Уравнения математической физики. Мицик М.Ф. - 16 стр.

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( )
4
2
ς
ς
−
= eK
,  (56)
2
π
=C
, 
( )
2
0
0
2
π
==
∫
∞
−
dzeK
z
.
 (56)  (52)  (51):
( ) ( )
( )
ξξϕ
π
ξ
de
ta
txu
ta
x
∫
∞
∞−
−
−
=
2
2
4
2
1
;
(57)
 (57) 
.
6. .

( )
x;y;zp
z
u
y
u
x
u
=
∂
∂
+
∂
∂
+
∂
∂
2
2
2
2
2
2
(58)
. 
(
)
0≡x;y;zp
, 

0
2
2
2
2
2
2
=
∂
∂
+
∂
∂
+
∂
∂
z
u
y
u
x
u
. (59)
, . , 
.

.
1) 
D
, 
Γ
, ,
(
)
(
)
zyxuMu ;;=
, 
.
2) 
Γ
:
( )
( )
Γ∈=
∂
∂
+ MM
n
u
Mu ,ψβα
, (60)

−
∂
∂
n
u

Γ
.
, 
0
=
β
, , 

( ) ( )
Γ
=
MMMu ,
1
ψ
. (61)

0
=
, , 
:
( )
Γ∈=
∂
∂
MM
n
u
,
2
ψ
. (62)
                                                        ς2
                                                    −
                                  K (ς ) = e             4
                                                             ,                                                         (56)
                                                    ∞
        π                                                                          π
                                        K (0 ) = ∫ e − z dz =
                                                                  2
     C=   ,                                                                          .
        2                                           0
                                                                                   2
                     (56)                    (52)                                             (51):
                                         ∞                   (ξ − x )     2
                                                         −
                u ( x; t ) =             ∫ ϕ (ξ )e
                                 1                               4 a 2t
                                                                              dξ                                       (57)
                               2a π t    −∞

                     (57)
                                                                                                .

           6.                                                                                                      .


                         ∂ 2u ∂ 2u ∂ 2u
                             +    +     = p ( x;y;z )                                                                  (58)
                         ∂ x2 ∂ y2 ∂ z2
                                    .       p( x;y;z ) ≡ 0 ,

                         ∂ 2u ∂ 2u ∂ 2u
                             +    +      = 0.                                                                          (59)
                         ∂ x2 ∂ y2 ∂ z 2
                                                                                                ,     .                ,
                                                .

       .
1)                D,                                                                     Γ,                                   ,
     u (M ) = u ( x; y; z ) ,
                                                                                                                   .
2)                                   Γ                                                                    :
                                                        ∂u
                                 α u (M ) + β              = ψ (M ), M ∈ Γ ,                                           (60)
                                                        ∂n
           ∂u
              −                                                                                                   Γ.
           ∂n
                         ,        β = 0,                                                                      ,

                                  u (M ) = ψ 1 (M ),                      M ∈Γ .                                       (61)
                    α = 0,                                                               ,
                                  :
                                  ∂u
                                     = ψ 2 (M ),                  M ∈Γ.                                                (62)
                                  ∂n

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