Элементы вариационного исчисления. Молчанова Л.А. - 7 стр.

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y(x)∈C
(1)
F
ϕ
0
(x), ϕ
1
(x), ϕ
2
(x), ··· , ϕ
n
(x), ··· ,
ϕ
k
(x) ∈ C
(1)
, (k = 0, 1, 2, ···),
ϕ
k
(a) = ϕ
k
(b) = 0, ( k = 1, 2, ···),
ϕ
0
(a) = y
a
, ϕ
0
(b) = y
b
.
y(x)
y(x) =
P
∞
n=0
C
n
ϕ
n
(x), C
0
= 1.
y
N
=
P
N
n=0
C
n
ϕ
n
(x), C
0
= 1,
N y
N
(x)∈C
(1)
y
N
(a)=y
a
y
N
(b)=y
b
C
n
y
N
(x)
J
J[y
N
(x)] =
Z
b
a
F (x,
N
X
n=0
C
n
ϕ
n
(x),
N
X
n=0
C
n
ϕ
0
n
(x))dx → min.
Φ(C
1
, C
2
, ··· , C
N
) C
i
Φ
Φ
∂Φ
∂C
n
= 0, n = 1, 2, ··· , N. (13)
C
1
, C
2
, ··· , C
N
C
n
= C
N
n
(n = 1, 2, ··· , N)
y
N
(x) N
y
1
(x), y
2
(x), ··· , y
N
(x)
y(x) = lim
N→∞
y
N
(x)
F
y y
0
ϕ
k
(x) = x
k−1
(x − a)(x − b), k = 1, 2, 3, ···
ϕ
k
(x) = sin[
kπ(x−a)
b−a
], k = 1, 2, 3, ···
Ïðåäïîëàãàåòñÿ, ÷òî y(x)∈C (1) è ôóíêöèÿ F íåïðåðûâíà ïî âñåì àðãóìåí-
òàì.
   Äëÿ ðåøåíèÿ ïîñòàâëåííîé çàäà÷è çàäàåòñÿ ïîëíàÿ ïîñëåäîâàòåëüíîñòü
ëèíåéíî íåçàâèñèìûõ ôóíêöèé
                     ϕ0 (x), ϕ1 (x), ϕ2 (x), · · · , ϕn (x), · · · ,
êîòîðûå âûáèðàþò òàê, ÷òîáû
                       ϕk (x) ∈ C (1) , (k = 0, 1, 2, · · · ),
                      ϕk (a) = ϕk (b) = 0, (k = 1, 2, · · · ),
                            ϕ0 (a) = ya , ϕ0 (b) = yb .
Ðåøåíèå y(x) ïðåäñòàâëÿåòñÿP     â âèäå
                                   ∞
                       y(x) = n=0 Cn ϕn (x), C0 = 1.
Âíà÷àëå èùåòñÿ ýêñòðåìóì äëÿ    PNôóíêöèè
                        yN = n=0 Cn ϕn (x), C0 = 1,
ãäå N - ôèêñèðîâàííîå ÷èñëî. ßñíî, ÷òî yN (x)∈C (1) , yN (a)=ya , yN (b)=yb .
Âûáèðàþò êîýôôèöèåíòû Cn òàê, ÷òîáû ôóíêöèÿ yN (x) äîñòàâëÿëà ìè-
íèìóì ôóíêöèîíàëó J , òî åñòü
                     Z b        XN                XN
         J[yN (x)] =      F (x,     Cn ϕn (x),        Cn ϕ0n (x))dx → min.
                        a       n=0            n=0

 èíòåãðàëå âñå ôóíêöèè çàäàíû. Âûïîëíèâ èíòåãðèðîâàíèå, ïîëó÷àþò
âïîëíå îïðåäåëåííóþ ôóíêöèþ Φ(C1 , C2 , · · · , CN ). Êîýôôèöèåíòû Ci íàäî
âûáðàòü òàê, ÷òîáû ôóíêöèÿ Φ ïðèíèìàëà ìèíèìàëüíîå çíà÷åíèå. Íåîá-
õîäèìûì óñëîâèåì ýêñòðåìóìà ôóíêöèè Φ áóäåò
                            ∂Φ
                                = 0, n = 1, 2, · · · , N.                    (13)
                            ∂Cn
   Ñèñòåìà óðàâíåíèé (13) - ýòî ñèñòåìà íåëèíåéíûõ óðàâíåíèé îòíî-
ñèòåëüíî íåèçâåñòíûõ C1 , C2 , · · · , CN . Ðåøèâ åå, íàõîäÿò êîýôôèöèåíòû
Cn = CnN (n = 1, 2, · · · , N ), êîòîðûå îïðåäåëÿþò ðåøåíèå ýêñòðåìàëüíîé
çàäà÷è yN (x). Áåðÿ ðàçíûå N , ñòðîÿò ìèíèìèçèðóþùóþ ïîñëåäîâàòåëü-
íîñòü y1 (x), y2 (x), · · · , yN (x). Åñëè ýòà ïîñëåäîâàòåëüíîñòü ñõîäèòñÿ ê íåêî-
òîðîé ïðåäåëüíîé ôóíêöèè, òî îíà ÿâëÿåòñÿ ðåøåíèåì èñõîäíîé çàäà÷è:
y(x) = lim yN (x).
       N →∞
   Ðåøåíèå çàäà÷è ñóùåñòâåííî óïðîùàåòñÿ, êîãäà F - êâàäðàòè÷íàÿ ôóíê-
öèÿ ïåðåìåííûõ y è y 0 .  ýòîì ñëó÷àå ïðèõîäèòñÿ ðåøàòü ñèñòåìó ëèíåé-
íûõ àëãåáðàè÷åñêèõ óðàâíåíèé.
   ×àùå âñåãî â êà÷åñòâå áàçèñíûõ ôóíêöèé áåðóòñÿ ñèñòåìû ôóíêöèé
                ϕk (x) = xk−1 (x − a)(x − b), k = 1, 2, 3, · · ·
èëè
                    ϕk (x) = sin[ kπ(x−a)
                                    b−a ], k = 1, 2, 3, · · ·


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