Уравнения математической физики. Мицик М.Ф. - 5 стр.

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( ) ( ) ( ) ( )
tTxXatTxX
2
⋅
′′
=
′′
⋅
. (12)
 (12) 
( ) ( )
tTxXa
2
⋅
:
( )
( )
( )
( )
xX
xX
tTa
tT
2
′
′
=
′
′
. (13)
 (13)  t,  – . 
    ,       
.
()
()
( )
( )
xX
xX
tTa
tT
2
−=
′
′
=
′
′
,
0
>
, (
0
<
).
:
0tTatT0,xXxX
2
=⋅+
′′
=⋅+
′′
(14)
 (7) :
( )
( )
tDsintCcostTx,BsinxAcosxX λλλλ +=+=
, (15)

DC,B,A,
– .  (15)  (11),
:
( )
( )
( )
( )
(
)
taDsintaCcosxBsinxAcostx;u λλλλ ++=
(16)
     ,    (4).  (4)
, 
( ) ( )
0X0,0X == l
, 
0BsinAcos,00B1A =+=⋅+⋅ ll λλ
.

,0sin0,B,0A =≠= lλ

,.....2,1n,n == lλ
(17)
 (17) , 
( )
( )
xnBsinxX lπ=
– .
 (14)
0
<
, :
( )
x-x-
BeAexX
λλ−
+=
,
  (4). , 
n c  (16), (17)  (11):
( )
( )
( )
( )
(
)
1,2,....n,tansinDtancosCxnsintx;u
nn
=+= lll πππ
 (1) , 
:
( )
( ) ( )( ) ( )
xnsintansinDtancosCtx;u
1
nn
lll
n
πππ ⋅+=
∑
∞
=
. (18)
 (18)  (5,5
/
) :
( ) ( )
,xnsinCxf
1
n
∑
∞
=
=
n
lπ
( )
( )( )
.xnsinanDx
1
n
∑
∞
=
=
n
ll ππϕ

n
C
n
D
:
( ) ( )
,xnsinxf
2
C
1
0
n
∑
∫
∞
=
=
n
l
l
l
π
( ) ( )
.xnsinx
an
2
D
1
0
n
∑
∫
∞
=
=
n
l
lπϕ
π
(19)
,  (1,4,5,5
/
)  (18,19).
                                  X (x ) ⋅ T ′′(t ) = a 2 X ′′(x ) ⋅ T (t ) .                                            (12)
                                                                                      T ′′(t ) X ′′(x )
                                                                  a X (x ) ⋅ T (t ) : 2
                                                                   2                            =
                                                                                     a T (t ) X (x )
                                                   (12)                                                 .                    (13)

                                       (13)                                  t,               –                          .
                                                             ,
                    T ′′(t ) X ′′(x )
                             =        =− ,                   >0,(                         <0
                   a 2 T (t ) X (x )
              .                                                                                                                  ).

                                                                                                :
             X ′′(x ) + ⋅ X (x ) = 0,           T ′′(t ) + a 2 ⋅ T (t ) = 0                                              (14)
                                              (7)                 :
X (x ) = Acos λ x + Bsin λ x,                     T (t ) = Ccos λ t + Dsin λ t ,                                         (15)
     A, B, C, D –                                               .                (15)                                      (11),
                              (                                     )(
             : u (x; t ) = Acos λ x + Bsin λ x Ccos a λ t + Dsin a λ t            (       )         (        ))          (16)
                                          ,                                                                       (4).           (4)
         ,           X (0 ) = 0, X (l ) = 0 ,               A ⋅ 1 + B ⋅ 0 = 0, Acos λ l + Bsin λ l = 0 .
         A = 0, B ≠ 0, sin λ l = 0,                                               λ = n l , n = 1, 2,.....                   (17)
              (17)                                                                          ,
X (x ) = Bsin (nπ l )x –                                                                                                 .
               (14) < 0 ,                                                                       :
                              X (x ) = Ae         -λ x
                                                         + Be −   -λ x
                                                                         ,
                                                                                  (4).                       ,
nc                         (16), (17)                                   (11):
u (x; t ) = sin (nπ l )x (C n cos (anπ l )t + D n sin (anπ l )t ), n = 1,2,....
                          (1)                                 ,
                                      ∞

                  : u (x; t ) = ∑ (C n cos (anπ l )t + D n sin (anπ l )t ) ⋅ sin (nπ l )x .                              (18)
                                     n =1

                                               (18)                                                 (5,5/)                   :
                          ∞                                              ∞
             f (x ) = ∑ C n sin (nπ l )x ,                ϕ (x ) = ∑ D n (anπ l )sin (nπ l )x.
                       n =1                                            n =1

                                                                                         Cn    Dn :
     2 ∞                                                                      ∞
              l                                                                   l
C n = ∑ ∫ f (x )sin (nπ l )x ,                                               ∑ ∫ ϕ (x )sin (nπ l )x.
                                                               2
                                                         Dn =                                                            (19)
     l n =1 0                                                 anπ            n =1 0

                      ,                                                                   (1,4,5,5/)              (18,19).

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