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1
MM .
.
ϕ
ϕ
ϕ ∆+
.
uO
,
1
MM 1.
( )
ϕ
ϕ
ϕ
sinsin TT
−
∆
+
.
ϕ
x
∆
( )
22
ϕ∆+∆xO
:
( ) ( )
( )
( )
=
∂
∂
−
∂
∆+∂
=⋅−∆+⋅=−∆+
x
txu
x
txxu
TtgTtgTTT
;;
sinsin ϕϕϕϕϕϕ
( )
( )
x
x
txu
Tx
x
txxu
T ∆
∂
∂
=∆
∂
∆+∂
=
;;
22
θ
,
10
<
<θ
.
, , -
, .
−
ρ
, –
x
∆
ρ
, :
x
x
u
T
t
u
x ∆
∂
∂
=
∂
∂
∆
2
2
2
2
ρ
.
2
a
T
=
ρ
, :
2
2
2
2
2
x
u
a
t
u
∂
∂
=
∂
∂
.
.
( )
tx;u
()
(
0
=
t
).
.
,
( ) ( )
0t;u,0t0;u == l
– . (4)
0
=
t
, , .
(
)
xf
,
( ) ( )
x|ux;0u
0t
f
=
=
=
. (5)
,
,
( )
xϕ
. ,
:
( )
x|
t
u
0t
ϕ=
∂
∂
=
.
(5) .
O
M
1
M
ϕ
ϕ
ϕ ∆+
x
xx
∆
+
X
u
l
( )
5
′
( )
5
′
u
MM 1 .
. M1 ϕ + ∆ϕ
M
ϕ
ϕ ϕ + ∆ϕ . O
x x+∆x l X
Ou ,
MM 1 1.
T sin (ϕ + ∆ϕ ) − T sin ϕ .
ϕ ∆x (
O ∆ x 2 + ∆ϕ 2
) :
∂u ( x + ∆x; t ) ∂u ( x; t )
T sin (ϕ + ∆ϕ ) − T sin ϕ = T ⋅ tg (ϕ + ∆ϕ ) − T ⋅ tgϕ = T − =
∂x ∂x
∂ 2 u ( x + θ ∆ x; t ) ∂ 2 u ( x; t )
=T ∆x = T ∆x , 0 < θ < 1.
∂x ∂x
, , -
, . ρ−
∂ 2u ∂ 2u
, – ρ ∆x , : ρ ∆x = T ∆x .
∂t2 ∂ x2
T ∂2u 2 ∂ u
2
= a2 , : = a .
ρ ∂ t2 ∂ x2
.
u (x; t )
( )
( t = 0 ).
.
,
u (0; t ) = 0, u (l ; t ) = 0 – . (4)
t =0
, , .
f (x) , u (x;0 ) = u | t =0 = f (x ) . (5)
,
, ϕ (x ) . ,
∂u
: | t = 0 = ϕ (x ) . (5′)
∂t
(5) (5′) .
