Контрольные работы по уравнениям математической физики. Ковтанюк А.Е. - 9 стр.

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13. u(0, t) = u(l, t) = 0,
u(x, 0) = sin
π
l
x, u
t
(x, 0) = sin
π
l
x + sin
3π
l
x.
14. u
x
(0, t) = u(l, t) = 0,
u(x, 0) = cos
π
2l
x + cos
3π
2l
x, u
t
(x, 0) = cos
3π
2l
x.
15. u(0, t) = u
x
(l, t) = 0,
u(x, 0) = sin
π
2l
x, u
t
(x, 0) = sin
π
2l
x + sin
3π
2l
x.
16. u
x
(0, t) = u
x
(l, t) = 0,
u(x, 0) = 2 + cos
π
l
x, u
t
(x, 0) = 1 + cos
2π
l
x.
17. u(0, t) = u(l, t) = 0,
u(x, 0) = sin
2π
l
x, u
t
(x, 0) = x.
18. u
x
(0, t) = u(l, t) = 0,
u(x, 0) = 0, u
t
(x, 0) = cos
3π
2l
x + cos
5π
2l
x.
19. u
x
(0, t) = u
x
(l, t) = 0,
u(x, 0) = 1 + cos
2π
l
x, u
t
(x, 0) = cos
π
l
x + cos
2π
l
x.
20. u(0, t) = u(l, t) = 0,
u(x, 0) = sin
2π
l
x + sin
3π
l
x, u
t
(x, 0) = sin
2π
l
x.
Задание 2. Решить методом разделения переменных следующую задачу для неоднород-
ного волнового уравнения:
1. u
tt
= a
2
u
xx
+ Ax + B, 0 < x < l, t > 0,
u(0, t) = U
1
, u(l, t) = U
2
,
u(x, 0) = U
1
(1 − l
−1
x) + U
2
l
−1
x, u
t
(x, 0) = 0,
а)A = 2, B = 1, U
1
= 1, U
2
= 0,
б)A = 1, B = 2, U
1
= 0, U
2
= 1,
в)A = 1, B = 0.
2. u
tt
= a
2
u
xx
+ Ax + B, 0 < x < l, t > 0,
u
x
(0, t) = 0, u(l, t) = U,
u(x, 0) = U, u
t
(x, 0) = V ,
а)A = 2, B = 1, U = 1, V = 0
б)A = 3, B = 1, U = 2, V = 1
в)A = 1, B = 0, U = 1, V = 2.
3. u
tt
= a
2
u
xx
+ Ax + B, 0 < x < l, t > 0,
u(0, t) = U, u
x
(l, t) = 0,
u(x, 0) = U, u
t
(x, 0) = V ,
а)A = 2, B = 1, U = 1, V = 0
б)A = 4, B = 1, U = 2, V = 1
в)A = 1, B = 0, U = 1, V = 2.
9
 13. u(0, t) = u(l, t) = 0,
     u(x, 0) = sin πl x, ut (x, 0) = sin πl x + sin 3πl x.

 14. ux (0, t) = u(l, t) = 0,
     u(x, 0) = cos 2lπ x + cos 3π
                               2l
                                  x, ut (x, 0) = cos 3π
                                                     2l
                                                        x.

 15. u(0, t) = ux (l, t) = 0,
     u(x, 0) = sin 2lπ x, ut (x, 0) = sin 2lπ x + sin 3π
                                                      2l
                                                         x.

 16. ux (0, t) = ux (l, t) = 0,
     u(x, 0) = 2 + cos πl x, ut (x, 0) = 1 + cos 2πl x.

 17. u(0, t) = u(l, t) = 0,
     u(x, 0) = sin 2πl x, ut (x, 0) = x.

 18. ux (0, t) = u(l, t) = 0,
     u(x, 0) = 0, ut (x, 0) = cos 3π
                                  2l
                                     x + cos 5π
                                             2l
                                                x.

 19. ux (0, t) = ux (l, t) = 0,
     u(x, 0) = 1 + cos 2πl x, ut (x, 0) = cos πl x + cos 2πl x.

 20. u(0, t) = u(l, t) = 0,
     u(x, 0) = sin 2πl x + sin 3πl x, ut (x, 0) = sin 2πl x.


   Задание 2. Решить методом разделения переменных следующую задачу для неоднород-
ного волнового уравнения:

  1. utt = a2 uxx + Ax + B, 0 < x < l, t > 0,
     u(0, t) = U1 , u(l, t) = U2 ,
     u(x, 0) = U1 (1 − l−1 x) + U2 l−1 x, ut (x, 0) = 0,
     а)A = 2, B = 1, U1 = 1, U2 = 0,
     б)A = 1, B = 2, U1 = 0, U2 = 1,
     в)A = 1, B = 0.

  2. utt = a2 uxx + Ax + B, 0 < x < l, t > 0,
     ux (0, t) = 0, u(l, t) = U ,
     u(x, 0) = U , ut (x, 0) = V ,
     а)A = 2, B = 1, U = 1, V = 0
     б)A = 3, B = 1, U = 2, V = 1
     в)A = 1, B = 0, U = 1, V = 2.

  3. utt = a2 uxx + Ax + B, 0 < x < l, t > 0,
     u(0, t) = U , ux (l, t) = 0,
     u(x, 0) = U , ut (x, 0) = V ,
     а)A = 2, B = 1, U = 1, V = 0
     б)A = 4, B = 1, U = 2, V = 1
     в)A = 1, B = 0, U = 1, V = 2.

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