Giải gần đúng hệ phương trình tích phân kì dị của một hệ phương trình cặp tích phân fourier

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼
✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖

◆●➷ ❚❍➚ ❚❍❆◆❍

●■❷■ ●❺◆ ✣Ĩ◆●
❍➏ P❍×❒◆● ❚❘➐◆❍ ❚➑❈❍ P❍❹◆ ❑➐ ❉➚
❈Õ❆ ▼❐❚ ❍➏ P❍×❒◆● ❚❘➐◆❍ ❈➄P
❚➑❈❍ P❍❹◆ ❋❖❯❘■❊❘
▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❚❤→✐ ◆❣✉②➯♥ ✲ ◆➠♠ ✷✵✶✺


✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼
✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖

◆●➷ ❚❍➚ ❚❍❆◆❍

●■❷■ ●❺◆ ✣Ĩ◆●
❍➏ P❍×❒◆● ❚❘➐◆❍ ❚➑❈❍ P❍❹◆ ❑➐ ❉➚
❈Õ❆ ▼❐❚ ❍➏ P❍×❒◆● ❚❘➐◆❍ ❈➄P
❚➑❈❍ P❍❹◆ ❋❖❯❘■❊❘
❈❤✉②➯♥ ♥❣➔♥❤✿ ❚❖⑩◆ ●■❷■ ❚➑❈❍
▼➣ sè✿

ữợ ồ
❚❍➚ ◆●❹◆

❚❤→✐ ◆❣✉②➯♥ ✲ ◆➠♠ ✷✵✶✺


✐

▲í✐ ❝❛♠ ✤♦❛♥
❚ỉ✐ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ♥ë✐ ❞✉♥❣ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ tr✉♥❣
t❤ü❝ ✈➔ ❦❤ỉ♥❣ trị♥❣ ❧➦♣ ✈ỵ✐ ❝→❝ ✤➲ t➔✐ ❦❤→❝✳ ❚ỉ✐ ❝ơ♥❣ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣
♠å✐ sü ❣✐ó♣ ✤ï ❝❤♦ ✈✐➺❝ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔② ✤➣ ✤÷đ❝ ❝↔♠ ì♥ ✈➔ ❝→❝
t❤ỉ♥❣ t✐♥ tr➼❝❤ tr ữủ ró ỗ ố
◆❣✉②➯♥✱ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✺
◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥

◆❣ỉ ❚❤à ❚❤❛♥❤


✐✐

▲í✐ ❝↔♠ ì♥
✣➸ ❤♦➔♥ t❤➔♥❤ ✤÷đ❝ ❧✉➟♥ ✈➠♥ ♠ët ❝→❝❤ tổ ổ ữủ
sỹ ữợ ú ✤ï ♥❤✐➺t t➻♥❤ ❝õ❛ ❚❙✳ ◆❣✉②➵♥ ❚❤à ◆❣➙♥✳ ❚æ✐ ①✐♥
❝❤➙♥ t❤➔♥❤ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ ❝ỉ ❣✐→♦ ✈➔ ①✐♥ ❣û✐ ❧í✐ tr✐ ➙♥
♥❤➜t ❝õ❛ tỉ✐ ✤è✐ ✈ỵ✐ ♥❤ú♥❣ ✤✐➲✉ ❝ỉ ❣✐→♦ ✤➣ ❞➔♥❤ ❝❤♦ tỉ✐✳
❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ●✐→♠ ❤✐➺✉ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✲
✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ❝ị♥❣ ❝→❝ P❤á♥❣✲ ❇❛♥ ❝❤ù❝ ♥➠♥❣ ❝õ❛ tr÷í♥❣ ✣↕✐ ❤å❝
❙÷ ♣❤↕♠ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ❦❤♦❛ ❚♦→♥ ✲ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠✱
❝→❝ ỵ ổ ợ ồ ✷✵✶✺✮ tr÷í♥❣ ✣↕✐
❤å❝ ❙÷ ♣❤↕♠ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ t t tr t ỳ
tự qỵ ụ ♥❤÷ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tỉ✐ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❤å❝✳
❚ỉ✐ ỷ ớ ỡ tợ trữớ r ồ ờ t❤ỉ♥❣ P→❝ ❑❤✉ỉ♥❣

t➾♥❤ ▲↕♥❣ ❙ì♥✱ ♥ì✐ tỉ✐ ❝ỉ♥❣ t→❝ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tæ✐ ❤♦➔♥ t❤➔♥❤ ❦❤â❛
❤å❝✳ ❚æ✐ ①✐♥ ❝↔♠ ì♥ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ❜➧✱ ♥❤ú♥❣ ♥❣÷í✐ t❤➙♥ ✤➣ ❧✉ỉ♥ ✤ë♥❣
✈✐➯♥✱ ❤é trđ ✈➔ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tæ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔
t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳ ❳✐♥ tr➙♥ trå♥❣ ❝↔♠ ì♥✦
❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✹ ♥➠♠ ✷✵✶✺
◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥

◆❣ỉ ❚❤à ❚❤❛♥❤


✐✐✐

▼ư❝ ❧ư❝
▲í✐ ❝❛♠ ✤♦❛♥

✐

▲í✐ ❝↔♠ ì♥

✐✐

▼ư❝ ❧ư❝

✐✐✐

▼ð ✤➛✉

✶

✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✸

✶✳✶ ▲ỵ♣ ❤➔♠ ❍♦❧❞❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷ ●✐→ trà ❝❤➼♥❤ ❝õ❛ t➼❝❤ ♣❤➙♥ ❦ý ❞à ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷✳✶ ●✐→ trà ❝❤➼♥❤ ❈❛✉❝❤② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷✳✷ ●✐→ trà ❝❤➼♥❤ ❝õ❛ t➼❝❤ ♣❤➙♥ ❦ý ❞à ✳ ✳ ✳
✶✳✸ ❚♦→♥ tû t➼❝❤ ♣❤➙♥ ❦ý ❞à tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ L2ρ
✶✳✸✳✶ ❑❤æ♥❣ ❣✐❛♥ L2ρ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✸✳✷ ❚♦→♥ tû t➼❝❤ ♣❤➙♥ ❦ý ❞à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✹ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦ý ❞à ❧♦↕✐ ♠ët ✳ ✳ ✳ ✳
✶✳✺ ❈→❝ ✤❛ t❤ù❝ ❈❤❡❜②✉s❤❡✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✺✳✶ ✣❛ t❤ù❝ ❈❤❡❜②✉s❤❡✈ ❧♦↕✐ ♠ët ✳ ✳ ✳ ✳ ✳
✶✳✺✳✷ ✣❛ t❤ù❝ ❈❤❡❜②✉s❤❡✈ ❧♦↕✐ ❤❛✐ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✻ ❍➺ ✈ỉ ❤↕♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè t✉②➳♥ t➼♥❤

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✳ ✸
✳ ✺
✳ ✺
✳ ✺
✳ ✻
✳ ✻
✳ ✼
✳ ✼
✳ ✽
✳ ✽
✳ ✶✵
✳ ✶✷


✐✈

✶✳✼ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❤➔♠ ❝ì ❜↔♥ ❣✐↔♠ ♥❤❛♥❤ ✳ ✳ ✳ ✳ ✳
✶✳✼✳✶ ❑❤æ♥❣ ❣✐❛♥ S ❝õ❛ ❝→❝ ❤➔♠ ❝ì ❜↔♥ ❣✐↔♠ ♥❤❛♥❤ ✳
✶✳✼✳✷ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❝→❝ ❤➔♠ ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✽ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❤➔♠ s✉② rë♥❣ t➠♥❣ ❝❤➟♠ ✳ ✳ ✳ ✳ ✳
✶✳✽✳✶ ❑❤æ♥❣ ❣✐❛♥ S ❝õ❛ ❝→❝ ❤➔♠ s✉② rë♥❣ t➠♥❣ ❝❤➟♠
✶✳✽✳✷ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❤➔♠ s✉② rë♥❣ t➠♥❣ ❝❤➟♠ ✳
✶✳✽✳✸ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ t➼❝❤ ❝❤➟♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✾ ❈→❝ ❦❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✾✳✶ ❑❤æ♥❣ ❣✐❛♥ H s(R) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✾✳✷ ❈→❝ ❦❤æ♥❣ ❣✐❛♥ Hos(Ω), Ho,os (Ω), H s(Ω) ✳

ỵ ú ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✶✵ ❈→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ✈❡❝tì ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✶✵✳✶ ❑❤→✐ ♥✐➺♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✶✶ P❤✐➳♠ ❤➔♠ t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✶✷ ❚♦→♥ tû ❣✐↔ ✈✐ ♣❤➙♥ ✈❡❝tì ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

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✷✷

✷ ●✐↔✐ ❣➛♥ ✤ó♥❣ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦➻ ❞à ❝õ❛ ♠ët ❤➺
♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r
✷✹
✷✳✶ ❚➼♥❤ ❣✐↔✐ ✤÷đ❝ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r
✷✳✶✳✶ P❤→t ❜✐➸✉ ❜➔✐ t♦→♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✶✳✷ ✣÷❛ ✈➲ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r ✳ ✳ ✳

✷✳✶✳✸ ❚➼♥❤ ❣✐↔✐ ✤÷đ❝ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥
✭✷✳✶✵✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✶✳✹ ✣÷❛ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r ❤➺ ♣❤÷ì♥❣
tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦ý ❞à ♥❤➙♥ ❈❛✉❝❤② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✶✳✺ ✣÷❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦➻ ❞à ♥❤➙♥ ❈❛✉❝❤②
✈➲ ❤➺ ✈ỉ ❤↕♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè t✉②➳♥ t➼♥❤ ✳ ✳

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✸✸


✈

✷✳✷ ●✐↔✐ ❣➛♥ ✤ó♥❣ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦➻ ❞à ❝õ❛ ♠ët ❤➺
♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✷✳✷✳✶ ✣÷❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦ý ❞à ✈➲ ❞↕♥❣ ❦❤æ♥❣
t❤ù ♥❣✉②➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✷✳✷✳✷ ❚➼♥❤ ❣➛♥ ✤ó♥❣ ♥❣❤✐➺♠ ❝õ❛ ♠ët ❤➺ ♣❤÷ì♥❣ tr➻♥❤
t➼❝❤ ♣❤➙♥ ❦ý ❞à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✻✵

ỵ tt ữỡ tr t ♥❤➙♥ ❈❛✉❝❤② ✤➣ ✤÷đ❝ ❤♦➔♥
t❤✐➺♥ ð ♥û❛ ✤➛✉ t❤➳ ❦➾ ✷✵✳ ❚r♦♥❣ ❜❛ t❤➟♣ ♥✐➯♥ ❣➛♥ ✤➙②✱ ♥❤✐➲✉ ♥❤➔ t♦→♥
❤å❝ q✉❛♥ t➙♠ ✤➳♥ ✈➜♥ ✤➲ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❞↕♥❣
b
a

ϕ(t)
dt +
x−t

b

ϕ(t)K(x, t)dt = f (x),
a

✭✶✮

tr♦♥❣ ✤â f (x) ✈➔ K(x, t) ❧➔ ♥❤ú♥❣ ❤➔♠ ✤➣ ❜✐➳t✱ ϕ(t) ❧➔ ❤➔♠ ❝➛♥ t➻♠✳ ❍➔♠
✭♥❤➙♥ ❤❛② ❤↕❝❤✮ K(x, t) t❤÷í♥❣ ❧➔ ❤➔♠ ❧✐➯♥ tư❝ tr➯♥ ❤➻♥❤ ❝❤ú ♥❤➟t
S = {(x, t) : (x, t) ∈ [a, b] × [a, b]}.

P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❞↕♥❣ ✭✶✮ ❣➦♣ ❤➛✉ ❤➳t tr♦♥❣ ❝→❝ ❜➔✐ t♦→♥ ❜✐➯♥ ❤é♥
❤ñ♣ ❝õ❛ ❱➟t ❧➼ t♦→♥ ✤è✐ ợ ổ trỡ ữ t ❤ð✱
✈➳t ♥ùt✱ ✈➳t r↕♥✱ ❝→❝ ❜➔✐ t♦→♥ ✈➲ t✐➳♣ ①ó❝ ừ tt ỗ
ữỡ ú ữỡ tr t
ỗ ữỡ ❝➛✉ ♣❤÷ì♥❣ trü❝ t✐➳♣✱ ♣❤÷ì♥❣ ♣❤→♣ ♥ë✐ s✉② ❜➡♥❣
♣❤÷ì♥❣ ♣❤→♣ ▲❛❣r❛♥❣❡✱ ♣❤÷ì♥❣ ♣❤→♣ s➢♣ ①➳♣ t❤ù tü✱ ♣❤÷ì♥❣ ♣❤→♣ ✤❛
t❤ù❝ trü❝ ❣✐❛♦✳ ❱✐➺❝ ❣✐↔✐ ♠ët sè ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦➻ ❞à ✤÷đ❝

t❤ü❝ ❤✐➺♥ t÷ì♥❣ tü ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦➻ ❞à✱ ❤➺ ♣❤÷ì♥❣ tr➻♥❤
t➼❝❤ ♣❤➙♥ ❦➻ ❞à ✤÷đ❝ ❜✐➳♥ ✤ê✐ tø ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥✳ ●➛♥ ✤➙②✱
◆❣✉②➵♥ ❱➠♥ ◆❣å❝ ✈➔ ◆❣✉②➵♥ ❚❤à ◆❣➙♥ ✤➣ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉ ✈➲ t➼♥❤
❣✐↔✐ ✤÷đ❝ ❝õ❛ ♠ët sè ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r ①✉➜t ❤✐➺♥ ❦❤✐
❣✐↔✐ ❜➔✐ t♦→♥ ❜✐➯♥ ❤é♥ ❤đ♣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤✐➲✉ ❤á❛ ✈➔ ữỡ tr
s ỏ ợ ố ữủ t ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥
❦➻ ❞à ✈➔ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦➻ ❞à✱ ❝❤ó♥❣ tỉ✐ ❝❤å♥ ✤➲
t➔✐ ✧●✐↔✐ ❣➛♥ ✤ó♥❣ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦➻ ❞à ❝õ❛ ♠ët ❤➺ ♣❤÷ì♥❣
tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r✧✳ ▲✉➟♥ ✈➠♥ ♥❣♦➔✐ ♣❤➛♥ ▼ð ✤➛✉✱ ❑➳t




t ỗ ữỡ ở
ữỡ ♠ët tr➻♥❤ ❜➔② tê♥❣ q✉❛♥ ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ❧ỵ♣ ❤➔♠
❍♦❧❞❡r✱ t➼❝❤ ♣❤➙♥ ❦➻ ❞à✱ ❣✐→ trà ❝❤➼♥❤ ❝õ❛ t➼❝❤ ♣❤➙♥ ❦➻ ❞à✱ t♦→♥ tû t➼❝❤ ♣❤➙♥
❦➻ ❞à tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ L2ρ✱ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦➻ ❞à✱ ❤➺ ✈ỉ ❤↕♥ ❝→❝
♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè t✉②➳♥ t➼♥❤✱ ❝→❝ ✤❛ t❤ù❝ ❈❤❡❜②✉s❤❡✈✱ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r
❝õ❛ ❝→❝ ❤➔♠ ❝ì ❜↔♥ ❣✐↔♠ ♥❤❛♥❤✱ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❝→❝ ❤➔♠ s✉② rë♥❣
t➠♥❣ ❝❤➟♠✱ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈✱ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ✈❡❝tì✱ ♣❤✐➳♠
❤➔♠ t✉②➳♥ t➼♥❤ ❧✐➯♥ tư❝✱ t♦→♥ tû ❣✐↔ ✈✐ ♣❤➙♥ ✈❡❝tì✳
❈❤÷ì♥❣ ❤❛✐ tr➻♥❤ ❜➔② ❝→❝ ❦➳t q✉↔ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥✳ ▼ư❝ ✷✳✶ tr➻♥❤
❜➔② ✈➲ t➼♥❤ ❣✐↔✐ ✤÷đ❝ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ①✉➜t ❤✐➺♥ ❦❤✐
❣✐↔✐ ❜➔✐ t♦→♥ ❜✐➯♥ ❤é♥ ❤đ♣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤✐➲✉ ❤á❛✱ ❝→❝
tr t tỗ t↕✐ ✈➔ ❞✉② ♥❤➜t ♥❣❤✐➺♠ ❝õ❛ ❤➺ ♣❤÷ì♥❣
tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r✱ ✤÷❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r
✈➲ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦➻ ❞à ♥❤➙♥ ❈❛✉❝❤②✱ s❛✉ ✤â ✤÷❛ ❤➺ ♣❤÷ì♥❣
tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦➻ ❞à ♥❤➙♥ ❈❛✉❝❤② ✈➲ ❤➺ ✈ỉ ❤↕♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè
t✉②➳♥ t➼♥❤✳ ▼ư❝ ✷✳✷ ❝❤ó♥❣ tỉ✐ t❤ü❝ ❤✐➺♥ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ❤➺ ♣❤÷ì♥❣ tr➻♥❤
❝➦♣ t➼❝❤ ♣❤➙♥ ❦➻ ❞à ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t rr ợ

ữợ ữ ữỡ tr t➼❝❤ ♣❤➙♥ ❦➻ ❞à ✈➲ ❞↕♥❣ ❦❤æ♥❣ t❤ù ♥❣✉②➯♥❀
t➼♥❤ ❣➛♥ ✤ó♥❣ ♠❛ tr➟♥ ❤↕❝❤ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦➻ ❞à❀ t❤ü❝
❤✐➺♥ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ❤➺ ✈ỉ ❤↕♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè t✉②➳♥ t➼♥❤ ✤➣ ✤÷đ❝
✧❝❤➦t ❝ưt✧ ✤➳♥ ◆❂✻ ✱ s❛✉ ✤â t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤
t➼❝❤ ♣❤➙♥ ❦➻ ❞à✳
▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ trữớ ồ ữ
ữợ sỹ ữợ ❦❤♦❛ ❤å❝ ❝õ❛ ❚❙✳ ◆❣✉②➵♥ ❚❤à ◆❣➙♥✳ ❚→❝ ❣✐↔ ①✐♥ ✤÷đ❝
❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t❤➔♥❤ ✈➔ s➙✉ s➢❝ t tợ ổ ữợ
trữớ ồ ữ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥
t❤✉➟♥ ❧đ✐ ✤➸ t→❝ ❣✐↔ ❤♦➔♥ t❤➔♥❤ ✤÷đ❝ ❦❤♦→ ❤å❝ ❝õ❛ ♠➻♥❤✳




ữỡ
tự
ợ r
❬✸❪✳ ●✐↔ sû L ❧➔ ✤÷í♥❣ ❝♦♥❣ trì♥ ✈➔ ϕ(ξ) ❧➔ ❤➔♠ ❝→❝

✤✐➸♠ ♣❤ù❝ ξ ∈ L. ◆â✐ r➡♥❣ ❤➔♠ ϕ(ξ) t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❍♦❧❞❡r ✭✤✐➲✉
❦✐➺♥ Hλ✮ tr➯♥ ✤÷í♥❣ ❝♦♥❣ L ♥➳✉ ✈ỵ✐ ❤❛✐ ✤✐➸♠ ❜➜t ❦ý ξ1, ξ2 ∈ L t❛ ❝â ❜➜t
✤➥♥❣ t❤ù❝
λ
|ϕ(ξ2 ) − ϕ(ξ1 )| < A |ξ2 − ξ1 | ,
✭✶✳✶✮
tr♦♥❣ ✤â A, λ ❧➔ ❝→❝ ❤➡♥❣ sè ❞÷ì♥❣✳
◆➳✉ λ > 1 t❤➻ tø ✤✐➲✉ ❦✐➺♥ ✭✶✳✶✮ s✉② r❛ ϕ (ξ) ≡ 0 tr➯♥ L ✈➔ ❞♦ ✤â
ϕ(ξ) ≡ const, ξ ∈ L. ❱➻ ✈➟② t❛ ❧✉æ♥ ❧✉æ♥ ❝❤♦ r➡♥❣ 0 < λ ≤ 1. ◆➳✉ λ = 1
t❤➻ ✤✐➲✉ ❦✐➺♥ ❍♦❧❞❡r trð t❤➔♥❤ ✤✐➲✉ ❦✐➺♥ ▲✐♣s❝❤✐t③✳ ❘ã r➡♥❣ λ ❝➔♥❣ ♥❤ä t❤➻
❧ỵ♣ ❤➔♠ Hλ ❝➔♥❣ rë♥❣✳ ▲ỵ♣ ❤➔♠ ❍♦❧❞❡r ❤➭♣ ♥❤➜t ❧➔ ❧ỵ♣ ❤➔♠ ▲✐♣s❝❤✐t③✳

❉➵ t❤➜② r➡♥❣✱ ♥➳✉ ❝→❝ ❤➔♠ ϕ1(ξ), ϕ2(ξ) t❤ä❛ r
tữỡ ự ợ số 1, 2 t tờ t tữỡ ợ
t❤ù❝ ❦❤→❝ ❦❤ỉ♥❣✮ ❝ơ♥❣ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❍♦❧❞❡r ✈ỵ✐ ❝❤➾ sè
λ = min(λ1 , λ2 )✳
◆➳✉ ❤➔♠ ϕ(ξ) ❝â ✤↕♦ ❤➔♠ ❤ú✉ ❤↕♥ tr➯♥ L t❤➻ ♥â t❤ä❛ ♠➣♥
st ữủ s r tứ ỵ ✈➲ sè ❣✐❛ ❤ú✉ ❤↕♥✳ ◆❣÷đ❝ ❧↕✐
♥â✐ ❝❤✉♥❣ ❦❤ỉ♥❣ ✤ó♥❣✳ ❚❤➼ ❞ư✱ ❤➔♠
ϕ(ξ) = |ξ|, ξ ∈ R,

t❤✉ë❝ ❧ỵ♣ ❤➔♠ ❍♦❧❞❡r tr➯♥ R✱ ♥❤÷♥❣ ❦❤ỉ♥❣ ❝â ✤↕♦ ❤➔♠ t↕✐ ξ = 0✳


✹

❱➼ ❞ö ✶✳✶✳✷✳ ❍➔♠ sè ϕ(x)

√

t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❍♦❧❞❡r ✈➔ ❝❤➾
sè λ = 1/2 tr➯♥ ♠å✐ ❦❤♦↔♥❣ ❝õ❛ trö❝ t❤ü❝✳ ◆➳✉ ♥❤÷ ❦❤♦↔♥❣ ✤â ❦❤ỉ♥❣
❝❤ù❛ ❣è❝ tå❛ ✤ë t❤➻ ϕ(x) ❝á♥ ❧➔ ❤➔♠ ❣✐↔✐ t➼❝❤✱ ❞♦ ✤â t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥
▲✐♣s❝❤✐t③✳
❱➼ ❞ö ✶✳✶✳✸✳ ❳➨t ❤➔♠ sè
ϕ(x) =

=




x

1
,
lnx

0 < x ≤ 21 ,

ϕ(0) = 0.

❉➵ t❤➜② r➡♥❣ ❤➔♠ sè ϕ(x) ❧➔ ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ 0 ≤ x ≤ 12 ✳ ◆❤÷♥❣ ✈➻
limx→0+ xλ lnx = 0, ∀λ > 0,

♥➯♥ ợ ồ A õ t t ữủ trà ❝õ❛ x s❛♦ ❝❤♦
|ϕ(x) − ϕ(0)| =

1
> Axλ .
lnx

◆❤÷ ✈➟②✱ ❤➔♠ ϕ(x) tr➯♥ ✤♦↕♥ ♥â✐ tr➯♥ ❦❤æ♥❣ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❍♦❧❞❡r✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✹✳ ❬✸❪✳ ❑➼ ❤✐➺✉ Hα(r), 0 < α ≤ 1, r ≥ 0 ❧➔ ❧ỵ♣ ❤➔♠ ①→❝
✤à♥❤ tr➯♥ ✤♦↕♥ [a, b] ❝â ✤↕♦ ❤➔♠ ❝➜♣ r t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❍♦❧❞❡r ✈ỵ✐ sè
♠ơ α✳
❑❤→✐ ♥✐➺♠ ✈➲ ✤✐➲✉ ❦✐➺♥ ❍♦❧❞❡r ❝â t❤➸ ♠ð rë♥❣ ❝❤♦ ❤➔♠ ♥❤✐➲✉ ❜✐➳♥ ✈ỵ✐
sè ❜✐➳♥ ❤ú✉ ❤↕♥ ❜➜t ❦ý✳ ✣➸ ✤ì♥ ❣✐↔♥ t❛ ①➨t tr÷í♥❣ ❤đ♣ ❤➔♠ ❤❛✐ ❜✐➳♥✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✺✳ ❬✸❪✳ ❍➔♠ ❤❛✐ ❜✐➳♥ ϕ(ξ, τ ) tr➯♥ D t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥
❍♦❧❞❡r ♥➳✉ ✈ỵ✐ ♠å✐ ξ1, ξ2, τ1, τ2 ∈ D ❝â ❜➜t ✤➥♥❣ t❤ù❝
|ϕ(ξ2 , τ2 ) − ϕ(ξ1 , τ1 )|

µ

ν

A |ξ2 − ξ1 | + B |τ2 − τ1 | ,

tr♦♥❣ ✤â A, B, µ, ν ❧➔ số ữỡ à,

1

= min(à, ) ✈➔ C = max(A, B)✱ t❤➻
|ϕ(ξ2 , τ2 ) − ϕ(ξ1 , τ1 )|

λ

λ

C[|ξ2 − ξ1 | + |τ2 − τ1 | ].

❘ã r➔♥❣ ❧➔✱ ♥➳✉ ϕ(ξ, τ ) t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❍♦❧❞❡r t❤❡♦ ❤é♥ ❤ñ♣ (ξ, τ )
t❤➻ ♥â t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❍♦❧❞❡r t❤❡♦ ξ ✤➲✉ t❤❡♦ τ ✈➔ t❤ä❛ ♠➣♥ ✤✐➲✉
❦✐➺♥ ❍♦❧❞❡r t❤❡♦ τ ✤➲✉ t❤❡♦ ξ ✳


✺

✶✳✷ ●✐→ trà ❝❤➼♥❤ ❝õ❛ t➼❝❤ ♣❤➙♥ ❦ý ❞à
✶✳✷✳✶ ●✐→ trà ❝❤➼♥❤ ❈❛✉❝❤②

●✐↔ sû a ✈➔ b ❧➔ ❤❛✐ ✤✐➸♠ ❤ú✉ ❤↕♥✳ ❳➨t t➼❝❤ ♣❤➙♥

b

dx
(a < c < b).
x−c

a

❈❤ó♥❣ t❛ ❤➣② t➼♥❤ t➼❝❤ ♣❤➙♥ tr➯♥ ✤➙② ♥❤÷ ❧➔ t➼♥❤ t➼❝❤ ♣❤➙♥ s✉② rë♥❣✱ t❛
❝â
b

 c−

dx
= lim 1 →0, 2 →0 
x−c

a

1

dx
+
x−c

a

= ln

b

c+

b−c
+ lim 1 →0, 2 →0 ln
c−a

1



dx 
x−c

2

.

2

✭✶✳✷✮

●✐ỵ✐ ❤↕♥ ❝õ❛ ❜✐➸✉ t❤ù❝ ❝✉è✐ ❝ò♥❣ tr♦♥❣ ✭✶✳✷✮ rã r➔♥❣ ❧➔ ♣❤ö t❤✉ë❝ ✈➔♦
❝→❝❤ t✐➳♥ ✤➳♥ ✵ ❝õ❛ 1 ✈➔ 2✳ ❱➻ ✈➟② t➼❝❤ ♣❤➙♥ ✭✶✳✷✮��✮ t❛ ♥❤➟♥ ✤÷đ❝
(1)

(1)

(1)

β0,0 = 0, β0,1 = 0.00409244, β0,2 = 0,
(1)

(1)

β0,3 = −1.69399 × 10−6 , β0,4 = 0,
(1)

(1)

β0,5 = 9.34376 × 10−10 , β0,6 = 0,
(1)

(1)

(1)

β1,0 = −0.00409585, β1,1 = 0, β1,2 = 5.10762 × 10−6 ,
(1)

(1)

(1)

(1)

(1)

(1)

β1,3 = 0, β1,4 = −4.73236 × 10−9 ,
β1,5 = 0, β1,6 = 4.2026 × 10−12 ,
(1)

β2,0 = 0, β2,1 = −5.11238 × 10−6 , β2,2 = 0,
(1)

(1)

β2,3 = 9.49869 × 10−9 , β2,4 = 0,
(1)

(1)

β2,5 = −1.26965 × 10−11 , β2,6 = 0,
(1)

(1)

(1)

β3,0 = 1.70254 × 10−6 , β3,1 = 0, β3,2 = −9.5072 × 10−9 ,
(1)

(1)

(1)

(1)

(1)

(1)

β3,3 = 0, β3,4 = 2.122 × 10−11 ,
β3,5 = 0, β3,6 = −3.30689 × 10−14 ,
(1)

β4,0 = 0, β4,1 = 0.0870666, β4,2 = 0,
(1)

(1)

β4,3 = −0.000110675, β4,4 = 0,
(1)

(1)

β4,5 = 4.21813 × 10−8 , β4,6 = 0,
(1)

(1)

β5,0 = −9.46472 × 10−10 , β5,1 = −1.09556 × 10−18 ,
(1)

(1)

β5,2 = 1.2732 × 10−11 , β5,3 = 3.78033 × 10−22 ,

(1)

(1)

β5,4 = −4.97309 × 10−14 , β5,5 = −1.3165 × 10−25 ,
(1)

β5,6 = 1.01924 × 10−16 ,
(1)

(1)

β6,0 = −3.83 × 10−19 , β6,1 = −4.23215 × 10−12 ,
(1)

(1)

(1)

(1)

β6,2 = 1.26341 × 10−21 , β6,3 = 3.31345 × 10−14 ,
β6,4 = 4.69903 × 10−26 , β6,5 = −1.0198 × 10−16 ,
(1)

β6,6 = −1.44635 × 10−27 ,


✺✸
(2)

(2)

(2)

β0,0 = 0, β0,1 = −0.0121449, β0,2 = 0,
(2)

(2)

β0,3 = 0.0000238998, β0,4 = 0,
(2)

(2)

β0,5 = −5.55185 × 10−8 , β0,6 = 0,
(2)

(2)

(2)

β1,0 = 0.0121941, β1,1 = 0, β1,2 = −0.0000733008,
(2)

(2)

(2)

(2)

(2)

(2)

β1,3 = 0, β1,4 = 2.92032 × 10−7 ,
β1,5 = 0, β1,6 = −9.84101 × 10−10 ,
(2)

β2,0 = 0, β2,1 = 0.0000735989, β2,2 = 0,
(2)

(2)

β2,3 = −5.92243 × 10−7 , β2,4 = 0,
(2)

(2)

(2)

(2)

β2,5 = 3.02848 × 10−9 , β2,6 = 0,
(2)

β3,0 = −0.0000244336, β3,1 = 0, β3,2 = 5.94296 × 10−7 ,
(2)

(2)

β3,3 = 0, β3,4 = −5.09864 × 10−9 ,
(2)

(2)

β3,5 = 0, β3,6 = 2.83912 × 10−11 ,
(2)

(2)

β4,0 = 1.81401 × 10−18 , β4,1 = −2.96121 × 10−7 ,
(2)

(2)

β4,2 = −8.1644 × 10−21 , β4,3 = 5.11302 × 10−9 ,
(2)

(2)

(2)

(2)

(2)

β4,4 = 2.1064 × 10−23 , β4,5 = −4.28717 × 10−11 , β4,6 = 0,
β5,0 = 5.84064 × 10−8 , β5,1 = 7.97836 × 10−19 ,
(2)

(2)

β5,2 = −3.05919 × 10−9 , β5,3 = −3.79152 × 10−23 ,
(2)

(2)

(2)

β5,4 = 4.29563 × 10−11 , β5,5 = 0, β5,6 = −2.94773 × 10−13 ,
(2)

(2)

β6,0 = −3.49786 × 10−18 , β6,1 = 1.00949 × 10−9 ,
(2)

(2)

β6,2 = 3.35852 × 10−20 , β6,3 = −2.85811 × 10−11 ,
(2)

(2)

β6,4 = −1.28721 × 10−22 , β6,5 = 2.95157 × 10−13 ,
(2)

β6,6 = 4.1552 × 10−25 .

❚❤❛② ❝→❝ βj,k(1) ✈➔ βj,k(2) t➼♥❤ ✤÷đ❝ ð tr➯♥ ✈➔♦ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✼✾✮ t
(j)
t ữủ ữỡ tr số t t ợ ❝→❝ ➞♥ ❧➔ A(j)
1 ✈➔ A2 ✈ỵ✐
j = 1, . . . , 6.


✺✹

●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✤â t❛ t➻♠ ✤÷đ❝ ♥❣❤✐➺♠ ♥❤÷ s❛✉✿

(1)


A1 = −0.996068a0 + 5.40169 × 10−25 a1 − 0.249017a2






−2.74236 × 10−21 a3 − 0.124509a4 + 1.05015 × 10−20 a5






−0.0120966b0 − 1.42396 × 10−27 b1 − 0.00301807b2







−2.25835 × 10−19 b3 − 0.00150752b4 − 1.16812 × 10−19 b5 ,




(1)


A2 = −2.333 × 10−25 a0 − 0.499997a1 − 5.73043 × 10−26 a2






−0.260881a3 + 6.84664 × 10−20 a4 − 0.167131a5






+3.19531 × 10−25 b0 − 0.0000367733b1 + 7.84839 × 10−26 b2







−0.00001915b3 − 4.98592 × 10−20 b4 − 0.0000122551b5 ,




(1)


A3 = −4.2008 × 10−6 a0 − 2.46536 × 10−27 a1 − 0.250001a2






+1.65392 × 10−23 a3 − 0.187501a4 − 3.08664 × 10−23 a5






+0.0000729508b0 − 2.71391 × 10−30 b1 + 0.0000180892b2





+1.01948 × 10−21 b + 9.00765 × 10−6 b − 3.0467 × 10−23 b ,
3

4

5

(1)


A4 = 1.12804 × 10−29 a0 − 4.72702 × 10−9 a1 + 2.77129 × 10−30 a2






−0.124986a3 − 2.35943 × 10−23 a4 − 0.124986a5 −






1.09833 × 10−28 b0 + 2.96087 × 10−7 b1 − 2.69771 × 10−29 b2







+1.53849 × 10−7 b3 + 2.32841 × 10−24 b4 + 9.83332 × 10−8 b5 ,




(1)


A5 = 1.18078 × 10−9 a0 + 6.42871 × 10−30 a1 + 2.91669 × 10−10 a2






−6.59398 × 10−26 a3 − 0.0625a4 − 3.56286 × 10−26 a5






−2.90826 × 10−7 b0 + 5.73221 × 10−32 b1 − 7.14321 × 10−8 b2







−2.63244 × 10−24 b3 − 3.54001 × 10−8 b4 + 1.39344 × 10−24 b5 ,




(1)


A6 = 8.15687 × 10−34 a0 + 6.23664 × 10−12 a1 + 2.00134 × 10−34 a2






−5.26882 × 10−9 a3 + 8.07526 × 10−27 a4 − 0.03125a5






+3.75721 × 10−32 b0 − 1.51422 × 10−9 b1 + 9.22835 × 10−33 b2




−7.84707 × 10−10 b + 2.08086 × 10−28 b − 5.008 × 10−10 b ,
3

4

5


✺✺

✈➔

(2)


A1 = −0.0120966a0 − 1.42396 × 10−27 a1 − 0.00301807a2






−2.25835 × 10−19 a3 − 0.00150752a4 − 1.16812 × 10−19 a5






−0.996068b0 + 5.40169 × 10−25 b1 − 0.249017b2







−2.74236 × 10−21 b3 − 0.124509b4 + 1.05015 × 10−20 b5 ,




(2)


A2 = 3.19531 × 10−25 a0 − 0.0000367733a1 + 7.84839 × 10−26 a2






−0.00001915a3 − 4.98592 × 10−20 a4 − 0.0000122551a5






−2.333 × 10−25 b0 − 0.499997b1 − 5.73043 × 10−26 b2







−0.260881b3 + 6.84664 × 10−20 b4 − 0.167131b5 ,




(2)


A3 = 0.0000729508a0 − 2.71391 × 10−30 a1 + 0.0000180892a2






+1.01948 × 10−21 a3 + 9.00765 × 10−6 a4 − 3.0467 × 10−23 a5






−4.2008 × 10−6 b0 − 2.46536 × 10−27 b1 − 0.250001b2





+1.65392 × 10−23 b − 0.187501b − 3.08664 × 10−23 b ,
3

4

5

(2)


A4 = −1.09833 × 10−28 a0 + 2.96087 × 10−7 a1 − 2.69771 × 10−29 a2






+1.53849 × 10−7 a3 + 2.32841 × 10−24 a4 + 9.83332 × 10−8 a5






+1.12804 × 10−29 b0 − 4.72702 × 10−9 b1 + 2.77129 × 10−30 b2







−0.124986b3 − 2.35943 × 10−23 .b4 − 0.124986.b5 ,




(2)


A5 = −2.90826 × 10−7 a0 + 5.73221 × 10−32 a1 − 7.14321 × 10−8 a2






−2.63244 × 10−24 a3 − 3.54001 × 10−8 a4 + 1.39344 × 10−24 a5






+1.18078 × 10−9 b0 + 6.42871 × 10−30 b1 + 2.91669 × 10−10 b2







−6.59398 × 10−26 b3 − 0.0625b4 − 3.56286 × 10−26 b5 ,




(2)


A6 = 3.75721 × 10−32 a0 − 1.51422 × 10−9 a1 + 9.22835 × 10−33 a2






−7.84707 × 10−10 a3 + 2.08086 × 10−28 a4 − 5.008 × 10−10 a5






+8.15687 × 10−34 b0 + 6.23664 × 10−12 b1 + 2.00134 × 10−34 b2




−5.26882 × 10−9 b + 8.07526 × 10−27 b − 0.03125b .
3

4

5

✭✷✳✽✹✮

❇➙② ❣✐í t❛ t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✺✵✮ ✈ỵ✐ N = 6.


✺✻

❚❛ ❝â
∗
(τ )
vm,6

1
=√
1 − τ2

6
(m)

Aj Tj (τ ), m = 1, 2,

✭✷✳✽✺✮

j=0

tr♦♥❣ ✤â✱ ❝→❝ A(1)

✈➔ A(2)
✤÷đ❝ t➼♥❤ tr♦♥❣ ✭✷✳✽✹✮ ✈➔ Tj (τ ) ❧➔ ❝→❝ ✤❛ t❤ù❝
j
j
❈❤❡❜②s❤❡✈ ❧♦↕✐ ♠ët✳
❘ót ❣å♥ t❛ ✤÷đ❝


∗

(τ )
v1,6


















1
(1)
(1)
(1)
(1)
(1)
(1)
(−A2 + A4 − A6 ) + (A1 − 3A3 + 5A5 )τ
2
1−τ
1
(1)
(1)
(1)
(1)
(1)
+√
(2A2 − 8A4 + 18A6 )τ 2 + (4A3 − 20A5 )τ 3
2
1−τ
1
(1)
(1)
(1)
(1)
(8A4 − 48A6 )τ 4 + 16A5 τ 5 + 32A6 τ 6 ,
+√
2
1−τ
1

(2)
(2)
(2)
(2)
(2)
(2)

∗

√
(−A2 + A4 − A6 ) + (A1 − 3A3 + 5A5 )τ
v
(τ
)
=

2,6

2

1−τ



1
(2)
(2)
(2)
(2)
(2)



√
+
(2A2 − 8A4 + 18A6 )τ 2 + (4A3 − 20A5 )τ 3


2

1−τ



1
(2)
(2)
(2)
(2)


(8A4 − 48A6 )τ 4 + 16A5 τ 5 + 32A6 τ 6 .
+√

2
1−τ
=√

❇➙② ❣✐í t❛ t➼♥❤ u1,6 ✈➔ u2,6 ✿
❉♦

(b − a)τ + b + a
),
2
(b − a)τ + b + a
v2∗ (τ ) = v2 (
),
2
(b − a)τ + b + a
t=
2
v1∗ (τ ) = v1 (t), v2∗ (τ ) = v2 (t).
v1∗ (τ ) = v1 (

♥➯♥ t❛ ❝â
▼➦t ❦❤→❝ t❛ ❝â




u∗1,6 (y)













u∗2,6 (y)










=
=
=
=

1 1 ∗
b−a
b−a
v1 (τ )s✐❣♥[
(y − τ )]
dτ
2 −1
2
2
b−a y ∗
b−a 1 ∗
v1 (τ )dτ −

v1 (τ )dτ,
4
4
−1
y
1 1 ∗
b−a
b−a
v2 (τ )s✐❣♥[
(y − τ )]
dτ
2 −1
2
2
b−a y ∗
b−a 1 ∗
v2 (τ )dτ −
v2 (τ )dτ,
4
4
−1
y

y ∈ (−1, 1),

y ∈ (−1, 1).


✺✼

❚➼♥❤ ✈➔ rót ❣å♥ t➼❝❤ ♣❤➙♥ tr➯♥ t❛ ✤÷đ❝
a − b√
(1)
(1)
(1)
u∗1,6 (y) =
1 − y 2 15A1 − 5A3 + 3A5
30
a − b√
(1)
(1)
(1)
(1)
(1)
+
1 − y 2 (15A2 − 15A4 + 15A6 )y + (20A3 − 36A5 )y 2
30
a − b√
(1)
(1)
(1)
(1)
+
1 − y 2 (30A4 − 80A6 )y 3 + 48A5 y 4 + 80A6 y 5 ,
30
a − b√

(2)
(2)
(2)

∗

(y)
=
u
1 − y 2 15A1 − 5A3 + 3A5

2,6


30


√
a
−
b

(2)
(2)
(2)
(2)
(2)


+
1 − y 2 (15A2 − 15A4 + 15A6 )y + (20A3 − 36A5 )y 2


30




a − b√
(2)
(2)
(2)
(2)

 +
1 − y 2 (30A4 − 80A6 )y 3 + 48A5 y 4 + 80A6 y 5 .
30
❚❛ ❝â uj,6(x) = uj,6( 2x b−−b a− a ), ợ j = 1, 2.

t ữủ


15(a + b) (1)
− (b − x)(x − a)

(1)
(1)


15A1 +
A2 − 5A3
u1,6 (x) =


15
a −3b


2

15(a
+
b)
30(a
+
b) (1)
20(a

+
b)

(1)
(1)
(1)

+
A
−
A4 + 3A5
A
+

3
4

2
3

(a − b)
a−b
(a − b)


4
2

48(a
+

b)
15(a
+ b) (1)
36(a
+
b)

(1)
(1)


A
+
A
+
−
A6

5
5
2
4

(a
−
b)
(a
−
b)
a

−
b




80(a + b)3 (1) 80(a + b)5 (1)


A6 +
A6
−


3
5

(a
−
b)
(a
−
b)



30 (1) 80(a + b) (1)
30 (1)




+[−
A
A
A4
−
+
2
3

2

a
−
b
(a
−
b)
a
−
b




180(a + b)2 (1) 144(a + b) (1) 384(a + b)3 (1)


−
A4 +

A5 −
A5

3
2
4

(a − b)
(a − b)
(a − b)
2
30
480(a
+
b)
800(a
+
b)4 (1)

(1)
(1)


A6 +
A6 −
A6 ]x
−

3
5


a
−
b
(a
−
b)
(a
−
b)




80
360(a + b) (1)
144
(1)
(1)


A
+
A
−
A5
+[

3
4

2
3
2

(a − b)
(a − b)
((a − b)



2

1152(a
+
b)
960(a
+
b)
3200(a + b)3 (1) 2

(1)
(1)

+
A5 −
A6 +
A6 ]x


4

3
5

(a
−
b)
(a
−
b)
(a
−
b)



−240 (1) 1536(a + b) (1)
640

(1)


+[
A
−
A
+
A6

4
5

3
4
3

(a − b)
(a − b)
(a − b)



2

6400(a
+
b)
768

(1)
(1)

−
A6 ]x3 + [
A5


5
4

(a − b)
(a − b)




6400(a + b) (1) 4
2560 5


+
A
]x
−
x ,
6
(a − b)5
(a − b)5


✺✽

✈➔

− (b − x)(x − a)
15(a + b) (2)

(2)
(2)


u
(x)

=
15A
+
A2 − 5A3
2,6

1

15
a −3b


2

20(a
+
b)
15(a
+
b)
30(a
+
b) (2)

(2)
(2)
(2)


+

A
−
A
+
A4 + 3A5

3
4
2
3

(a − b)
a−b
(a − b)



4
2

36(a + b) (2) 48(a + b) (2) 15(a + b) (2) 80(a + b)3 (2)


A5 +
A5 +
A6 −
A6
−



2
4
3

(a
−
b)
(a
−
b)
a
−
b
(a
−
b)



30 (2) 80(a + b) (2)
30 (2)
80(a + b)5 (2)



A
+
[−
A
−

A
+
A4
+
6
2
3

5
2

(a
−
b)
a
−
b
(a
−
b)
a
−
b




180(a + b)2 (2) 144(a + b) (2) 384(a + b)3 (2)



−
A4 +
A5 −
A5

3
2
4

(a − b)
(a − b)
(a − b)
2
480(a
+
b)
800(a
+
b)4 (2)
30

(2)
(2)


A6 +
A6 −
A6 ]x
−


3
5

a
−
b
(a
−
b)
(a
−
b)




80
360(a + b) (2)
144
1152(a + b)2 (2)
(2)
(2)


+[
A
+
A
−
A

+
A5

4
2 3
3
2 5
4

(a
−
b)
(a
−
b)
((a
−
b)
(a
−
b)




−240 (2)
960(a + b) (2) 3200(a + b)3 (2) 2


A

A
A
+
]x
+
[
−

6
6

3
5
3 4

(a
−
b)
(a
−
b)
(a
−
b)




1536(a + b) (2)
640

6400(a + b)2 (2) 3
(2)


−
A
A
A6 ]x
+
−

5
4
3 6
5

(a
−
b)
(a
−
b)
(a
−
b)




768

6400(a + b) (2) 4
2560 5

(2)

A
A
x .
+
]x
−
+[
6
(a − b)4 5
(a − b)5
(a − b)5


✺✾

❑➳t ❧✉➟♥ ❝❤✉♥❣
▲✉➟♥ ✈➠♥ ✤➣ tr➻♥❤ ❜➔② ✈➔ ✤↕t ✤÷đ❝ ♠ët sè ❦➳t q✉↔ s❛✉ ✤➙②✿
✶✳ ❚r➻♥❤ ❜➔② tê♥❣ q✉❛♥ ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ❧ỵ♣ ❤➔♠ ❍♦❧❞❡r✱ t➼❝❤
♣❤➙♥ ❦➻ ❞à✱ ❣✐→ trà ❝❤➼♥❤ ❝õ❛ t➼❝❤ ♣❤➙♥ ❦➻ ❞à✱ t♦→♥ tû t➼❝❤ ♣❤➙♥ ❦➻ ❞à tr♦♥❣
❦❤æ♥❣ ❣✐❛♥ L2ρ ✱ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦➻ ❞à✱ ❤➺ ✈ỉ ❤↕♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤
✤↕✐ sè t✉②➳♥ t➼♥❤✱ ❝→❝ ✤❛ t❤ù❝ ❈❤❡❜②✉s❤❡✈✱ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❝→❝ ❤➔♠
❝ì ❜↔♥ ❣✐↔♠ ♥❤❛♥❤✱ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❝→❝ ❤➔♠ s✉② rë♥❣ t➠♥❣ ❝❤➟♠✱
❝→❝ ❦❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈✱ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ✈❡❝tì✱ ♣❤✐➳♠ ❤➔♠ t✉②➳♥
t➼♥❤ ❧✐➯♥ tư❝✱ t♦→♥ tû ❣✐↔ ✈✐ ♣❤➙♥ ✈❡❝tì✳
✷✳ ❚r➻♥❤ ❜➔② t➼♥❤ ❣✐↔✐ ✤÷đ❝ ❝õ❛ ♠ët ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥

❋♦✉r✐❡r✳
✸✳ ❚❤ü❝ ❤✐➺♥ ✈✐➺❝ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ♠ët ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦ý ❞à
❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r ❣➦♣ tr♦♥❣ ❜➔✐ t♦→♥ ❜✐➯♥ ❤é♥
❤đ♣ ❝õ❛ ♣❤÷ì♥❣ tr ợ ữợ s
ữ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦ý ❞à ✈➲ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥
❦ý ❞à ✈➲ ❞↕♥❣ ❦❤æ♥❣ t❤ù ♥❣✉②➯♥✳
✰ ❚❤ü❝ ❤✐➺♥ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ❤➺ ✈ỉ ❤↕♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè t✉②➳♥
t➼♥❤ ✤➣ ✤÷đ❝ ✧❝❤➦t ❝ưt✧ ✤➳♥ N = 6 ✈➔ s❛✉ ✤â t➻♠ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛
❤➺ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦ý ❞à✳


✻✵

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦
❚➔✐ ❧✐➺✉ ❚✐➳♥❣ ❱✐➺t
❬✶❪ ◆❣✉②➵♥ ❱➠♥ ❚❤❛♥❤ ✭✷✵✶✵✮ ✧ ●✐↔✐ ❣➛♥ ✤ó♥❣ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤
♣❤➙♥ ❦➻ ❞à ♥❤➙♥ ❈❛✉❝❤② ✈➔ ù♥❣ ❞ö♥❣✧✳

❚➔✐ ❧✐➺✉ ❚✐➳♥❣ ❆♥❤
❬✷❪ ❇r②❝❤❦♦✈ ❯✳ ❆✳ ❛♥❞ Pr✉❞♥✐❦♦✈ ❆✳ P✳ ✭✶✾✾✼✮✱ ●❡♥❡r❛❧✐③❡❞ ✐♥t❡❣r❛❧
tr❛♥s❢♦r♠❛t✐♦♥s✱ ◆❛✉❦❛✱ ▼♦s❝♦✇✳
❬✸❪ ❉✉❞✉❝❤❛✈❛ ❘✳ ✭✶✾✼✾✮ ✱ ■♥t❡❣r❛❧ ❊q✉❛t✐♦♥s ✇✐t❤ ❋✐①❡❞ ❙✐♥❣❧❛r✐t❡s✱
❚❡✉❜♥❡r ❱❡r❧❛❣s❣❡s❡❧❧s❝♦❤❛❢t✱ ▲❡✐♣③✐❣✳
❬✹❪ ❊s❦✐♥ ●✳■ ✭✶✾✼✸✮✱ ❇♦✉♥❞❛r② ❱❛❧✉❡ Pr♦❜❧❡♠s ❢♦r ❊❧❧✐♣t✐❝ Ps❡✉❞♦❞✐❢✲
❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ ◆❛✉❦❛✱ ▼♦s❝♦✇✱ ✭✐♥ ❘✉ss✐❛♥✮✳
❬✺❪ ❑❛♥t♦r♦✈✐❝❤ ▲✳❱✳✱ ❑r②❧♦✈ ❨✉✳❆✳✭✶✾✻✷✮✱❆♣♣r♦①✐♠❛t❡ ▼❡t❤♦❞s ✐♥
❍✐❣❤❡r ❆♥❛❧②s✐s✱ ❋✐③♠❛t❣✐③✱ ▼♦s❝♦✇✱ ✭✐♥ ❘✉ss✐❛✮
❬✻❪ ❑r②❧♦✈ ❱✳■ ✭✷✵✵✻✮✱ ❆♣♣r♦①✐♠❛t❡ ❈❛❧❝✉❧❛t✐♦♥ ♦❢ ■♥t❡❣r❛❧s✱ ❉♦✈❡r P✉❜❧✐✲
❝❛t✐♦♥ ■◆❈
❬✼❪ ▲✐♦♥s ❏✳▲✳✱ ▼❛❣❡♥❡s ❊✳ ✭✶✾✻✽✮ ✱ Pr♦❜❧❡♠s ❛✉① ❧✐♠✐t❡s ♥♦♥ ❤♦♠♦❣❡♥❡s

❡t ❛♣♣❧✐❝❛t✐♦♥s✱ ❱♦❧✉♠❡ ✶✱ ❉✉♥♦❞✲ Pr✐s✳
❬✽❪ ◆❣✉②❡♥ ❱❛♥ ◆❣♦❝ ✭✶✾✽✽✮✱ ✧❖♥ t❤❡ s♦❧✈❛❜✐❧✐t② ♦❢ ❞✉❛❧ ✐♥t❡❣r❛❧ ❡q✉❛✲
t✐♦♥s ✐♥✈♦❧✈✐♥❣ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠s✧✱ ❆❝t❛ ▼❛t❤✳ ❱✐❡t♥❛♠✐❝❛✱ ✶✸✭✷✮✱
♣♣✳ ✷✶✲✸✵✳
❬✾❪ ◆❣✉②❡♥ ❱❛♥ ◆❣♦❝ ✭✷✵✵✾✮✱ ✧❉✉❛❧ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ✐♥✈♦❧✈✐♥❣ ❋♦✉r✐❡r
tr❛♥s❢♦r♠❛t✐♦♥s ✇✐t❤ ✐♥❝r❡❛s✐♥❣ s②♠❜♦❧s✧✱ ❆❝t❛ ▼❛t❤✳ ❱✐❡t♥❛♠✐❝❛✱
✸✹✭✸✮♣♣✳✸✵✺✲✸✶✽✳


✻✶

❬✶✵❪ ◆❣✉②❡♥ ❱❛♥ ◆❣♦❝ ❛♥❞ ◆❣✉②❡♥ ❚❤✐ ◆❣❛♥ ✭✷✵✵✾✮✱ ✧❖♥ ❛ s②st❡♠ ♦❢
❞✉❛❧ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ✐♥✈♦❧✈✐♥❣ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠s✧✱ ❚↕♣ ❝❤➼ ❑❤♦❛
❤å❝ ✈➔ ❈æ♥❣ ♥❣❤➺✱ ✣↕✐ ❤å❝ ❚❤→✐ ♥❣✉②➯♥✱ ✺✹✭✻✮✱ ♣♣✳ ✶✵✼✲✶✶✷✳
❬✶✶❪ ◆❣✉②❡♥ ❱❛♥ ◆❣♦❝ ❛♥❞ ◆❣✉②❡♥ ❚❤✐ ◆❣❛♥ ✭✷✵✶✶✮✱ ✧❖♥ s♦♠❡ s②st❡♠s
♦❢ ❞✉❛❧ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ✐♥✈♦❧✈✐♥❣ ❋♦✉r✐❡r ❚r❛♥s❢♦r♠s✧✱ ❆❧❣❡❜r❛✐❝
❙tr✉❝t✉r❡s ✐♥ P❛rt✐❛❧ ❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s ❘❡❧❛t❡❞ t♦ ❈♦♠♣❧❡① ❛♥❞
❈❧✐❢❢♦r❞ ❆♥❛❧②s✐s✱ ❍♦ ❈❤✐ ▼✐♥❤ ❈✐t② ❯♥✐✈❡rs✐t② ♦❢ ❊❞✉❝❛t✐♦♥ Pr❡ss✱
♣♣✳ ✷✷✺✲✷✹✽✱ ✭❇❛s❡❞ ♦♥ t❤❡ s❡❧❡❝t❡❞ ❧❡❝t✉r❡s ♦❢ t❤❡ 17th ■♥t❡r♥❛t✐♦♥❛❧
❈♦♥❢❡r❡♥❝❡ ♦♥ ❋✐♥✐t❡ ❛♥❞ ■♥❢✐♥✐t❡ ❉✐♠❡♥s✐♦♥❛❧ ❈♦♠♣❧❡① ❆♥❛❧②s✐s ❛♥❞
❆♣♣❧✐❝❛t✐♦♥s✱ ❍♦ ❈❤✐ ▼✐♥❤ ❈✐t②✱ ❆✉❣✉st ✶✲✸✱ ✷✵✵✾✮✳
❬✶✷❪ P♦♣♦✈ ●✳ ■❛✳ ✭✶✾✽✷✮✱ ❈♦♥t❛❝t Pr♦❜❧❡♠s ❢♦r ❛ ▲✐♥❡❛r❧② ❉❡❢♦r♠❡❞ ❇❛s❡✱
❱➼❤❝❤❛ ❙❤❦♦❧❛✱ ❑✐❡✈ ✭✐♥ ❘✉ss✐❛♥✮✳
❬✶✸❪ ❱❧❛❞✐♠✐r♦✈ ❱✳❙✳ ✭✶✾✼✾✮ ●❡♥❡r❛❧✐③❡❞ ts tt
Pỵs s r ss
P❛♥❡❦❤ ❇✳P✳ ✭✶✾✻✺✮ ✱✧❙♦♠❡ s♣❛❝❡s ♦❢ ❣❡♥❡r❛❧✐③❡❞
❢✉♥❝t✐♦♥s ❛♥❞ ✐♠❜❡❞❞✐♥❣ t❤❡♦r❡♠✧ ❯s♣❡❦❤✐ ▼❛t❤✳ ◆❛✉❦❛✱ ✷✵✭✶✮✱ ♣♣✳
✸✲✼✹ ✭✐♥ ❘✉ss✐❛♥✮✳