Уравнения математической физики. Уравнение теплопроводности. Хуснутдинов Н.Р. - 10 стр.

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f(x) = T
x(l/4−x)
l
2
, 0 ≤ x ≤ l/4
0 , l/4 ≤ x ≤ l
.
f(x) = T
(
3x
2
l
2
, 0 ≤ x ≤ l/2
3(x−l)
2
l
2
, l/2 ≤ x ≤ l
.
G = g
x
l
, f(x) = T
0 , 0 ≤ x ≤ l/4
(x−l)(l/4−x)
l
2
, l/4 ≤ x ≤ l
.
G = g
e
x
2l
−
3x
2
4l
2
cos
πvt
l
, f(x) = 0, F (x) =
(
3x
2
l
, 0 ≤ x ≤ l/4
(x−l)
2
3l
, l/4 ≤ x ≤ l
φ = g
e
−
πvt
l
− 1
ψ = 0
, F (x) = 0, f(x) =
0, 0 ≤ x ≤ l/4
(x−
l
4
)(
3l
4
−x)
h
, l/4 ≤ x ≤ 3 l/4
0, 3l/4 ≤ x ≤ l
f(x) = T
x−l
l
sin
2πx
l
.
f(x) = T
(
4(1 − cos
πx
l
) , 0 ≤ x ≤ l/2
16(x−l)
2
l
2
, l/2 ≤ x ≤ l
.
G = g
x
2
l
2
, f(x) = T
x
2
l
2
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) = T
0 , 0 ≤ x ≤ l/4
(x−l)(l/4−x)
l
2
, l/4 ≤ x ≤ l
.
f(x) = T
(
−
3x
2
l
2
, 0 ≤ x ≤ l/3
−
3(x−l)
2
4l
2
, l/3 ≤ x ≤ l
.
G = ge
x/l
, f(x) = T
0 , 0 ≤ x ≤ l/2
(x−l)(l/2−x)
l
2
, 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) = T
x
2
l
2
sin
πx
l
.
f(x) = T
(
4(1 − cos
πx
l
) , 0 ≤ x ≤ 3l/4
64(x−l)
2
l
2
(1 +
√
2
2
) , 3l/4 ≤ x ≤ l
.
G = ge
2x/l
, f(x) = T
(x−l)
2
l
2
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) = T
0 , 0 ≤ x ≤ l/2
(x−l)(l/2−x)
l
2
, l/2 ≤ x ≤ l
.
f(x) = T
(
3x
2
l
2
, 0 ≤ x ≤ l/4
(x−l)
2
3l
2
, l/4 ≤ x ≤ l
.
G = g
2x
l
+ 4
, f(x) = T
0, 0 ≤ x ≤ l/4
(x−l/4)(3l/4−x)
l
2
, l/4 ≤ x ≤ 3 l/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) = T
(x−l)
2
l
2
sin
πx
2l
.
f(x) = T
(
4(1 − cos
πx
l
) , 0 ≤ x ≤ l/3
9(x−l)
2
2l
2
, l/3 ≤ x ≤ l
.
                                                                                                                                                                    (       2
                                                     1
                                                                                                              
                                                                                                                  x       6x2
                                                                                                                                
                                                                                                                                          3πvt                    − 3xl ,    0 ≤ x ≤ l/5
                                                                                                  4.   G=g        l   −    l2       sin     l , f (x)
                                                                                                                                                   = 0, F (x) =     3(x−l)2
                        x(l/4−x)                                                                                                                                  −         , l/5 ≤ x ≤ l
                                                                                                                                                           x(3l/4−x) 16l
                    
                                   ,     0 ≤ x ≤ l/4
1.   f (x) = T              l2                       .                                                        2 2
                                                                                                       φ = g vl2t e−3πvt/l                                           , 0 ≤ x ≤ 3l/4
                        0,               l/4 ≤ x ≤ l                                              5.                                 , F (x) = 0, f (x) =       h                       .
                    (
                        3x2
                                                                                                       ψ=0                                                   0,          3l/4 ≤ x ≤ l
2.   f (x) = T           l2 ,  0 ≤ x ≤ l/2
                                             .
                        3(x−l)2
                           l2 ,
                               l/2 ≤ x ≤ l                                                                                                               4

                            0,               0 ≤ x ≤ l/4
     G = g xl , f (x) = T
                                                                                                                      2
3.                                                         .                                           f (x) = T x(      πx
                            (x−l)(l/4−x)
                                         ,   l/4 ≤ x ≤ l                                          1.
                                                                                                                  l2 sin l .
                                  l2                ( 2
                                                      3x                                                              4(1 − cos πxl )√,                    0 ≤ x ≤ 3l/4
                                                        l ,                        0 ≤ x ≤ l/4         f (x) = T                                                        .
            x           
                       2                                                                          2.
4.   G = g e 2l − 3x         πvt
                     4l2 cos l , f (x) = 0, F (x) =   (x−l)2
                                                                                                                      64(x−l)2
                                                                                                                               (1 + 22 ) ,                 3l/4 ≤ x ≤ l
                                                                                , l/4 ≤ x ≤ l                            l2
                                                        3l                                                                                      2
           πvt                         
                                           0,                                    0 ≤ x ≤ l/4      3.   G = ge2x/l , f (x) = T (x−l)
                                                                                                                                l2  sin πx
                                                                                                                                        2l .
     φ = g e− l − 1
                                                                                                                                                 l
                                                                                                                                                  − 3 (1 − cos πx
                                         
                                                                                                                                                                  l ), 0 ≤ x ≤ l/2
                                           (x− 4l )( 3l
                                                     4 −x)
5.                  , F (x) = 0, f (x) =           h       , l/4 ≤ x ≤ 3l/4                       4.   G = g 4x cos  2πvt
                                                                                                                          , F (x)  = 0, f (x) =     4
     ψ=0                                 
                                           0,                 3l/4 ≤ x ≤ l
                                                                                                              l        l                          − 3l (x − l)2 ,      l/2 ≤ x ≤ l
                                                                                                                πvt/l
                                                                                                                           
                                                                                                       ψ=g e          −1                                   2
                                                                                                  5.                           , f (x) = 0 , F (x) = (x−l)
                                                                                                                                                        h    sin πx
                                                                                                                                                                  4l .
                                                                                                       φ=0
                                                     2

                                                                                                                                                         5
1.   f (x) = T x−l     2πx
               (l sin l .                                                                                             
                   4(1 − cos πx                                                                                           0,                        0 ≤ x ≤ l/2
                              l ) ,                 0 ≤ x ≤ l/2                                   1.   f (x) = T          (x−l)(l/2−x)                          .
2.   f (x) = T     16(x−l)2                                     .                                                                      , l/2 ≤ x           ≤l
                      l2    ,                       l/2 ≤ x ≤ l                                                       (         l2
                                                                                                                          3x2
                                                                                                                           l2 ,      0 ≤ x ≤ l/4
              2
                                x2
3.   G=   g xl2 ,   f (x) = T   l2     sin   πx
                                              l .                                                 2.   f (x) = T          (x−l)2                              .
                                                          (
                                                                            πx                                                     , l/4 ≤ x ≤ l
                                                              4l(1 − cos     l ),  0 ≤ x ≤ 3l/4                             3l2
     G = g xl cos πvt
                   l , F (x) = 0 , f (x) =
                                                                               √
                                                                                                                                         
4.                                                            64(x−l)2                                                              0,                    0 ≤ x ≤ l/4
                                                                  l    (1   + 22 ),3l/4 ≤ x ≤ l                2x
                                                                                                                                      (x−l/4)(3l/4−x)
                                                                                                  3.   G=g     + 4 , f (x) = T
                                                                                                                l                             l2        ,  l/4 ≤ x ≤ 3l/4
     ψ = g vtl e−2πvt/l
                        , f (x) = 0 , F (x) = |x − 2l | sin 2πx                                                                       0,                   3l/4 ≤ x ≤ l
                                                                                                                                   
5.
     φ=0                                                     l .                                                                                        3 2
                                                                                                            3x      −lx     2πvt                          x ,          0 ≤ x ≤ l/6
                                                                                                  4. G = g
                                                                                                             l −e        sin l , f (x) = 0, F (x) = l3               2
                                                                                                                                                          25l (x − 2)  ,l/6 ≤x≤l
                                                     3
                                                                                                           vt − πvt
                                                                                                                                          
                                                                                                     φ=g l e     l                            0,                 0 ≤  x  ≤ 3l/4
                                                                                                  5.                 , F (x) = 0, f (x) =      (x−l)(3l/4−x)
                                                                                                     ψ=0                                                      , 3l/4 ≤ x ≤ l
                    
                        0,                      0 ≤ x ≤ l/4                                                                                          h
1.   f (x) = T          (x−l)(l/4−x)                        .
                              l2         ,      l/4 ≤ x ≤ l
                    (                                                                                                                                    6
                          3x2
                        − l2 ,            0 ≤ x ≤ l/3
2.   f (x) = T                   2                     .                                                                   2
                        − 3(x−l)
                            4l2 ,        l/3 ≤ x ≤ l                                             1.   f (x) = T (x−l)     πx
                                                                                                                 (l2 sin 2l .
                                        0,              0 ≤ x ≤ l/2                                                 4(1 − cos πx
                                                                                                                               l ) ,                    0 ≤ x ≤ l/3
3.   G = gex/l , f (x) = T              (x−l)(l/2−x)                .                             2.   f (x) = T    9(x−l)2                                         .
                                             l2      , l/2 ≤ x ≤ l                                                          ,                           l/3 ≤ x ≤ l
                                                                                                                       2l2


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