Уравнения математической физики. Уравнение колебаний струны. Хуснутдинов Н.Р. - 20 стр.

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F (x) = 0 , f(x) =
0 , 0 ≤ x ≤ l/4
(x−l/4)(l/2−x)
h
, l/4 ≤ x ≤ l/2
0 , l/2 ≤ x ≤ l
.
f(x) = 0 , F (x) =
3
l
x
2
, 0 ≤ x ≤ l/8
3
49l
(x − l)
2
, l/8 ≤ x ≤ l
.
G = g
x
l
+ e
−
x
l
, F (x) = 0, f(x) =
0, 0 ≤ x ≤ l/8
(x−
l
8
)(
l
2
−x)
h
, l/8 ≤ x ≤ l/2
0, l/2 ≤ x ≤ l
G = g
x
l
+ 3
2
cos
πvt
l
, f(x) = 0, F (x) =
1
7l
x
2
, 0 ≤ x ≤ l/2
1
7l
(x − l)
2
, l/2 ≤ x ≤ l
φ = g
vt
l
e
−
πvt
l
− e
πvt
l
ψ = 0
, F (x) = 0, f(x) =
(
x(
l
8
−x)
h
, 0 ≤ x ≤ l/8
0, l/8 ≤ x ≤ l
f(x) = 0 , F (x) =
x(x−l)
h
cos
2πx
l
.
F (x) = 0 , f(x) =
4l(1 − cos
πx
l
) , 0 ≤ x ≤ l/5
25
4l
(1 − cos
π
5
)(x − l)
2
, l/5 ≤ x ≤ l
.
G = ge
x/4l
, f(x) = 0 , F (x) =
x(x−l)
h
cos
πx
l
.
G = g
3x
2l
sin
πvt
l
, F (x) = 0, f(x) =
4l(1 − cos
πx
l
), 0 ≤ x ≤ l/6
9
25l
(1 −
√
3
2
)(x − l)
2
,l/6 ≤ x ≤ l
ψ = g
e
−2πvt/l
− e
2πvt/l
φ = 0
, f(x) = 0 , F (x) = (x − l) sin
πx
2l
.
F (x) = 0 , f(x) =
0 , 0 ≤ x ≤ l/8
(x−l/8)(l/2−x)
h
, l/8 ≤ x ≤ l/2
0 , l/2 ≤ x ≤ l
.
f(x) = 0 , F (x) =
3
l
x
2
, 0 ≤ x ≤ l/9
3
64l
(x − l)
2
, l/9 ≤ x ≤ l
.
G = g
1 −
4x
l
2
, F (x) = 0, f(x) =
0, 0 ≤ x ≤ l/8
(x−
l
8
)(
l
4
−x)
h
, l/8 ≤ x ≤ l/4
0, l/4 ≤ x ≤ l
G = g
x
l
+ e
−
x
l
cos
3πvt
l
, f(x) = 0, F (x) =
1
7l
x
2
, 0 ≤ x ≤ l/3
1
28l
(x − l)
2
,l/3 ≤ x ≤ l
φ = g
vt
l
e
2πvt
l
ψ = 0
, F (x) = 0, f(x) =
(
0, 0 ≤ x ≤ l/8
(x−l)(
l
8
−x)
h
, l/8 ≤ x ≤ l
f(x) = 0 , F (x) =
x(x−l)
h
cos
πx
l
.
F (x) = 0 , f(x) =
−7l(1 − cos
πx
l
) , 0 ≤ x ≤ l/3
−
63
18l
(x − l)
2
, l/3 ≤ x ≤ l
.
G = g
3x
2l
, f(x) = 0 , F (x) = |x −
l
2
| sin
πx
l
.
G = ge
x
l
cos
πvt
l
, F (x) = 0, f(x) =
−
l
6
(1 − cos
πx
l
), 0 ≤ x ≤ 2l/3
−
9
4l
(x − l)
2
, 2l/3 ≤ x ≤ l
ψ = g
e
−3πvt/l
− e
3πvt/l
φ = 0
, f(x) = 0 , F (x) = x sin
2πx
l
.
F (x) = 0 , f(x) =
0 , 0 ≤ x ≤ l/8
(x−l/8)(l/4−x)
h
, l/8 ≤ x ≤ l/4
0 , l/4 ≤ x ≤ l
.
f(x) = 0 , F (x) =
1
7l
x
2
, 0 ≤ x ≤ l/2
1
7l
(x − l)
2
, l/2 ≤ x ≤ l
.
G = g
3x
2
4l
2
, F (x) = 0 , f(x) =
x(l/8−x)
h
, 0 ≤ x ≤ l/8
0 , l/8 ≤ x ≤ l
.
G = g
e
x
l
+ 1
cos
2πvt
l
, f(x) = 0, F (x) =
1
7l
x
2
, 0 ≤ x ≤ l/4
1
63l
(x − l)
2
, l/4 ≤ x ≤ l
φ = g
vt
l
e
4πvt
l
ψ = 0
, F (x) = 0, f(x) =
(
x(x−
3l
8
)
h
, 0 ≤ x ≤ 3l/8
0, 3l/8 ≤ x ≤ l
f(x) = 0 , F (x) = |x −
l
2
| sin
πx
l
.
F (x) = 0 , f(x) =
4l(1 − cos
πx
l
) , 0 ≤ x ≤ l/6
9
25l
(1 −
√
3
2
)(x − l)
2
, l/6 ≤ x ≤ l
.
                                                                                                                                             1 2
                                     ÂÀÈÀÍÒ 13                                                                                                 7l x ,        0 ≤ x ≤ l/3
                                                                                                     x   x
                                                                                          4. G = g  + e− l cos 3πvt
                                                                                                     l            l  , f (x) = 0, F (x) =         1         2
                                                                                                                                                28l (x − l)  ,l/3 ≤x≤l
                                0,               0 ≤ x ≤ l/4                                                                  (
                                     (x−l/4)(l/2−x)
                                                                                                 vt 2πvt
                                                                                             φ=g l e l                            0,                 0 ≤ x ≤ l/8
1. F (x) = 0 , f (x) =                     h  , l/4 ≤ x ≤ l/2 .                           5.             , F (x) = 0, f (x) =     (x−l)( 8l −x)
                                                                                             ψ=0                                                , l/8 ≤ x ≤ l
                              0,                  l/2 ≤ x ≤ l
                                                                                                                                       h
                           3 2
                                x   ,          0 ≤  x ≤ l/8
2. f (x) = 0 , F (x) =        l
                                3        2                       .                                                      ÂÀÈÀÍÒ 16
                              49l (x − l) , l/8 ≤ x ≤ l
                                              0, l l               0 ≤ x ≤ l/8
                                                                                          1. f (x) = 0 , F (x) =
                                                                                                                   x(x−l)
                                                                                                                           cos πx
           x    − xl                             (x− 8 )( 2 −x)                                                      h          l .
3. G = g l + e
                     
                       , F (x) = 0, f (x) =                     , l/8 ≤ x ≤ l/2                                      −7l(1 − cos πx
                                                                                                                   
                                                                                                                                     l ) ,     0 ≤ x ≤ l/3
                                                       h
                                                 0,                 l/2 ≤ x ≤ l           2. F (x) = 0 , f (x) =                                               .
                                             
                                                                                                                         63
                                                    1 2                                                             − 18l  (x − l)2 ,         l/3 ≤ x ≤ l
                  2                                       x  ,         0 ≤ x ≤ l/2        3. G = g 3x                                 l       πx
4. G = g xl + 3 cos πvt                                                                            2l , f (x) = 0 , F (x) = |x −2 | sin l .
                
                                                      7l
                           l , f (x) = 0, F (x) =      1           2
                                                      7l (x − l) , l/2 ≤ x ≤ l                                                          − 6l (1 − cos πx l ), 0 ≤ x ≤ 2l/3
                                                                                                    x
                                                                                          4. G = ge l cos πvt
                                                                                                           l , F (x) = 0, f (x) =
                                                    (
                    πvt        πvt                            x( 8l −x)
     φ = g vtl e−    l    −e    l
                                                                        ,   0 ≤ x ≤ l/8                                                    9
                                                                                                                                        − 4l  (x − l)2 ,      2l/3 ≤ x ≤ l
5.                                     , F (x) = 0, f (x) =        h
                                                              0,            l/8 ≤ x ≤ l                           3πvt/l
                                                                                                      −3πvt/l
                                                                                                                         
     ψ=0                                                                                     ψ=g e            −e
                                                                                          5.                                 , f (x) = 0 , F (x) = x sin 2πx  l .
                                                                                             φ=0
                                     ÂÀÈÀÍÒ 14
                                                                                                                        ÂÀÈÀÍÒ 17
                               x(x−l)
1. f (x) = 0 , F (x) =
                               h cos 2πx
                                        l .                                                                        
                            4l(1 − cos πx                                                                           0,                       0 ≤ x ≤ l/8
2. F (x) = 0 , f (x) =                     l ) ,               0 ≤ x ≤ l/5
                                                                             .                                         (x−l/8)(l/4−x)
                            25
                                (1 −  cos  π
                                             )(x  −  l)2
                                                          ,    l/5 ≤ x ≤ l                1. F (x) = 0 , f (x) =             h           ,    l/8 ≤ x ≤ l/4 .
                             4l            5
                                                                                                                        0,                    l/4 ≤ x ≤ l
                                                                                                                    
                                          x(x−l)
3. G = gex/4l , f (x) = 0 , F (x) =                cos πxl .
                                            h
                                                                                                                     1 2
                                               4l(1   −  cos  πx
                                                                 ),     0 ≤ x ≤ l/6       2. f (x) = 0 , F (x) =        7l x ,          0 ≤ x ≤ l/2
                                                                                                                                                          .
                                                                                                                        1         2
4. G = g 3x
         2l sin πvt
                 l  , F (x) =  0, f (x)  =       9
                                                            √ l
                                                             3        2                                                 7l (x − l)  ,   l/2 ≤ x ≤ l
                                               25l (1 − 2 )(x − l) ,l/6 ≤ x ≤ l
                                                                                                                                    x(l/8−x)
                                                                                                       2                                         , 0 ≤ x ≤ l/8
   ψ = g e−2πvt/l − e2πvt/l                                                               3. G = g 3x
                                
                                                                                                    4l2 , F (x) = 0 , f (x) =                                       .
                                                                                                                                          h
5.                                  , f (x) = 0 , F (x) = (x − l) sin πx   2l .                                                       0,              l/8 ≤ x ≤ l
   φ=0                                                                                                                                          1 2
                                                                                                                                                   7l x ,         0 ≤ x ≤ l/4
                                                                                                       x
                                                                                          4. G = g e l + 1 cos 2πvt
                                                                                                              
                                                                                                                      l , f (x) = 0, F (x) =        1
                                                                                                                                                       (x − l)2
                                                                                                                                                                , l/4 ≤ x ≤ l
                                     ÂÀÈÀÍÒ 15                                                                                    (              63l
                                                                                                         4πvt                                 3l
                                                                                                                                        x(x− 8 )
                                                                                            φ = g vtl e l                                        , 0 ≤ x ≤ 3l/8
                                0,                0 ≤ x ≤ l/8                            5.                  , F (x) = 0, f (x) =          h
                                     (x−l/8)(l/2−x)                                          ψ=0                                        0,           3l/8 ≤ x ≤ l
1. F (x) = 0 , f (x) =                     h  , l/8 ≤ x ≤ l/2 .
                             0,                    l/2 ≤ x ≤ l
                         
                          3 2                                                                                          ÂÀÈÀÍÒ 18
                               x   ,           0 ≤  x ≤ l/9
2. f (x) = 0 , F (x) =       l
                               3         2                      .
                                 (x  − l)  ,   l/9  ≤  x≤l
                             64l                                                          1. f (x) = 0 , F (x) = |x − 2l | sin πx
                                                                                                                                l .
                                              0, l l             0 ≤ x ≤ l/8
                                             
                                                                                                                                  πx
                                                                                                                       4l(1 − cos    )   ,         0 ≤ x ≤ l/6
                                                                                                                   
                  2                                                                                                            √ l
3. G = g 1 − 4x      , F (x) =   0, f (x) =     (x− 8 )( 4 −x)
                                                               , l/8 ≤ x ≤ l/4            2. F (x) = 0 , f (x) =        9        3                             .
                l                                     h
                                                                                                                       25l (1 − 2 )(x    − l)2 ,   l/6 ≤ x ≤ l
                                                0,                l/4 ≤ x ≤ l
                                             


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