Классические методы математической физики - 200 стр.

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y B
at
(x)
∂u(x, t)
∂t
=
u(x, t)
t
+
I(x, t)
4πat
.
t
∂
2
u(x, t)
∂t
2
= −
u
t
2
+
1
t
u
t
+
I
4πat
−
I
4πat
2
+
1
4πat
∂I
∂t
=
1
4πat
∂I(x, t)
∂t
.
∂I(x, t)
∂t
= a
Z
S
at
(x)
∆ϕ(y)|
y=x+atn
dσ
r
.
I B
at
(x)
(ρ, θ, ψ) x y = x + ρn 0 ≤ ρ < at
I(x, t) =
at
Z
0
2π
Z
0
π
Z
0
∆ϕ(y)ρ
2
sin θdθdψdρ, y = x + ρn ∈ B
at
(x).
t
S
1
S
at
(x)
∂I(x, t)
∂t
= a
2π
Z
0
π
Z
0
∆ϕ(y)a
2
t
2
sin θdθdψ = a
3
t
2
Z
S
1
∆ϕ(y)dσ
1
= a
Z
S
at
(x)
∆ϕ(y)dσ
r
,
y = x + atn
∂
2
u(x, t)
∂t
2
=
a
4π
Z
S
at
(x)
∆ϕ(y)
r
y=x+atn
dσ
r
.
u
ϕ ∈ C
2
(R
3
)
u
u|
t=0
= 0,
∂u
∂t
t=0
= ϕ(x)
R
3
.
ãäå òî÷êà y âî âòîðîì è òðåòüåì èíòåãðàëàõ ïðîáåãàåò âåñü øàð Bat (x),
ïåðåïèøåì (3.11) â âèäå
                      ∂u(x, t) u(x, t) I(x, t)
                              =        +       .                 (3.13)
                        ∂t         t     4πat
Äèåðåíöèðóÿ ýòî âûðàæåíèå ïî t, ïîëó÷èì
 ∂ 2u(x, t)
                             
                u 1 u      I        I     1 ∂I      1 ∂I(x, t)
            = −   +    +        −      +         =             . (3.14)
    ∂t2         t2 t t 4πat       4πat2 4πat ∂t    4πat ∂t
  Íåòðóäíî âèäåòü, ÷òî
                           ∂I(x, t)
                                            Z
                                    =a                ∆ϕ(y)|y=x+atn dσr .                    (3.15)
                             ∂t
                                          Sat (x)

 ñàìîì äåëå, ïåðåõîäÿ â èíòåãðàëå I ïî øàðó Bat(x) ê ñåðè÷åñêèì êîîð-
äèíàòàì (ρ, θ, ψ) ñ öåíòðîì â òî÷êå x, ïîëàãàÿ y = x + ρn, ãäå 0 ≤ ρ < at,
èìååì
                      Zat Z2π Zπ
       I(x, t) =                   ∆ϕ(y)ρ2 sin θdθdψdρ, y = x + ρn ∈ Bat (x).
                      0    0   0

Äèåðåíöèðóÿ ïî âðåìåíè t, âõîäÿùåìó â ïåðåìåííûé âåðõíèé ïðåäåë
âíåøíåãî èíòåãðàëà, è ïåðåõîäÿ ïîñëåäîâàòåëüíî ê èíòåãðàëàì ïî åäèíè÷-
íîé ñåðå S1 è ñåðå Sat (x), ïîëó÷àåì ñîîòíîøåíèå
              Z2π Zπ
∂I(x, t)
                                                                 Z                   Z
                                    2 2                    3 2
         =a               ∆ϕ(y)a t sin θdθdψ = a t                    ∆ϕ(y)dσ1 = a         ∆ϕ(y)dσr ,
  ∂t
              0   0                                              S1              Sat (x)

ãäå y = x + atn. Òåì ñàìûì (3.15) äîêàçàíî. Èç (3.14) è (3.15) âûòåêàåò, â
ñâîþ î÷åðåäü, ÷òî
                  ∂ 2u(x, t)    a   ∆ϕ(y)
                                  Z
                             =                      dσr .           (3.16)
                     ∂t2       4π      r    y=x+atn
                                            Sat (x)

  Ïîäñòàâëÿÿ (3.9) è (3.16) â (3.1), ïðèõîäèì ê âûâîäó, ÷òî óíêöèÿ u,
îïðåäåëÿåìàÿ îðìóëîé (3.4), óäîâëåòâîðÿåò âîëíîâîìó óðàâíåíèþ (3.1)
äëÿ ëþáîé óíêöèè ϕ ∈ C 2 (R3 ). Èç (3.7) è (3.10), êðîìå òîãî, ñëåäóåò, ÷òî
óíêöèÿ u óäîâëåòâîðÿåò íà÷àëüíûì óñëîâèÿì
                                            ∂u
                               u|t=0 = 0,                 = ϕ(x) â R3 .                      (3.17)
                                            ∂t      t=0

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