Классические методы математической физики. Алексеев Г.В. - 23 стр.

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ρ = 1 p = 1 q = 0
a = 1
X
′′
+ λX = 0 (0, l).
α = γ = 1 β = 0 δ = 0
X(0) = X(l) = 0.
λ
k
=
kπ
l
2
, X
k
(x) = sin
kπ
l
x k = 1, 2, ... .
α = γ = 0 β = −1 δ = 1
X
′
(0) = X
′
(l) = 0.
λ
k
X
k
λ
k
=
kπ
l
2
, X
k
(x) = cos
kπ
l
x, k = 0, 1, 2, ... .
λ
0
= 0, X
0
(x) =
α = δ = 1 β = γ = 0
X(0) = 0, X
′
(l) = 0.
λ
k
=
(2k + 1)π
2l
2
, X
k
(x) = sin
(2k + 1)π
2l
x, k = 1, 2, ... .
′
α = δ = 0 β = −1 γ = 1
X
′
(0) = 0, X(l) = 0.
λ
k
=
(2k + 1)π
2l
2
, X
k
(x) = cos
(2k + 1)π
2l
x, k = 1, 2, ... .
    çàêëþ÷åíèå ýòîãî ïóíêòà ïðèâåäåì ÿâíûå îðìóëû äëÿ ñîáñòâåííûõ
çíà÷åíèé è ñîáñòâåííûõ óíêöèé ñïåêòðàëüíîé çàäà÷è (2.7), (2.8) â òîì
÷àñòíîì ñëó÷àå, êîãäà ρ = 1, p = 1, q = 0, òàê ÷òî (2.1) ïåðåõîäèò â
âîëíîâîå óðàâíåíèå (1.1) ïðè a = 1, à (2.7) ïðèíèìàåò âèä
                           X ′′ + λX = 0 â (0, l).
àññìîòðèì íåñêîëüêî òèïîâ êðàåâûõ óñëîâèé â (2.8).
  1) α = γ = 1, β = 0, δ = 0. Óñëîâèÿ (2.8) ïðèíèìàþò âèä
                              X(0) = X(l) = 0.
 Ÿ 1 áûëî ïîêàçàíî, ÷òî ñîáñòâåííûå çíà÷åíèÿ è óíêöèè ñîîòâåòñòâóþ-
ùåé ñïåêòðàëüíîé çàäà÷è îïðåäåëÿþòñÿ ñîîòíîøåíèÿìè
                    2
                     kπ                  kπ
              λk =         , Xk (x) = sin x k = 1, 2, ... .
                      l                   l
  2) α = γ = 0, β = −1, δ = 1. Óñëîâèÿ (2.8) ïðèíèìàþò âèä
                             X ′ (0) = X ′ (l) = 0.
Ïðîñòîé àíàëèç (ñì., íàïðèìåð, [21℄) ïîêàçûâàåò, ÷òî ñîáñòâåííûå çíà÷åíèÿ
λk è óíêöèè Xk èìåþò âèä
                   2
                    kπ                   kπ
            λk =          , Xk (x) = cos x, k = 0, 1, 2, ... .
                     l                    l
Îòìåòèì åùå ðàç, ÷òî èìåííî â ñëó÷àå êðàåâûõ óñëîâèé Íåéìàíà ïåðâîå
ñîáñòâåííîå çíà÷åíèå ðàâíî íóëþ, à îòâå÷àþùàÿ åìó ñîáñòâåííàÿ óíêöèÿ
åñòü êîíñòàíòà: λ0 = 0, X0 (x) = onst.
   3) α = δ = 1, β = γ = 0. Óñëîâèÿ (2.8) ïðèíèìàþò âèä
                            X(0) = 0, X ′ (l) = 0.
Àíàëîãè÷íûé àíàëèç (ñì., íàïðèìåð, [21, ñ. 127℄) ïîêàçûâàåò, ÷òî
                       2
              (2k + 1)π                   (2k + 1)π
       λk =                , Xk (x) = sin           x, k = 1, 2, ... .
                  2l                          2l
  3′ ) α = δ = 0, β = −1, γ = 1. Óñëîâèÿ (2.8) ïðèíèìàþò âèä
                            X ′ (0) = 0, X(l) = 0.
Ñîáñòâåííûå çíà÷åíèÿ è óíêöèè èìåþò âèä
                       2
              (2k + 1)π                   (2k + 1)π
       λk =                , Xk (x) = cos           x, k = 1, 2, ... .
                  2l                          2l
                                      23

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