Типовые

21

Задача 17. Решить смешанную задачу.

17.1.

utt=9uxx;

u(0,t)=─8

,

u(2,t)=2;

u(x,0)=sin6πx─8+5x,

ut(x,0)=0.

17.2.

utt=9uxx;

u(0,t)=7 ,

u(1,t)=2;

u(x,0)= 7─5x,

ut(x,0)=12πsin4πx

17.3.

utt=9uxx;

u(0,t)=─6

,

u(3,t)=6;

u(x,0)=3sin3πx─6+4x,

ut(x,0)=0.

17.4.

utt=9uxx;

u(0,t)= 5t ,

u(2,t)=─3t;

u(x,0)=0,

ut(x,0)=12πsin4πx+5─4x

17.5.

utt=16uxx;

u(0,t)=─4,

u(1,t)=─1;

u(x,0)=5sin2πx─4+3x, ut(x,0)=0.

17.6.

utt=16uxx;

u(0,t)=3 ,

u(2,t)=7 ;

u(x,0)=3+2x, ut(x,0)=12πsin3πx

17.7.

utt=16uxx;

u(0,t)=─2

,

u(3,t)=1;

u(x,0)=7sin4πx─2+x, ut(x,0)=0.

17.8.

utt=16uxx;

u(0,t)=t ,

u(2,t)=─t;

u(x,0)=,

ut(x,0)=28πsin7πx+1─x

17.9.

utt=25uxx;

u(0,t)=─1,

u(1,t)=─3;

u(x,0)=9sin3πx─1─2x, ut(x,0)=0.

17.10.

utt=25uxx;

u(0,t)=3 ,

u(2,t)=─5;

u(x,0)= 3─4x,

ut(x,0)=20πsin4πx

17.11.

utt=25uxx;

u(0,t)=─5

,

u(1,t)=1;

u(x,0)=11sin3πx─5+6x, ut(x,0)=0.

17.12.

utt=25uxx;

u(0,t)= 7 ,

u(2,t)=─3;

u(x,0)=0,

ut(x,0)=10πsin2πx+7─5x

17.13.

utt=36uxx;

u(0,t)=─9

,

u(3,t)=3;

u(x,0)=13sin3πx─9+4x, ut(x,0)=0.

17.14.

utt=36uxx;

u(0,t)=8 ,

u(2,t)=2;

u(x,0)=8─3x,

ut(x,0)=18πsin3πx

17.15.

utt=36uxx;

u(0,t)=─6

,

u(3,t)=0;

u(x,0)=15sin2πx─6+2x, ut(x,0)=0.

17.16.

utt=36uxx;

u(0,t)=4t

,

u(4,t)=8;

u(x,0)=0,

ut(x,0)=24πsin4πx+4+x

17.17.

utt=49uxx;

u(0,t)=─2

,

u(3,t)=─5;

u(x,0)=17sin3πx─2─x, ut(x,0)=0.

17.18.

utt=49uxx;

u(0,t)= 3 ,

u(2,t)=─1;

u(x,0)= 3─2x,

ut(x,0)=35πsin5πx

17.19.

utt=49uxx;

u(0,t)=─1

,

u(1,t)=─4;

u(x,0)=19sin7πx─1─3x,

ut(x,0)=0.

17.20.

utt=49uxx;

u(0,t)=2t

,

u(2,t)=─6t;

u(x,0)=0,

ut(x,0)=28πsin4πx+2─4x

17.21.

utt=64uxx;

u(0,t)=─4

,

u(1,t)=─9;

u(x,0)=─21sin3πx─4─5x, ut(x,0)=0.

17.22.

utt=64uxx;

u(0,t)=2 ,

u(3,t)= ─7 ;

u(x,0)=2─3x, ut(x,0)=24πsin3πx

17.23.

utt=64uxx;

u(0,t)=-3

,

u(1,t)=1;

u(x,0)=23sin2πx─3+4x,

ut(x,0)=0.

17.24.

utt=64uxx;

u(0,t)=4t

,

u(2,t)=─6t;

u(x,0)= 0,

ut(x,0)=32πsin4πx+4─5x

17.25.

utt=81uxx;

u(0,t)=─5

,

u(3,t)=1;

u(x,0)=25sin3πx─5+2x,

ut(x,0)=0.

17.26.

utt=81uxx;

u(0,t)=6 ,

u(4,t)=─2;

u(x,0)= 6─2x,

ut(x,0)=27πsin3πx

17.27.

utt=81uxx;

u(0,t)=─7

,

u(3,t)=2;

u(x,0)=27sin2πx─7+3x,

ut(x,0)=0.

17.28.

utt=81uxx;

u(0,t)=8t

,

u(4,t)=─4t;

u(x,0)=0,

ut(x,0)=27πsin3πx+8─3x

17.29.

utt=4uxx;

u(0,t)=─9

,

u(2,t)=1;

u(x,0)=29sin4πx─9+5x, ut(x,0)=0.

17.30.

utt=4uxx;

u(0,t)=9 ,

u(3,t)=─3;

u(x,0)=9─4x,

ut(x,0)=10πsin5πx

17.31.

utt=4uxx;

u(0,t)=─1

,

u(2,t)=5;

u(x,0)=31sin2πx─1+3x,

ut(x,0)=0.

18.1.

utt=uxx+65e─3tsin(x)

18.2.

utt=

1

uxx+3sin(2t)sin(2x)

4

18.3.

utt=uxx+16cos(8t)sin(8x)

18.4. utt=

1

uxx+8sin(3t)sin(3x)

9

1

uxx+50e─7tsin(4x)

1

18.5.

utt=

18.6.

utt=

uxx+3cos(2t)sin(5x)

16

25

18.7.

utt=4uxx+28cos(14t)sin(7x)

18.8.

utt=

1

uxx+8cos(3t)sin(6x)

36

1

uxx+37e─6tsin(7x)

1

18.9.

utt=

18.10. utt=

uxx+15sin(4t)sin(8x)

49

64

18.11. utt=9uxx+36cos(18t)sin(6x)

18.12. utt=

1

uxx+15cos(4t)sin(9x)

81

18.13. utt=uxx+26e─5tsin(x)

18.14. utt=

1

uxx+24sin(5t)sin(2x)

4

18.15. utt=16uxx+40cos(20t)sin(5x)

18.16. utt=

1

uxx+24cos(5t)sin(3x)

9

1

uxx+17e─4tsin(4x)

1

18.17. utt=

18.18. utt=

uxx+35sin(6t)sin(5x)

16

25

18.19. utt=25uxx+40cos(20t)sin(4x)

18.20. utt=

1

uxx+35cos(6t)sin(5x)

36

1

uxx+10e─3tsin(7x)

1

18.21. utt=

18.22. utt=

uxx+48sin(7t)sin(8x)

49

64

18.23. utt=36uxx+36cos(18t)sin(3x)

18.24. utt=

1

uxx+48cos(7t)sin(9x)

81

18.25. utt=uxx+5e─2tsin(x)

18.26. utt=

1

uxx+63sin(8t)sin(2x)

4

18.27. utt=49uxx+28cos(14t)sin(2x)

18.28. utt=

1

uxx+63cos(8t)sin(3x)

9

18.29. utt=

1

uxx+2e─tsin(4x)

18.30. utt=

1

uxx+80sin(9t)sin(5x)

16

25

23

Задача 19. Решить задачу Дирихле для уравнения Лапласа в круге.

19.1

Du=0, 0£r<1

19.2

Du=0,

0£r <2

19.3

Du=0, 0£r<1

u½

=2cos3j

u

2=sin3j

u½

=2sin3j

r=1

r=

r=1

19.4

Du=0, 0£r<2

19.5

Du=0,

0£r<3

19.6

Du=0, 0£r<1

u

2=cos3j

u½

r =3

=2sin3j

u

1=4sin3j

r=

r=

19.7

Du=0, 0£r<4

19.8

Du=0,

0£r<2

19.9

Du=0, 0£r<1

u½

r =4

=2cos3j

u

2=3sin3j

u½

r =1

=8sin3j

r=

19.10

Du=0, 0£r<2

19.11

Du=0,

0£r<1

19.12

Du=0, 0£r<2

u

=5cos3j

u½

r =1

=7cos3j

u

=9sin3j

r=2

r=2

19.13

Du=0, 0£r<2

19.14

Du=0,

0£r<2

19.15

Du=0, 0£r<3

u½

=12sin3j

u

=4cos3j

u½

=5cos3j

r=2

r=2

r=3

19.16

Du=0, 0£r<3

19.17

Du=0,

0£r<2

19.18

Du=0, 0£r<1

u

=2 cos3j

u½

r =2

=6sin3j

u

=14sin3j

r=3

r=1

19.19

Du=0, 0£r<2

19.20

Du=0,

0£r<1

19.21

Du=0, 0£r<1

u½

r =2

= 5sin3j

u

=14cos3j

u½

r =1

=11cos3j

r=1

19.22

Du=0, 0£r<3

19.23

Du=0,

0£r<4

19.24

Du=0, 0£r<1

u =sin3j

u½

r =4

=5 sin3j

u

=17 cos3j

r=3

r=1

19.25

Du=0, 0£r<2

19.26

Du=0,

0£r<1

19.27

Du=0, 0£r<2

u½

=14cos3j

u

=15sin3j

u½

r =2

=21 sin3j

r=2

r=1

19.28

Du=0, 0£r<1

19.29

Du=0,

0£r<2

19.30

Du=0, 0£r<4

u

=9cos3j

u½

r =2

=19cos3j

u

=17sin3j

r=1

r=4

24

Задача 20. Решить краевую задачу для уравнения Пуассона в кольце.

20.1

uxx+uyy=

x 2

- y 2

,

1£r£2,

20.2

uxx+uyy=

x 2

- y 2

,

1/2£r£1,

x 2

+ y 2

x 2

+ y 2

u½r =1=0,

¶u

r = 2

= 0

u½r =1/2=0,

¶u

r =

= 1

¶r

¶r

1

20.3

uxx+uyy=

x 2

- y 2

,

1£r£2,

20.4

uxx+uyy=

x 2

- y 2

,

1£r£2,

x 2

+ y 2

x 2

+ y 2

u½r =1=1,

¶u

r = 2

= 0

u½r =1=1, ¶u

= 1

¶r

¶r

r = 2

20.5

uxx+uyy=

x 2

- y 2

,

1/2£r£1,

20.6

uxx+uyy=

x 2

- y 2

,

1/2£r£1,

x 2

+ y 2

x 2

+ y 2

u½r =1/2=1,

¶u

= 1

u½r =1/2=2,

¶u

r =

= 0

¶r

r = 1

¶r

1

20.7

uxx+uyy=

x 2

- y 2

,

1£r£2,

20.8

uxx+uyy=

x 2

- y 2

,

1£r£2,

x 2

+ y 2

x 2

+ y 2

u½r =1=0,

¶u

r = 2

= 2

u½r =1=2, ¶u

= 0

¶r

¶r

r = 2

20.9

uxx+uyy=

x 2

- y 2

,

1/2£r£1,

20.10

uxx+uyy=

x 2

- y 2

,

1/2£r£1,

x 2

+ y 2

x 2

+ y 2

u½r =1/2=0,

¶u

= 0

u½r =1/2=1,

¶u

r =

= 0

¶r

r = 1

¶r

1

20.11

uxx+uyy=

x 2

- y 2

,

1/2£r£1,

20.12

uxx+uyy=

x 2

- y 2

,

1£r£3,

x 2

+ y 2

x 2

+ y 2

u½r =1/2=0,

¶u

= 2

u½r =1=1, ¶u

= 0

¶r

r = 1

¶r

r = 3

20.13

uxx+uyy=

x 2

- y 2

,

1£r£3,

20.14

uxx+uyy=

x 2

- y 2

,

1£r£3,

x 2

+ y 2

x 2

+ y 2

u½r =1=0,

¶u

r = 3

= 1

u½r =1=1, ¶u

= 1

¶r

¶r

r = 3

20.15

uxx+uyy=

x 2

- y 2

,

1£r£3,

20.16

uxx+uyy=

x 2

- y 2

,

1£r£3,

x 2

+ y 2

x 2

+ y 2

u½r =1=1,

¶u

r = 3

= 2

u½r =1=2, ¶u

= 1

¶r

¶r

r = 3

26

Задание 21. Решить задачу для уравнения Пуассона в шаровом слое.

21.1

uxx+uyy+uzz = xz,

1£ r £ 2,

21.2

uxx+uyy+uzz = xz,

1£ r £ 2,

u½r =1=0,

u½r =2=0

u½r =1=0,

u½r =2=1

21.3

uxx+uyy+uzz = xz,

1£ r < 2,

21.4

uxx+uyy+uzz = xz,

1< r < 2,

u½r =1=1,

u½r =2=0

u½r =1=21,

u½r =2=1

21.5

uxx+uyy+uzz = xz,

1< r < 2,

21.6

uxx+uyy+uzz = xz,

1< r < 2,

u½r =1=1,

u½r =2=2

u½r =1=2,

u½r =2=1

21.7

uxx+uyy+uzz = xz,

1< r < 2,

21.8

uxx+uyy+uzz = xz,

1< r < 2,

u½r =1=2,

u½r =2=2

u½r =1=3,

u½r =2=1

21.9

uxx+uyy+uzz = xz,

1/2£ r £ 1,

21.10

uxx+uyy+uzz = xz,

1/2<r<1,

u½r =1/2=0,

u½r =1= -1

u½r =1/2=1,

u½r =1=0

21.11

uxx+uyy+uzz = xz,

1/2£ r < 1,

21.12

uxx+uyy+uzz = xz,

1/2<r<1,

u½r =1/2=1,

u½r =1=1

u½r =1/2=0,

u½r =1=0

21.13

uxx+uyy+uzz = xz,

1/2< r < 1,

21.14

uxx+uyy+uzz = xz,

1/2<r<1,

u½r =1/2= -1,

u½r =1=0

u½r =1/2= -1,

u½r =1= -1

21.15

uxx+uyy+uzz = xz,

1/2< r < 1,

21.16

uxx+uyy+uzz = xz,

1/2<r<1,

u½r =1/2=1,

u½r =1= -1

u½r =1/2= -1,

u½r =1=1

21.17

uxx+uyy+uzz = xz,

1/2< r < 1,

21.18

uxx+uyy+uzz = xz,

2< r < 3,

u½r =1/2=1,

u½r =1= -2

u½r =2=0,

u½r =3=0

21.19

uxx+uyy+uzz = xz,

2< r < 3,

21.20

uxx+uyy+uzz = xz,

2< r < 3,

u½r =2=0,

u½r =3= -1

u½r =2= -1,

u½r =3=0

21.21

uxx+uyy+uzz = xz,

2< r < 3,

21.22

uxx+uyy+uzz = xz,

2< r < 3,

u½r =2=1,

u½r =3=3

u½r =2=1,

u½r =3= -3

21.23

uxx+uyy+uzz = xz,

2< r < 3,

21.24

uxx+uyy+uzz = xz,

2< r < 3,

u½r =2=0,

u½r =3=4

u½r =2=2,

u½r =3=0

21.25

uxx+uyy+uzz = xz,

2< r < 3,

21.26

uxx+uyy+uzz = xz,

1< r < 3,

u½r =2=1,

u½r =3= -2

u½r =1=1,

u½r =1=1

21.27

uxx+uyy+uzz = xz,

1< r < 3,

21.28

uxx+uyy+uzz = xz,

1< r < 3,

u½r =1=0,

u½r =3=1

u½r =1=1,

u½r =3=0

21.29

uxx+uyy+uzz = xz,

1< r < 3,

21.30

uxx+uyy+uzz = xz,

1< r < 3,

u½r =1= -1,

u½r =3=1

u½r =1=1,

u½r =3= -1

21.31

uxx+uyy+uzz = xz,

1< r < 3,

u½r =1=2,

u½r =3=1

22.1

Du+k2u=0,

0£ r < 1

22.2

Du+k2u=0,

0£ r < 1,5

u½

r =1

=25sin3j

u½

r =1,5

=24sin3j

22.3

Du+k2u=0,

0£ r < 2

22.4

Du+k2u=0,

0£ r < 2,5

u½

r =2

=23sin3j

u½

=22sin3j

r =2,5

22.5

Du+k2u=0,

0£ r < 20

22.6

Du+k2u=0,

0£ r < 3

u½

r =20

=21sin3j

u½

r =3

=20sin3j

22.7

Du+k2u=0,

0£ r < 4

22.8

Du+k2u=0,

0£ r < 15

u½

r =4

=19sin3j

u½

r =15

=18sin3j

22.9

Du+k2u=0,

0£ r < 3,5

22.10

Du+k2u=0,

0£ r < 4,5

u½

r =3,5

=17sin3j

u½

r =4,5

=16sin3j

22.11