Уравнения математической физики. Уравнение колебаний струны. Хуснутдинов Н.Р. - 18 стр.

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F (x) = 0 , f(x) =
(
4l(1 − cos
πx
l
) , 0 ≤ x ≤ l/2
16(x−l)
2
l
, l/2 ≤ x ≤ l
.
G = g
x
2
l
2
, f(x) = 0 , F (x) =
x
2
h
sin
πx
l
.
G = g
x
l
cos
πvt
l
, F (x) = 0 , f(x) =
(
4l(1 − cos
πx
l
), 0 ≤ x ≤ 3l/4
64(x−l)
2
l
(1 +
√
2
2
),3l/4 ≤ x ≤ l
ψ = g
vt
l
e
−2πvt/l
φ = 0
, f(x) = 0 , F (x) = |x −
l
2
| sin
2πx
l
.
F (x) = 0 , f(x) =
0 , 0 ≤ x ≤ l/4
(x−l)(l/4−x)
h
, l/4 ≤ x ≤ l
.
f(x) = 0 , F (x) =
(
−
3x
2
l
, 0 ≤ x ≤ l/3
−
3(x−l)
2
4l
, l/3 ≤ x ≤ l
.
G = ge
x/l
, F (x) = 0 , f(x) =
0 , 0 ≤ x ≤ l/2
(x−l)(l/2−x)
h
, l/2 ≤ x ≤ l
.
G = g
x
l
−
6x
2
l
2
sin
3πvt
l
, f(x) = 0, F (x) =
(
−
3x
2
l
, 0 ≤ x ≤ l/5
−
3(x−l)
2
16l
, l/5 ≤ x ≤ l
φ = g
v
2
t
2
l
2
e
−3πvt/l
ψ = 0
, F (x) = 0, f(x) =
x(3l/4−x)
h
, 0 ≤ x ≤ 3l/4
0, 3l/4 ≤ x ≤ l
.
f(x) = 0 , F (x) =
x
2
h
sin
πx
l
.
F (x) = 0 , f(x) =
(
4l(1 − cos
πx
l
) , 0 ≤ x ≤ 3l/4
64(x−l)
2
l
(1 +
√
2
2
) , 3l/4 ≤ x ≤ l
.
G = ge
2x/l
, f(x) = 0 , F (x) =
(x−l)
2
h
sin
πx
2l
.
G = g
4x
l
cos
2πvt
l
, F (x) = 0, f(x) =
−
l
3
(1 − cos
πx
l
), 0 ≤ x ≤ l/2
−
4
3l
(x − l)
2
, l/2 ≤ x ≤ l
ψ = g
e
πvt/l
− 1
φ = 0
, f(x) = 0 , F (x) =
(x−l)
2
h
sin
πx
4l
.
F (x) = 0 , f(x) =
0 , 0 ≤ x ≤ l/2
(x−l)(l/2−x)
h
, l/2 ≤ x ≤ l
.
f(x) = 0 , F (x) =
(
3x
2
l
, 0 ≤ x ≤ l/4
(x−l)
2
3l
, l/4 ≤ x ≤ l
.
G = g
2x
l
+ 4
, F (x) = 0, f(x) =
0, 0 ≤ x ≤ l/4
(x−l/4)(3l/4−x)
h
, l/4 ≤ x ≤ 3l/4
0, 3l/4 ≤ x ≤ l
G = g
3x
l
− e
−
x
l
sin
2πvt
l
, f(x) = 0, F (x) =
3
l
x
2
, 0 ≤ x ≤ l/6
3
25l
(x − 2)
2
,l/6 ≤ x ≤ l
φ = g
vt
l
e
−
πvt
l
ψ = 0
, F (x) = 0, f(x) =
0, 0 ≤ x ≤ 3l/4
(x−l)(3l/4−x)
h
, 3l/4 ≤ x ≤ l
f(x) = 0 , F (x) =
(x−l)
2
h
sin
πx
2l
.
F (x) = 0 , f(x) =
(
4l(1 − cos
πx
l
) , 0 ≤ x ≤ l/3
9(x−l)
2
2l
, l/3 ≤ x ≤ l
.
G = g
4x
l
, f(x) = 0 , F (x) = |x −
l
2
| sin
2πx
l
.
G = g
x
3l
cos
3πvt
l
, F (x) = 0, f(x) =
−
l
3
(1 − cos
πx
l
), 0 ≤ x ≤ l/2
−
4
3l
(x − l)
2
, l/2 ≤ x ≤ l
ψ = g
v
2
t
2
l
2
e
2πvt/l
φ = 0
, f(x) = 0 , F (x) =
x(x−l)
h
cos
πx
4l
.
F (x) = 0 , f(x) =
0 , 0 ≤ x ≤ l/4
(x−l/4)(3l/4−x)
h
, l/4 ≤ x ≤ 3l/4
0 , 3l/4 ≤ x ≤ l
.
f(x) = 0 , F (x) =
(
−
3x
2
l
, 0 ≤ x ≤ l/5
−
3(x−l)
2
16l
, l/5 ≤ x ≤ l
.
G = g
2x
2
l
2
, F (x) = 0 , f(x) =
x(3l/4−x)
h
, 0 ≤ x ≤ 3l/4
0 , 3l/4 ≤ x ≤ l
.
                                      (
                                          4l(1 − cos πx
                                                      l ) ,        0 ≤ x ≤ l/2                                                        ÂÀÈÀÍÒ 5
2. F (x) = 0 , f (x) =                    16(x−l)2                             .
                                              l    ,               l/2 ≤ x ≤ l                                                 
                                                                                                                                     0,                 0 ≤ x ≤ l/2
3. G =
              2
          g xl2     , f (x) = 0 , F (x) =          x2
                                                        sin(πx                                       1. F (x) = 0 , f (x) =                                         .
                                                             l .
                                                                                                                                     (x−l)(l/2−x)
                                                   h
                                                                                                                                           h      , l/2 ≤ x    ≤l
                                                                 4l(1 − cos πxl ),  0 ≤ x ≤ 3l/4
                                                                                                                               (
                                                                                                                                     3x2
4. G = g xl cos                                                                                                                        l ,      0 ≤ x ≤ l/4
                      πvt
                       l    , F (x) = 0 , f (x) =                64(x−l)2
                                                                                √
                                                                                  2                  2. f (x) = 0 , F (x) =                                          .
                                                                     l    (1 + 2 ),3l/4 ≤ x ≤ l                                      (x−l)2
                                                                                                                                             , l/4 ≤ x ≤ l
                                                                                                                                        3l
     ψ = g vtl e−2πvt/l                                                                                                                           
5.                      , f (x) = 0 , F (x) = |x − 2l | sin 2πx
                                                             l .
                                                                                                                                                0,                     0 ≤ x ≤ l/4
     φ=0                                                                                             3. G = g    2x
                                                                                                                      
                                                                                                                 + 4 , F (x) = 0, f (x) =          (x−l/4)(3l/4−x)
                                                                                                                  l                                       h          ,  l/4 ≤ x ≤ 3l/4
                                                                                                                                                  0,                    3l/4 ≤ x ≤ l
                                                                                                                                               
                                            ÂÀÈÀÍÒ 3                                                                                                     3 2
                                                                                                                         x                                     x   ,         0 ≤ x ≤ l/6
                                                                                                     4. G = g 3x             sin 2πvt
                                                                                                                           
                                                                                                                      −l                                     l
                                                                                                              l −e                l , f (x) = 0, F (x) =      3
                                                                                                                                                                 (x  −  2)2
                                                                                                                                                                            ,l/6 ≤x≤l
                                          0,                     0 ≤ x ≤ l/4                                                                                 25l
1. F (x) = 0 , f (x) =                    (x−l)(l/4−x)                       .                               vt − πvt
                                                                                                                                              
                                                                                                                                                  0,                  0 ≤ x ≤ 3l/4
                                                          ,      l/4 ≤ x ≤ l                            φ=g l e l
                                                h                                                    5.                 , F (x) = 0, f (x) =      (x−l)(3l/4−x)
                                                                                                        ψ=0                                                      ,    3l/4 ≤x≤l
                                      (
                                            3x2
                                          − l ,           0 ≤ x ≤ l/3                                                                                   h
2. f (x) = 0 , F (x) =                             2                   .
                                          − 3(x−l)
                                               4l    ,    l/3 ≤ x ≤ l                                                                 ÂÀÈÀÍÒ 6
                                                     
                                                         0,              0 ≤ x ≤ l/2
3. G = gex/l , F (x) = 0 , f (x) =                       (x−l)(l/2−x)                .                                      (x−l)     2
                                                                                                                                       πx
                                                                                                     1. f (x) = 0 , F (x) =
                                                               h   ( , l/2 ≤ x ≤ l 2
                                                                                                                            ( h sin 2l .
            
                x       6x2
                                                    − 3xl ,     0 ≤ x ≤ l/5                                                   4l(1 − cos πx
                                                                                                                                           l ) ,             0 ≤ x ≤ l/3
4. G = g            −sin 3πvt
                            l , f (x) = 0, F (x) =            2                                      2. F (x) = 0 , f (x) =    9(x−l)2                                   .
                l        l2
                                                     − 3(x−l)   , l/5 ≤ x ≤ l                                                     2l    ,                    l/3 ≤ x ≤ l
                                              x(3l/4−x) 16l
        v 2 t2 −3πvt/l
   φ = g l2 e                                           , 0 ≤ x ≤ 3l/4                               3. G = g 4x                            l
                                                                                                               l , f (x) = 0 , F (x) = |x − 2 | sin
                                                                                                                                                             2πx
                                                                                                                                                              l .
5.                     , F (x) = 0, f (x) =        h                        .
                                                                                                                                                        − 3l (1 − cos πx
                                                                                                                                                    
   ψ=0                                          0,          3l/4 ≤ x ≤ l                                      x        3πvt                                              l ),   0 ≤ x ≤ l/2
                                                                                                     4. G = g 3l cos     l , F (x)   = 0, f (x) =          4
                                                                                                                                                        − 3l  (x − l)2 ,        l/2 ≤ x ≤ l
                                            ÂÀÈÀÍÒ 4                                                           2 2
                                                                                                          ψ = g vl2t e2πvt/l                                x(x−l)
                                                                                                     5.                      , f (x) = 0 , F (x) =            h      cos πx
                                                                                                                                                                         4l .
                       x     πx       2                                                                   φ=0
1. f (x) = 0 , F (x) =
                       (h sin l .
                          4l(1 − cos πx  ),
                                       l √                            0 ≤ x ≤ 3l/4                                                    ÂÀÈÀÍÒ 7
2. F (x) = 0 , f (x) =    64(x−l)2                                                 .
                              l    (1 + 22 ) ,                        3l/4 ≤ x ≤ l                                             
                                                        (x−l)2       πx
                                                                                                                                0,                          0 ≤ x ≤ l/4
             2x/l
3. G = ge             , f (x) = 0 , F (x) =               h     sin 2l .                            1. F (x) = 0 , f (x) =          (x−l/4)(3l/4−x)
                                                                                                                                                        ,    l/4 ≤ x ≤ 3l/4 .
                                                                                                                                            h
                                                                 − 3l (1 − cos πx l ), 0 ≤ x ≤ l/2
                                                                                                                            
                                                                                                                               0,                            3l/4 ≤ x ≤ l
4. G = g 4x
          l cos
                        2πvt
                          l , F (x)       = 0, f (x) =              4
                                                                 − 3l  (x − l)2 ,      l/2 ≤ x ≤ l                          (       2
                                                                                                                               − 3xl ,               0 ≤ x ≤ l/5
   ψ = g eπvt/l − 1                                                                                  2. f (x) = 0 , F (x) =                                      .
                                  
                                                                    (x−l)2         πx                                                   2
5.                                    , f (x) = 0 , F (x) =           h      sin   4l .                                        − 3(x−l)   ,          l/5 ≤ x ≤ l
   φ=0                                                                                                                              16l 
                                                                                                                                                    x(3l/4−x)
                                                                                                                 2                                              ,        0 ≤ x ≤ 3l/4
                                                                                                     3. G = g 2x
                                                                                                               l2 , F (x) = 0 , f (x) =
                                                                                                                                                        h                             .
                                                                                                                                                 0,                      3l/4 ≤ x ≤ l


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