Ôóíêöèîíàëüíî- äèåðåíöèàëüíûåîïåðàòîðûòàêîãîðîäàâîçíèêàþòïðèèçó÷åíèèêîëåáàíèéáàëîêèìîñòîâ,ñî- ñòàâëåííûõèçìàòåðèàëîâðàçëè÷íîéïëîòíîñòè.×òîáûðåøèòüóíêöèîíàëüíî-äèåðåíöèàëüíîå óðàâíåíèå, çàäàþùåå äèåðåíöèàëüíûéîïåðàòîð,ïðèìåíÿåòñÿìåòîäâàðèàöèè ïîñòîÿííûõ.å-

2019,Òîì21,Âûïóñê 2,Ñ.3857

ÓÄÊ517.927

DOI10.23671/VNC.2019.2.32116

ÎÁÈÑÑËÅÄÎÂÀÍÈÈ ÑÏÅÊÒÀ

ÔÓÍÊÖÈÎÍÀËÜÍÎ-ÄÈÔÔÅÅÍÖÈÀËÜÍÎ Î ÎÏÅÀÒÎÀ

ÑÑÓÌÌÈÓÅÌÛÌ ÏÎÒÅÍÖÈÀËÎÌ

Ñ. È. Ìèòðîõèí

1

1

ÍÈÂÖÌÓèì.Ì.Â.Ëîìîíîñîâà,

îññèÿ,119991,Ìîñêâà,Ëåíèíñêèåãîðû,1

E-mail:mitrokhin-sergeyyandex.ru

Àííîòàöèÿ.  ðàáîòå èçó÷àåòñÿ óíêöèîíàëüíî-äèåðåíöèàëüíûé îïåðàòîð âîñüìîãî ïîðÿä-

êà ñ ñóììèðóåìûì ïîòåíöèàëîì. ðàíè÷íûå óñëîâèÿ ÿâëÿþòñÿ ðàçäåëåííûìè. Ôóíêöèîíàëüíî-

äèåðåíöèàëüíûåîïåðàòîðûòàêîãîðîäàâîçíèêàþòïðèèçó÷åíèèêîëåáàíèéáàëîêèìîñòîâ,ñî-

ñòàâëåííûõèçìàòåðèàëîâðàçëè÷íîéïëîòíîñòè.×òîáûðåøèòüóíêöèîíàëüíî-äèåðåíöèàëüíîå

óðàâíåíèå, çàäàþùåå äèåðåíöèàëüíûéîïåðàòîð,ïðèìåíÿåòñÿìåòîäâàðèàöèè ïîñòîÿííûõ.å-

øåíèåèñõîäíîãîóíêöèîíàëüíî-äèåðåíöèàëüíîãîóðàâíåíèÿñâåäåíîêðåøåíèþèíòåãðàëüíîãî

óðàâíåíèÿÂîëüòåððû.Ïîëó÷èâøååñÿèíòåãðàëüíîåóðàâíåíèåÂîëüòåððûðåøàåòñÿìåòîäîìïîñëå-

äîâàòåëüíûõïðèáëèæåíèéÏèêàðà.Âðåçóëüòàòåèññëåäîâàíèÿèíòåãðàëüíîãîóðàâíåíèÿïîëó÷åíû

àñèìïòîòè÷åñêèåîðìóëûèîöåíêèäëÿðåøåíèéóíêöèîíàëüíî-äèåðåíöèàëüíîãî óðàâíåíèÿ,

çàäàþùåãî äèåðåíöèàëüíûé îïåðàòîð. Ïðè áîëüøèõ çíà÷åíèÿõ ñïåêòðàëüíîãî ïàðàìåòðà âû-

âåäåíà àñèìïòîòèêà ðåøåíèé äèåðåíöèàëüíîãîóðàâíåíèÿ, îïðåäåëÿþùåãîäèåðåíöèàëüíûé

îïåðàòîð. Àíàëîãè÷íîàñèìïòîòè÷åñêèìîöåíêàì ðåøåíèé äèåðåíöèàëüíîãîîïåðàòîðà âòîðîãî

ïîðÿäêàñãëàäêèìèèêóñî÷íî-ãëàäêèìèêîýèöèåíòàìèóñòàíàâëèâàþòñÿàñèìïòîòè÷åñêèåîöåí-

êè ðåøåíèé èñõîäíîãî óíêöèîíàëüíî-äèåðåíöèàëüíîãî óðàâíåíèÿ. Ïîëó÷åííûå àñèìïòîòè÷å-

ñêèåîðìóëûïðèìåíÿþòñÿäëÿèçó÷åíèÿãðàíè÷íûõóñëîâèé.Âðåçóëüòàòåïðèõîäèìêèçó÷åíèþ

êîðíåéóíêöèè,ïðåäñòàâëåííîéââèäå îïðåäåëèòåëÿâîñüìîãîïîðÿäêà.×òîáûíàéòèêîðíèýòîé

óíêöèè,íåîáõîäèìîèçó÷èòüèíäèêàòîðíóþäèàãðàììó.Êîðíèóðàâíåíèÿíàñîáñòâåííûåçíà÷åíèÿ

íàõîäÿòñÿ â âîñüìè ñåêòîðàõ áåñêîíå÷íî ìàëîãî ðàñòâîðà, îïðåäåëÿåìûõ èíäèêàòîðíîéäèàãðàì-

ìîé.Èçó÷åíûïîâåäåíèåêîðíåéýòîãîóðàâíåíèÿâêàæäîìèçñåêòîðîâèíäèêàòîðíîéäèàãðàììûè

àñèìïòîòèêàñîáñòâåííûõçíà÷åíèéèññëåäóåìîãîäèåðåíöèàëüíîãîîïåðàòîðà.

Êëþ÷åâûå ñëîâà: óíêöèîíàëüíî-äèåðåíöèàëüíûé îïåðàòîð, êðàåâàÿ çàäà÷à, ñóììèðóåìûé

ïîòåíöèàë,ãðàíè÷íûåóñëîâèÿ,ñïåêòðàëüíûéïàðàìåòð,èíäèêàòîðíàÿäèàãðàììà,àñèìïòîòèêàñîá-

ñòâåííûõçíà÷åíèé.

Mathematial Subjet Classiation (2010):34K08.

Îáðàçåö öèòèðîâàíèÿ: Ìèòðîõèí Ñ. È. Îá èññëåäîâàíèè ñïåêòðà óíêöèîíàëüíî-äèåðåí-

öèàëüíîãîîïåðàòîðàññóììèðóåìûìïîòåíöèàëîì//Âëàäèêàâê.ìàò.æóðí.2019.Ò.21,âûï.2.

Ñ.3857.DOI:10.23671/VNC.2019.2.321 16.

1.Ïîñòàíîâêà çàäà÷è

Èññëåäóåì óíêöèîíàëüíî-äèåðåíöèàëüíûé îïåðàòîð (ÔÄÎ), çàäàâàåìûé óðàâ-

íåíèåì

y ⁽⁸⁾ (x) + q(x)y(x) = λa ⁸ y(x) + αr(x)y(b), 0 6 x 6 π, 0 < b < π, a > 0,

⁽¹⁾

^{2019 ÌèòðîõèíÑ.È.}

ãäå

λ ∈ C

ñïåêòðàëüíûé ïàðàìåòð,

α ∈ C

, ñðàçäåëåííûìè ãðàíè÷íûìè óñëîâèÿìè

y ^{(m 1 )} (0) = y ^{(m 2 )} (0) = · · · = y ^{(m 6 )} (0) = y ^{(n 1 )} (π) = y ^{(n 2 )} (π) = 0,

⁽²⁾

m ₁ < m ₂ < · · · < m ₆

^,

n ₁ < n ₂

^;

m _k , n ₁ , n ₂ ∈ { 0, 1, 2, . . . , 7 }

^.

Âóðàâíåíèè (1)

q(x)

^{ïîòåíöèàë,}

ρ(x) = a ⁸ > 0

^{âåñîâàÿóíêöèÿ. Ìûïðåäïîëà-}

ãàåì, ÷òî

q(x)

^è

r(x)

ñóììèðóåìûåóíêöèè íà îòðåçêå

[0; π]

^:

q(x) ∈ L ₁ [0; π] ⇔

Z ^x

0

q(t)dt ′

x

= q(x)

^{ïî÷òèäëÿ âñåõçíà÷åíèé}

x

^íà

[0; π];

r(x) ∈ L ₁ [0; π] ⇔ Z ^x

0

r(t)dt ′

x

= r(x)

^{ïî÷òèäëÿ âñåõçíà÷åíèé}

x

^íà

[0; π].

(3)

2. Èñòîðè÷åñêèé îáçîð

Ñïåêòðàëüíûå ñâîéñòâà îáûêíîâåííûõ äèåðåíöèàëüíûõ îïåðàòîðîâ â ñëó÷àå äî-

ñòàòî÷íî ãëàäêèõ êîýèöèåíòîâ èçó÷àþòñÿ óæå äîñòàòî÷íî äàâíî.  ìîíîãðàèè [1,

ãë. 2℄èçó÷åíîàñèìïòîòè÷åñêîåïîâåäåíèåñîáñòâåííûõçíà÷åíèéèñîáñòâåííûõóíêöèé

äèåðåíöèàëüíûõ îïåðàòîðîâ ñ ãëàäêèìè, íåñêîëüêî ðàç äèåðåíöèðóåìûìè êîý-

èöèåíòàìè. Òàì æå îïèñàíà ìåòîäèêà íàõîæäåíèÿ àñèìïòîòèêè ðåøåíèé äèåðåí-

öèàëüíûõóðàâíåíèé,çàäàþùèõäèåðåíöèàëüíûé îïåðàòîðñ ãëàäêèìè êîýèöèåí-

òàìè, ïðè áîëüøèõ çíà÷åíèÿõ ñïåêòðàëüíîãî ïàðàìåòðà. Ïîëüçóÿñü òàêîé àñèìïòîòè-

êîé ðåøåíèé, â ðàáîòå [2℄ áûëè íàéäåíû àñèìïòîòè÷åñêèå îðìóëûäëÿ êîðíåé îäíîãî

êëàññà öåëûõ óíêöèé, êîòîðûå âîçíèêàþò â ñëó÷àå èçó÷åíèÿ îáûêíîâåííûõ äèå-

ðåíöèàëüíûõîïåðàòîðîâñðåãóëÿðíûìè ãðàíè÷íûìèóñëîâèÿìèñäîñòàòî÷íî ãëàäêèìè

êîýèöèåíòàìè.

Ñ ïîìîùüþ àñèìïòîòè÷åñêèõ îðìóë, íàéäåííûõ â ðàáîòå [2℄, â ðàáîòàõ [3℄ è [4℄

áûëèíàéäåíû îðìóëûäëÿâû÷èñëåíèÿ ðåãóëÿðèçîâàííûõ ñëåäîâäèåðåíöèàëüíûõ

îïåðàòîðîââûñøèõ ïîðÿäêîâñ äîñòàòî÷íî ãëàäêèìè êîýèöèåíòàìè.

Âðàáîòå[5℄àâòîð ïðîäåìîíñòðèðîâàëìåòîäèêóíàõîæäåíèÿîðìóëðåãóëÿðèçîâàí-

íûõñëåäîâäëÿäèåðåíöèàëüíûõîïåðàòîðîââòîðîãîïîðÿäêàñðàçðûâíûìèêîýè-

öèåíòàìè. Ñïåêòðàëüíûå ñâîéñòâà äèåðåíöèàëüíûõ îïåðàòîðîâ ñ êóñî÷íî-ãëàäêèìè

êîýèöèåíòàìè èçó÷àëèñü àâòîðîì â ðàáîòå [6℄,à â ðàáîòå [7℄ áûë èçó÷åí äèåðåí-

öèàëüíûéîïåðàòîð ñêóñî÷íî-ãëàäêîé âåñîâîéóíêöèåé.

 ðàáîòàõ [8, 9℄ áûëà èçó÷åíà êðàåâàÿ çàäà÷à äëÿ óíêöèîíàëüíî-äèåðåíöèàëü-

íîãîîïåðàòîðàâòîðîãîïîðÿäêàñãëàäêèìïîòåíöèàëîì,áûëèâû÷èñëåíûîðìóëûðåãó-

ëÿðèçîâàííûõñëåäîâòàêîãîîïåðàòîðà.Âðàáîòå[10℄àâòîðîìáûëèâû÷èñëåíûîðìóëû

ðåãóëÿðèçîâàííûõ ñëåäîâ äëÿ óíêöèîíàëüíî-äèåðåíöèàëüíûõ îïåðàòîðîâ âòîðîãî

ïîðÿäêàñ êóñî÷íî-ãëàäêèì ïîòåíöèàëîì.

Âðàáîòå[11℄áûëñîâåðøåíáîëüøîéïðîãðåññâèçó÷åíèèäèåðåíöèàëüíûõ îïåðà-

òîðîâñíåãëàäêèìèêîýèöèåíòàìè,áûëèçó÷åíîïåðàòîðâòîðîãîïîðÿäêàññóììèðóå-

ìûìïîòåíöèàëîì,íàéäåíûàñèìïòîòèêè ñîáñòâåííûõçíà÷åíèéèñîáñòâåííûõóíêöèé

îïåðàòîðàØòóðìàËèóâèëëÿ íàîòðåçêå. Âðàáîòàõ[12, 13℄áûëèèçó÷åíûîïåðàòîðû

âòîðîãîïîðÿäêà,óêîòîðûõïîòåíöèàë ÿâëÿåòñÿ

δ

^{-óíêöèåé.}

Âðàáîòàõ[1417℄ àâòîð ïðîäåìîíñòðèðîâàë íîâóþ ìåòîäèêó èçó÷åíèÿäèåðåíöè-

÷òî ìåòîäèêà ðàáîòû [11℄íå ïåðåíîñèòñÿíà îïåðàòîðû,óêîòîðûõïîðÿäîêðàâåí ÷åòû-

ðåìèëèâûøå.Îòìåòèìòàêæå,÷òîñëîæíîñòüèçó÷åíèÿäèåðåíöèàëüíûõîïåðàòîðîâ

âîçðàñòàåòñóâåëè÷åíèåìïîðÿäêàäèåðåíöèàëüíûõóðàâíåíèé,çàäàþùèõäèåðåí-

öèàëüíûé îïåðàòîð.

ðàíè÷íûå óñëîâèÿ (2) ïîêàçûâàþò, ÷òî ìû èçó÷àåì ñðàçó öåëîå ñåìåéñòâîóíêöè-

îíàëüíî-äèåðåíöèàëüíûõ îïåðàòîðîâ âîñüìîãî ïîðÿäêà ñ ñóììèðóåìûìè êîýèöè-

åíòàìè. Ôóíêöèîíàëüíî-äèåðåíöèàëüíûå îïåðàòîðû (òàê íàçûâàåìûå íàãðóæåííûå

îïåðàòîðû)ïîðÿäêàâûøå âòîðîãîðàíååàêòè÷åñêè íåèçó÷àëèñü (äàæå âñëó÷àå ãëàä-

êèõêîýèöèåíòîâ).

3. Àñèìïòîòèêà ðåøåíèé âñïîìîãàòåëüíîãî óðàâíåíèÿ ïðè

λ → ∞

àññìîòðèìâñïîìîãàòåëüíîåäèåðåíöèàëüíîåóðàâíåíèå,ïîëó÷àþùååñÿèç(1)ïðè

α = 0

^{(èëè ïðè}

r(x) ≡ 0

^):

y ⁽⁸⁾ (x) + q(x)y(x) = λa ⁸ y(x), 0 6 x 6 π, a > 0.

⁽⁴⁾

Îáîçíà÷èì

λ = s ⁸

^,

s = √ ⁸

λ

^{, ïðè÷åì äëÿ} êîððåêòíîñòè äàëüíåéøèõ âûêëàäîê âûáåðåì îñíîâíóþ âåòâü àðèìåòè÷åñêîãî êîðíÿ, äëÿ êîòîðîé

√ 8

1 = +1

^{. Ïóñòü}

w _k (k = 1, 2, . . . , 8)

^{ðàçëè÷íûå êîðíè âîñüìîé ñòåïåíèèç åäèíèöû:}

w ⁸ _k = 1, w _k = e ^{2πi 8 (k−1)} , k = 1, 2, . . . , 8; w ₁ = 1;

w ₂ = e ^{2πi 8} = cos 2π

8

+ i sin 2π

8

=

√ 2 2 + i

√ 2

2 = z 6 = 0, w ₃ = e ^{4πi 8} = i = z ² ; . . . ; w _m = z ^m−1 , m = 1, 2, . . . , 8.

(5)

×èñëà

w _k (k = 1, 2, . . . , 8)

^{èç (5) äåëÿò åäèíè÷íóþ îêðóæíîñòü íà âîñåìü ðàâíûõ}

÷àñòåé. Äëÿ íèõñïðàâåäëèâû ñëåäóþùèåñîîòíîøåíèÿ:

8

X

k=1

w _k ^m = 0, m = 1, 2, . . . , 7;

8

X

k=1

w _k ^m = 8, m = 0, m = 8.

⁽⁶⁾

Ìåòîäàìè ðàáîò [1416℄ óñòàíàâëèâàåòñÿ ñëåäóþùàÿ òåîðåìà.

Òåîðåìà 1. Îáùåå ðåøåíèå äèåðåíöèàëüíîãî óðàâíåíèÿ (4) èìååò ñëåäóþùèé

âèä:

y(x, s) =

8

X

k=1

C _k y _k (x, s); y ^(m) (x, s) =

8

X

k=1

C _k y ^(m) _k (x, s), m = 1, 2, . . . , 7,

⁽⁷⁾

ãäå

C _k (k = 1, 2, . . . , 8)

ïðîèçâîëüíûå ïîñòîÿííûå, ïðè ýòîì ïðè áîëüøèõ çíà÷åíèÿõ ñïåêòðàëüíîãî ïàðàìåòðà

λ

^äëÿ óíäàìåíòàëüíîé ñèñòåìû

{ y _k (x, s) } ⁸ k=1

ñïðàâåäëèâû

ñëåäóþùèåàñèìïòîòè÷åñêèå îðìóëûè îöåíêè:

y _k (x, s) = e ^{aw k sx} − A _7k (x, s) 8a ⁷ s ⁷ + O

e ^{| Im s|ax} s ¹⁴

, k = 1, 2, . . . , 8,

⁽⁸⁾

y ^(m) _k (x, s) = (as) ^m

w ^m _k e ^{aw k sx} − A ^m _7k (x, s) 8a ⁷ s ⁷ + O

e ^{|Im s|ax} s ¹⁴

,

⁽⁹⁾

k = 1, 2, . . . , 8; m = 1, 2, . . . , 7;

A _7k (x, s) = w _e ^{aw 1 sx}

x

Z

0

q(t)e ^{a(w k −w 1 )st} dt _ak1 + w ₂ e ^{aw 2 sx}

x

Z

0

q(t)e ^{a(w k −w 2 )st} dt _ak2

+ · · · + w ₈ e ^{aw 8 sx}

x

Z

0

q(t)e ^{a(w k −w 8 )st} dt _ak8 , k = 1, 2, . . . , 8,

⁽¹⁰⁾

A ^m _7k (x, s) =

8

X

n=1

w _n w ^m _n e ^{aw n sx}

x

Z

0

q(t)e ^{a(w k −w n )st} dt _akn , m = 1, 2, . . . , 7.

⁽¹¹⁾

Ïðèâûâîäå îðìóë(8)(11)ìûòðåáîâàëè âûïîëíåíèÿñëåäóþùèõíà÷àëüíûõóñëî-

âèé:

A _7k (0, s) = 0; A ^m _7k (0, s) = 0; y _k (0, s) = 1; y ^(m) _k (0, s) = (as) ^m w _k ^m , k = 1, 2, . . . , 8; m = 1, 2, . . . , 7.

(12)

4. åøåíèå óíêöèîíàëüíî-äèåðåíöèàëüíîãîóðàâíåíèÿ (1)

Îáîçíà÷èì ÷åðåç

∆ ₀ (x, s)

îïðåäåëèòåëü Âðîíñêîãî óíäàìåíòàëüíîé ñèñòåìû ðåøå- íèé

{ y _k (x, s) } ⁸ k=1

âñïîìîãàòåëüíîãî äèåðåíöèàëüíîãîóðàâíåíèÿ(4):

∆ ₀ (x, s) = det Wr[y ₁ (x, s), y ₂ (x, s), . . . , y ₈ (x, s)]

=

y ₁ (x, s) y ₂ (x, s) . . . y ₇ (x, s) y ₈ (x, s) y ₁ ^′ (x, s) y ^′ ₂ (x, s) . . . y ₇ ^′ (x, s) y ^′ ₈ (x, s) . . . . y ₁ ⁽⁶⁾ (x, s) y ⁽⁶⁾ ₂ (x, s) . . . y ⁽⁶⁾ ₇ (x, s) y ₈ ⁽⁶⁾ (x, s) y ₁ ⁽⁷⁾ (x, s) y ⁽⁷⁾ ₂ (x, s) . . . y ⁽⁷⁾ ₇ (x, s) y ₈ ⁽⁷⁾ (x, s)

.

⁽¹³⁾

Èçîáùåéòåîðèèäèåðåíöèàëüíûõóðàâíåíèéñëåäóåò,÷òîîïðåäåëèòåëüÂðîíñêîãî

∆ ₀ (x, s)

^{íå çàâèñèòîò ïàðàìåòðà}

x

^:

∆ 0 (x, s) = ∆ 0 (s) = ∆ 0 (0; s).

⁽¹⁴⁾

Èñïîëüçóÿ îðìóëû (8)(12),íàõîäèì

∆ ₀ (x, s) = ∆ ₀ (0, s) = ∆ ₀ (s)

=

1 1 . . . 1 1

(as)w 1 (as)w 2 . . . (as)w 7 (as)w 8

. . . . (as) ⁶ w ⁶ ₁ (as) ⁶ w ₂ ⁶ . . . (as) ⁶ w ₇ ⁶ (as) ⁶ w ⁶ ₈ (as) ⁷ w ⁷ ₁ (as) ⁷ w ₂ ⁷ . . . (as) ⁷ w ₇ ⁷ (as) ⁷ w ⁷ ₈

= ∆ ₀₀ (as) ²⁸ ,

⁽¹⁵⁾

ãäå

∆ ₀₀

îïðåäåëèòåëü Âàíäåðìîíäà ÷èñåë

w ₁ , w ₂ , . . . , w ₈

^:

∆ ₀₀ = det Wandermond ^′ s(w ₁ , w ₂ , . . . , w ₈ ) =

1 1 . . . 1 1

w ₁ w ₂ . . . w ₇ w ₈ w ⁶ ₁ w ⁶ ₂ . . . w ⁶ ₇ w ⁶ ₈ w ⁷ ₁ w ⁷ ₂ . . . w ⁷ ₇ w ⁷ ₈

= Y

k>m k,m=1,2,...,8

(w _k − w m ) = ∆ ₀₀ 6 = 0.

⁽¹⁶⁾

àçëîæèâîïðåäåëèòåëü

∆ ₀ (x, s)

^{èç (13)ïî ïîñëåäíåéñòðîêå, ïîëó÷èì}

∆ ₀ (x, s) = − y ₁ (x, s)D ₈₁ (x, s) + y ₂ (x, s)D ₈₂ − · · · − y ₇ (x, s)D ₈₇ (x, s) + y ₈ (x, s)D ₈₈ (x, s),

⁽¹⁷⁾

ãäå

D _8k (x, s) (k = 1, 2, . . . , 8)

àëãåáðàè÷åñêèå ìèíîðû ê ýëåìåíòàì âîñüìîé ñòðîêè è

k

^{-ãîñòîëáöà} îïðåäåëèòåëÿ

∆ ₀ (x, s)

^{èç (13):}

D ₈₁ (x, s) =

y ₂ (x, s) . . . y ₇ (x, s) y ₈ (x, s) y ^′ ₂ (x, s) . . . y ₇ ^′ (x, s) y ^′ ₈ (x, s) . . . . y ⁽⁶⁾ ₂ (x, s) . . . y ⁽⁶⁾ ₇ (x, s) y ₈ ⁽⁶⁾ (x, s) ,

. . . .,

D ₈₈ (x, s) =

y ₁ (x, s) y ₂ (x, s) . . . y ₇ (x, s) y ^′ ₁ (x, s) y ₂ ^′ (x, s) . . . y ^′ ₇ (x, s) . . . . y ⁽⁶⁾ ₁ (x, s) y ₂ ⁽⁶⁾ (x, s) . . . y ₇ ⁽⁶⁾ (x, s) .

(18)

Ïåðåïèøåì óðàâíåíèå(1)ââèäå

y ⁽⁸⁾ (x) + q(x)y(x) − λa ⁸ y(x) = αr(x)y(b)

^{èðåøèìåãî}

ìåòîäîì âàðèàöèè ïîñòîÿííûõ: áóäåì èñêàòü ðåøåíèå â âèäå

y = P ₈

k=1 C k (x, s)y k (x, s)

^,

ãäå

C _k (x, s)

íåèçâåñòíûå óíêöèè,

y _k (x, s) (k = 1, 2, . . . , 8)

ëèíåéíî-íåçàâèñèìûå ðå- øåíèÿâñïîìîãàòåëüíîãî óðàâíåíèÿ(4). Âðåçóëüòàòå äîêàæåì ñëåäóþùååóòâåðæäåíèå.

Òåîðåìà2.åøåíèå

y(x, s)

óíêöèîíàëüíî-äèåðåíöèàëüíîãîóðàâíåíèÿ(1)ïðåä- ñòàâëÿåòñÿâ âèäå

y(x, s) =

8

X

k=1

C _k y _k (x, s) + αy (b, s)

∆ 0 (s) H ₈ (x, s),

⁽¹⁹⁾

ãäå

C _k (k = 1, 2, . . . , 8)

ïðîèçâîëüíûå ïîñòîÿííûå,

{ y _k (x, s) } ⁸ k=1

óíäàìåíòàëüíàÿ ñèñòåìàðåøåíèéóðàâíåíèÿ (4), îïðåäåëÿåìàÿ îðìóëàìè (7)(12),

H ₈ (x, s) =

8

X

k=1

( − 1) ^k y _k (x, s)

x

Z

0

r(t)D _8k (t, s) dt _rk .

⁽²⁰⁾

Ïðè ýòîì â ñèëó ñâîéñòâ ñóììèðóåìîñòè (3), ñâîéñòâ îïðåäåëèòåëåé è îðìóë (6)

ïîëó÷àåì

y ^(m) (x, s) =

8

X

k=1

C _k y ^(m) _k (x, s) + αy(b, s)

∆ ₀ (s) H ₈ ^(m) (x, s), m = 1, 2, . . . , 7,

⁽²¹⁾

H ^(m) (x, s) =

8

X

k=1

( − 1) ^k y _k ^(m) (x, s)

x

Z

0

r(t)D _8k (t, s)dt rk , m = 1, 2, . . . , 7,

⁽²²⁾

âåëè÷èíà

∆ ₀ (s)

^{îïðåäåëåíà îðìóëàìè (14)(16).}

Ñïðàâåäëèâîñòü îðìóë (19)(22) ìîæíî ïåðåïðîâåðèòü íåïîñðåäñòâåííîé ïîäñòà-

íîâêîé ýòèõ îðìóëâ óðàâíåíèå (1).

Ïîäñòàâëÿÿ

x = b

^{â óðàâíåíèå (19), (20), íàõîäèì}

y(b, s) =

8

X

k=1

C _k y _k (b, s) + αy(b, s)

∆ ₀ (s) H ₈ (b, s), ∆ ₀ (s) 6 = 0,

îòêóäàïîëó÷àåì

y(b, s) = P 8

k=1 C _k y _k (b, s)

ψ ₈ (b, s) , ψ ₈ (b, s) = 1 − α

∆ ₀ (s) H ₈ )b, s) 6 = 0.

⁽²³⁾

Ïîñòàâèì

y(b, s)

^{èç (23) â (19), ñäåëàåì} íåîáõîäèìûå ïðåîáðàçîâàíèÿ, ïðèäåì ê âû- âîäó îñïðàâåäëèâîñòè ñëåäóþùåãî óòâåðæäåíèÿ.

Òåîðåìà 3.Îáùååðåøåíèåóíêöèîíàëüíî-äèåðåíöèàëüíîãîóðàâíåíèÿ(1)èìå-

åòñëåäóþùèé âèä:

y(x, s) =

8

X

k=1

C k h k (x, s); y ^(m) (x, s) =

8

X

k=1

C k h ^(m) _k (x, s), m = 1, 2, . . . , 7,

⁽²⁴⁾

C k (k = 1, 2, . . . , 8)

ïðîèçâîëüíûå ïîñòîÿííûå,

h _k (x, s) = y _k (x, s) + α

∆ 0 (s)

y _k (b, s)

ψ ₈ (b, s) H ₈ (x, s), k = 1, 2, . . . , 8,

⁽²⁵⁾

h ^(m) _k (x, s) = y ^(m) _k (x, s) + α

∆ ₀ (s)

y k (b, s)

ψ ₈ (b, s) H ₈ ^(m) (x, s),

⁽²⁶⁾

k = 1, 2, . . . , 8, m = 1, 2, . . . , 7,

óíêöèè

y _k (x, s)

^,

y _k ^(m) (x, s)

^{îïðåäåëåíû îðìóëàìè(7)(12),}

H ₈ (x, s)

^,

H ₈ ^(m) (x, s)

^{îïðåäå-}

ëåíûâ (20)(22),

ψ ₈ (b, s)

^{îïðåäåëåíàâ (23).}

Ïðèýòîì ñïðàâåäëèâû ñëåäóþùèåíà÷àëüíûå óñëîâèÿ:

H ₈ (0, s) = 0; H ₈ ^(m) (0, s) = 0; h _k (0, s) = y _k (0, s) = 1;

h ^(m) _k (0, s) = y ^(m) _k (0, s) = (as) ^m w _k ^m , k = 1, 2, . . . , 8; m = 1, 2, . . . , 7.

(27)

5. Èçó÷åíèå ãðàíè÷íûõ óñëîâèé (2)

Ïîäñòàâëÿÿ îðìóëû (24)(27)âãðàíè÷íûå óñëîâèÿ (2),èìååì

y ^{(m p )} (0, s) = 0 ⇔

8

X

k=1

C _k h ^(m _k ^{p )} (0, s) = 0 ⇔

8

X

k=1

C _k y ^(m _k ^{p )} (0, s) = 0

⇔

8

X

k=1

C _k (as) ^{m p} w ^m _k ^p = 0, p = 1, 2, . . . , 6;

⁽²⁸⁾

y ^{(n j )} (π, s) = 0 ⇔

8

X

k=1

C _k h ⁽ⁿ _k ^{j )} (π, s) = 0, j = 1, j = 2.

⁽²⁹⁾

Ñèñòåìà (28), (29) ñèñòåìà èç âîñüìè óðàâíåíèé ñ âîñåìüþ íåèçâåñòíûìè

C ₁ , C ₂ , . . . , C ₈

^{.Ýòàñèñòåìàèìååòíåíóëåâûåðåøåíèÿòîëüêîâòîìñëó÷àå,êîãäàååîïðå-}

äåëèòåëüðàâåí íóëþ.Ïîýòîìóâåðíî ñëåäóþùåå óòâåðæäåíèå.

Òåîðåìà 4. Óðàâíåíèåíà ñîáñòâåííûåçíà÷åíèÿ ÔÄÎ (1)(3) èìååò âèä

f (s) =

y ₁ ^{(m 1 )} (0, s) y ₂ ^{(m 1 )} (0, s) . . . y ^(m ₇ ^{1 )} (0, s) y ₈ ^{(m 1 )} (0, s) y ₁ ^{(m 2 )} (0, s) y ₂ ^{(m 2 )} (0, s) . . . y ^(m ₇ ^{2 )} (0, s) y ₈ ^{(m 2 )} (0, s) . . . . y ₁ ^{(m 6 )} (0, s) y ₂ ^{(m 6 )} (0, s) . . . y ^(m ₇ ^{6 )} (0, s) y ₈ ^{(m 6 )} (0, s) h ⁽ⁿ ₁ ^{1 )} (π, s) h ⁽ⁿ ₂ ^{1 )} (π, s) . . . h ⁽ⁿ ₇ ^{1 )} (π, s) h ⁽ⁿ ₈ ^{1 )} (π, s) h ⁽ⁿ ₁ ^{2 )} (π, s) h ⁽ⁿ ₂ ^{2 )} (π, s) . . . h ⁽ⁿ ₇ ^{2 )} (π, s) h ⁽ⁿ ₈ ^{2 )} (π, s)

.

⁽³⁰⁾

Ó÷èòûâàÿ íà÷àëüíûå óñëîâèÿ(27), ïåðåïèøåì óðàâíåíèå(30) â ñëåäóþùåìâèäå:

f (s) = (as) ^{m 1} (as) ^{m 2} (. . . )(as) ^{m 6}

×

w ₁ ^{m 1} w ^m ₂ ¹ . . . w ^m ₇ ¹ w ^m ₈ ¹ w ₁ ^{m 2} w ^m ₂ ² . . . w ^m ₇ ² w ^m ₈ ² . . . .

w ₁ ^{m 6} w ^m ₂ ⁶ . . . w ^m ₇ ⁶ w ^m ₈ ⁶ h ⁽ⁿ ₁ ^{1 )} (π, s) h ⁽ⁿ ₂ ^{1 )} (π, s) . . . h ⁽ⁿ ₇ ^{1 )} (π, s) h ⁽ⁿ ₈ ^{1 )} (π, s) h ⁽ⁿ ₁ ^{2 )} (π, s) h ⁽ⁿ ₂ ^{2 )} (π, s) . . . h ⁽ⁿ ₇ ^{2 )} (π, s) h ⁽ⁿ ₈ ^{2 )} (π, s)

= 0.

⁽³¹⁾

àçëîæèâîïðåäåëèòåëü

f (s)

^{èçîðìóëû(31)ïîïîñëåäíèìäâóìñòðî÷êàì,ïîëó÷èì}

f (s) = H ₁₂ W ₃₄₅₆₇₈ + H ₂₃ W ₁₄₅₆₇₈ − H ₃₄ W ₁₂₅₆₇₈ + · · · + H ₇₈ W ₁₂₃₄₅₆

+H ₁₈ W ₂₃₄₅₆₇ − H ₁₃ W ₂₄₅₆₇₈ + H ₁₄ W ₂₃₅₆₇₈ = · · · = 0,

⁽³²⁾

H _mk =

h ⁽ⁿ m ^{1 )} (π, s) h ⁽ⁿ _k ^{1 )} (π, s) h ⁽ⁿ m ^{2 )} (π, s) h ⁽ⁿ _k ^{2 )} (π, s)

, m, k = 1, 2, . . . , 8;

⁽³³⁾

W _{j 1 ,j 2 ,j 3 ,j 4 ,j 5 ,j 6}

⁽

j _k = 1, 2, . . . , 8

^;

k = 1, 2, . . . , 6

⁾ àëãåáðàè÷åñêèå ìèíîðûêýëåìåíòó

H _mk

â îïðåäåëèòåëå

f (s)

^{èç (31),}

j n 6 = m

^,

j n 6 = k

^{, çíàê ¾+¿ â îðìóëå (32) ñòàâèòñÿ â òîì}

ñëó÷àå,åñëè ïåðåñòàíîâêà

(m, k, j ₁ , j ₂ , j ₃ , j ₄ , j ₅ , j ₆ )

^{÷åòíàÿ,çíàê¾}

−

^¿åñëèïåðåñòàíîâêà

Àëãåáðàè÷åñêèå ìèíîðû

W j ₁ ,j ₂ ,j ₃ ,j ₄ ,j ₅ ,j ₆

^{áëàãîäàðÿ óäîáíûì} îáîçíà÷åíèÿì ëåãêî âû-

÷èñëÿþòñÿ:

W ₁₂₃₄₅₆ =

w ₁ ^{m 1} w ^m ₂ ¹ . . . w ^m ₆ ¹ w ₁ ^{m 2} w ^m ₂ ² . . . w ^m ₆ ² . . . . w ₁ ^{m 6} w ^m ₂ ⁶ . . . w ^m ₆ ⁶

=

1 ^{m 1} z ^{m 1} . . . z ^{5m 1} 1 ^{m 2} z ^{m 2} . . . z ^{5m 2} . . . . 1 ^{m 6} z ^{m 6} . . . z ^{5m 6}

= Y

k>n;

k,n=1,2,...,6

(z ^{m k} − z ^{m n} ) = W ₆ 6 = 0,

⁽³⁴⁾

òàê êàê îïðåäåëèòåëü

W ₁₂₃₄₅₆

ïðåäñòàâëÿåò ñîáîé îïðåäåëèòåëü Âàíäåðìîíäà ÷èñåë

z ^{m 1} , z ^{m 2} , . . . , z ^{m 6}

^.

Äàëåå èìååì

W ₂₃₄₅₆₇ =

w ₂ ^{m 1} w ^m ₃ ¹ . . . w ^m ₇ ¹ w ₂ ^{m 2} w ^m ₃ ² . . . w ^m ₇ ² . . . . w ₂ ^{m 6} w ^m ₃ ⁶ . . . w ^m ₇ ⁶

=

z ^{m 1} z ^{2m 1} . . . z ^{6m 1} z ^{m 2} z ^{2m 2} . . . z ^{6m 2} . . . . z ^{m 6} z ^{2m 6} . . . z ^{6m 6}

= z ^{m 1} z ^{m 2} (. . . )z ^{m 6} W ₁₂₃₄₅₆ = z ^{M 6} W ₆ , M ₆ =

6

X

k=1

m _k .

⁽³⁵⁾

Àíàëîãè÷íûì îáðàçîì âûâîäèì

W ₃₄₅₆₇₈ = z ^{2M 6} W ₆ ; W ₁₄₅₆₇₈ = ( − 1)z ^{3M 6} W ₆ ; W ₁₂₅₆₇₈ = z ^{4M 6} W ₆ ; W ₁₂₃₆₇₈ = ( − 1)z ^{5M 6} W ₆ ; W ₁₂₃₄₇₈ = z ^{6M 6} W ₆ ;

W ₁₂₃₄₅₈ = ( − 1)z ^{7M 6} W ₆ ; W ₁₂₃₄₅₆ = z ^{8M 6} W ₆ = W ₆ .

(36)

Ïîäñòàâèì îðìóëû (34)(36)âóðàâíåíèå (33), ïîäåëèì íà

z ^{2M 6} W ₆ 6 = 0

^{, ïîëó÷èì}

f (s) =

h ⁽ⁿ ₁ ^{1 )} (π, s) h ⁽ⁿ ₂ ^{1 )} (π, s) h ⁽ⁿ ₁ ^{2 )} (π, s) h ⁽ⁿ ₂ ^{2 )} (π, s)

−

h ⁽ⁿ ₂ ^{1 )} (π, s) h ⁽ⁿ ₃ ^{1 )} (π, s) h ⁽ⁿ ₂ ^{2 )} (π, s) h ⁽ⁿ ₃ ^{2 )} (π, s)

z ^{M 6} + z ^{2M 6}

h ⁽ⁿ ₃ ^{1 )} (π, s) h ⁽ⁿ ₄ ^{1 )} (π, s) h ⁽ⁿ ₃ ^{2 )} (π, s) h ⁽ⁿ ₄ ^{2 )} (π, s)

− . . .

= { φ ₁₂ − φ ₂₃ z ^{M 6} + φ ₃₄ z ^{2M 6} − . . . } (as) ^{n 1} (as) ^{n 2} = 0,

⁽³⁷⁾

ïðè ýòîì êàæäûé èç îïðåäåëèòåëåé

φ _mk

^{ìîæíî âûïèñàòü áîëåå ïîäðîáíî ñ ïîìîùüþ}

îðìóë (25),(26):

φ ₁₂ =

h ⁽ⁿ ₁ ¹⁾ (π,s) (as) ^{n 1}

h ⁽ⁿ ₂ ¹⁾ (π,s) (as) ^{n 1} h ⁽ⁿ ₁ ²⁾ (π,s)

(as) ^{n 2}

h ⁽ⁿ ₂ ²⁾ (π,s) (as) ^{n 2}

=

u ₁₁ u ₁₂ u ₂₁ u ₂₂

=

y ⁽ⁿ ₁ ¹⁾ (π,s) (as) ^{n 1} + _∆ ^α

0 (s) y ₁ (b,s) ψ ₈ (b,s)

H ₈ ^{(n 1)} (π,s) (as) ^{n 1}

y ⁽ⁿ ₂ ¹⁾ (π,s) (as) ^{n 1} + _∆ ^α

0 (s) y ₂ (b,s) ψ ₈ (b,s)

H ₈ ^{(n 1)} (π,s) (as) ^{n 1} y ⁽ⁿ ₁ ²⁾ (π,s)

(as) ^{n 2} + _∆ ^α

0 (s) y 1 (b,s) ψ 8 (b,s)

H ₈ ^{(n 2)} (π,s) (as) ^{n 2}

y ⁽ⁿ ₂ ²⁾ (π,s) (as) ^{n 2} + _∆ ^α

0 (s) y 2 (b,s) ψ 8 (b,s)

H ₈ ^{(n 2)} (π,s) (as) ^{n 2}

,

(38)

φ ₂₃ =

h ⁽ⁿ ₂ ¹⁾ (π,s) (as) ^{n 1}

h ⁽ⁿ ₃ ¹⁾ (π,s) (as) ^{n 1} h ⁽ⁿ ₂ ²⁾ (π,s)

(as) ^{n 2}

h ⁽ⁿ ₃ ²⁾ (π,s) (as) ^{n 2}

=

u ₁₂ u ₁₃ u ₂₂ u ₂₃

=

y ⁽ⁿ ₂ ¹⁾ (π,s) (as) ^{n 1} + _∆ ^α

0 (s) y 2 (b,s) ψ ₈ (b,s)

H ₈ ^{(n 1)} (π,s) (as) ^{n 1}

y ₃ ^{(n 1)} (π,s) (as) ^{n 1} + _∆ ^α

0 (s) y 3 (b,s) ψ ₈ (b,s)

H ₈ ^{(n 1)} (π,s) (as) ^{n 1} y ⁽ⁿ ₂ ²⁾ (π,s)

(as) ^{n 2} + _∆ ^α

0 (s) y 2 (b,s) ψ 8 (b,s)

H ₈ ^{(n 2)} (π,s) (as) ^{n 2}

y ₃ ^{(n 2)} (π,s) (as) ^{n 2} + _∆ ^α

0 (s) y 3 (b,s) ψ 8 (b,s)

H ₈ ^{(n 2)} (π,s) (as) ^{n 2}

, . . .

(39)

Ïîäñòàâëÿÿ îðìóëû(8)(11) è (17)(22) â (38), (39), âèäèì, ÷òî îïðåäåëèòåëè

φ ₁₂

^,

φ ₁₃

ïðåäñòàâëÿþò ñîáîé êâàçèïîëèíîìû. Òàêèì îáðàçîì, óíêöèÿ

f (s)

^{èç (37) òàêæå}

ïðåäñòàâëÿåò ñîáîé êâàçèïîëèíîì.

Äëÿ íàõîæäåíèÿ êîðíåé óðàâíåíèÿ (37) íåîáõîäèìî èçó÷èòü òàê íàçûâàåìóþ èíäè-

êàòîðíóþ äèàãðàììó ýòîãî óðàâíåíèÿ (ñì. [18, ãë. 12℄), ò. å. âûïóêëóþ îáîëî÷êó ïîêà-

çàòåëåé ýêñïîíåíò, âõîäÿùèõ â ýòî óðàâíåíèå. àñêëàäûâàÿ îïðåäåëèòåëè

φ ₁₂ , φ ₁₃ , . . .

ïî ñòîëáöàì, ïðèìåíÿÿ îðìóëû (8)(11), âèäèì, ÷òî â îïðåäåëèòåëü

φ ₁₂

^{âõîäÿò ýêñ-}

ïîíåíòû

e ^{a(w 1 +w 2 )sπ}

^{, â} îïðåäåëèòåëü

φ ₂₃

^{âõîäÿò ýêñïîíåíòû}

e ^{a(w 2 +w 3 )sπ} , . . .

^{, â îïðå-}

äåëèòåëü

φ _mk

^{ýêñïîíåíòû}

e ^{a(w m +w k )sπ}

^{. Çíà÷èò,} èíäèêàòîðíàÿ äèàãðàììà óðàâíå- íèÿ(37)(39) èìååò ñëåäóþùèéâèä:

èñ. 1.

Íà ðèñ. 1 âíóòðåííÿÿ åäèíè÷íàÿ îêðóæíîñòü äåëèòñÿ ÷èñëàìè

w _k (k = 1, 2, . . . , 8)

èç (5) íà âîñåìü ðàâíûõ ÷àñòåé, äëÿ âòîðîé îêðóæíîñòè ââåäåíû îáîçíà÷åíèÿ

w _km = w _k + w _m (k, m = 1, 2, . . . , 8)

^{. Íà íàðóæíþþîêðóæíîñòü} (èíäèêàòîðíóþ äèàãðàììó) ïî- ïàäàþòòîëüêî òî÷êè

w ₁ + w ₂

^,

w ₂ + w ₃

^,

w ₃ + w ₄ , . . .

^,

w ₇ + w ₈

^,

w ₈ + w ₉ = w ₈ + w ₁

^{, òî÷êè}

w _k + w _m

^,

(m − k) > 2

^{ïîïàäàþòâíóòðü}èíäèêàòîðíîé äèàãðàììûèíà àñèìïòîòèêóêîð- íåéóðàâíåíèÿ(37)(39)íåâëèÿþò. Êîðíèóðàâíåíèÿ(37)(39)ìîãóòíàõîäèòüñÿòîëüêî

ââîñüìèçàøòðèõîâàííûõñåêòîðàõðèñ.1,áåñêîíå÷íî ìàëîãîðàñòâîðà,áèññåêòðèñûêî-

òîðûõÿâëÿþòñÿñåðåäèííûìè ïåðïåíäèêóëÿðàìèêñòîðîíàì ïðàâèëüíîãîâîñüìèóãîëü-

íèêà

w ₁₂ w ₂₃ w ₃₄ . . . w ₇₈ w ₈₁ w ₁₂

^.

6. Óðàâíåíèå íà ñîáñòâåííûå çíà÷åíèÿ ÔÄÎ (1)(3) â ñåêòîðå 1)

èíäèêàòîðíîé äèàãðàììû

Äëÿ òîãî, ÷òîáû èçó÷èòüêîðíè óðàâíåíèÿ(37)(39) â ñåêòîðå1) èíäèêàòîðíîé äèà-

ãðàììûíàðèñ.1,íàäîîñòàâèòü òîëüêîýêñïîíåíòûñïîêàçàòåëÿìè

w ¯ ₈₁ = w ₁₂ = w ₁ + w ₂

è

w ¯ ₇₈ = w ₂₃ = w ₂ + w ₃

^{, ò. å. ýêñïîíåíòû}

e ^{a(w 1 +w 2 )sπ}

^è

e ^{a(w 2 +w 3 )sπ}

^{. Ïîýòîìó} ñïðàâåäëèâî óòâåðæäåíèå.

Òåîðåìà 5. Óðàâíåíèå íà ñîáñòâåííûå çíà÷åíèÿ ÔÄÎ (1)(3) â ñåêòîðå 1) èíäèêà-

òîðíîé äèàãðàììûíà ðèñ.1 èìååòñëåäóþùèé âèä:

g ₁ (s) =

u ₁₁ u ₁₂ u ₂₁ u ₂₂ −

u ₁₂ u ₁₃ u ₂₂ u ₂₃

z ^{M 6} = 0,

⁽⁴⁰⁾

ïðè÷åìâîâñåõàñèìïòîòè÷åñêèõîðìóëàõíåîáõîäèìîîñòàâèòüòîëüêîýêñïîíåíòûñïî-

êàçàòåëÿìè

w ₁ + w ₂

^,

w ₂ + w ₃

^{,âåëè÷èíû}

u _mk

^{îïðåäåëåíû â (38), (39).}

Èçó÷èì ñíà÷àëà àñèìïòîòè÷åñêîå ïîâåäåíèå óíêöèé

D _8k (x, s) (k = 1, 2, . . . , 8)

èç(17), (18).

Ïðèìåíÿÿ îðìóëû (8)(12)èñâîéñòâà îïðåäåëèòåëåé, èç (18) èìååì

D ₈₁ (x, s) =

v ₂ − ^{A 72} _R ^(x,s) ₇ + . . . . . . v ₇ − ^{A 77} _R ^(x,s) ₇ + . . . v ₈ − ^{A 78} _R ^(x,s) ₇ + . . . w ₂ v ₂ − ^{A 1 72} R ^(x,s) 7 + . . . . . . w ₇ v ₇ − ^{A 1 77} R ^(x,s) 7 + . . . w ₈ v ₈ − ^{A 1 78} R ^(x,s) 7 + . . . . . . . w ₂ ⁶ v ₂ − ^{A 6 72} _R ^(x,s) ₇ + . . . . . . w ₇ ⁶ v ₇ − ^{A 6 77} _R ^(x,s) ₇ + . . . w ⁶ ₈ v ₈ − ^{A 6 78} _R ^(x,s) ₇ + . . .

× (as)(as) ² (. . . )(as) ⁶ = (as) ²¹

D _81,0 (x, s) − D _81,7 (x, s) 8a ⁷ s ⁷ + O

1 s ¹⁴

,

(41)

ãäå ââåäåíû îáîçíà÷åíèÿ

v _k = e ^{aw k sx} (k = 1, 2, . . . , 8)

^,

R ₇ = 8a ⁷ s ⁷

^{, ¾}

+ . . .

^{¿ =¾}

+O( _s ¹ 14 )

^¿,

D _81,0 (x, s) =

v ₂ . . . v ₇ v ₈ w ₂ v ₂ . . . w ₇ v ₇ w ₈ v ₈ . . . . w ⁶ ₂ v ₂ . . . w ⁶ ₇ v ₇ w ⁶ ₈ v ₈

=

8

Y

k=2

v _k δ ₈₁ ,

⁽⁴²⁾

ïðèýòîì â ñèëóîðìóë (5),(6) èìååì

8

Y

k=2

v _k =

8

Y

k=2

e ^{aw k sx} = exp(a(w ₂ + w ₃ + · · · + w ₈ )sx) = e ^{−aw 1 sx} ,

⁽⁴³⁾

δ ₈₁

àëãåáðàè÷åñêèé ìèíîð ê ýëåìåíòó âîñüìîé ñòðîêè è ïåðâîãî ñòîëáöà îïðåäåëèòå- ëÿ

∆ ₀₀

^èç(16):

δ ₈₁ =

1 . . . 1 1

w ₂ . . . w ₇ w ₈ . . . . w ₂ ⁶ . . . w ₇ ⁶ w ⁶ ₈

.

⁽⁴⁴⁾

Ëåììà 1. Ìàòðèöà

δ _kn (k, n = 1, 2, . . . , 8)

àëãåáðàè÷åñêèõ ìèíîðîâ ê ýëåìåíòàì

b _kn (k, n = 1, 2, . . . , 8)

îïðåäåëèòåëÿ

∆ 00

^{èç (16)èìååò ñëåäóþùèé âèä:}

(δ _kn ) =

















δ ₁₁ δ ₁₂ δ ₁₃ . . . δ ₁₇ δ ₁₈ δ ₂₁ δ ₂₂ δ ₂₃ . . . δ ₂₇ δ ₂₈ δ ₃₁ δ ₃₂ δ ₃₃ . . . δ ₃₇ δ ₃₈ . . . . δ ₇₁ δ ₇₂ δ ₇₃ . . . δ ₇₇ δ ₇₈ δ ₈₁ δ ₈₂ δ ₈₃ . . . δ ₈₇ δ ₈₈

















= ∆ ₀₀ 8



















1 − 1 1 . . . 1 − 1

− w ₁ ⁻¹ w ⁻¹ ₂ − w ⁻¹ ₃ . . . − w ⁻¹ ₇ w ⁻¹ ₈ w ⁻² ₁ − w ₂ ⁻² w ⁻² ₃ . . . w ⁻² ₇ − w ₈ ⁻² . . . . w ⁻⁶ ₁ − w ₂ ⁻⁶ w ⁻⁶ ₃ . . . w ⁻⁶ ₇ − w ₈ ⁻⁶

− w ₁ ⁻⁷ w ⁻⁷ ₂ − w ⁻⁷ ₃ . . . − w ⁻⁷ ₇ w ⁻⁷ ₈



















(45)

 ñïðàâåäëèâîñòè ëåììû ìîæíî óáåäèòüñÿ, ðàñêëàäûâàÿ îïðåäåëèòåëü

∆ ₀₀

^{èç (16)}

ïî ñòðî÷êàì èëè ïîñòîëáöàì, èñïîëüçóÿîðìóëû(45).

Ñòðîãîå äîêàçàòåëüñòâî ëåììû ïðèâåäåíîàâòîðîì âðàáîòå [19℄.

Ñ ó÷åòîì(43)(45), îðìóëà (42)ïðèìåò âèä

D _81,0 (x, s) =

8

Y

k=2

v _k δ ₈₁ = e ^{−aw 1 sx} ( − w ⁻⁷ ₁ ) = ( − 1)w ₁ e ^{−aw 1 sx} ,

⁽⁴⁶⁾

òàê êàê

w ₁ ⁸ = w _k ⁸ (k = 1, 2, . . . , 8)

^{, ïðè ýòîìâ îðìóëå (41)èìååì}

D _81,7 (x, s) =

A ₇₂ (x, s) v ₃ . . . v ₈ A ¹ ₇₂ (x, s) w ₃ v ₃ . . . w ₈ v ₈ . . . . A ⁶ ₇₂ (x, s) w ⁶ ₃ v ₃ . . . w ⁶ ₈ v ₈

+ · · · +

v ₂ . . . v ₇ A ₇₈ (x, s) w ₂ v ₂ . . . w ₂ v ₇ A ¹ ₇₈ (x, s) . . . . w ⁶ ₂ v ₂ . . . w ⁶ ₂ v ₇ A ⁶ ₇₈ (x, s)

.

⁽⁴⁷⁾

Àíàëîãè÷íî îðìóëàì (41)(47) ìîæíî âû÷èñëèòü îïðåäåëèòåëè

D _8k (x, s) (k = 1, 2, . . . , 8)

^{èç (17)(18):}

D _8k (x, s) = (as) ²¹

D _8k,0 (x, s) − D _8k,7 (x, s) 8a ⁷ s ⁷ + O

1 s ¹⁴

, k = 1, 2, . . . , 8,

⁽⁴⁸⁾

D _8k,0 (x, s) = ( − 1) ^k w k e ^{−aw k sx} , k = 1, 2, . . . , 8,

⁽⁴⁹⁾

ïðè ýòîì âåëè÷èíû

D _8k,7 (x, s)

âûïèñûâàþòñÿ â âèäå ñóììû îïðåäåëèòåëåé àíàëîãè÷íî âåëè÷èíå

D _81,7 (x, s)

^èç(47).

Èñïîëüçóÿ îðìóëû (7)(12), (42)(50), (23)(27), (37), (39), âåêòîðû-ñòîëáöû

(u _1k ; u _2k ) ^∗ (k = 1, 2, 3)

^{èç îðìóëû(40)ïðèâåäåì êñëåäóþùåìó âèäó:}

u ₁₁ u ₂₁

=





w ⁿ ₁ ¹ e ^{aw 1 sπ} − ^A

n 1 71 (π,s)

8a ⁷ s ⁷ + ^αe _8a ^aw 7 s ^{1 7 sb} R(π; s; n ₁ ) w ⁿ ₁ ² e ^{aw 1 sπ} − ^A

n 2 71 (π,s)

8a ⁷ s ⁷ + ^αe _8a ^aw 7 s ^{1 7 sb} R(π; s; n ₂ )



 ,

⁽⁵⁰⁾

R(π; s; n, k) =

8

X

m=1

w _m w ⁿ _m ^k e ^{aw m sx}

x

Z

0

r(t)e ^{−aw m st} dt _rm , k = 1, 2;

⁽⁵¹⁾

u ₁₂ u ₂₂

= 

w ₂ ^{n 1} e ^{aw 2 sπ} − ^{A 72} _8a 7 ^(π,s) s ⁷ + ^αe _{8a 7 s} ² ₇ R(π; s; n ₁ ) w ₂ ^{n 2} e ^{aw 2 sπ} − ^A

n 2 72 (π,s)

8a ⁷ s ⁷ + ^αe _8a ^aw 7 s ^{2 7 sb} R(π; s; n ₂ )

 ,

⁽⁵²⁾

u ₁₃ u ₂₃

=





w ₃ ^{n 1} e ^{aw 3 sπ} − ^A

n 1 73 (π,s)

8a ⁷ s ⁷ + ^αe _8a ^aw 7 s ^{3 7 sb} R(π; s; n ₁ ) w ₃ ^{n 2} e ^{aw 3 sπ} − ^A

n 2 73 (π,s)

8a ⁷ s ⁷ + ^αe _8a ^aw 7 s ^{3 7 sb} R(π; s; n ₂ )



 ,

⁽⁵³⁾

ïðè÷åì äëÿ ñåêòîðà 1) â âåëè÷èíàõ

A ⁿ _7m ^k (π, s)

^{èç (10), (11) è}

R(π; s; n k ) (k = 1, 2

^;

m = 1, 2, 3)

^{èç(51) íåîáõîäèìî îñòàâëÿòü òîëüêî ýêñïîíåíòû}

e ^{aw 1 sπ}

^,

e ^{aw 2 sπ}

^è

e ^{aw 3 sπ}

^.

Ïðèìåíÿÿ îðìóëû (50)(53), óðàâíåíèå(40) ìîæíî ïåðåïèñàòüâ ñëåäóþùåì âèäå:

g ₁ (s) = g _1,0 (s) − g _1,7,1 (s)

8a ⁷ s ⁷ + g _1,7,2 (s) 8a ⁷ s ⁷ + O

1 s ¹⁴

= 0,

⁽⁵⁴⁾

g _1,0 (s) =

w ⁿ ₁ ¹ e ^{aw 1 sπ} w ⁿ ₂ ¹ e ^{aw 2 sπ} w ⁿ ₁ ² e ^{aw 1 sπ} w ⁿ ₂ ² e ^{aw 2 sπ}

−

w ⁿ ₂ ¹ e ^{aw 2 sπ} w ₃ ^{n 1} e ^{aw 3 sπ} w ⁿ ₂ ² e ^{aw 2 sπ} w ₃ ^{n 2} e ^{aw 3 sπ}

z ^{M 6} ,

⁽⁵⁵⁾

g _1,7,1 (s) =

A ⁿ ₇₁ ¹ (π, s) w ₂ ^{n 1} e ^{aw 2 sπ} A ⁿ ₇₁ ² (π, s) w ₂ ^{n 2} e ^{aw 2 sπ} 1

+

w ₁ ^{n 1} e ^{aw 2 sπ} A ⁿ ₇₂ ¹ (π, s) w ₁ ^{n 2} e ^{aw 2 sπ} A ⁿ ₇₂ ² (π, s) 2

−

A ⁿ ₇₂ ¹ (π, s) w ⁿ ₃ ¹ e ^{aw 3 sπ} A ⁿ ₇₂ ² (π, s) w ⁿ ₃ ² e ^{aw 3 sπ} 3

z ^{M 6} −

w ⁿ ₂ ¹ e ^{aw 2 sπ} A ⁿ ₇₃ ¹ (π, s) w ⁿ ₂ ² e ^{aw 2 sπ} A ⁿ ₇₃ ² (π, s) 4

z ^{M 6} ,

⁽⁵⁶⁾

g _1,7,2 (s) =

αe ^{aw 1 sb} R(π; s; n ₁ ) w ⁿ ₂ ¹ e ^{aw 2 sπ} αe ^{aw 1 sb} R(π; s; n ₂ ) w ⁿ ₂ ² e ^{aw 2 sπ} 5

+

w ₁ ^{n 1} e ^{aw 1 sπ} αe ^{aw 2 sb} R(π; s; n ₁ ) w ₁ ^{n 2} e ^{aw 1 sπ} αe ^{aw 2 sb} R(π; s; n ₂ ) 6

−

αe ^{aw 2 sb} R(π; s; n ₁ ) w ₃ ^{n 1} e ^{aw 3 sπ} αe ^{aw 2 sb} R(π; s; n ₂ ) w ₃ ^{n 2} e ^{aw 3 sπ} 7

z ^{M 6} −

w ⁿ ₂ ¹ e ^{aw 2 sπ} αe ^{aw 3 sb} R(π; s; n ₁ ) w ⁿ ₂ ² e ^{aw 2 sπ} αe ^{aw 3 sb} R(π; s; n ₂ ) 8

z ^{M 6} .

⁽⁵⁷⁾

Ïðèìåíÿÿ ñâîéñòâà îïðåäåëèòåëåé, óíêöèþ

g _1,0 (s)

^{èç(55) ïðèâåäåì êâèäó}

g _1,0 (s) =

w ⁿ ₁ ¹ w ₂ ^{n 1} w ⁿ ₁ ² w ₂ ^{n 2}

e ^{a(w 1 +w 2 )sπ} −

w ⁿ ₂ ¹ w ⁿ ₃ ¹ w ⁿ ₂ ² w ⁿ ₃ ²

e ^{a(w 2 +w 3 )sπ} z ^{M 6} ,

⁽⁵⁸⁾

ïðèýòîì áëàãîäàðÿîðìóëàì(5)èìååì

w ₁ ^{n 1} w ⁿ ₂ ¹ w ₁ ^{n 2} w ⁿ ₂ ²

=

1 ^{n 1} z ^{n 1} 1 ^{n 2} z ^{n 2}

= z ^{n 2} − z ^{n 1} = E ₂ ;

w ⁿ ₂ ¹ w ⁿ ₃ ¹ w ⁿ ₂ ² w ⁿ ₃ ²

=

z ^{n 1} z ^{2n 1} z ^{n 2} z ^{2n 2}

= z ^{n 1} z ^{n 2} E ₂ = z ^{N 2} E ₂ , N ₂ = n ₁ + n ₂ .

(59)

Âû÷èñëèì îïðåäåëèòåëü

| . . . | 1

^{èç(56), ïðèìåíÿÿîðìóëû(10),(11), (59)èñâîéñòâà}

îïðåäåëèòåëåé

| . . . | 1 =

w ₁ w ⁿ ₁ ¹ v _{1 π}

R

0

. . .

a11

+ w ₂ w ⁿ ₂ ¹ v _{2 π}

R

0

. . .

a12

+ w ₃ w ⁿ ₃ ¹ v _{3 π}

R

0

. . .

a13

w ⁿ ₂ ¹ w ₁ w ⁿ ₁ ² v ₁

_π R

0

. . .

a11

+ w ₂ w ⁿ ₂ ² v _{2 π}

R

0

. . .

a12

+ w ₃ w ⁿ ₃ ² v _{3 π}

R

0

. . .

a13

w ⁿ ₂ ²

e ^{aw 2 sπ}

= e ^{aw 2 sπ}





 w ₁ v ₁

Z ^π

0

. . .

a11

w ⁿ ₁ ¹ w ⁿ ₂ ¹ w ⁿ ₁ ² w ⁿ ₂ ²

+ w ₂ v ₂ Z ^π

0

. . .

a12

w ⁿ ₂ ¹ w ⁿ ₂ ¹ w ⁿ ₂ ² w ⁿ ₂ ²

+ w ₃ v ₃ Z ^π

0

. . .

a13

w ⁿ ₃ ¹ w ₂ ^{n 1} w ⁿ ₃ ² w ₂ ^{n 2}







(60) = w ₁ E ₂ e ^{a(w 1 +w 2 )sπ} Z ^π

0

. . .

a11

− w ₃ E ₂ z ^{N 2} e ^{a(w 2 +w 3 )sπ} Z ^π

0

. . .

a13

,

(60)

ãäåáûëè ââåäåíû îáîçíà÷åíèÿ

v k = e ^{aw k sπ} (k = 1, 2, . . . , 8)

^.

Àíàëîãè÷íûì îáðàçîì âûâîäÿòñÿ îðìóëû äëÿ îïðåäåëèòåëåé

| . . . | 2 , | . . . | 3

^è

| . . . | 4

èç (56):

| . . . | 2 = e ^{aw 1 sπ}

w ⁿ ₁ ¹ w ₁ w ₁ ^{n 1} v _{1 π}

R

0

. . .

a21

+ w ₂ w ⁿ ₂ ¹ v _{2 π}

R

0

. . .

a22

+ w ₃ w ⁿ ₃ ¹ v _{3 π}

R

0

. . .

a23

w ⁿ ₁ ² w ₁ w ₁ ^{n 2} v _{1 π}

R

0

. . .

a21

+ w ₂ w ⁿ ₂ ² v _{2 π}

R

0

. . .

a22

+ w ₃ w ⁿ ₃ ² v _{3 π}

R

0

. . .

a23

= w ₂ E ₂ e ^{a(w 1 +w 2 )sπ}

π

Z

0

q(t)dt _a22 ;

(61)

| . . . | 3 = w ₂ E ₂ z ^{N 2} e ^{a(w 2 +w 3 )sπ} Z ^π

0

. . .

a22

;

| . . . | 4 = − w ₁ E ₂ e ^{a(w 1 +w 2 )sπ} Z ^π

0

. . .

a31

+ w ₃ E ₂ z ^{N 2} e ^{a(w 2 +w 3 )sπ} Z ^π

0

. . .

a33

.

(62)

Äàëåå âû÷èñëÿåì îïðåäåëèòåëè

| . . . | m (m = 5, 6, 7, 8)

^{èç (57):}

| . . . | 5 = α

w ₁ w ⁿ ₁ ¹ v _{1 π}

R

0

. . .

r1

+ w ₂ w ⁿ ₂ ¹ v _{2 π}

R

0

. . .

r2

+ w ₃ w ⁿ ₃ ¹ v _{3 π}

R

0

. . .

r3

w ₂ ^{n 1} w ₁ w ⁿ ₁ ² v ₁

_π R

0

. . .

r1

+ w ₂ w ⁿ ₂ ² v _{2 π}

R

0

. . .

r2

+ w ₃ w ⁿ ₃ ² v _{3 π}

R

0

. . .

r3

w ₂ ^{n 2}

× e ^{aw 1 sb} e ^{aw 2 sπ} = w ₁ E ₂ αe ^{aw 1 sb} e ^{a(w 1 +w 2 )sπ} Z ^π

0

. . .

r1

− αw ₃ E ₂ z ^{N 2} e ^{aw 1 bs} e ^{a(w 2 +w 3 )sπ} Z ^π

0

. . .

r3

;

(63)