Уравнения математической физики. Уравнение теплопроводности. Хуснутдинов Н.Р. - 13 стр.

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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 3 l/8
0, 3l/8 x l
f(x) = T
|x
l
2
|
l
sin
πx
l
.
f(x) = T
4(1 cos
πx
l
) , 0 x l/6
9
25l
2
(1
3
2
)(x l)
2
, l/6 x l
.
G = g
2x
2
l
2
+ e
x/l
, f(x) = (
x
l
1) sin
πx
2l
.
G = ge
2x
l
sin
πvt
l
, F (x) = 0, f (x) =
4l(1 cos
πx
l
), 0 x l/7
49
9l
(1 cos
π
7
)(x l)
2
,l/7 x l
ψ = g
e
πvt/2l
e
πvt/2l
φ = 0
, f(x) = 0 , F (x) =
x(xl)
h
cos
πx
2l
.
f(x) = T
x(l/8x)
l
2
, 0 x l/8
0 , l/8 x l
.
f(x) = T
1
7l
2
x
2
, 0 x l/3
1
28l
2
(x l)
2
, l/3 x l
.
G = g
1 e
2x
l
, f(x) = T
(
0, 0 x l/8
(xl)(
l
8
x)
l
2
, l/8 x l
G = g
3x
2
l
2
x
l
cos
2πvt
l
, f(x) = 0, F (x) =
1
7l
x
2
, 0 x 2l/3
4
7l
(x l)
2
,2l/3 x l
φ = g
vt
l
e
πvt
2l
ψ = 0
, F (x) = 0, f(x) =
(
0, 0 x 3l/8
(xl)(x
3l
8
)
h
, 3l/8 x l
f(x) = T
xl
l
sin
πx
2l
.
f(x) = T
1
6
(1 cos
πx
l
) , 0 x 2l/3
9
4l
2
(x l)
2
, 2l/3 x l
.
G = g
x
l
6x
2
l
2
, f(x) =
x
l
sin
2πx
l
.
G = g
1
x
l
2
cos
3πvt
l
, F (x) = 0,f(x) =
2l(1 cos
πx
l
), 0 x 2l/3
27
l
(x l)
2
, 2l/3 x l
ψ = g
vt
l
e
πvt/3l
φ = 0
, f(x) = 0 , F (x) = (x l) sin
2πx
l
.
f(x) = T
0 , 0 x l/8
(xl)(l/8x)
l
2
, l/8 x l
.
f(x) = T
1
7l
2
x
2
, 0 x l/4
1
63l
2
(x l)
2
, l/4 x l
.
G = g
x
l
1
, f(x) = T
x(x3l/8)
l
2
, 0 x 3l/8
0 , 3l/8 x l
.
G = g
x
l
1
sin
2πvt
l
, f(x) = 0, F (x) =
4l(1 cos
πx
l
),0 x 2l/3
54
l
(x l)
2
, 2l/3 x l
φ = g
e
πvt
4l
e
πvt
4l
ψ = 0
, F (x) = 0, f(x) =
(
0, 0 x l/4
(xl)(
l
4
x)
h
, l/4 x l
f(x) = T
x
l
sin
2πx
l
.
f(x) = T
4(1 cos
πx
l
) , 0 x l/7
49
9l
2
(1 cos
π
7
)(x l)
2
, l/7 x l
.
G = g
x
l
+ 3
2
, f(x) = T
x(xl)
l
2
cos
πx
2l
.
G = g
3x
2
4l
2
sin
3πvt
l
, F (x) = 0, f(x) =
(
4l(1 cos
πx
l
), 0 x l/2
16(xl)
2
l
, l/2 x l
ψ = g
v
2
t
2
l
2
e
πvt/4l
φ = 0
, f(x) = 0 , F (x) =
x
2
h
sin
πx
l
.
                                                 1 2                                                              
                                                     7l x ,         0 ≤ x ≤ l/4                       x       6x2                   x
           x
   G = g e l + 1 cos 2πvt                                                             3.   G=g            −             , f (x) =       sin 2πx
                                                                                                                                             l .
                
4.
                        l , f (x) = 0, F (x) =        1         2                                     l        l2                   l
                                                         (x − l)  , l/4 ≤ x ≤ l
                                                     63l                                                                                               2l(1 − cos πx
                                                                                                                                                      
                                                                                                          x 2  3πvt                                                  l ), 0 ≤ x ≤ 2l/3
                                     (                                                                     
        vt 4πvt
   φ=g l e l                              x(x− 3l
                                               8  )
                                                    , 0 ≤ x ≤ 3l/8                    4.   G =g 1−        l cos l , F (x)               = 0,f (x) =    27        2
5.              , F (x) = 0, f (x) =         h                                                                                                          l (x − l)  ,     2l/3 ≤ x ≤ l
   ψ=0                                    0,           3l/8 ≤ x ≤ l                        ψ = g vtl e−πvt/3l
                                                                                      5.                      , f (x) = 0 , F (x) = (x − l) sin 2πx
                                                                                                                                                 l .
                                                                                           φ=0
                                         18

                                                                                                                                          21
                 |x− 2l |
1.   f (x) = T          sin πx
                             l .
                 l                                                                                       
                                                                                                              0,        0 ≤ x ≤ l/8
                     4(1 − cos√πxl ) ,        0 ≤ x ≤ l/6                             1.   f (x) = T                                   .
2. f (x) = T          9          3      2                    .                                                (x−l)(l/8−x)
                                                                                                                    , l/8 ≤ x ≤ l
                     25l2 (1
                           − 2 )(x − l) ,    l/6 ≤ x ≤ l                                                     1 2
                                                                                                                    l2
                                                                                                                  x   , 0 ≤ x ≤ l/4
           2
            2x
3. G = g
             l2 + e
                    x/l
                         , f (x) = ( xl − 1) sin πx
                                                 2l .                                 2. f (x) = T            7l2
                                                                                                               1                       .
                                                                                                              63l2 (x  − l)2
                                                                                                                    , l/4 ≤ x ≤ l
                                            4l(1 − cos πx
                                           
                                                        l ),      0 ≤ x ≤ l/7
           2x
                                                                                                                      x(x−3l/8)
                 πvt                                                                                                               , 0 ≤ x ≤ 3l/8
4. G = ge l sin
                  l , F (x) = 0, f (x) = 49 (1 − cos π )(x − l)2 ,l/7 ≤ x ≤ l                    x
                                                                                                       
                                                                                      3. G = g
                                                                                                 l − 1 , f (x) = T                                        .
                                                                                                                            l2
                                             9l         7                                                               0,             3l/8 ≤ x ≤ l
   ψ = g e−πvt/2l − eπvt/2l
                               
                                  , f (x) = 0 , F (x) = x(x−l)                                                                        4l(1 − cos πx
                                                                                                                                     
5.                                                             cos πx
                                                                    2l .                        x
                                                                                                           2πvt                                     l ),0 ≤ x ≤ 2l/3
                                                                                      4. G = g
                                                                                                 l − 1 sin l , f (x) = 0, F (x) = 54 (x − l)2 ,
   φ=0                                                       h
                                                                                                                                                         2l/3 ≤ x ≤ l
                                                                                                πvt       πvt
                                                                                                                                     (l
                                         19                                             φ=g e    − 4l
                                                                                                        − e 4l                           0,                 0 ≤ x ≤ l/4
                                                                                      5.                         , F (x) = 0, f (x) =     (x−l)( 4l −x)
                                                                                         ψ=0                                                   h        , l/4 ≤ x ≤ l
                      x(l/8−x)
                 
                                 ,
                               0 ≤ x ≤ l/8
1.   f (x) = T            l2                   .
                     0,        l/8 ≤ x ≤ l
                1 2                                                                                                                      22

2.   f (x) = T   7l2 x ,           0 ≤ x ≤ l/3
                                                   .
                  1          2
                 28l2 (x − l) ,    l/3 ≤ x ≤ l                                             f (x) = T x       2πx
                                    (                                                 1.
                                                                                                       l sin l .
                                       0,               0 ≤ x ≤ l/8                                        4(1 − cos πx  l ) ,               0 ≤ x ≤ l/7
                  2x
                      
3.   G = g 1 − e− l , f (x) = T        (x−l)( 8l −x)                                  2.   f (x) = T       49             π            2                    .
                                               l2  , l/8 ≤ x ≤ l                                           9l2 (1  − cos   7 )(x  −  l)  ,   l/7 ≤ x ≤ l
                                                     1 2                                                   2
                                                                                           G = g xl + 3 , f (x) = T x(x−l)                 πx
                                                                                                           
                                                          7l x ,     0 ≤ x ≤ 2l/3                                                      cos(2l .
          2           
             3x      x      2πvt                                                      3.
4. G = g
              l2 − l cos l , f (x) = 0, F (x) =
                                                                                                                                 l2
                                                          4         2
                                                             (x − l)  ,2l/3 ≤ x ≤ l                                                           4l(1 − cos πx
                                                          7l                                         2
                                                                                                                                                          l ), 0 ≤ x ≤ l/2
                                                                                           G = g 3x    sin 3πvt
                                                                                                                 , F (x)  =   0, f (x)  =
                                         (
                πvt                                                                   4.
   φ = g vtl e 2l                          0,                 0 ≤ x ≤ 3l/8                        4l 2       l                                16(x−l)2
                                                                                                                                                       ,       l/2 ≤ x ≤ l
5.                  , F (x) = 0, f (x) =   (x−l)(x− 3l )                                                                                          l
   ψ=0                                              8
                                                         , 3l/8 ≤ x ≤ l                            2 2
                                                h                                          ψ = g vl2t eπvt/4l                                         x2
                                                                                      5.                      , f (x) = 0 , F (x) =                   h    sin πx
                                                                                                                                                                l .
                                                                                           φ=0
                                         20


1.   f (x) = T x−l
               l sin
                        πx
                         2l .
                    1
                  − 6 (1 − cos πxl ) ,   0 ≤ x ≤ 2l/3
2.   f (x) = T                                        .
                  − 4l92 (x − l)2 ,      2l/3 ≤ x ≤ l


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