Уравнения математической физики. Мицик М.Ф. - 10 стр.

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
( )
( )
tyxguuau
yyxxtt
;;
2
−=
′′
+
′′
−
′′
, 
ρ
T
a =
. (31)
 (31) . 
.

( )
0;;
≡
tyxg
, 
.
( )
yyxxtt
uuau
′′
+
′′
=
′′
2
. (32)
 (31) 
.
.
      
D

,  
−
Γ
 .  
:
( ) ( )
yxFuyxfu
tt
;|,;|
00
=
′
=
==
, (33)

D

yxFyxf ;,;


( )
yx;
.

, 
0| =
Γ
u
, (34)

Γ
|u

u

Γ
.
.
.

D

0
r

 (. 3).
 3.
Y
X
O
ϕ
r
D
              utt′′ − a 2 (u ′xx′ +u ′yy′ ) = − g (x; y; t ) ,
                                                                              T
                                                                         a=     .                   (31)
                                                                              ρ
                     (31)                                                                       .
                                                                 .
              g (x; y; t ) ≡ 0 ,
                                               .
                       utt′′ = a 2 (u ′xx′ +u ′yy′ ) .                                              (32)
                                       (31)
      .

                                                                                            .

                                                                                            D
,                                                   Γ−                              .
                           :
                      u |t =0 = f (x; y ), u ′ |t =0 = F (x; y ) ,                                  (33)
                     D                     f (x; y ), F ( x; y )
                                                                               (x; y )                  .

          ,
                                u |Γ = 0 ,                                                          (34)
    u |Γ                                                             u                      Γ.

                                                                                        .
                                                                                                    .
                                              D                  r0
                       (       . 3).

                                                         Y


                                                         r ϕ
                                                         O                              X
                                           D



                                                         3.

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