:
1) ρ;
2) , ;
3) - , .
[ , + ∆ ] (. . 6) :
[ , + ∆ ] = , + , .
. 6
, , ∆U, : ∆Q=CρS∆x∆U, - (= , 1 , 1), S - .
, ∆t ( ) : Q1 = -kSUx(x, t)∆t, k - (= , , 1). . , , , , , , , , Ux < 0. , Q1 , .
, : Q2 = -kSUx(x +∆x,t)∆t.
, , , :
∆Q = Q1 - Q2 => CpS∆x∆U = kSUx(x + ∆, t) ∆t - kSUx(x, t)∆t.
S∆x∆t ∆ ∆t , :
Ut=a2Uxx,
- .
, , q(x,t),
Ut = a2Uxx + f(x,t),
.
.
U|t=0 = φ() ( U(x,0) = φ()) , φ(). , φ , , .
, , . (5) , . , g1(t) ≡ 1 g2(t) ≡ 2, 1 2 - . , 1= 2 = 0 . (6) . , g1(t) = g2(t) = 0, . , ( ). , (7) , (, ). , (7) -:
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(14)
( ) , . , h1 > 0 - , g1(t) - . , . , , :
(14) , λ2 , , , , , .
(14) . , - ( ), . , , h1, .
(14) = 0 . :
. , . , , , , , .
- .
- :
Ut = Uxx, 0<x<l, t>0,
(15)
U(0,t) = U(l,t)=0, t>0,
(16)
(17)
.
1. (15) U(x,t) = X(x)T(t).
:
:
. , (16), ), , :
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2. -
-, 3. , λ>0.
(. )
3. ) :
4. (15):
(18)
(15)
(19)
, U(x,t) (16).
5. An (19), (17):
(20)
, φ(x) -. , .
φ(x) (19), (15), (16), (17).
. (19), , 3, - Ut = a2Uxx.