Классические методы математической физики. Алексеев Г.В. - 21 стр.

UptoLike

ВУЗ: 

ДВФУ | Владивосток

Составители: 

Рубрика: 

(λ
k
− λ
m
)ρ(x)X
k
(x)X
m
(x) +
d
dx
{p(x)[X
m
(x)X
′
k
(x) − X
k
(x)X
′
m
(x)]} = 0.
x l
(λ
m
− λ
k
)
l
Z
0
ρ(x)X
k
(x)X
m
(x)dx = p(x)[X
m
(x)X
′
k
(x) − X
k
(x)X
′
m
(x)] |
x=l
x=0
.
(λ
k
− λ
m
)
l
Z
0
ρ(x)X
k
(x)X
m
(x)dx = 0.
λ
m
6= λ
k
λ
1
< λ
2
< ...
X
1
, X
2
, ...
ρ
[p(x)X
′
k
(x)]
′
− q( x)X
k
(x) = −λ
k
ρ(x)X
k
(x).
X
k
λ
k
= −
l
Z
0
{[p(x)X
′
k
(x)]
′
− q(x)X
k
(x)}X
k
(x)dx.
λ
k
=
l
Z
0
[p(x)(X
′
k
(x))
2
+ q(x)X
2
k
(x)]dx −[p(x)X
k
(x)X
′
k
(x)] |
x=l
x=0
.
[p(x)X
k
(x)X
′
k
(x)] |
x=l
x=0
≤ 0.
p(x) ≥ p
0
> 0 q( x) ≥ 0
λ
k
≥ 0 k = 1, 2, ...
0 ≤ λ
1
< λ
2
< ....
êîòîðîå, êàê íåòðóäíî âèäåòü, ìîæíî ïåðåïèñàòü â âèäå
                              d
 (λk − λm )ρ(x)Xk (x)Xm(x) +    {p(x)[Xm(x)Xk′ (x) − Xk (x)Xm
                                                            ′
                                                              (x)]} = 0.
                             dx
Èíòåãðèðóÿ ýòî ðàâåíñòâî ïî x â ïðåäåëàõ îò 0 äî l, ïîëó÷èì
               Zl
 (λm − λk )         ρ(x)Xk (x)Xm(x)dx = p(x)[Xm(x)Xk′ (x) − Xk (x)Xm
                                                                   ′
                                                                     (x)] |x=l
                                                                           x=0 .
               0
                                                                    (2.12)
Ïðèíèìàÿ âî âíèìàíèå ãðàíè÷íûå óñëîâèÿ (2.8), ëåãêî óáåæäàåìñÿ, ÷òî
ïðàâàÿ ÷àñòü â (2.12) ðàâíà íóëþ.  òàêîì ñëó÷àå èç (2.12) ñëåäóåò, ÷òî
                                      Zl
                         (λk − λm )        ρ(x)Xk (x)Xm(x)dx = 0.
                                      0

Îòñþäà â ñèëó óñëîâèÿ λm 6= λk âûòåêàåò (2.11).
   Èç ñâîéñòâà îðòîãîíàëüíîñòè ñîáñòâåííûõ óíêöèé ëåãêî ñëåäóåò, ÷òî
âñå ñîáñòâåííûå çíà÷åíèÿ çàäà÷è (2.7), (2.8) âåùåñòâåííû. Ïîêàæåì, áîëåå
òîãî, ÷òî âñå îíè íåîòðèöàòåëüíû. Äåéñòâèòåëüíî, ïóñòü λ1 < λ2 < ... 
âñå ñîáñòâåííûå çíà÷åíèÿ çàäà÷è (2.7), (2.8), à X1 , X2 , ...  îòâå÷àþùàÿ èì
îðòîíîðìèðîâàííàÿ (ñ âåñîì ρ) ñèñòåìà ñîáñòâåííûõ óíêöèé. Ñîãëàñíî
îïðåäåëåíèþ, èìååì
                      [p(x)Xk′ (x)]′ − q(x)Xk (x) = −λk ρ(x)Xk (x).           (2.13)
Óìíîæàÿ îáå ÷àñòè íà Xk , èíòåãðèðóÿ è ïðèíèìàÿ âî âíèìàíèå (2.10),
ïîëó÷èì
                  Zl
            λk = − {[p(x)Xk′ (x)]′ − q(x)Xk (x)} Xk (x)dx.
                            0
Îòñþäà, ïîñëå èíòåãðèðîâàíèÿ ïåðâîãî ñëàãàåìîãî ïî ÷àñòÿì, áóäåì èìåòü
          Zl
   λk =        [p(x)(Xk′ (x))2 + q(x)Xk2(x)]dx − [p(x)Xk (x)Xk′ (x)] |x=l
                                                                      x=0 .   (2.14)
          0

Ïðåäïîëîæèì, ÷òî
                         [p(x)Xk (x)Xk′ (x)] |x=l
                                              x=0 ≤ 0.                  (2.15)
Òàê êàê p(x) ≥ p0 > 0, q(x) ≥ 0, òî èç îðìóëû (2.14) íåïîñðåäñòâåííî
ñëåäóåò, ÷òî λk ≥ 0, k = 1, 2, .... Òàêèì îáðàçîì, â äîïîëíåíèå ê (2.9) èìååò
ìåñòî
                               0 ≤ λ1 < λ2 < ....                       (2.16)

                                               21

Страницы