Уравнения математической физики. Сборник задач. Даишев Р.А - 71 стр.

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r
(0, R
0
), c
k
c
k
=
R
0
R
0
r
2
J
0
³
µ
k
r
R
0
´
rdr
R
0
R
0
J
2
0
³
µ
k
r
R
0
´
rdr
=
2
(R
0
J
2
(µ
k
))
R
0
Z
0
r
3
J
0
µ
µ
k
r
R
0
dr.
((xJ
1
(x))
0
= xJ
0
(x)) J
0
1
(x) = −J
0
(x).
U(r, φ, t) =
∞
X
n=1
2A
µ
0
n
J
1
(µ
0
n
)
exp
Ã
−
a
2
[µ
0
n
]
2
t
l
2
!
J
0
Ã
µ
0
n
r
l
!
.
U(r, t) = U
0
1 + 2
∞
X
n=1
J
0
³
µ
n
r
R
´
µ
n
J
0
0
(µ
n
)
exp
Ã
−
µ
2
n
a
2
R
2
t
!
,
µ
1
, µ
2
, µ
3
, ... J
0
(µ) = 0.
U(r, φ, t) = U
0
exp
Ã
−
a
2
[µ
0
1
]
2
t
l
2
!
J
0
Ã
µ
0
1
r
l
!
.
U(r, t) =
2
R
2
∞
X
n=1
µ
2
n
µ
2
n
+ H
2
R
2
×
×exp
Ã
−
µ
2
n
a
2
R
2
t
!
J
0
³
µ
n
r
R
´
J
2
0
(µ
n
)
∞
Z
0
ρf(ρ)J
0
µ
µ
n
ρ
R
dρ,
µ
1
, µ
2
, ...
µJ
0
0
(µ) + HRJ
0
(µ) = 0.
ñîáñòâåííûå ôóíêöèè çàäà÷è îðòîãîíàëüíû ñ âåñîì r íà ïðî-
ìåæóòêå (0, R0 ), òî êîýôôèöèåíòû ck ìîãóò áûòü íàéäåíû ïî
ôîðìóëàì:
              R
              R0           ³         ´
                   r2 J0 µk Rr0 rdr                           Z        R0          µ        ¶
              0                                       2                    r
       ck =            ³             ´         =      2
                                                                r 3 J 0 µk    dr.
               RR0                               (R0 J (µk ))              R0
                     J02 µk Rr0 rdr                                   0
               0

Îñòàâøèéñÿ èíòåãðàë ëåãêî âû÷èñëÿåòñÿ ñ èñïîëüçîâàíèåì ñî-
îòíîøåíèé ((xJ1 (x))0 = xJ0 (x)) è J10 (x) = −J0 (x).
123.
                               ∞
                                                          Ã                   !     Ã       !
                               X
                              2A             a2 [µ0n ]2 t     µ0n r
         U (r, φ, t) =      0      0
                                       exp −              J 0       .
                       n=1 µn J1 (µn )           l2             l
124.
                                                 ³     ´                 Ã                !
                                             ∞ J      r
                                             X   0 µn R
                                                     µ2n a2 
          U (r, t) = U0 1 + 2         0
                                               exp −       t ,
                               n=1 µn J0 (µn )        R2

ãäå µ1 , µ2 , µ3 , ... -ïîëîæèòåëüíûå êîðíè óðàâíåíèÿ J0 (µ) = 0.
125.                               Ã               !    Ã       !
                                      a2 [µ01 ]2 t        µ01 r
               U (r, φ, t) = U0 exp −                J0           .
                                          l2               l
126.                                              ∞
                                              2 X         µ2n
                               U (r, t) =                         ×
                                              R2 n=1 µ2n + H 2 R2
                       Ã                 !     ³          ´
                                             J0 µn Rr Z∞                      µ        ¶
                         µ2 a2                                                    µn ρ
              × exp     − n2 t                                    ρf (ρ)J0             dρ,
                          R                   J02 (µn )                            R
                                                              0
ãäå µ1 , µ2 , ... -ïîëîæèòåëüíûå êîðíè óðàâíåíèÿ

                                  µJ00 (µ) + HRJ0 (µ) = 0.

                                                   71

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