Контрольные работы по уравнениям математической физики. Ковтанюк А.Е. - 11 стр.

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10. u
tt
= a
2
u
xx
+ (Ax + B)sint + Cx + D, 0 < x < l, t > 0,
u
x
(0, t) = 0, u(l, t) = U(t),
u(x, 0) = U(0), u
t
(x, 0) = V ,
а)A = 2, B = 1, C = 4, D = 0, U = const,
б)A = 1, B = 0, C = 2, D = 1, U = sint,
в)A = 2, B = 0, C = 1, D = 0, U = sint + 1,
г)A = 4, B = 1, C = 0, D = 0, U = sin2t − 1,
д)A = 2, B = 0, C = 0, D = 1, U = sin2t.
11. u
tt
= a
2
u
xx
+ (Ax + B)sint + Cx + D, 0 < x < l, t > 0,
u(0, t) = U(t), u
x
(l, t) = 0,
u(x, 0) = U(0), u
t
(x, 0) = V ,
а)A = 2, B = 1, C = 3, D = 1, U = const,
б)A = 1, B = 1, C = 0, D = 1, U = 2sint,
в)A = 4, B = 0, C = 1, D = 0, U = 2sint + 1,
г)A = 3, B = 2, C = 0, D = 0, U = sin2t + 1,
д)A = 2, B = 0, C = 0, D = 1, U = sin2t .
12. u
tt
= a
2
u
xx
+ (Ax + B)sint + (Cx + D)cos2t, 0 < x < l, t > 0,
u
x
(0, t) = u
x
(l, t) = 0,
u(x, 0) = 0, u
t
(x, 0) = V ,
а)A = 2, B = 1, C = 0, D = 1,
б)A = 1, B = 2, C = 1, D = 0,
в)A = 1, B = 0, C = 0, D = 2,
г)A = 3, B = 1, C = 2, D = 1,
д)A = 2, B = 0. C = 1, D = 1.
13. u
tt
= a
2
u
xx
+ (Ax + B)cost + Csinx + D, 0 < x < l, t > 0,
u(0, t) = U
1
(t), u(l, t) = U
2
(t),
u(x, 0) = l
−1
(U
2
(0) − U
1
(0))x + U
1
(0), u
t
(x, 0) = 0,
а)A = 2, B = 1, C = 3, D = 0, U
1
, U
2
= const,
б)A = 0, B = 1, C = 4, D = 1, U
1
= cost, U
2
= 2,
в)A = 0, B = 0, C = 0, D = 1, U
1
= sint, U
2
= cost,
г)A = 0, B = 0, C = 2, D = 1, U
1
= cost, U
2
= 1,
д)A = −1, B = 0, C = 1, D = 0, U
1
= sint, U
2
= lcost.
14. u
tt
= a
2
u
xx
+ (Ax + B)cost + Csinx + D, 0 < x < l, t > 0,
u
x
(0, t) = 0, u(l, t) = U(t),
u(x, 0) = U(0), u
t
(x, 0) = 0,
а)A = 2, B = 1, C = 3, D = 1, U = const,
б)A = 1, B = 0, C = 4, D = 1, U = cost,
в)A = 3, B = 0, C = 1, D = 0, U = 2cost + 1,
г)A = 2, B = 1, C = 0, D = 0, U = cos2t,
д)A = 2, B = 0, C = 1, D = 0, U = cos2t − 1.
11
10. utt = a2 uxx + (Ax + B)sint + Cx + D,        0 < x < l, t > 0,
    ux (0, t) = 0, u(l, t) = U (t),
    u(x, 0) = U (0), ut (x, 0) = V ,
   а)A = 2, B = 1, C = 4, D = 0, U = const,
   б)A = 1, B = 0, C = 2, D = 1, U = sint,
   в)A = 2, B = 0, C = 1, D = 0, U = sint + 1,
   г)A = 4, B = 1, C = 0, D = 0, U = sin2t − 1,
   д)A = 2, B = 0, C = 0, D = 1, U = sin2t.

11. utt = a2 uxx + (Ax + B)sint + Cx + D,        0 < x < l, t > 0,
    u(0, t) = U (t), ux (l, t) = 0,
    u(x, 0) = U (0), ut (x, 0) = V ,
   а)A = 2, B = 1, C = 3, D = 1, U = const,
   б)A = 1, B = 1, C = 0, D = 1, U = 2sint,
   в)A = 4, B = 0, C = 1, D = 0, U = 2sint + 1,
   г)A = 3, B = 2, C = 0, D = 0, U = sin2t + 1,
   д)A = 2, B = 0, C = 0, D = 1, U = sin2t .

12. utt = a2 uxx + (Ax + B)sint + (Cx + D)cos2t,        0 < x < l, t > 0,
    ux (0, t) = ux (l, t) = 0,
    u(x, 0) = 0, ut (x, 0) = V ,
   а)A = 2, B = 1, C = 0, D = 1,
   б)A = 1, B = 2, C = 1, D = 0,
   в)A = 1, B = 0, C = 0, D = 2,
   г)A = 3, B = 1, C = 2, D = 1,
   д)A = 2, B = 0. C = 1, D = 1.

13. utt = a2 uxx + (Ax + B)cost + Csinx + D, 0 < x < l, t > 0,
    u(0, t) = U1 (t), u(l, t) = U2 (t),
    u(x, 0) = l−1 (U2 (0) − U1 (0))x + U1 (0), ut (x, 0) = 0,
   а)A = 2, B = 1, C = 3, D = 0, U1 , U2 = const,
   б)A = 0, B = 1, C = 4, D = 1, U1 = cost, U2 = 2,
   в)A = 0, B = 0, C = 0, D = 1, U1 = sint, U2 = cost,
   г)A = 0, B = 0, C = 2, D = 1, U1 = cost, U2 = 1,
   д)A = −1, B = 0, C = 1, D = 0, U1 = sint, U2 = lcost.

14. utt = a2 uxx + (Ax + B)cost + Csinx + D,        0 < x < l, t > 0,
    ux (0, t) = 0, u(l, t) = U (t),
    u(x, 0) = U (0), ut (x, 0) = 0,
   а)A = 2, B = 1, C = 3, D = 1, U = const,
   б)A = 1, B = 0, C = 4, D = 1, U = cost,
   в)A = 3, B = 0, C = 1, D = 0, U = 2cost + 1,
   г)A = 2, B = 1, C = 0, D = 0, U = cos2t,
   д)A = 2, B = 0, C = 1, D = 0, U = cos2t − 1.


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