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✿ ✻✵✳✹✻✳✵✶✳✵✷ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ữợ ✲ ◆➠♠ ✷✵✶✺ ✐ ▲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 t❤→♥❣ ✹ ♥➠♠ ✷✵✶✺ ◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥ ◆❣ỉ ❚❤à ❚❤❛♥❤ ✐✐ ▲í✐ ❝↔♠ ì♥ ✣➸ ❤♦➔♥ t❤➔♥❤ ✤÷đ❝ ❧✉➟♥ ✈➠♥ ♠ët ❝→❝❤ ❤♦➔♥ ❝❤➾♥❤✱ tỉ✐ ❧✉ỉ♥ ♥❤➟♥ ✤÷đ❝ sü ữợ ú ù t t ❚❤à ◆❣➙♥✳ ❚æ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ ❝ỉ ❣✐→♦ ✈➔ ①✐♥ ❣û✐ ❧í✐ tr✐ ➙♥ ♥❤➜t ❝õ❛ tỉ✐ ✤è✐ ✈ỵ✐ ♥❤ú♥❣ ✤✐➲✉ ❝ỉ ❣✐→♦ ✤➣ ❞➔♥❤ ❝❤♦ tæ✐✳ ❚æ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ●✐→♠ ❤✐➺✉ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ❝ò♥❣ ❝→❝ P❤á♥❣✲ ❇❛♥ ❝❤ù❝ ♥➠♥❣ ❝õ❛ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ❦❤♦❛ ❚♦→♥ ✲ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠✱ ❝→❝ ỵ ổ ợ ✷✵✶✺✮ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✲ ✣↕✐ ❤å❝ ❚❤→✐ t t tr t ỳ tự qỵ ❜→✉ ❝ơ♥❣ ♥❤÷ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tỉ✐ ❤♦➔♥ t❤➔♥❤ õ ổ ỷ ỡ tợ trữớ ❚r✉♥❣ ❤å❝ ♣❤ê t❤ỉ♥❣ P→❝ ❑❤✉ỉ♥❣ t➾♥❤ ▲↕♥❣ ❙ì♥✱ ♥ì✐ tæ✐ ❝æ♥❣ t→❝ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tæ✐ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❤å❝✳ ❚ỉ✐ 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➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✳ ✺ ✳ ✺ ✳ ✺ ✳ ✻ ✳ ✻ ✳ ✼ ✳ ✼ ✳ ✽ ✳ ✽ ✳ ✶✵ ✳ ✶✷ ✐✈ ✶✳✼ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❤➔♠ ❝ì ❜↔♥ ❣✐↔♠ ♥❤❛♥❤ ✳ ✳ ✳ ✳ ✳ ✶✳✼✳✶ ❑❤ỉ♥❣ ❣✐❛♥ 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 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✹ ✶✹ ✶✺ ✶✺ ✶✻ ✶✼ ✶✼ ✶✼ ✶✽ ✶✾ ✶✾ ✶✾ ✷✶ ✷✷ ✷ ●✐↔✐ ❣➛♥ ✤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➼♥❤ ✳ ✳ ✷✹ ✷✹ ✷✺ ✷✻ ✷✾ ✸✸ ✈ ✷✳✷ ●✐↔✐ ❣➛♥ ✤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 ❞à ♥❤➙♥ ❈❛✉❝❤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] ì [a, b]} Pữỡ tr t ❞↕♥❣ ✭✶✮ ❣➦♣ ❤➛✉ ❤➳t tr♦♥❣ ❝→❝ ❜➔✐ t♦→♥ ❜✐➯♥ ❤é♥ ❤đ♣ ❝õ❛ ❱➟t ❧➼ t♦→♥ ✤è✐ ✈ỵ✐ ♠✐➲♥ ❦❤ỉ♥❣ trì♥ ♥❤÷ ❝→❝ ❜➔✐ t♦→♥ ✈➲ ❦❤❡ ❤ð✱ ✈➳t ♥ùt✱ ✈➳t r↕♥✱ ❝→❝ ❜➔✐ t♦→♥ ✈➲ t✐➳♣ ①ó❝ ❝õ❛ ❧➼ 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❛ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ s♦♥❣ ỏ ợ ố ữủ t ữỡ 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 ❈❤÷ì♥❣ ♠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♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈✱ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ✈❡❝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❛✱ ❝→❝ ✣à♥❤ ❧➼ ✷✳✶✳✶✱ ✣à♥❤ ❧➼ ✷✳✶✳✸ tr➻♥❤ ❜➔② ✈➲ t➼♥❤ tỗ t t ữỡ 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➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r ợ ữợ ữ ữỡ tr t ❞à ✈➲ ❞↕♥❣ ❦❤ỉ♥❣ 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 ữ ữợ sỹ ữợ ❚→❝ ❣✐↔ ①✐♥ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝❤➙♥ t s s t tợ ổ ữợ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ✤➸ t→❝ ❣✐↔ ❤♦➔♥ t❤➔♥❤ ✤÷đ❝ ❦❤♦→ ❤å❝ ❝õ❛ ♠➻♥❤✳ ✸ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶ ▲ỵ♣ ❤➔♠ ❍♦❧❞❡r ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❬✸❪✳ ●✐↔ sû L ❧➔ ✤÷í♥❣ ❝♦♥❣ trì♥ ✈➔ ϕ(ξ) ❧➔ ❤➔♠ ❝→❝ ✤✐➸♠ ♣❤ù❝ ξ ∈ L ◆â✐ r➡♥❣ ❤➔♠ ϕ(ξ) t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❍♦❧❞❡r ✭✤✐➲✉ ❦✐➺♥ H tr ữớ L ợ 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 > t❤➻ tø ✤✐➲✉ ❦✐➺♥ ✭✶✳✶✮ s✉② r❛ ϕ (ξ) ≡ tr➯♥ L ✈➔ ❞♦ ✤â ϕ(ξ) ≡ const, ξ ∈ L ❱➻ ✈➟② t❛ ❧✉æ♥ ❧✉æ♥ ❝❤♦ r➡♥❣ < λ ≤ ◆➳✉ λ = t❤➻ ✤✐➲✉ ❦✐➺♥ ❍♦❧❞❡r trð t❤➔♥❤ ✤✐➲✉ ❦✐➺♥ ▲✐♣s❝❤✐t③✳ ❘ã r➡♥❣ λ ❝➔♥❣ ♥❤ä t❤➻ ❧ỵ♣ ❤➔♠ Hλ ❝➔♥❣ rë♥❣✳ ▲ỵ♣ ❤➔♠ ❍♦❧❞❡r ❤➭♣ ♥❤➜t ❧➔ ❧ỵ♣ ❤➔♠ ▲✐♣s❝❤✐t③✳ ❉➵ t❤➜② r➡♥❣✱ ♥➳✉ ❝→❝ ❤➔♠ 1(), 2() tọ r tữỡ ự ợ ❝→❝ ❝❤➾ sè λ1, λ2✱ t❤➻ tê♥❣✱ t➼❝❤ ✈➔ ❝↔ tữỡ ợ tự ổ ụ tọ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❍♦❧❞❡r ✈ỵ✐ ❝❤➾ sè λ = min(λ1 , λ2 )✳ ◆➳✉ ❤➔♠ ϕ(ξ) ❝â ✤↕♦ ❤➔♠ ❤ú✉ ❤↕♥ tr➯♥ L t❤➻ ♥â t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ▲✐♣s❝❤✐t③✳ ữủ s r tứ ỵ số ❣✐❛ ❤ú✉ ❤↕♥✳ ◆❣÷đ❝ ❧↕✐ ♥â✐ ❝❤✉♥❣ ❦❤ỉ♥❣ ✤ó♥❣✳ ❚❤➼ ❞ư✱ ❤➔♠ ϕ(ξ) = |ξ|, ξ ∈ R, t❤✉ë❝ ❧ỵ♣ ❤➔♠ ❍♦❧❞❡r tr➯♥ R✱ ♥❤÷♥❣ ❦❤ỉ♥❣ ❝â ✤↕♦ ❤➔♠ t↕✐ ξ = 0✳ ✹✼ tr♦♥❣ ✤â ∗ (y − τ )Tj (τ ) K11 √ = dτ, − τ2 −1 ∗ K12 (y − τ )Tj (τ ) (2) √ γj (y) = dτ − τ2 −1 (1) γj (y) ❳➜♣ ①➾ γj(1)(y) ✈➔ γj(2)(y) ❜ð✐ ❤➺ trü❝ ❣✐❛♦ {U0(y), = (1) ✭✷✳✻✾✮ (2) ✭✷✳✼✵✮ βj,k Uk (y), k=0 N −1 (2) βj,k Uk (y), γj (y) = ✭✷✳✻✽✮ U1 (y), , UN −1 (x)} N −1 (1) γj (y) ✭✷✳✻✼✮ k=0 tr♦♥❣ ✤â (1) βj,k (2) βj,k = π = π (1) ✭✷✳✼✶✮ (2) ✭✷✳✼✷✮ − y γj (y)Uk (y), −1 1 − y γj (y)Uk (y), k = 0, 1, , N − −1 ❙û ❞ö♥❣ ✭✷✳✻✾✮ ✈➔ ✭✷✳✼✵✮ t❛ ❝â N −1 N N (1) (1) Aj γj (y) = j=1 (1) ✭✷✳✼✸✮ (2) (2) ✭✷✳✼✹✮ (1) (2) ✭✷✳✼✺✮ (2) (1) ✭✷✳✼✻✮ k=0 j=1 N −1 N N (2) (2) Aj γj (y) = j=1 Aj βj,k Uk (y), k=0 j=1 N −1 N N (1) (2) Aj γj (y) = Aj βj,k Uk (y), k=0 j=1 j=1 N −1 N N (2) (1) Aj γj (y) j=1 (1) Aj βj,k Uk (y), = Aj βj,k Uk (y) k=0 j=1 ✹✽ ❚ø ✭✷✳✺✾✮✱ ✭✷✳✻✵✮✱ ✭✷✳✼✸✮✱ ✭✷✳✼✹✮✱ ✭✷✳✼✺✮ ✈➔ ✭✷✳✼✻✮ t❛ t❤✉ ✤÷đ❝     −              N −1 N N (1) Aj Uj−1 (y) j=1 (1) Aj βj,k Uk (y) k=0 j=1 N −1 N N −1 (2) (2) Aj βj,k Uk (y) + ∗(1) = k=0 j=1 fk Uk (y), k=0 N −1 N N     −             (1) + (1) (2) ✭✷✳✼✼✮ (2) Aj βj,k Uk (y) Aj Uj−1 (y) + k=0 j=1 j=1 N −1 N N −1 (2) (1) Aj βj,k Uk (y) + ∗(2) = k=0 j=1 fk Uk (y), k=0 tr♦♥❣ ✤â = π (2) βj,k = π ∗(1) fk = π ∗(2) fk = π (1) βj,k ∗ (y − τ )Tj (τ ) K11 √ dτ dy, − τ2 −1 ∗ (y − τ )Tj (τ ) K12 √ dτ dy, − τ2 −1 1 1− y Uk (y) −1 1 − y Uk (y) −1 −1 −1 − y f1∗ (y)Uk (y)dy, − y f2∗ (y)Uk (y)dy (2) ❚ø ✭✷✳✼✼✮ t❛ ❝â ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè t✉②➳♥ t➼♥❤ ①→❝ ✤à♥❤ ❤➺ sè A(1) j , Aj ❝õ❛ v1∗(τ ) ✈➔ v2∗(τ )    (1)  −A1 +        ✳✳ ✳✳         (1)    −AN +   (2)   −A1 +       ✳✳ ✳✳         (2)    −AN + N N (1) (1) Aj βj,0 (2) + ✳✳ j=1 j=1 N N (1) (1) Aj βj,N −1 j=1 N (2) + ∗(1) (2) Aj βj,N −1 = fN −1 , j=1 N (1) (2) (2) Aj βj,0 + ✳✳ j=1 ∗(2) (1) ✭✷✳✼✽✮ Aj βj,0 = f0 , j=1 N N (1) (2) Aj βj,N −1 j=1 ∗(1) (2) Aj βj,0 = f0 , (2) + (1) ∗(2) Aj βj,N −1 = fN −1 j=1 ❚❛ s➩ ❣✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✼✽✮ ✈ỵ✐ N = ❙û ❞ư♥❣ ✭✷✳✻✷✮ ✈➔ ✭✷✳✻✸✮ ợ N =6 t t ữủ ữỡ tr s❛✉   (1)   −A1 +          (1)   −A2 +         (1)   −A3 +          (1)   −A4 +          (1)  −A5 +          (1)   −A  +    (2)  −A1 +          (2)   −A2 +          (2)   −A3 +         (2)   −A4 +          (2)   −A5 +         (2)    −A6 + 6 (1) (1) Aj βj,0 + j=1 (2) (2) (2) (2) (2) (2) (2) (2) (2) (2) (2) (2) Aj βj,0 = j=1 (1) (1) Aj βj,1 + j=1 Aj βj,1 = j=1 (1) (1) Aj βj,2 + j=1 Aj βj,2 = j=1 (1) (1) Aj βj,3 + j=1 Aj βj,3 = j=1 (1) (1) Aj βj,4 + j=1 Aj βj,4 = j=1 (1) (1) Aj βj,5 + j=1 Aj βj,5 = j=1 (1) (2) Aj βj,0 (2) (1) Aj βj,0 + j=1 j=1 (1) (2) Aj βj,1 + j=1 (2) (1) (2) (1) (2) (1) (2) (1) (2) (1) j=1 (1) (2) j=1 Aj βj,2 = j=1 (1) (2) Aj βj,3 + j=1 Aj βj,3 = j=1 (1) (2) Aj βj,4 + j=1 Aj βj,4 = j=1 (1) (2) Aj βj,5 + j=1 Aj βj,5 = j=1 16a1 + 8a3 + 5a5 , 32 4a2 + 3a4 , 16 a3 + a5 , a4 , 16 a5 , 32 8b0 + 2b2 + b4 , = Aj βj,1 = Aj βj,2 + 8a0 + 2a2 + a4 , ✭✷✳✼✾✮ 16b1 + 8b3 + 5b5 , 32 4b2 + 3b4 , 16 b3 + b5 , b4 , 16 b5 32 ❇➙② ❣✐í✱ t❛ t➼♥❤ ❝→❝ ❤➺ sè βj,k(1) ✈➔ βj,k(2) ✣➦t ∗ K11 (y − τ )Tj (τ ) √ dτ, − τ2 −1 ∗ K12 (y − τ )Tj (τ ) (2) √ Jj (y) = dτ, j = 0, 1, 2, − τ2 −1 (1) Jj (y) = ✭✷✳✽✵✮ ✭✷✳✽✶✮ ✺✵ ❚r÷í♥❣ ❤đ♣ λ = 101 t❛ t➼♥❤ ✤÷đ❝ (1) J0 (y) = 0.183745 sin(0.0322548y) + 0.0127688 sin(0.174576y) + 0.0000790727 sin(0.453662y) + 7.09061 × 10−9 sin(0.939507y), (1) J1 (y) = −0.00296372 cos(0.0322548y) − 0.00111883 cos(0.174576y) − 0.000018414 cos(0.453662y) − 3.76209 × 10−9 cos(0.939507y), (1) J2 (y) = −0.0000238996 sin(0.0322548y) − 0.0000488923 sin(0.174576y) − 2.10657 × 10−6 sin(0.453662y) − 9.1803 × 10−10 sin(0.939507y), (1) J3 (y) = 1.28482 × 10−7 cos(0.032254y) + 1.42347 × 10−6 cos(0.174576y) + 1.59966 × 10−7 cos(0.453662y) + 1.46473 × 10−10 cos(0.93950y), (1) J4 (y) = 3.86526 × 10−19 cos(0.032254y) + 1.93263 × 10−19 cos(0.17457y) + 1.50987 × 10−21 cos(0.45366y) − 7.37239 × 10−25 cos(0.939507y) + 5.18027 × 10−10 sin(0.032254y) + 3.10749 × 10−8 sin(0.174576y) + 9.09477 × 10−9 sin(0.453662y) + 1.73949 × 10−11 sin(0.93950y), (1) J5 (y) = −1.6709 × 10−12 cos(0.032254y) − 5.4263 × 10−10 cos(0.17457y) − 4.13305 × 10−10 cos(0.453662y) − 1.64644 × 10−12 cos(0.93950y) − 5.12423 × 10−17 sin(0.032254y) − 3.0922 × 10−18 sin(0.174576y), (1) J6 (y) = −3.9342 × 10−19 cos(0.17457y) + 9.16704 × 10−21 cos(0.45366y) − 2.61983 × 10−24 cos(0.93950y − 3.54529 × 10−15 sin(0.0322548y) − 7.8956 × 10−12 sin(0.174576y) − 1.56442 × 10−11 sin(0.453662y) − 1.29587 × 10−13 sin(0.939507y), ✈➔ (2) J0 (y) = −0.253686 sin(0.0322548y) − 0.0731685 sin(0.174576y) − 0.00738339 sin(0.453662y) − 0.0000852925 sin(0.939507y), ✺✶ (2) J1 (y) = 0.00409183 cos(0.0322548y) + 0.00641119 cos(0.174576y) + 0.0017194 cos(0.453662y + 0.0000452539 cos(0.939507y), (2) J2 (y) = 0.0000329967 sin(0.0322548y) + 0.000280166 sin(0.174576y) + 0.000196701 sin(0.453662y) + 0.0000110429 sin(0.939507y), (2) J3 (y) = −1.77388 × 10−7 cos(0.032254y) − 8.1569 × 10−6 cos(0.17457y) − 0.000014936 cos(0.453662y) − 1.76192 × 10−6 cos(0.93950y), (2) J4 (y) = 8.006 × 10−19 cos(0.0322548y) + 8.2826 × 10−19 cos(0.174576y) + 1.93263 × 10−19 cos(0.45366y) − 7.15209 × 10−10 sin(0.032254y) − 1.7806 × 10−7 sin(0.174576y) − 8.49221 × 10−7 sin(0.45366y) − 2.09242 × 10−7 sin(0.939507y), (2) J5 (y) = 2.3069 × 10−12 cos(0.0322548y) + 3.10943 × 10−9 cos(0.17457y) + 3.85922 × 10−8 cos(0.45366y) + 1.98049 × 10−8 cos(0.93950y) + 4.94753 × 10−17 sin(0.0322548y), (2) J6 (y) = 4.0033 × 10−19 cos(0.032254y) − 3.13362 × 10−18 cos(0.17457y) − 7.7995 × 10−19 cos(0.453662y) − 1.83341 × 10−20 cos(0.93950y) + 4.83562 × 10−15 sin(0.032254y) + 4.52441 × 10−11 sin(0.17457y) + 1.46078 × 10−9 sin(0.45366y) + 1.5587 × 10−9 sin(0.939507y) ✣➸ t➼♥❤ ❝→❝ t➼❝❤ ♣❤➙♥ (1) π = π βj,k = (2) βj,k (1) − y Uk (y).Jj (y)dy, −1 (2) − y Uk (y).Jj (y)dy −1 ✭✷✳✽✷✮ ✭✷✳✽✸✮ ✺✷ t❛ t❤❛② ❝→❝ Jj(1)(y) ✈➔ Jj(2)(y) ✈ø❛ 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❤✉ ✤÷đ❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè t✉②➳♥ t➼♥❤ ✈ỵ✐ ❝→❝ ➞♥ ❧➔ A(j) ✈➔ A2 ✈ỵ✐ j = 1, , ✺✹ ●✐↔✐ ❤➺ ♣❤÷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 , (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 , ✺✺ ✈➔  (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 , (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 ✭✷✳✽✹✮ ❇➙② ❣✐í t❛ t➻♠ ♥❣❤✐➺♠ ❣➛♥ ú ữỡ tr ợ N = ✺✻ ❚❛ ❝â ∗ (τ ) vm,6 =√ − τ2 (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) (−A2 + A4 − A6 ) + (A1 − 3A3 + 5A5 )τ 1−τ (1) (1) (1) (1) (1) +√ (2A2 − 8A4 + 18A6 )τ + (4A3 − 20A5 )τ 1−τ (1) (1) (1) (1) (8A4 − 48A6 )τ + 16A5 τ + 32A6 τ , +√ 1−τ (2) (2) (2) (2) (2) (2)  ∗  √ (−A2 + A4 − A6 ) + (A1 − 3A3 + 5A5 )τ v (τ ) =  2,6   1−τ    (2) (2) (2) (2) (2)   √ + (2A2 − 8A4 + 18A6 )τ + (4A3 − 20A5 )τ    1−τ    (2) (2) (2) (2)   (8A4 − 48A6 )τ + 16A5 τ + 32A6 τ +√  1−τ =√ ❇➙② ❣✐í t❛ t➼♥❤ u1,6 ✈➔ u2,6 ✿ ❉♦ (b − a)τ + b + a ), (b − a)τ + b + a v2∗ (τ ) = v2 ( ), (b − a)τ + b + a t= 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 ∗ b−a b−a v1 (τ )s✐❣♥[ (y − τ )] dτ −1 2 b−a y ∗ b−a ∗ v1 (τ )dτ − v1 (τ )dτ, 4 −1 y 1 ∗ b−a b−a v2 (τ )s✐❣♥[ (y − τ )] dτ −1 2 b−a y ∗ b−a ∗ v2 (τ )dτ − v2 (τ )dτ, 4 −1 y y ∈ (−1, 1), y ∈ (−1, 1) ✺✼ ❚➼♥❤ ✈➔ rót ❣å♥ t➼❝❤ ♣❤➙♥ tr➯♥ t❛ ✤÷đ❝ a − b√ (1) (1) (1) u∗1,6 (y) = − y 15A1 − 5A3 + 3A5 30 a − b√ (1) (1) (1) (1) (1) + − y (15A2 − 15A4 + 15A6 )y + (20A3 − 36A5 )y 30 a − b√ (1) (1) (1) (1) + − y (30A4 − 80A6 )y + 48A5 y + 80A6 y , 30 a − b√  (2) (2) (2) ∗  (y) = u − y 15A1 − 5A3 + 3A5  2,6   30   √ a − b  (2) (2) (2) (2) (2)   + − y (15A2 − 15A4 + 15A6 )y + (20A3 − 36A5 )y   30    a − b√ (2) (2) (2) (2)   + − y (30A4 − 80A6 )y + 48A5 y + 80A6 y 30 ❚❛ ❝â uj,6(x) = uj,6( 2x b−−b a− a ), ✈ỵ✐ j = 1,                   ❚❛ t➼♥❤ ✤÷đ❝  15(a + b) (1) − (b − x)(x − a)  (1) (1)   15A1 + A2 − 5A3 u1,6 (x) =   15 a −3b    15(a + b) 30(a + b) (1) 20(a + b)  (1) (1) (1)  + A − A4 + 3A5 A +    (a − b) a−b (a − b)    48(a + b) 15(a + b) (1) 36(a + b)  (1) (1)   A + A + − A6  5  (a − b) (a − b) a − b     80(a + b)3 (1) 80(a + b)5 (1)   A6 + A6 −    (a − b) (a − b)    30 (1) 80(a + b) (1) 30 (1)    +[− A A A4 − +   a − b (a − b) a − b     180(a + b)2 (1) 144(a + b) (1) 384(a + b)3 (1)   − A4 + A5 − A5  (a − b) (a − b) (a − b) 30 480(a + b) 800(a + b)4 (1)  (1) (1)   A6 + A6 − A6 ]x −   a − b (a − b) (a − b)     80 360(a + b) (1) 144 (1) (1)   A + A − A5 +[   (a − b) (a − b) ((a − b)     1152(a + b) 960(a + b) 3200(a + b)3 (1)  (1) (1)  + A5 − A6 + A6 ]x    (a − b) (a − b) (a − b)    −240 (1) 1536(a + b) (1) 640  (1)   +[ A − A + A6   (a − b) (a − b) (a − b)     6400(a + b) 768  (1) (1)  − A6 ]x3 + [ A5    (a − b) (a − b)    6400(a + b) (1) 2560   + A ]x − x , (a − b)5 (a − b)5 ✺✽ ✈➔  − (b − x)(x − a) 15(a + b) (2)  (2) (2)   u (x) = 15A + A2 − 5A3 2,6   15 a −3b    20(a + b) 15(a + b) 30(a + b) (2)  (2) (2) (2)   + A − A + A4 + 3A5   (a − b) a−b (a − b)     36(a + b) (2) 48(a + b) (2) 15(a + b) (2) 80(a + b)3 (2)   A5 + A5 + A6 − A6 −    (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 +   (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  (a − b) (a − b) (a − b) 480(a + b) 800(a + b)4 (2) 30  (2) (2)   A6 + A6 − A6 ]x −   a − b (a − b) (a − b)     80 360(a + b) (2) 144 1152(a + b)2 (2) (2) (2)   +[ A + A − A + A5  3  (a − b) (a − b) ((a − b) (a − b)     −240 (2) 960(a + b) (2) 3200(a + b)3 (2)   A A A + ]x + [ −  6   (a − b) (a − b) (a − b)     1536(a + b) (2) 640 6400(a + b)2 (2) (2)   − A A A6 ]x + −   (a − b) (a − b) (a − b)     768 6400(a + b) (2) 2560  (2)  A A x + ]x − +[ (a − b)4 (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♦♥❣ ❦❤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♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈✱ ❝→❝ ❦❤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 ộ ủ ữỡ tr ợ ữợ s ữ ữỡ 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 = ✈➔ 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❡❞ ❋✉♥❝t✐♦♥s ✐♥ tt Pỵs s r ss P❛♥❡❦❤ ❇✳P✳ ✭✶✾✻✺✮ ✱✧❙♦♠❡ s♣❛❝❡s ♦❢ ❣❡♥❡r❛❧✐③❡❞ ❢✉♥❝t✐♦♥s ❛♥❞ ✐♠❜❡❞❞✐♥❣ t❤❡♦r❡♠✧ ❯s♣❡❦❤✐ ▼❛t❤✳ ◆❛✉❦❛✱ ✷✵✭✶✮✱ ♣♣✳ ✸✲✼✹ ✭✐♥ ❘✉ss✐❛♥✮✳