Уравнения математической физики. Мицик М.Ф. - 14 стр.

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( )
2
2
2
;
x
txu
a
t
u
∂
∂
=
∂
∂
, 
ρc
k
a =
. (43)
        
).

, 
(
)
txu ;
,
.
:
(
)
(
)
−= xxu ϕ0;
, 

x

0
=
t
;
(
)
(
)
ttu
1
;0 ψ=
,
(
)
(
)
−= ttlu
2
; ψ
, , 
, 
(
)
t
1
ψ

(
)
t
2
ψ
.
.


x
.

. :
 (43) 
Rx
∈

0
>
t
,

( ) ( )
xxu
=
0;
(44)
,
. 2.
( ) ( ) ( )
tTxXtxu
⋅
=
;
. (45)
 (45)  (43), :
( ) ( ) ( ) ( )
tTxXatTxX ⋅
′′
=
′
⋅
2

( )
()
( )
( )
2
2
λ−=
′
′
=
′
xX
xX
tTa
tT
.


( ) ( )
0
22
=⋅+
′
tTatT λ
,
( ) ( )
0
2
=+
′′
xXxX λ
. (46)
 (46), :
ta
eT
22
λ−
=
,
xBxAX
λ
λ
sincos
+
=
.
 (45).
(
)
(
)
(
)
(
)
xBxAetxu
ta
λλλλ
λ
λ
sincos;
22
+=
−
(47)

A

B

λ

.  (47) .
                                    ∂u    2 ∂ u ( x; t )
                                             2
                                                                                  k
                                       =a                ,             a=           .                             (43)
                                    ∂t         ∂x 2                              cρ

                                                      ).

                      ,                   u (x; t )                                                               ,
                                                                                   .
                                                                                           :
           u (x;0 ) = ϕ ( x ) −                                        ,
                                  x                                         t = 0;
           u (0; t ) = ψ 1 (t ) , u (l ; t ) = ψ 2 (t ) −                                        ,                ,
                                                                                    ,
ψ 1 (t )      ψ 2 (t ) .

                                                                                                         .


                                                                                                             x.

              .                                                                                      :
                                                             (43)          x∈R          t > 0,

                                    u (x;0 ) = ϕ ( x )                                                            (44)
                                                                                                              ,
                             . 2.
                                    u (x; t ) = X ( x ) ⋅ T (t ) .                                                (45)
                          (45)                        (43),                 :
                                                                                 T ′(t )   X ′′(x )
                          X ( x ) ⋅ T ′(t ) = a 2 X ′′( x ) ⋅ T (t )                     =          = −λ2 .
                                                                                a T (t ) X (x )
                                                                                  2




                      T ′(t ) + a 2 λ2 ⋅ T (t ) = 0 ,                      X ′′( x ) + λ2 X (x ) = 0 .            (46)
                                    (46),              :
                          T = e − a λ t , X = A cos λx + B sin λx .
                                          2 2



                                                 (45).
                                    u λ ( x; t ) = e − a λ t ( A(λ ) cos λx + B(λ ) sin λx )
                                                       2 2
                                                                                                                  (47)
                                                    A B                        λ
                  .                                                 (47)                                      .

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