Уравнения математической физики

Ɉɬɜɟɬ: u x,t

l

sin 2Sat sin 2Sx .

2Sa

l

l

utt

uxx, 0 x 2, t ! 0,

utt

uxx, 0 x 1, t ! 0,

u x,0

0,ut x,0

x 2 x ,

u x,0

x x 1 ,ut x,0 0,

u 0,t u 2,t 0.

u 0,t u 1,t 0.

u

2u

, 0 x 1, t ! 0,

1

utt x,0

xx0,ut x,0

x 1 x ,

utt

4 uxx, 0 x 1, t ! 0,

u 0,t u 1,t 0.

u x,0

x x 1 ,ut x,0 0,

u 0,t u 1,t 0.

utt

3uxx, 0 x 3, t ! 0,

utt

4uxx, 0 x 3, t ! 0,

u x,0

0,ut x,0

x 3 x ,

u x,0

x x 3 ,ut x,0 0,

u 0,t u 3,t 0.

u 0,t u 3,t 0.

utt

4uxx, 0 x 2, t ! 0,

u

1 u

, 0

x 1 , t ! 0,

u x,0

0,ut x,0

x 2 x ,

tt

9

xx

2

u 0,t u 2,t 0.

u x,0

x x 12

,ut x,0 0,

u 0,t u 12 ,t

0.

u

tt

1 u

xx

, 0

x 4, t ! 0,

u

tt

9u

xx

, 0 x 3 , t ! 0,

4

x x

2

u x,0

0,ut

x,0

x 4 x ,

u x,0

3

,ut x,0 0,

u 0,t u 4,t 0.

u 0,t u 23

,t

2

0.

1

utt

16uxx, 0 x 1, t ! 0,

utt

9 uxx, 0 x 3, t ! 0,

u x,0

x x 1 ,ut x,0 0,

u x,0 0,ut x,0

x 3 x ,

u 0,t u 1,t 0.

u 0,t u 3,t 0.

utt

16uxx, 0 x 2, t ! 0,

utt

16uxx, 0 x 1, t ! 0,

u x,0

0,ut x,0

x 2 x ,

u x,0

x x 1 ,ut x,0 0,

u 0,t u 2,t 0.

u 0,t u 1,t 0.

utt

uxx, 0 x 3, t ! 0,

utt

uxx, 0 x 3, t ! 0,

u x,0

0,ut x,0

x 3 x ,

u x,0

x x 3 ,ut x,0 0,

u 0,t u 3,t 0.

u 0,t u 3,t 0.

u

tt

9u

xx

, 0 x 1 ,

t ! 0,

u

tt

4u

xx

, 0 x 1 , t ! 0,

2

x 12 x ,

x x 12

2

u x,0 0,ut x,0

u x,0

,ut x,0 0,

u 0,t u 12 ,t 0.

u 0,t u 12 ,t

0.

utt

9uxx

, 0 x 2, t ! 0,

u

4u

, 0 x 2 , t ! 0,

u x,0

0,ut x,0

x 2 x ,

tt

9

xx

32

3

u 0,t u 2,t 0.

u x,0

x x

,ut x,0 0,

u 0,t u 32 ,t

0.

1

u

4u

, 0 x 1, t ! 0,

utt

4uxx, 0 x 3, t ! 0,

utt x,0

xxx x 1 ,ut x,0 0,

u x,0

0,ut x,0

x 3 x ,

u 0,t u 1,t 0.

u 0,t u 3,t 0.

u

tt

4 u

xx

, 0

x 1,

t ! 0,

u

tt

1 u

xx

, 0 x 1 , t ! 0,

9

4

2

u x,0

0,ut

x,0

x 1 x ,

u x,0

x x

1

,ut x,0 0,

u 0,t u 1,t

0.

u 0,t u 12 ,t

2

0.

u

1 u

, 0

x

3

, t ! 0,

utt

4uxx, 0 x 2, t ! 0,

tt

4

xx

2

x 23 x ,

u x,0

x x 2 ,u

x,0 0,

u x,0 0,ut x,0

u 0,t u 2,t 0. t

u 0,t u 23 ,t 0.

1

u

9u

, 0 x 3, t ! 0,

utt

9 uxx, 0 x 2,

t ! 0,

utt x,0 xx x x 3 ,ut x,0 0,

u x,0 0,ut x,0

x 2 x ,

u 0,t u 3,t 0.

u 0,t u 2,t 0.

u

tt

9 u

xx

, 0

x 1,

t ! 0,

u

tt

u

xx

, 0 x

3 ,

t ! 0,

4

x x

2

u x,0

0,ut

x,0

x 21 x ,

u x,0

3

,ut x,0 0,

u 0,t u 1,t 0.

u 0,t u 23 ,t

2

0.

u

r

R

f (M) ,

(1)

ɝɞɟ f (M) – ɡɚɞɚɧɧɚɹ ɮɭɧɤɰɢɹ, ɧɟɩɪɟɪɵɜɧɚɹ ɧɚ ɨɤɪɭɠɧɨɫɬɢ.

ɉɪɢɦɟɪ 4.1. Ɋɟɲɢɬɶ ɤɪɚɟɜɭɸ ɡɚɞɚɱɭ ɞɥɹ ɭɪɚɜɧɟɧɢɹ 'u

0 ɜ ɤɪɭ-

ɝɟ x2 y2 d1, ɟɫɥɢ ɧɚ ɝɪɚɧɢɰɟ ɤɪɭɝɚ u sin3 M .

Ɋɟɲɟɧɢɟ: ɍɪɚɜɧɟɧɢɟ Ʌɚɩɥɚɫɚ ɜ ɩɨɥɹɪɧɵɯ ɤɨɨɪɞɢɧɚɬɚɯ r,M

ɢɦɟɟɬ ɜɢɞ

1 w

wu

1 w2u

r

r wr

0

(2)

wr

r 2

wM2

r

d

r dRdr

d 2Ɏ

dM2

dr

O const .

R

Ɏ

ɛ) r

2 cc

c

OR

0,

R 0

f

R

rR

Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ )cc O)

0 ɢɦɟɟɬ ɜɢɞ

) M

Ae O M Be

O M ,

(3)

ɝɞɟ A ɢ B – ɤɨɧɫɬɚɧɬɵ.

O t 0

ɢ ɢɦɟɟɬ

ɩɟɪɢɨɞ 2ʌ ɩɪɢ

ɗɬɨ ɪɟɲɟɧɢɟ

ɩɟɪɢɨɞɢɱɧɨ ɩɪɢ

O n2 n

0,1,... .

ȿɫɥɢ O

O0

0,)0 M

A0 .

ȿɫɥɢ O

O

n

n2, )

n

M

A cosnM B

n

sin nM, (n

1,2,...) .

n

ȿɫɥɢ O

O

0 ,

r

2 cc

c

0 . Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ

0

R

rR

R0 r C0 D0 lnr . Ɍɚɤ ɤɚɤ

R0 0

f , ɬɨ D0 0 .

ȿɫɥɢ O

O

n

2

,

r

2 cc

c

2

R

0 .

n

R

rR

n

Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ

R

n

r

C

r n

D

r n ,

n 1,2,... .

n

n

Ɍɚɤ ɤɚɤ

Rn 0

f, ɬɨ Dn

0 .

-0 r,M C0 A0

a0 ,

-n r,M Cnr n

An cosnM Bn sin nM r n an cosnM bn sin nM .

f

f

u r,M ¦-n

r,M

a0 ¦r n an cosnM bn sinnM .

n 0

n 1

6. ɂɫɩɨɥɶɡɭɹ ɝɪɚɧɢɱɧɨɟ ɭɫɥɨɜɢɟ u 1,M

sin3 M

3sinM 1 sin3M ,

4

4

f

3 sinM 1 sin3M . Ɉɬɫɸɞɚ

ɢɦɟɟɦu 1,M

a0 ¦an cosnM bn sin nM

n 1

4

4

a

0

0, a

n

0,b

3

,b

0, b

1 ,b

0 n t 4 . ȼ ɪɟɡɭɥɶɬɚɬɟ

1

4

2

3

4

n

u r,M

3 rsinM 1 r3 sin3M .

4

4

Ɉɬɜɟɬ: u r,M

3 rsinM

1 r3 sin3M .

4

4

ɉɪɢɦɟɪ 4.2. Ɋɟɲɢɬɶ ɤɪɚɟɜɭɸ ɡɚɞɚɱɭ 'u 0, 0 d r r0 ,

u

r r

acos3 Mbsin3 M pcosM qsinM c .

0

u r ,M

c 3a p cosM 3b q sinM a cos3M a sin3M.

0

4

4

4

4

Ɍɨɝɞɚ

u r0,M

f

an cosnM bn sinnM

a0 ¦r0n

n 1

c

3a p cosM 3b q sinM a cos3M a sin3M

4

4

4

4

1

3a

a

ɋɥɟɞɨɜɚɬɟɥɶɧɨ,

a0

c, a1

4 p , a2

0, a3

,

r0

4r0

3

1

3b

b

an 0, ɩɪɢn t 4 ,

b1

4 q ,b2 0 , b3

,bn

0 ɩɪɢn t 4 .

r0

4r0

3

Ɉɬɜɟɬ:

p cosM 3b

q sinMº»

a cos3M b sin3M .

u r,M

c

r

ª« 3a

r3

3

r0 ¬

4

4

¼

r0

4

4

1.

'u

0, 0 d r 2,

2.

'u

0,0 d r 2,

u

r

2

2cos3 M sin3 M sinM

u

r

2

4cos3 M sinM 7

3.

'u

0,0 d r 1,

4.

'u

0,0 d r 4,

u

r

1

4cos3 M 4sin3 M

u

r

4

4cos3 M sin3 M

cosM 2

2cosM 3

5.

'u

0,0 d r 3,

6.

'u

0,0 d r 1,

u

r

3

12sin3 M cosM sinM

u

r

1

2cos3 M 4sin3 M 2cosM

7.

'u

0,0 d r 3,

8.

'u

0,0 d r 1,

u

r

3

3cos3 M 2sin3 M 3cosM

u

r

1

cos3 M 2sin3 M sinM

9.

'u

0,0 d r 2,

10.

'u

0,0 d r 3,

u

r

2

cos3 M 3sin3 M

u

r

3

9cos3 M 4sin3 M 2cosM

3cosM 2sinM

11.

'u

0,0 d r 2,

12.

'u

0,0 d r 4,

u

r

2

2cos3 M 2sin3 M cosM

u

r

4

4cos3 M 2sin3 M 8

13.

'u

0, 0 d r 2,

14.

'u

0,0 d r 1,

u

r

2

3cos3 M 5sin3 M 2cosM

u

r

1

5cos3 M 3sin3 M 2cosM

15.

'u

0,0 d r 1,

16.

'u

0,0 d r 1,

u

r

1

cos3 M 4sin3 M

u

r

1

cos3 M 2sin3 M

5cosM sinM

cosM sinM

17.

'u

0, 0 d r 1,

18.

'u

0, 0 d r 2,

u

r

1

3cos3 M 2sin3 M

u

r

2

cos3 M 4sin3 M

5sinM 6

5sinM 2

19.

'u

0, 0 d r 4,

20.

'u

0, 0 d r 2,

u

r

4

6cos3 M 5sin3 M sinM

u

r

2

2cos3 M sin3 M sinM

21.

'u

0, 0 d r 1,

22.

'u

0,0 d r 1,

u

r

1

3cos3 M 5sin3 M

u

r

1

cos3 M 2sin3 M

2sinM 5

cosM sinM

23.

'u

0, 0 d r 1,

24.

'u

0, 0 d r 3,

u

r

1

4cos3 M 4sin3 M 2sinM

u

r

3

2cos3 M 4sin3 M 6

25.

'u

0, 0 d r 1,

26.

'u

0, 0 d r 3,

u

r

1

12cos3 M sin3 M 3cosM

u

r

3

3cos3 M sin3 M cosM 5

27.

'u

0, 0 d r 2,

28.

'u

0, 0 d r 2,

u

r

2

cos3 M 3sin3 M sinM

u

r

2

4cos3 M sinM 7

29.

'u

0, 0 d r 3,

30.

'u

0, 0 d r 2,

u

r

3

cos3 M 2sin3 M 3sinM

u

r

2

4cos3 M 3sin3 M cosM

ɪɹɸɳɟɣ ɭɪɚɜɧɟɧɢɸ ɫɬɚɰɢɨɧɚɪɧɨɣ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ

'T

w2T w2T

0

(4)

wx2

wy2

ɢ ɞɜɭɦ ɩɚɪɚɦ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ

T

x 0

0, T

x a

f y ,

(5)

T

y

0

0, T

y b

0.

(6)

Ɋɟɲɟɧɢɟ ɢɳɟɦ ɦɟɬɨɞɨɦ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ

T x, y

X x Y y .

(7)

ɉɨɞɫɬɚɜɥɹɹ (7) ɜ ɭɪɚɜɧɟɧɢɟ (4), ɩɨɥɭɱɢɦ ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ

X x

ɭɪɚɜɧɟɧɢɟ

cc

2

X(x)

0,

(8)

X (x) O

ɪɟɲɟɧɢɟ ɤɨɬɨɪɨɝɨ

X (x)

AeOx BeOx .

(9)

cc

2

0 ,

(10)

Y (y) O Y(y)

Y(0) Y(b) 0 .

(11)

Ɋɟɲɚɹ ɭɪɚɜɧɟɧɢɟ (10), ɩɨɥɭɱɚɟɦ

Y(y)

C cosOy DsinOy .

(12)

Ʉɨɧɫɬɚɧɬɵ ɋ, D, O ɧɚɯɨɞɢɦ ɢɡ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ (5)

Y(0) C 0 , Y(b) DsinOb 0 .

Ɉɬɫɸɞɚ ɩɨɥɭɱɚɟɦ, ɭɱɢɬɵɜɚɹ, ɱɬɨ D z 0, Ob Sn ɢɥɢ

On

Sn ,

n 1,2,3,...

(13)

b

T

x, y

§

e

Snx

N

e

Snx

·

Sny

(14)

b

b

¨M

¸sin

b

n

©

n

n

¹

f

§

Snx

Nne

Snx

·

Sny

.

(15)

b

b

T x, y ¦¨M ne

¸sin

b

n 1

©

¹

Ʉɨɧɫɬɚɧɬɵ M n

ɢ Nn ɧɚɯɨɞɢɦ ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɱɬɨɛɵ ɧɚɣɞɟɧɧɨɟ ɪɟ-

ɲɟɧɢɟ (15) ɭɞɨɜɥɟɬɜɨɪɹɥɨ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ (5):

f

Sny

f

§

Sna

Sna

·

Sny

0 ¦ M n Nn sin

,

b

Nne

b

.

b

f y ¦¨M n e

¸sin

b

n 1

n 1

©

¹

­M

n

N

n

0,

°

Sna

Sna

b

S

.

®

b2 ³ f y sin

ny

e b

Nn e b

dy

°M n

b

¯

0

Sn

Sn

Ɉɬɫɸɞɚ, ɭɱɢɬɵɜɚɹ, ɱɬɨ sh Sn a

e b a e

b a

, ɧɚɯɨɞɢɦ

2

b

2

f

§b

Sny

·sh

Snx

Sny

b

T x, y

b

¦¨³ f y sin

b

dy¸

sin

b .

Sna

n 1

©0

¹sh

b

2

f

§b

Sny

·sh

Snx

Sny

b

Ɉɬɜɟɬ: T x, y

b

¦¨³ f y sin

b

dy¸

sin

b .

Sna

n 1

©0

¹sh

b