Уравнения математической физики. Уравнение колебаний струны. Хуснутдинов Н.Р. - 15 стр.

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w(x, t)
w(x, 0) = u(x, 0) − U(x, 0) = f(x) − φ(0) − [ψ(0) − φ(0)]
x
l
,
w
′
t
(x, 0) = u
′
t
(x, 0) − U
′
t
(x, 0) = F (x) − φ
′
(0) − [ψ
′
(0) − φ
′
(0)]
x
l
.
w(x, t)
∂
2
w
∂t
2
− v
2
∂
2
w
∂x
2
= G − U
′′
tt
(x, t).
f
a
(x) = f(x) − φ(0) − [ψ(0) − φ(0)]
x
l
,
F
a
(x) = F (x) − φ
′
(0) − [ψ
′
(0) − φ
′
(0)]
x
l
,
G
a
(x, t) = G(x, t) − φ
′′
(t) − [ψ
′′
(t) − φ
′′
(t)]
x
l
.
w(x, t)
G
a
∂
2
w
∂t
2
− v
2
∂
2
w
∂x
2
= G
a
.
w(x, 0) = f
a
(x),
w
′
t
(x, 0) = F
a
(x),
w(0, t) = 0, w(l, t) = 0.
w(x, t)
u(x, t)
u(x, t) = w(x, t) + U(x, t)
U(x, t) w(x, t)
U(x, t) = α(t)x + [β(t) − α(t)]
x
2
2l
w
′
x
(0, t) = u
′
x
(0, t) − U
′
x
(0, t) = α(t) − α(t) = 0,
w
′
x
(l, t) = u
′
x
(l, t) − U
′
x
(l, t) = β(t) − β(t) = 0,
w(x, t)
f
b
(x) = f(x) − α(0)x − [β(0) − α(0)]
x
2
2l
,
F
b
(x) = F (x) − α
′
(0)x − [β
′
(0) − α
′
(0)]
x
2
2l
,
G
b
(x, t) = G(x, t) − α
′′
(t)x − [β
′′
(t) − α
′′
(t)]
x
2
2l
.
w(x, t)
G
b
∂
2
w
∂t
2
− v
2
∂
2
w
∂x
2
= G
b
.
w(x, 0) = f
b
(x),
w
′
t
(x, 0) = F
b
(x),
óíêöèÿ w(x, t). Ïîëó÷àåì                                                              2.9    Âûíóæäåííûå êîëåáàíèÿ ñòðóíû êîíå÷íîé äëèíû.
                                                                                                  Íåîäíîðîäíûå ãðàíè÷íûå óñëîâèÿ (3b).
                                                                  x
       w(x, 0) = u(x, 0) − U (x, 0) = f (x) − φ(0) − [ψ(0) − φ(0)] ,
                                                                  l
                                                                               x       Ñäåëàåì çàìåíó èñêîìîé óíêöèè
     wt′ (x, 0) = u′t (x, 0) − Ut′ (x, 0) = F (x) − φ′ (0) − [ψ ′ (0) − φ′ (0)] .
                                                                               l
                                                                                                               u(x, t) = w(x, t) + U (x, t)
   Íîâàÿ óíêöèÿ w(x, t) óäîâëåòâîðÿåò óðàâíåíèþ êîëåáàíèÿ ñòðó-
íû, íî ñ èçìåíåííîé âûíóæäàþùåé ñèëîé                                                  è ïîäáåðåì óíêöèþ U (x, t) òàê, ÷òîáû íîâàÿ óíêöèÿ w(x, t) óäî-
                                                                                       âëåòâîðÿëà áû îäíîðîäíûì ãðàíè÷íûì óñëîâèÿì (2b). Ëåãêî âèäåòü,
                         ∂2w       ∂2w                                                 ÷òî óíêöèÿ
                              − v 2 2 = G − Utt′′ (x, t).
                         ∂t 2      ∂x                                                                                                    x2
                                                                                                        U (x, t) = α(t)x + [β(t) − α(t)]
                                                                                                                                         2l
     Îáîçíà÷èì
                                                                                       óäîâëåòâîðÿåò òðåáóåìûì óñëîâèÿì. Äåéñòâèòåëüíî
                                                          x
                fa (x)   =    f (x) − φ(0) − [ψ(0) − φ(0)] ,
                                                          l                                     wx′ (0, t) = u′x (0, t) − Ux′ (0, t) = α(t) − α(t) = 0,
                                         ′         ′         x
                                                             ′
                Fa (x)   =    F (x) − φ (0) − [ψ (0) − φ (0)] ,                                 wx′ (l, t) = u′x (l, t) − Ux′ (l, t) = β(t) − β(t) = 0,
                                                             l
                                                               x
                                                                                       ò.å. óíêöèÿ w(x, t) óäîâëåòâîðÿåò îäíîðîäíûì ãðàíè÷íûì óñëîâèÿì
                                         ′′       ′′      ′′
              Ga (x, t) =     G(x, t) − φ (t) − [ψ (t) − φ (t)] .
                                                                l
                                                                                       (2b).
Òàêèì îáðàçîì, óíêöèÿ w(x, t) óäîâëåòâîðÿåò óðàâíåíèþ êîëåáà-                             Îáîçíà÷èì
íèÿ ñòðóíû ñ ïëîòíîñòüþ âûíóæäàþùåé ñèëû Ga
                                                                                                                                                x2
                                2            2                                                    fb (x)    = f (x) − α(0)x − [β(0) − α(0)]          ,
                               ∂ w       ∂ w                                                                                                     2l
                                    − v 2 2 = Ga .
                               ∂t 2      ∂x                                                                                                          x2
                                                                                                  Fb (x)    = F (x) − α′ (0)x − [β ′ (0) − α′ (0)] ,
                                                                                                                                                     2l
åøåíèå óäîâëåòâîðÿåò ñëåäóþùèì íà÷àëüíûì óñëîâèÿì:
                                                                                                                                                        x2
                                                                                                Gb (x, t)   = G(x, t) − α′′ (t)x − [β ′′ (t) − α′′ (t)] .
                                 w(x, 0) = fa (x),                                                                                                      2l
                                wt′ (x, 0) = Fa (x),                                      Òàêèì îáðàçîì, óíêöèÿ w(x, t) óäîâëåòâîðÿåò óðàâíåíèþ êîëå-
                                                                                       áàíèÿ ñòðóíû ñ ïëîòíîñòüþ âûíóæäàþùåé ñèëû Gb
è   îäíîðîäíûì    ãðàíè÷íûì óñëîâèÿì
                                                                                                                  ∂2w       2
                                                                                                                          2∂ w
                             w(0, t) = 0, w(l, t) = 0.                          (40)                                  − v      = Gb .
                                                                                                                  ∂t2      ∂x2
Ïîýòîìó äîñòàòî÷íî ðåøèòü çàäà÷ó ñ îäíîðîäíûìè ãðàíè÷íûìè óñ-                          åøåíèå óäîâëåòâîðÿåò ñëåäóþùèì íà÷àëüíûì óñëîâèÿì:
ëîâèÿìè (40), ïîëó÷èòü âûðàæåíèå äëÿ w(x, t), à çàòåì ïî îðìóëå
(39) âû÷èñëèòü èñêîìóþ óíêöèþ u(x, t).                                                                             w(x, 0) = fb (x),
                                                                                                                    wt′ (x, 0) = Fb (x),


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