Уравнения математической физики. Мицик М.Ф. - 4 стр.

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   .    
.
2. .
 (1) 
 
( )
55,
′
.    
     , 
.

   
( ) ( )
ηξ
21
ΨΨ
. 

() ( ) ( )
atxatxtxu +Ψ+−Ψ=
21
;
(6)
   (1),   
 (6) (1).
 
( ) ( )
ηξ
21
ΨΨ
, -
,  (6) 
( )
55,
′
.

0
=
t
:
( ) ( ) ( ) ( ) ( ) ( )
xxaxaxfxx ϕ=
′
Ψ+
′
Ψ−=Ψ+Ψ
2121
,
. (7)
 (7):
( ) ( ) ( )
,
1
0
21
Cd
a
xx
x
x
+−=Ψ−Ψ
∫
ξξϕ

( ) ( )
.
0102
xxC
Ψ
Ψ
(8)
(7) (8) :
( )
( ) ( )
,
1
2
1
0
1
+−=Ψ
∫
Cd
a
xfx
x
x
ξξϕ
(9)
( )
( ) ( )
.
1
2
1
0
2
−+=Ψ
∫
Cd
a
xfx
x
x
ξξϕ
(10)
 (9) t,  (10) t  (6), 
:
( )
( ) ( )
( )
.
2
1
2
2
ξξϕ d
a
atxfatxf
x
atx
atx
∫
+
−
+
++−
=Ψ

.
:
2
2
2
2
2
x
u
a
t
u
∂
∂
=
∂
∂
(1)
 (4)  (5), .
      (1),
 (4) :
( ) ( ) ( )
tTxXtx;u ⋅=
. (11)
 (11)  (1):
( )
5
′
                                                         .
                                                                .

2.                                                                                                                 .

                                                                                                             (1)
                             (5, 5′) .
                                                                                                       ,
                                                                                              .

                                                                                 Ψ1 (ξ ) Ψ2 (η ) .
                             u( x; t ) = Ψ1 ( x − at ) + Ψ2 ( x + at )                                        (6)
                                             (1),
                  (6) (1).
                                                                                          Ψ1 (ξ ) Ψ2 (η ) , -
              ,                       (6)                                                           (5, 5′) .
                                                             ′            ′
     t =0             : Ψ1 ( x ) + Ψ2 ( x ) = f ( x ), − a Ψ1 ( x ) + a Ψ2 ( x ) = ϕ ( x ) .                   (7)
                                                                                        (7):
                               x
      Ψ1 (x ) − Ψ2 (x ) = − ∫ ϕ (ξ )dξ + C ,
                           1
                                                                C = Ψ2 ( x0 ) − Ψ1 ( x0 ).                   (8)
                           a x0

                                      1                                    
                                                           x
                              (   )           (   )    1
                                                               (  )
                                                       a x∫0
(7)     (8)            :   Ψ1   x   =      f   x   −        ϕ  ξ  d ξ + C  ,                                (9)
                                      2                                   
                                      1                                 
                                                         x
                          Ψ2 (x ) =  f ( x ) + ∫ ϕ (ξ )dξ − C  .
                                                     1
                                                                                                              (10)
                                      2            a x0                
            (9)          t,      (10)                 t                                               (6),
                                            f ( x − at ) + f ( x + at ) 1
                                                                                 x + at
                             Ψ2 (x ) =                                           ϕ (ξ )dξ .
                                                                         2a x −∫at
                       :                                               +
                                                         2


                                                                                                  .
                                                                                  ∂ u     2
                                                                                             2 ∂ u
                                                                                                  2

                                                                                : 2 =a                (1)
                                                                                  ∂t           ∂ x2
                                          (4)                                    (5), (5′) .
                                                                                                        (1),
                                                      (4)           : u (x; t ) = X (x ) ⋅ T (t ) . (11)
                                   (11)                      (1):

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