L ! " # $ % $ & #' &(
) $ ! " # $ * x ∈ (0, L) +%
& $, t > 0 +,) ' " #-# u(x, t))
$ 2δ >
0 x) . Δt $
u(x, t + Δt) − u(x, t) ∼
=A
Δt
ΔQ
Δt
1
,
2δ
ΔQ
Δt / 0& $ $ A ' 1 ' ) 2 &(
" & #" 0& $ $ #(
)
3 h # $ +h << δ, ΔQ ∼ ΔQE,δ,h ΔQD,δ,h
−
,
=
Δt
Δt
Δt
ΔQE,δ,h ∼ u(x − δ − h, t) − u(x − δ + h, t)
=B
Δt
2h
' 0& $ & $ +"
x − δ, ΔQD,δ,h ∼ u(x + δ − h, t) − u(x + δ + h, t)
=B
Δt
2h
' 0& $ & +" x+δ,)
4 &5 6 ' 1 # '
)
2 h $ &"5
ΔQE,δ,h ∼
= −Bux (x − δ, t)
Δt
ΔQD,δ,h ∼
= −Bux (x + δ, t).
Δt
7
8
ΔQ ∼
= B(ux (x + δ, t) − ux (x − δ, t)).
Δt
ux (x + δ, t) − ux (x − δ, t) ∼
= 2δ uxx (x, t),
497 ut = kuxx k = AB $
u(x, t + Δt) − u(x, t)
.
Δt→0
Δt
ut = lim
! * # ' k $ & Λ ⊂ Rn ) $ ∂Λ ' / -# + / " #,)
9 " 0& ' Λ ΦΛ ( #' &½
ΦΛ (t) = −k
∂Λ
∇u(t, ξ) · n(ξ)dn−1 ξ,
+:,
ΦΛ (t) ' 0& Λ t k ' # ' ' 1 ∂Λ ξ ∈ ∂Λ ∇u(t, ξ) ' )
2 " #" '(
' n(ξ)
∂u
∂Q
(t, x) = c ρ
(t, x),
∂t
∂t
+,
Q ' ' c ' / ρ )¾
!
¾
"
# k c ρ $% &
! '
() # *+ , !
½
n(ξ)
Λ
∂Λ
ξ
3 $" $"5 +:, +, #)
3 #" $ # τ0 τ1 ' #' 1 #" ' ' $ )
2 $ ' τ0 τ1 ' τ
ΦΛ (t) dt.
+;,
1
τ0
9 ' ' $ ' "
#" #" ' +,) 2 " +, Λ
# (τ0 , τ1) ' τ1
τ0
Λ
∂Q
(t, x)dn x dt
∂t
#" ' # Λ) 7 #" $
τ1
τ0
Λ
∂Q
(t, x)dn x dt +
∂t
τ1
τ0
ΦΛ (t) dt = 0
$$ # (τ0 , τ1) $$ Λ)
+:, +, +<, τ1
τ0
∂u
(t, x)dn x dt −
cρ
∂t
Λ
τ1
τ0
k
;
∂Λ
∇u(t, ξ) · n(ξ)dn−1 ξ dt = 0
+<,
# $"
τ1
τ0
$ ∂u
(t, x)dn x dt −
cρ
∂t
Λ
τ1
τ0
Λ
τ1
τ0
k
Λ
Δu(t, x)dn x dt = 0
∂u
(t, x) − kΔu(t, x) dn x dt = 0,
cρ
∂t
Δu ' u + R3 Δu =
∂2u ∂2u ∂2u
+
+
,) 8 ' 5 ∂x2 ∂y 2 ∂z 2
" $ $"
k
∂u
=
Δu,
∂t
cρ
$ ' $" )
<