Уравнения математической физики. Мицик М.Ф. - 6 стр.

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2. .
  ,   
, ,
   –   .   
     ,   
  .  ,    
 .      ,
, .
      
,     :  L,
,  R  G.
       
( )
−x;ti


(
)
−x;tU
      
x

 
t
.      

x

xx
∆
+
. 
(
)
xo ∆
     
xR
∆


xL
∆
. 
( ) ( ) ( )
( )
t
txi
xLtxxiRtxxUtxU
∂
∂
∆+∆=∆+−
;
;;;
(20)
, 
(
)
xo ∆
( ) ( )
x
x
txU
txUtxxU ∆
∂
∂
=−∆+
;
;;
(21)
 (20) (21), :
( )
0; =+
∂
∂
+
∂
∂
txiR
t
i
L
x
U
(22)

[ ]
xxx ∆+;

()( )
( )
x
x
txi
txxitxi ∆
∂
∂
−=∆+−
;
;;
(23)
     , 
(
)
txxUG ;∆
 
    , 
( )
t
txU
x
∂
∂
∆
;

( ) ( ) ( )
( )
t
txU
xtxxUGtxxitxi
∂
∂
∆+∆=∆+−
;
;;;
. (24)
 (23) (24), :
0=+
∂
∂
+
∂
∂
GU
t
U
x
i
. (25)
   2.                                                                                                             .

                                                                    ,
                                      ,                                                                               ,
                     –                                                         .
                                                                                    ,
                                  .                  ,
                                      .                                                                               ,
                              ,                                                         .

                 ,                                                             :                                  L,
             ,                            R                                                 G.
                                                                                                 i (x;t ) −
         U (x;t ) −                                                                                           x
                           t.
             x           x + ∆x .
o(∆x )                                                                                                 R∆x
                         L∆x .
                                                                         ∂i ( x; t )
             U (x; t ) − U (x + ∆x; t ) = R ∆xi(x; t ) + L∆x                                            (20)
                                                                            ∂t
                          ,                  o(∆x )
                                            ∂U ( x; t )
             U ( x + ∆x; t ) − U ( x; t ) =             ∆x                                              (21)
                                               ∂x
            (20)          (21),                                     :
              ∂U             ∂i
                      + L + R i( x; t ) = 0                                                             (22)
                ∂x           ∂t
                                              [x; x + ∆x]
                                               ∂i ( x; t )
              i ( x; t ) − i ( x + ∆x; t ) = −             ∆x                                           (23)
                                                  ∂x
                                                                ,                  G∆xU ( x; t )
                                                                                                       ∂U ( x; t )
                                                                           ,                      ∆x
                                                                                                         ∂t

                                                                        ∂U (x; t )
              i ( x; t ) − i ( x + ∆x; t ) = G∆xU ( x; t ) + ∆x                    .                    (24)
                                                                          ∂t
            (23) (24),        :
              ∂i   ∂U
                 +     + GU = 0 .                                                                       (25)
             ∂x     ∂t

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