 4.32 | 4.34 . : ( " )-

8##$ - ; fnkm t, gnkm, pnkm ) * " / #$ 4.26 | 4.28 4- /-5 4.29 | 4.31 .

259

4.2.35 sin x cos 4y cos 3z=2 = v121 x y z | -

, :

f121 t = e

fnkm t 0

n k m 6= 1 2 1:

% 4.2.33 , 4.2.34

sin 2x cos 2y cos 3z=2 = v211 x y z

sin x cos 2y cosz=2

v110 x y z |

, -

g211 = 3

gnkm = 0

n k m 6= 2 1 1

p110 = 2

pnkm = 0

n k m 6= 1 1 0:

& ' , ( ) (

* +

4.32 | 4.34 #

n k m 6=

6= 1 2 1 n k m 6= 2 1 1 , n k m 6= 1 1 0

, ' -.

( , :

unkm t

0 #

n k m 6=1 2 1

n k m 6=2 1 1

n k m 6=1 1 0:

4:2:48

& ) (

* + , '

+ a2 121u121 t = e 2t

u1210 = 0

0 = 01

4:2:49

d u

121

dt2

d2u211 t

u2110 = 3

du2110

= 01

4:2:50

dt2

+ a 211u211 t = 0

+ a2 110u110 t = 0

u1100 = 0

= 2:

4:2:51

d u

110

dt2

3 + ,

) ( 4.2.49 | 4.2.51 '

sinaq 121t cosaq 121t + e

u121 t =

121

4 + a2 121

4:2:52

u211 t = 3 cosap

211

2 sinap

t :

u110 t =

110

ap 110

260

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