Классические методы математической физики. Алексеев Г.В. - 15 стр.

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∞
X
k=1
[T
′′
k
(t) + ω
2
k
T
k
(t)]sin
kπx
l
= f(x, t),
ω
k
= kπa/l f
(0, l)
f(x, t) =
∞
X
k=1
f
k
(t)sin
kπx
l
.
f
k
(t) t
f
k
(t) =
2
l
l
Z
0
f(x, t)sin
kπx
l
dx.
f sin(kπx/l)
T
′′
k
(t) + ω
2
k
T
k
(t) = f
k
(t), k = 1, 2, ... .
k = 1, 2, ...
T
k
T
k
T
k
(0) = 0, T
′
k
(0) = 0, k = 1, 2, ... .
T
k
(t) =
1
ω
k
t
Z
0
f
k
(τ)sinω
k
(t − τ)dτ
f
k
(τ)
T
k
(t) =
2
lω
k
t
Z
0
[sinω
k
(t − τ)
l
Z
0
f(x, τ)sin
kπx
l
dx]dτ.
(1.35) è îáîèì íà÷àëüíûì óñëîâèÿì â (1.37). Ïîäñòàâëÿÿ (1.41) â (1.35) è
ðàññóæäàÿ îðìàëüíî, ïîëó÷èì
                  ∞
                  X                                     kπx
                         [Tk′′(t) + ωk2Tk (t)]sin           = f (x, t),             (1.42)
                                                         l
                   k=1

ãäå ωk = kπa/l. Ïðåäïîëîæèì, ÷òî óíêöèþ f ìîæíî ðàçëîæèòü â ðÿä
Ôóðüå ïî ñèíóñàì (1.11) â èíòåðâàëå (0, l):
                                           ∞
                                           X                 kπx
                           f (x, t) =            fk (t)sin       .                  (1.43)
                                                              l
                                           k=1

Çäåñü êîýèöèåíòû fk (t), çàâèñÿùèå îò t êàê îò ïàðàìåòðà, îïðåäåëÿþòñÿ
àíàëîãè÷íî (1.17) îðìóëîé

                                       Zl
                                  2                       kπx
                         fk (t) =           f (x, t)sin       dx.                   (1.44)
                                  l                        l
                                       0

  Ñîîòíîøåíèÿ (1.42) è (1.43) àêòè÷åñêè ïðåäñòàâëÿþò ñîáîé ðàçëîæå-
íèÿ îäíîé è òîé æå óíêöèè f â ðÿä Ôóðüå ïî ñèíóñàì sin(kπx/l). Ïðè-
ðàâíèâàÿ ñîîòâåòñòâóþùèå êîýèöèåíòû îáîèõ ðàçëîæåíèé, ïðèõîäèì ê
ðàâåíñòâàì
                  Tk′′ (t) + ωk2 Tk (t) = fk (t), k = 1, 2, ... . (1.45)
Ïðè êàæäîì k = 1, 2, ... (1.45) ïðåäñòàâëÿåò ñîáîé äèåðåíöèàëüíîå
óðàâíåíèå âòîðîãî ïîðÿäêà îòíîñèòåëüíî óíêöèè Tk . ×òîáû îäíîçíà÷íî
îïðåäåëèòü Tk , çàäàäèì ñ ó÷åòîì îäíîðîäíîñòè íà÷àëüíûõ óñëîâèé â (1.37)
íà÷àëüíûå óñëîâèÿ

                   Tk (0) = 0, Tk′ (0) = 0, k = 1, 2, ... .                         (1.46)

åøåíèå óðàâíåíèÿ (1.45) ïðè íà÷àëüíûõ óñëîâèÿõ (1.46) èìååò âèä
                                      Zt
                             1
                    Tk (t) =                fk (τ )sinωk (t − τ )dτ                 (1.47)
                             ωk
                                      0

èëè (ïîñëå ïîäñòàíîâêè âìåñòî fk (τ ) èõ âûðàæåíèé èç (1.44))

                           Zt                      Zl
                      2                                                kπx
            Tk (t) =            [sinωk (t − τ )         f (x, τ )sin       dx]dτ.   (1.48)
                     lωk                                                l
                            0                      0


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