Классические методы математической физики - 199 стр.

UptoLike

ВУЗ: 

ДВФУ | Владивосток

Составители: 

Рубрика: 

S
1
u(x, t) =
t
4π
Z
S
1
ϕ(y)dσ
1
≡
t
4π
Z
S
1
ϕ(x + αat, y + βat, z + γat)dσ
1
.
u ∈ C
l
(R
4
) ϕ ∈ C
l
(R
3
) l =
1, 2, 3...
∆u(x, t) =
t
4π
Z
S
1
∂
2
ϕ(y)
∂ξ
2
+
∂
2
ϕ(y)
∂η
2
+
∂
2
ϕ(y)
∂ζ
2
dσ
1
≡
t
4π
Z
S
1
∆ϕ(y)dσ
1
,
y = x + atn
∆u(x, t) ≡
1
4πa
2
t
Z
S
at
(x)
∆ϕ(y)|
y=x+atn
dσ
r
, y = ( ξ, η, ζ) ∈ S
at
(x).
t
∂u(x, t)
∂t
=
1
4π
Z
S
1
ϕ(y)dσ
1
+
at
4π
Z
S
1
α
∂ϕ(y)
∂ξ
+ β
∂ϕ(y)
∂η
+ γ
∂ϕ(y)
∂ζ
dσ
1
,
y = x + atn
∂u(x, t)
∂t
=
u(x, t)
t
+
1
4πat
Z
S
at
(x)
α
∂ϕ(y)
∂ξ
+ β
∂ϕ(y)
∂η
+ γ
∂ϕ(y)
∂ζ
dσ
r
=
=
u(x, t)
t
+
1
4πat
Z
S
at
(x)
gradϕ(y) · n(y)dσ
r
.
Z
S
at
(x)
gradϕ(y) ·n(y)dσ
r
=
Z
B
at
(x)
div [gradϕ(y)] dξdηdζ =
=
Z
B
at
(x)
∆ϕ(y)dξdηdζ ≡ I(x, t),
ïðåîáðàçóåì åãî ê ñîîòâåòñòâóþùåìó èíòåãðàëó ïî åäèíè÷íîé ñåðå S1 .
Ïîëó÷èì
              t              t
                Z              Z
   u(x, t) =      ϕ(y)dσ1 ≡      ϕ(x + αat, y + βat, z + γat)dσ1. (3.7)
             4π             4π
                   S1                          S1

Èç (3.7) âûòåêàåò, â ÷àñòíîñòè, ÷òî u ∈ C l (R4 ), åñëè ϕ ∈ C l (R3 ), l =
1, 2, 3... .
    Äèåðåíöèðóÿ äâàæäû (3.7) ïîä çíàêîì èíòåãðàëà, èìååì
                 Z  2
                    ∂ ϕ(y) ∂ 2ϕ(y) ∂ 2ϕ(y)
                                              
               t                                        t
                                                          Z
   ∆u(x, t) =              +        +           dσ1 ≡       ∆ϕ(y)dσ1,
              4π      ∂ξ 2     ∂η 2    ∂ζ 2            4π
                    S1                                                              S1
                                                                     (3.8)
ãäå y = x + atn. Äåëàÿ â (3.8) çàìåíó, îáðàòíóþ ê (3.6), ïåðåïèøåì (3.8) â
âèäå
                 1
                     Z
   ∆u(x, t) ≡            ∆ϕ(y)|y=x+atndσr , y = (ξ, η, ζ) ∈ Sat (x). (3.9)
              4πa2 t
                         Sat (x)

Äèåðåíöèðóÿ äàëåå (3.7) ïî t, ïîëó÷èì
                                Z                           
  ∂u(x, t)    1             at       ∂ϕ(y)    ∂ϕ(y)    ∂ϕ(y)
                Z
           =      ϕ(y)dσ1 +        α       +β       +γ         dσ1,
    ∂t       4π             4π        ∂ξ       ∂η       ∂ζ
                    S1                             S1
                                                                   (3.10)
ãäå y = x + atn. Ó÷èòûâàÿ (3.7) è äåëàÿ âî âòîðîì èíòåãðàëå îáðàòíóþ ê
(3.6) çàìåíó, ïåðåïèøåì (3.10) â âèäå
                              Z                             
    ∂u(x, t) u(x, t)     1           ∂ϕ(y)    ∂ϕ(y)    ∂ϕ(y)
             =       +             α       +β       +γ         dσr =
       ∂t         t    4πat           ∂ξ       ∂η       ∂ζ
                                         Sat (x)

                      u(x, t)    1
                                                     Z
                    =         +                              gradϕ(y) · n(y)dσr .        (3.11)
                        t       4πat
                                                   Sat (x)

  Ïðèìåíÿÿ îðìóëó àóññàÎñòðîãðàäñêîãî âèäà
        Z                        Z
            gradϕ(y) · n(y)dσr =   div [gradϕ(y)] dξdηdζ =
         Sat (x)                                        Bat (x)
                                   Z
                           =              ∆ϕ(y)dξdηdζ ≡ I(x, t),                         (3.12)
                               Bat (x)

                                                        199

Страницы