Контрольные работы по уравнениям математической физики. Ковтанюк А.Е. - 10 стр.

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4. u
tt
= a
2
u
xx
+ Ax + B, 0 < x < l, t > 0,
u
x
(0, t) = 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 = 1, B = 1, U = 2, V = 1
в)A = 1, B = 0, U = 1, V = 2.
5. u
tt
= a
2
u
xx
+ Asinx + 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) = V ,
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.
6. u
tt
= a
2
u
xx
+ Acosx + 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 = 3, B = 1, U = 1, V = 0
б)A = 1, B = 2, U = 2, V = 3
в)A = 1, B = 0, U = 1, V = 2.
7. u
tt
= a
2
u
xx
+ Asinx + 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 = 1, B = 3, U = 1, V = 0
б)A = 2, B = 1, U = 2, V = 1
в)A = 1, B = 0, U = 1, V = 2.
8. u
tt
= a
2
u
xx
+ Acosx + B, 0 < x < l, t > 0,
u
x
(0, t) = u
x
(l, t) = 0,
u(x, 0) = U, u
t
(x, 0) = V ,
а)A = 3, B = 1, U = 1, V = 0
б)A = 1, B = 2, U = 2, V = 3
в)A = 1, B = 0, U = 1, V = 2.
9. u
tt
= a
2
u
xx
+ (Ax + B)sint + Cx + 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) = V ,
а)A = 2, B = 1, C = 4, D = 3, U
1
, U
2
= const,
б)A = 0, B = 2, C = 2, D = 1, U
1
= sint, U
2
= 1,
в)A = 0, B = 0, C = 0, D = 1, U
1
= sint, U
2
= cost,
г)A = 1, B = 0, C = 2, D = 1, U
1
= sint, U
2
= 2,
д)A = 0, B = −1, C = 1, D = 0, U
1
= cost, U
2
= lsint.
10
4. utt = a2 uxx + Ax + B, 0 < x < l, t > 0,
   ux (0, t) = ux (l, t) = 0,
   u(x, 0) = U , ut (x, 0) = V ,
   а)A = 2, B = 1, U = 1, V = 0
   б)A = 1, B = 1, U = 2, V = 1
   в)A = 1, B = 0, U = 1, V = 2.

5. utt = a2 uxx + Asinx + 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) = V ,
   а)A = 2, B = 1, U1 = 1, U2 = 0,
   б)A = 1, B = 2, U1 = 0, U2 = 1,
   в)A = 1, B = 0.

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

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

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

9. utt = a2 uxx + (Ax + B)sint + Cx + 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) = V ,
   а)A = 2, B = 1, C = 4, D = 3, U1 , U2 = const,
   б)A = 0, B = 2, C = 2, D = 1, U1 = sint, U2 = 1,
   в)A = 0, B = 0, C = 0, D = 1, U1 = sint, U2 = cost,
   г)A = 1, B = 0, C = 2, D = 1, U1 = sint, U2 = 2,
   д)A = 0, B = −1, C = 1, D = 0, U1 = cost, U2 = lsint.



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