Hệ phương trình cặp tích phân fourier của bài toán biên hỗn hợp đối với dải đàn hồi

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ✣■◆❍ ❚❍➚ ❚❍❷❖ ❍➏ P❍×❒◆● ❚❘➐◆❍ ❈➄P ❚➑❈❍ P❍❹◆ ❋❖❯❘■❊❘ ❈Õ❆ ❇⑨■ ❚❖⑩◆ ❇■➊◆ ❍➱◆ ❍ÑP ✣➮■ ❱❰■ ❉❷■ ✣⑨◆ ❍➬■ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✷✵ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ✣■◆❍ ❚❍➚ ❚❍❷❖ ❍➏ P❍×❒◆● ❚❘➐◆❍ ❈➄P ❚➑❈❍ P❍❹◆ ❋❖❯❘■❊❘ ❈Õ❆ ❇⑨■ ❚❖⑩◆ ❇■➊◆ ❍➱◆ ❍ÑP ✣➮■ ❱❰■ ❉❷■ ✣⑨◆ ❍➬■ ❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ❣✐↔✐ t➼❝❤ ▼➣ sè✿ ✽✳✹✻✳✵✶✳✵✷ ▲❯❾◆ ❱❿◆ ữớ ữợ ◆●❯❨➍◆ ❚❍➚ ◆●❹◆ ❚❤→✐ ◆❣✉②➯♥ ✲ ✷✵✷✵ ▲í✐ ❝❛♠ ✤♦❛♥ ❚æ✐ 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 ❧✉➟♥ ✈➠♥ ✣■◆❍ ❚❍➚ ❚❍❷❖ ✐ ▲í✐ ❝↔♠ ì♥ ✣➸ ❤♦➔♥ t❤➔♥❤ ✤÷đ❝ ❧✉➟♥ ✈➠♥ ♠ët ❝→❝❤ ❤♦➔♥ ❝❤➾♥❤✱ tỉ✐ ❧✉ỉ♥ ♥❤➟♥ ✤÷đ❝ sü ❣✐ó♣ ✤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 ❣✐→♦ ✤➣ ❞➔♥❤ ❝❤♦ tæ✐✳ ❚æ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ●✐→♠ ❤✐➺✉ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✕ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ❝→❝ P❤á♥❣ ❝❤ù❝ ♥➠♥❣ ❝õ❛ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✕ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ❝→❝ ỵ ổ ợ ✕ ✷✵✷✵✮ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✕ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t➟♥ t➻♥❤ tr✉②➲♥ ✤↕t ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ qỵ ụ ữ t tổ t❤➔♥❤ ❦❤â❛ ❤å❝✳ ❇↔♥ ❧✉➟♥ ✈➠♥ ❝❤➢❝ ❝❤➢♥ s➩ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ ❦❤✐➳♠ ❦❤✉②➳t ✈➻ ✈➟② r➜t ♠♦♥❣ ♥❤➟♥ ữủ sỹ õ õ ỵ t ổ ❣✐→♦ ✈➔ ❝→❝ ❜↕♥ ❤å❝ ✈✐➯♥ ✤➸ ❧✉➟♥ ✈➠♥ ♥➔② ✤÷đ❝ ❤♦➔♥ ❝❤➾♥❤ ❤ì♥✳ ❈✉è✐ ❝ị♥❣ ①✐♥ ❝↔♠ ì♥ ❣✐❛ ✤➻♥❤ ✈➔ ❜↕♥ ❜➧ ✤➣ ✤ë♥❣ ✈✐➯♥✱ ❦❤➼❝❤ ❧➺ tæ✐ tr♦♥❣ t❤í✐ ❣✐❛♥ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✻ ♥➠♠ ✷✵✷✵ ❚→❝ ❣✐↔ ✣■◆❍ ❚❍➚ ❚❍❷❖ ✐✐ ▼ư❝ ❧ư❝ ▲í✐ ❝❛♠ ✤♦❛♥ ▲í✐ ❝↔♠ ì♥ ▼ư❝ ❧ư❝ ▼ð ✤➛✉ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✐ ✐✐ ✐✐✐ ✶ ✷ ✶✳✶ ❍➺ ✈ỉ ❤↕♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✷ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ✶✳✷✳✶ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❝→❝ ❤➔♠ ❝ì ❜↔♥ ❣✐↔♠ ♥❤❛♥❤ ✳ ✳ ✳ ✹ ✶✳✷✳✷ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❤➔♠ s✉② rë♥❣ t➠♥❣ ❝❤➟♠ ✳ ✳ ✳ ✳ ✳ ✺ ❑❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✸ H s (R) ✶✳✸✳✶ ❑❤æ♥❣ ❣✐❛♥ ✶✳✸✳✷ ❈→❝ ❦❤æ♥❣ ❣✐❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ s (Ω) , H s (Ω) Hos (Ω) , Ho,o ✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✹ ❑❤ỉ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ✈❡❝tì ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✶✳✺ ❚♦→♥ tû ❣✐↔ ✈✐ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r ❝õ❛ t ộ ủ ố ợ ỗ ✶✹ ✷✳✶ ❇➔✐ t♦→♥ ❜✐➯♥ ❤é♥ ❤đ♣ ✤è✐ ✈ỵ✐ ❞↔✐ ỗ ✳ ✳ ✳ ✳ ✶✹ ✣➦t ❜➔✐ t♦→♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✐✐✐ ✷✳✶✳✷ ✣÷❛ ❜➔✐ t♦→♥ ❜✐➯♥ ❤é♥ ❤đ♣ ✈➲ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ ❚➼♥❤ ❣✐↔✐ ✤÷đ❝ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r ✳ ✳ ✶✺ ✶✼ ✷✳✷✳✶ ❚➼♥❤ ❣✐↔✐ ✤÷đ❝ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r ✶✼ ✷✳✷✳✷ ❇✐➳♥ ✤ê✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r ✈➲ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦ý ❞à ✷✳✷✳✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ❇✐➳♥ ✤ê✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦ý ❞à ✈➲ ❤➺ ✈ỉ ❤↕♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè t✉②➳♥ t➼♥❤ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✸✵ ✸✶ ✐✈ ▼ð ✤➛✉ P❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ✈➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ t❤÷í♥❣ ①✉➜t ❤✐➺♥ tr♦♥❣ ❝→❝ ❜➔✐ t♦→♥ ✈➲ ❞à t➟t✱ tr♦♥❣ ♠ỉ✐ tr÷í♥❣ ♥❤÷ ❝→❝ ❜➔✐ t♦→♥ ✈➲ ✈➳t ♥ùt✱ ❝→❝ ❜➔✐ t♦→♥ ✈➲ ❞↔✐ ✤➔♥ ỗ tỗ t t ❜➔✐ t♦→♥ ♥➔② ✤➣ ✤÷đ❝ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉✳ ❚➼♥❤ ❣✐↔✐ ✤÷đ❝ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t ợ rr ữủ ởt sè ♥❤➔ t♦→♥ ❤å❝ ♥❤÷ P♦♣♦✈✳●✳❨❛✱ ❉✉❞✉❝❤❛✈❛r✳❘✱ ◆❣✉②➵♥ ❱➠♥ ◆❣å❝✱✳✳✳ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉✳ ❚➼♥❤ ❣✐↔✐ ✤÷đ❝ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r ✤÷đ❝ ◆❣✉②➵♥ ❱➠♥ ◆❣å❝✱ ◆❣✉②➵♥ ự ợ ố ữủ t t➼♥❤ ❣✐↔✐ ✤÷đ❝ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ✈ỵ✐ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ①✉➜t ❤✐➺♥ ❦❤✐ ❣✐↔✐ ❜➔✐ t♦→♥ ❜✐➯♥ ❤é♥ ❤đ♣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ s♦♥❣ ✤✐➲✉ ❤á❛ tr ỗ tổ t ữỡ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r ❝õ❛ ❜➔✐ t♦→♥ ❜✐➯♥ ❤é♥ ủ ố ợ ỗ ▼ð ✤➛✉✱ ❑➳t ❧✉➟♥✱ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✱ ❧✉➟♥ ✈➠♥ ❝â ✷ ❝❤÷ì♥❣ ♥ë✐ ❞✉♥❣✿ ❈❤÷ì♥❣ ✶ tr➻♥❤ ❜➔② tê♥❣ q✉❛♥ ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ❤➺ ✈ỉ ❤↕♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè t✉②➳♥ t➼♥❤✱ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❝→❝ ❤➔♠ ❝ì ❜↔♥✱ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈✱ ❦❤ỉ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ✈❡❝tì✱ t♦→♥ tû ❣✐↔ ✈✐ ♣❤➙♥✳ ❈❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ✈➲ t➼♥❤ ❣✐↔✐ ✤÷đ❝ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r ①✉➜t ❤✐➺♥ ❦❤✐ ❣✐↔✐ ❜➔✐ t ộ ủ ố ợ ỗ ữ ❝→❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r ✈➲ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦ý ❞à✱ ✤÷❛ t✐➳♣ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦ý ❞à ✈➲ ❤➺ ✈æ ❤↕♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè t✉②➳♥ t➼♥❤✳ ✶ ❈❤÷ì♥❣ ✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔ ✈➲ ❤➺ ✈ỉ ❤↕♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè t✉②➳♥ t➼♥❤✱ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❝→❝ ❤➔♠ ❝ì ❜↔♥ ❣✐↔♠ ♥❤❛♥❤✱ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❝→❝ ❤➔♠ s✉② rë♥❣ t➠♥❣ ❝❤➟♠✱ ❦❤ỉ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈✱ ❦❤ỉ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ✈❡❝tì✱ t♦→♥ tû ❣✐↔ ✈✐ ♣❤➙♥✳ ❈→❝ ❦➳t q✉↔ tr➻♥❤ ❜➔② tr♦♥❣ ❝❤÷ì♥❣ ♥➔② ✤÷đ❝ t❤❛♠ ❦❤↔♦ tø t➔✐ ❧✐➺✉ [2, 3, 4] ✶✳✶ ❍➺ ✈ỉ ❤↕♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè t✉②➳♥ t➼♥❤ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ ∞ xi = ci,k xk + bi , (i = 1, 2, ) , ✭✶✳✶✮ k=1 tr♦♥❣ ✤â xi ❧➔ ❝→❝ sè ❝➛♥ ①→❝ ✤à♥❤✱ ci,k ✈➔ bi ❧➔ ❝→❝ ❤➺ sè ✤➣ ❜✐➳t✱ ✤÷đ❝ ❣å✐ ❧➔ ❤➺ ✈ỉ ❤↕♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè t✉②➳♥ t➼♥❤✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✳ ❚➟♣ ❤đ♣ ♥❤ú♥❣ sè x1 , x2 , ✤÷đ❝ ❣å✐ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ✭✶✳✶✮ ♥➳✉ ❦❤✐ t❤❛② ✤ê✐ ♥❤ú♥❣ sè ✤â ✈➔♦ ✈➳ ♣❤↔✐ ❝õ❛ ✭✶✳✶✮ t❛ ❝â ❝→❝ ❝❤✉é✐ ❤ë✐ tö ✈➔ t➜t ❝↔ ♥❤ú♥❣ ✤➥♥❣ t❤ù❝ ✤÷đ❝ t❤♦↔ ♠➣♥✳ ◆❣❤✐➺♠ ✤÷đ❝ ❣å✐ ❧➔ ♥❣❤✐➺♠ ❝❤➼♥❤ ♥➳✉ ♥â ✤÷đ❝ t➻♠ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ❧✐➯♥ t✐➳♣ ✈ỵ✐ ❣✐→ trà ❜❛♥ ✤➛✉ ❜➡♥❣ ❦❤ỉ♥❣✳ ✷ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✸✳ ❍➺ ✈ỉ ❤↕♥ ✭✶✳✶✮ ✤÷đ❝ ❣å✐ ❧➔ ❤➺ ❝❤➼♥❤ q✉② ♥➳✉ ∞ |ci,k | < 1, (i = 1, 2, ) ✭✶✳✷✮ |ci,k | ≤ − θ, < θ < 1, (i = 1, 2, ) , ✭✶✳✸✮ k=1 ◆➳✉ ❝â t❤➯♠ ✤✐➲✉ ❦✐➺♥ ∞ k=1 t❤➻ ❤➺ ♥➔② ✤÷đ❝ ❣å✐ ❧➔ ❤♦➔♥ t♦➔♥ ❝❤➼♥❤ q✉②✳ ◆➳✉ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✷✮ ✭t÷ì♥❣ ù♥❣ ✭✶✳✸✮✮ ✤ó♥❣ ✈ỵ✐ i = N + 1, N + 2, , t❤➻ ❤➺ ✭✶✳✶✮ ✤÷đ❝ ❣å✐ ❧➔ tü❛ ❝❤➼♥❤ q✉② ✭t÷ì♥❣ ù♥❣✱ tü❛ ❤♦➔♥ t♦➔♥ ❝❤➼♥❤ q✉②✮✳ ❚❛ ❦➼ ❤✐➺✉ ∞ ρi = − |ci,k |, (i = 1, 2, ) k=1 ❍➺ ❝❤➼♥❤ q✉② ρ i > 0✱ ❤➺ ❤♦➔♥ t♦➔♥ ❝❤➼♥❤ q✉② ❝❤♦ ρi ≥ θ > 0✳ ●✐↔ sû ❤➺ ✭✶✳✶✮ ❧➔ ❤➺ ❝❤➼♥❤ q✉② ✈➔ ❝→❝ ❤➺ sè tü ❞♦ bi t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ |bi | ≤ Kρi , (K = const > 0) ỵ ỹ tỗ t ❝❤➦♥✮✳ ◆➳✉ ❝→❝ ❤➺ sè tü ❞♦ ❝õ❛ ❤➺ ✈æ ❤↕♥ ❝❤➼♥❤ q✉② t❤♦↔ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ✭✶✳✹✮ t❤➻ ♥â ❝â ♥❣❤✐➺♠ ❜à ❝❤➦♥ |xi| ≤ K ✈➔ ♥❣❤✐➺♠ ♥➔② ❝â t❤➸ t➻♠ ✤÷đ❝ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ❧✐➯♥ t ỵ ỹ t ửt x ❝õ❛ ❤➺ ❝❤➼♥❤ q✉② ∞ xi = ci,k xk + bi , (i = 1, 2, 3, ) , k=1 ❝ị♥❣ ✈ỵ✐ ❝→❝ ❤➺ sè tü ❞♦ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ |bi| ≤ Kρi ❝â t❤➸ t➻♠ ✤÷đ❝ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ✧❝❤➦t ❝öt✧✱ ♥❣❤➽❛ ❧➔ ♥➳✉ xNi ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ❤ú✉ ❤↕♥ N xi = ci,k xk + bi , (i = 1, 2, 3, , N ) , k=1 t❤➻ x∗i = lim xN i N →∞ ỵ q õ t õ ❦❤æ♥❣ q✉→ ♠ët ♥❣❤✐➺♠ t✐➳♥ ✤➳♥ ❦❤æ♥❣✱ ♥❣❤➽❛ ❧➔ lim xi = i→∞ ✶✳✷ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ✶✳✷✳✶ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❝→❝ ❤➔♠ ❝ì ❜↔♥ ❣✐↔♠ ♥❤❛♥❤ ❑❤ỉ♥❣ ❣✐❛♥ S ❝õ❛ ❝→❝ ❤➔♠ ❝ì ❜↔♥ ❣✐↔♠ ♥❤❛♥❤ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳ f ∈ C ∞ (R) ❑➼ ❤✐➺✉ S = S (R) ❧➔ t➟♣ ❤ñ♣ ❝õ❛ ❝→❝ ❤➔♠ ❦❤↔ ✈✐ ✈æ ❤↕♥ t❤♦↔ ♠➣♥ ✤✐➲✉ ❦✐➺♥ p p Dk f < ∞, p = 0, 1, 2, , m, |[f ]|p = sup (1 + |x|) x∈R tr♦♥❣ ✤â ❦➼ ❤✐➺✉ {fk } ⊂ S D = k=0 d ✳ dx ❉➣② |[f ]|p ✤÷đ❝ ❣å✐ ❧➔ ❤ë✐ tư ✤➳♥ ❤➔♠ k → ∞, p = 0, 1, 2, , m✳ ❚➟♣ ❤ñ♣ S k ❧➔ ♠ët ❤å ❝→❝ ♥û❛ ❝❤✉➞♥✳ ❉➣② f ∈ S✱ ♥➳✉ |[fk − f ]|p ợ tử tr ữủ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❝ì ❜↔♥ ❣✐↔♠ ♥❤❛♥❤✳ ❱➼ ❞ö✿ ❍➔♠ f (x) = e−x ∈ C (R) ỵ ❤đ♣ C0∞ (R) ❝õ❛ ❝→❝ ❤➔♠ ❦❤↔ ✈✐ ✈ỉ ❤↕♥ ❝â ❣✐→ ❝♦♠♣❛❝t tr♦♥❣ S ❧➔ trò ♠➟t tr♦♥❣ S t❤❡♦ t♦♣♦ ❝õ❛ S ❇✐➳♥ ✤ê✐ ❋♦✉rr✐❡r ❝õ❛ ❝→❝ ❤➔♠ ❝ì ❜↔♥ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✸✳ ❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❝→❝ ❤➔♠ ❝ì ❜↔♥ f ∈S ✤÷đ❝ ①→❝ ✤à♥❤ t❤❡♦ ❝ỉ♥❣ t❤ù❝ +∞ f (x)eixt dx F [f ] (t) = −∞ ✹ ✭✶✳✺✮ (1 − 2ν) sinh2 (|t| h) + |t|2 h2 a21 (t) = a12 (t) = , 4(1 − ν)2 + |t|2 h2 + (3 − 4ν) sinh2 (|t| h) (1 − ν) [cosh (|t| h) sinh (|t| h) − |t| h] 4(1 − ν)2 + |t|2 h2 + (3 − 4ν) sinh2 (|t| h) a22 (t) = ✭✷✳✶✵✮ ✭✷✳✶✶✮ ✷✳✷ ❚➼♥❤ ❣✐↔✐ ✤÷đ❝ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r ✷✳✷✳✶ ❚➼♥❤ ❣✐↔✐ ✤÷đ❝ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r ❚❛ ❜✐➸✉ ữợ s pF [|t| A0 (t) u (t)] (x) = f (x) , x ∈ Ω, p F −1 [u (t)] (x) = 0, x ∈ Ω := R\Ω, tr♦♥❣ ✤â ✭✷✳✶✷✮ f (x) = (ϕ1 (x) , ϕ2 (x))T , u (t) = F [u] = (u1 (t) , u2 (t))T , F −1 ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ❝ỉ♥❣ t❤ù❝ ✭✶✳✶✽✮ ✈➔ a11 (t) A0 (t) = ✈ỵ✐ i.sign (t) a12 (t) , −i.sign (t) a21 (t) a22 (t) a11 (t)✱ a21 (t)✱ a12 (t)✱ a22 (t) ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ❝→❝ ❝ỉ♥❣ t❤ù❝ ✭✷✳✾✮✱ ✭✷✳✶✵✮✱ ✭✷✳✶✶✮✱ ❚❛ ❝â p ✈➔ p ❧➛♥ ❧÷đt ❧➔ ❝→❝ t♦→♥ tû ❤↕♥ ❝❤➳ t÷ì♥❣ ù♥❣ tr➯♥ |t| Ao (t) ∈ → − α − ,→ α = (1, 1) γ11 = γ22 = ✈➔ lim|t|→∞ aij (t) = γij , Ω tr♦♥❣ ✤â (1 − ν) − 2ν , γ12 = γ21 = − 4ν − 4ν ❚❛ ❝â ❦➳t q✉↔ s❛✉✿ aij (t) − γij = |t|−∞ , |t| → ∞ ❇ê ✤➲ ✷✳✷✳✶✳ ▼❛ tr➟♥ Ao (t) ❧➔ ữỡ ợ t = ự ❚❛ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤ a11 a22 − a12 a21 > 0, ∀t = ✶✼ ✈➔ R\Ω ❚ù❝ ❧➔ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤ ∆ = 4(1 − ν)2 cosh2 (|t| h) sinh2 (|t| h) − |t|2 h2 − (1 − 2ν) sinh2 (|t| h) + |t|2 h2 ✈ỵ✐ ♠å✐ >0 t = 0✳ ξ = |t| h✱ ❚❤➟t ✈➟②✱ ✤➦t t❛ ❝â ∆ = − 8ν + 4ν sinh4 ξ + sinh2 ξ − ξ − − 4ν + 4ν sinh4 ξ −2 (1 − 2ν) ξ sinh2 ξ − ξ = (3 − 4ν) sinh4 ξ + 4(1 − ν)2 sinh2 ξ − ξ − (2 − 4ν) ξ sinh2 ξ − ξ = (2 − 4ν) sinh4 ξ − ξ sinh2 ξ + sinh4 ξ − ξ + 4(1 − v)2 sinh2 ξ − ξ = sinh2 ξ − ξ (2 − 4ν) sinh2 ξ + sinh2 ξ + ξ + 4(1 − ν)2 ∆>0 ❍✐➸♥ ♥❤✐➯♥ ❧➔ ✈ỵ✐ ♠å✐ ξ > 0✳ ❚ø ❇ê ✤➲ ✭✷✳✷✳✶✮✱ t❛ s✉② r❛ ❇ê ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ A (t) := |t| A0 (t) ∈ → − α , → − α = (1, 1) ỵ ỹ t ●✐↔ sû f ∈ H →−α /2 (Ω)✳ ❑❤✐ ✤â ♥❣❤✐➺♠ → − α /2 u ∈ H0 (Ω) ❝õ❛ ữỡ tr tỗ t t ❈❤ù♥❣ ♠✐♥❤✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ✤à♥❤ ❧➼✱ t❛ ❝❤ù♥❣ ♠✐♥❤ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✷✮ ❝❤➾ ❝â ♥❣❤✐➺♠ t➛♠ t❤÷í♥❣ pF −1 [|t| A0 (t) u (t)] (x) = f (x) , x ∈ Ω, p F −1 [u (t)] (x) = 0, x ∈ Ω := R\Ω ❚ø → − α /2 u ∈ H0 (Ω) ❤➺ tr➯♥ ✈✐➳t ❧↕✐ (Au) (x) = 0, x ∈ Ω, ✭✷✳✶✸✮ (Au)(x) = pF −1 [A(t)u(t)] (x) ✭✷✳✶✹✮ tr♦♥❣ ✤â ❱➻ → − Au ∈ H − α /2 (Ω) → − α /2 H0 1/2 (Ω) ✱ tø ✭✷✳✶✹✮ t❛ ❝â ✶✽ +∞ uT (t) F lpF −1 [Au] (t) dt [Au, u] = −∞ ❱➻ t➼❝❤ ♣❤➙♥ tr➯♥ ❦❤ỉ♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ ❝→❝❤ ❝❤å♥ lpF −1 [Au]✱ t❛ ❝â t❤➸ ✈✐➳t t❤→❝ tr✐➸♥ ð ❞↕♥❣ s❛✉ lpF −1 [Au] = F −1 [Au] ❚ø ✤➙② t❛ ❝â t❤➸ +∞ uT (t)A (t) u (t) dt, [Au, u] = ✭✷✳✶✺✮ −∞ ❦❤✐ ✤â tø ✭✷✳✶✸✮ ✈➔ ✭✷✳✶✹✮ t❛ ❝â ∞ uT (t)A (t) u (t) dt = 0, [Au, u] = −∞ ✈➻ Re uT (t) A (t) u (t) ≥ 0, s✉② r❛ u (t) = 0, u (x) = ❉♦ ✤â ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t ❝❤➾ ❝â ♥❣❤✐➺♠ t➛♠ t❤÷í♥❣✳ ❱➟② ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ✭✷✳✶✷✮ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t✳ ✣à♥❤ ❧➼ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❑➼ ❤✐➺✉ (Au)(x) = pF −1 [A(t)u(t)] (x) ✭✷✳✶✻✮ ❚❛ ❜✐➸✉ ❞✐➵♥ ❤➺ ✭✷✳✺✮ ữợ (Au) (x) = f (x) , x tt sỹ tỗ t ✭✷✳✶✼✮ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ → − α = (1, 1)T ✳ ✶✾ ✭✷✳✶✼✮ → − α /2 H0 (Ω)✱ ❚❛ ✤➦t A+ (t) = |t| coth (|t| h) α iβ.signt −iβ.signt α , B (t) = |t| A0 (t) − A+ (t) a11 (t) − α coth (|t| h) i.signt (a12 (t) − β coth (|t| h)) = |t| −i.signt (a12 (t) − β coth (|t| h)) a22 (t) − α coth (|t| h) tr♦♥❣ ✤â (1 − ν) − 2ν , β= , α−β = − 4ν − 4ν − 4ν → − − ❚❛ ❝â A (t) ∈ α , → = (1, 1) = ỵ + + ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû u1 = a1 + ib1 , u2 = a2 + ib2 , a1 , b1 , a2 , b2 ∈ R ❚❛ ❝â |u1 |2 = a21 + b21 , |u2 |2 = a22 + b22 , ✈➔ T u A+ u = t coth (th) α a21 + b21 + a22 + b22 − 2βsign (t) (a1 b2 − a2 b1 ) ≥ t coth (th) α a21 + b21 + a22 + b22 − 2β (a21 + b21 ) (a22 + b22 ) ≥ (α − β) t coth (th) a21 + b21 + a22 + b22 ⇔ βt coth (th) |u1 |2 + |u2 |2 − |u1 | |u2 | ≥ ❉♦ ✤â T u A+ u ≥ (α − β) t coth (th) |u1 |2 + |u2 |2 = t coth (t) h |u1 |2 + |u2 |2 − 4ν ✭✷✳✶✽✮ ❙û ❞ö♥❣ ❇ê ✤➲ ✭✶✳✺✳✸✮ t❛ ❝â t❤➸ ❝❤➾ r❛ r➡♥❣ t coth (th) ∈ + (R) õ õ tỗ t ởt sè ❈ ❞÷ì♥❣ s❛♦ ❝❤♦ t coth (th) ≥ C (1 + |t|) , ∀t ∈ R ❚ø ✭✷✳✶✽✮ ✈➔ ✭✷✳✷✵✮ ❦➨♦ t❤❡♦ A+ (t) ∈ ♠✐♥❤✳ ✷✵ → − α + , → − α = (1, 1)✳ ✭✷✳✶✾✮ ✣à♥❤ ❧➼ ✤÷đ❝ ❝❤ù♥❣ , ❚❛ ❝â ❦➳t q✉↔ s❛✉✿ → − −β B (ξ) ∈ → − , β = (, ) , > ỵ ỹ tỗ t sỷ (x) (x) s❛♦ ❝❤♦ ❤➔♠ f (x) − ①→❝ ✤à♥❤ t❤✉ë❝ ✈➔♦ H−→−α /2 (Ω)✱ → α = (1, 1) ❑❤✐ ✤â ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ → − ♣❤➙♥ ✭✷✳✶✷✮ ❝â ♠ët ♥❣❤✐➺♠ ❞✉② ♥❤➜t u = F −1 [u] ∈ Hoα /2 (Ω) , tù❝ ❧➔ → − → − u (x, 0) ∈ Hoα /2 (Ω) , v (x, 0) ∈ Hoα /2 (Ω) , tr♦♥❣ ✤â u (x, 0) ✈➔ v (x, 0) ❧➔ ♥❤ú♥❣ ❝❤✉②➸♥ ✈à tr➯♥ trö❝ y = ❈❤ù♥❣ ♠✐♥❤✳ ❇✐➸✉ ❞✐➵♥ t♦→♥ tû ❆ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ❤➔♠ ✭✷✳✶✻✮ tr♦♥❣ ❞↕♥❣ A = A+ + B, ✈ỵ✐ A+ u = pF −1 [A+ u] , B+ u = pF −1 [B+ u] , u = [F u] , ✭✷✳✷✵✮ ✈➔ s❛✉ ✤â ①➨t ❤➺ ♣❤÷ì♥❣ tr➻♥❤ → − A+ u (x) = g (x) , u (x) ∈ Hoα /2 (Ω) , ✈ỵ✐ − −→ α /2 g (x) ∈ Ho (Ω) ❧➔ ♠ët ❤➔♠ ✈❡❝tì ✤➣ ❝❤♦✳ ❚ø ∞ lj ϕj (t) uj (t)dt, [f , u] := j=1 −∞ ✈➔ ∞ F [vT ] (t)A+ (t) F [u] (t) dt, − (u, v)A+ ,→ α /2 = −∞ t❛ ❝â ∞ − F [vT ] (t)A+ (t) F (u) (t) dt = (u, v)A+ ,→ α /2 [A+ u, v] = −∞ ✷✶ ✭✷✳✷✶✮ ❝❤♦ u v tỡ tũ ỵ tở α /2 Ho (Ω) ❉♦ ✤â✱ ♥➳✉ → − α /2 u ∈ Ho (Ω) t❤ä❛ ♠➣♥ ✭✷✳✷✶✮ t❤➻ → − α /2 − (u, v)A+ ,→ (Ω) , α /2 = [g, v] , ∀v ∈ Ho ✈➻ ✭✷✳✷✷✮ → − α /2 [g, v] ❧➔ ♠ët ❤➔♠ t✉②➳♥ t➼♥❤ ❧✐➯♥ tư❝ tr➯♥ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt Ho t➼♥❤ ❝❤➜t ❝õ❛ ✤à♥❤ ❧➼ ❘✐❡s③✱ ❝â ♠ët ♥❣❤✐➺♠ ❞✉② ♥❤➜t uo ∈ → − α /2 − [u, v] = (uo , v)A+ ,→ (Ω) , α /2 , v ∈ Ho u = uo tø ✭✷✳✷✷✮ ✈➔ ✭✷✳✷✸✮ ❦➨♦ t❤❡♦ uo ✭✷✳✷✸✮ ❤ì♥ ♥ú❛ = A−1 g − A+ ,→ α /2 (Ω) ❜ð✐ → − α /2 Ho (Ω) s❛♦ ❝❤♦ − A+ ,→ α /2 ❈ ❧➔ ♠ët ❤➡♥❣ sè ❞÷ì♥❣✳ ❉♦ ✤â t♦→♥ tû A−1 ≤C g → − H− α /2 (Ω) , ❧➔ ❜à õ t ữợ A+ u + Bu = f , t❛ t❤✉ ✤÷đ❝ −1 u + A−1 + Bu = A+ f ❚❤❡♦ ❇ê ✤➲ ✭✶✳✺✳✽✮ t♦→♥ tû → − α /2 tö❝ tø Ho (Ω) ✈➔♦ − −→ α /2 H Bu (Ω)✳ ✭✷✳✷✹✮ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✷✵✮ ❧➔ ❤♦➔♥ t♦➔♥ ❧✐➯♥ ❉♦ ✤â t♦→♥ tû A−1 + B ❧➔ ❤♦➔♥ t♦➔♥ ❧✐➯♥ tư❝✳ ❚❤❡♦ ✤â ❤➺ ✭✷✳✷✹✮ ❧➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❋r❡❞❤♦❧♠✳ ❉♦ ✤â ❤➺ ♥➔② ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t ♥❣❤✐➺♠ → − α /2 u ∈ Ho (Ω)✳ ✷✳✷✳✷ ❇✐➳♥ ✤ê✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r ✈➲ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦ý ❞à ✣à♥❤ ♥❣❤➽❛ ✷✳✷✳✺✳ ρ (x) = (x − a) (b − x) (a < x < b) ●✐↔ sû L2ρ±1 (Ω) ✳ ❚❛ ❣å✐ ổ rt t ổ ữợ ✈➔ ❝❤✉➞♥ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ❝ỉ♥❣ t❤ù❝ b ρ±1 (x) u (x) v (x)dx, (u, v)L2±1 = ρ a ✷✷ u = L2ρ±1 (u, v)L2±1 < +∞ ρ ❇ê ✤➲ ✷✳✷✳✻✳ ●✐↔ sû ϕ ∈ L2ρ (Ω)✳ ❑➼ ❤✐➺✉ ϕo ❧➔ t❤→❝ tr✐➸♥ ✲ ❦❤æ♥❣ ❝õ❛ ❤➔♠ ϕ tr➯♥ R✳ ❑❤✐ ✤â ϕo ∈ Ho−1/2 (Ω)✳ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ L2ρ±1 (Ω) t❛ ①➨t t♦→♥ tû t➼❝❤ ♣❤➙♥ ❦➻ ❞à b SΩ [ϕ] (x) = πi ϕ (ξ) dξ, x−ξ x ∈ Ω, a tr♦♥❣ ✤â t➼❝❤ ♣❤➙♥ ✤÷đ❝ ❤✐➸✉ t❤❡♦ ♥❣❤➽❛ ❣✐→ trà ❝❤➼♥❤ ❈❛✉❝❤②✳ ❍➺ ✭✷✳✺✮ ❝â t ữủ t ữợ F [|t| a11 (t) u1 (t) + i |t| sign (t) a12 (t) u2 (t)] (x) = ϕ1 (x) , x ∈ (Ω) , −1 F [(−i) |t| sign (t) a21 (t) u1 (t) + |t| a22 (t) u2 (t)] (x) = ϕ2 (x) , ✭✷✳✷✺✮ tr♦♥❣ ✤â a11 (t)✱ a21 (t)✱ a12 (t)✱ a22 (t) ✭✷✳✶✵✮✱ ✭✷✳✶✶✮✳ ◆❣❤✐➺♠ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ❝→❝ ❝æ♥❣ t❤ù❝ ✭✷✳✾✮✱ u1 (x) = F −1 [u1 ] (x) ✈➔ u2 (x) = F −1 [u2 ] (x) ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ✭✷✳✺✮ ữủ ữợ b um (x) = vm (ξ)sign (x − ξ) dξ, ✭✷✳✷✻✮ a ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ −1/2 vm ∈ L2ρ (Ω) ⊂ Ho (Ω) ♥❣❤➽❛ ❧➔ b vm (x) dx = 0, (m = 1, 2) ✭✷✳✷✼✮ a ❙û ❞ö♥❣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r✱ ❜✐➳♥ ✤ê✐ ❤➺ ✭✷✳✷✻✮ t❛ ✤÷đ❝ b um (t) = (−it) vm (ξ) eitξ dξ = vm (t) , (m = 1, 2) (−it) ✭✷✳✷✽✮ a ❚❤➳ ✭✷✳✷✽✮ ✈➔♦ ✭✷✳✷✺✮ t❛ t❤✉ ✤÷đ❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ F −1 [i.sign (t) a11 (t) v1 (t) − a12 (t) v2 (t)] (x) = ϕ1 (x) , x ∈ (Ω) F −1 [a21 (t) v1 (t) + i.sign (t) a22 (t) v2 (t)] (x) = ϕ2 (x) , ✷✸ ✭✷✳✷✾✮ ❙û ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ b F −1 [sign (t) F [v]] (x) = πi v (ξ) dξ , v ∈ L2ρ±1 (Ω) x−ξ a ❚❛ ❜✐➳♥ ✤ê✐ ✈➳ tr→✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✾✮ t❛ ✤÷đ❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦➻ ❞à    α      π     b b v1 (ξ) dξ + x−ξ a v1 (ξ) m11 (x − ξ) dξ − a v2 (ξ) m12 (x − ξ) dξ − βv2 (x) a x ∈ (Ω) , = ϕ1 (x) , b   α      π     b b b v2 (ξ) dξ + x−ξ a v2 (ξ) m22 (x − ξ) dξ + βv1 (x) v1 (ξ) m21 (x − ξ) dξ + a a = ϕ2 (x) , x ∈ (Ω) , ✭✷✳✸✵✮ tr♦♥❣ ✤â ∞ m11 (x) = π (a11 (t) − α) sin txdt, ∞ m22 (x) = π (a22 (t) − α) sin txdt, ∞ m12 (x) = m21 (x) = π (a12 (t) − β) cos txdt ✷✳✷✳✸ ❇✐➳♥ ✤ê✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦ý ❞à ✈➲ ❤➺ ✈æ ❤↕♥ ❝→❝ ữỡ tr số t t ỵ tû t➼❝❤ ♣❤➙♥ SΩ ❜à ❝❤➦♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ L2ρ (Ω)✿ ±1 SΩ [ϕ] L2ρ±1 (Ω) ≤C ϕ ✷✹ L2ρ±1 (Ω) ●✐↔ sû Tk (x) ✈➔ Uk (x) ❧➔ ❝→❝ ✤❛ t❤ù❝ ❈❤❡❜②s❤❡✈ ❧♦↕✐ ♠ët ✈➔ ❧♦↕✐ ❤❛✐✱ t÷ì♥❣ ù♥❣✳ ❚❛ ❝â ❝→❝ ♠è✐ q✉❛♥ ❤➺ s❛✉ ✤➙②✿ Tn (cos θ) = cos nθ, sin (n + 1) θ Un (cos θ) = , sin θ ✭✷✳✸✶✮ b Tk [η (x)]Tj [η (x)] dx = αk δkj , ρ (x) ✭✷✳✸✷✮ Uk [η (x)] Uj [η (x)] ρ (x) dx = βδkj , ✭✷✳✸✸✮ Tk [η (y)] dy −2π = Uk−1 [η (x)] , k = 0, 1, , (x − y) ρ (y) b − a ✭✷✳✸✹✮ π (b − a) ρ (y) Uk−1 [η (y)] dy = Tk [η (x)] , k = 1, 2, , x−y ✭✷✳✸✺✮ a b a b a b a tr♦♥❣ ✤â δkj ❧➔ ❑r♦♥❡❝❦❡r ✈➔ ❦➼ ❤✐➺✉ αk = π, π , k = 0, k = 1, 2, , π(b − a)2 β= , 2x − (a + b) η (x) = b−a ❚r♦♥❣ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ✭✷✳✸✵✮ t❤❛② ❝→❝ ❤➔♠ tr♦♥❣ ✤â ψm (ξ) ∈ L2ρ−1 (Ω)✱ vm (ξ) = t❛ t❤✉ ✤÷đ❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✉✿ ✷✺ ψm (ξ) ✱ ρ (ξ)    α      π         b b ψ1 (ξ) dξ + ρ (ξ) (x − ξ) a a ψ2 (ξ) m12 (x − ξ) dξ ρ (ξ) a ψ2 (x) = ϕ1 (x) , −β ρ (x) b   α      π         b ψ1 (ξ) m11 (x − ξ) dξ − ρ (ξ) b ψ2 (ξ) dξ + ρ (ξ) (x − ξ) a x ∈ (Ω) , b ψ1 (ξ) m21 (x − ξ) dξ + ρ (ξ) a ψ2 (ξ) m22 (x − ξ) dξ ρ (ξ) a ψ1 (x) +β = ϕ2 (x) , ρ (x) x ∈ (Ω) ✭✷✳✸✻✮ ❇✐➸✉ ❞✐➵♥ ❝→❝ ❤➔♠ ψ1 (ξ) ✈➔ ❤➔♠ ψ2 (ξ) ữợ ộ s () = A1j Tj [η (ξ)] , ✭✷✳✸✼✮ A2j Tj [η (ξ)] , ✭✷✳✸✽✮ j=1 ∞ ψ2 (ξ) = j=1 tr♦♥❣ ✤â A1j ✈➔ A2j ❧➔ ❝→❝ ❤➡♥❣ sè ❝❤÷❛ ❜✐➳t✱ ♥❣♦➔✐ r❛ t❛ ❝á♥ ❝â Am j ∞ j=1 ∈ l2 ❚❤➳ ✭✷✳✸✼✮ ✈➔ ✭✷✳✸✽✮ ✈➔♦ ✭✷✳✸✻✮✱ t❤❛② ✤ê✐ t❤ù tü ❝õ❛ t➼❝❤ ♣❤➙♥ ✈➔ tê♥❣ t❛ t❤✉ ✤÷đ❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✉✿  b b  ∞ ∞   T [η (ξ)] Tj [η (ξ)] α j  1  + m11 (x − ξ) dξ A A  j j  π ρ (ξ) (x − ξ) ρ (ξ)   j=1 j=1  a a   b  ∞ ∞   Tj [η (ξ)] Tj [η (ξ)]   − Aj m12 (x − ξ) dξ − β A2j = ϕ1 (x),    ρ (ξ) ρ (ξ) j=1 j=1 a b b  ∞ ∞  α T [η (ξ)] Tj [η (ξ)]  j   A2j + A1j m21 (x − ξ) dξ   π ρ (ξ) (x − ξ) ρ (ξ)   j=1 j=1  a a   b  ∞ ∞   T j [η (ξ)]  Tj [η (ξ)]  + A m (x − ξ) + β A = ϕ2 (x)  22 j j  ρ (ξ) ρ (ξ)  j=1 j=1 a ✷✻ ✭✷✳✸✾✮ ❚❤❡♦ ✣à♥❤ ▲➼ ✭✷✳✷✳✼✮✱ s❛✉ ♠ët sè ❜✐➳♥ ✤ê✐ ❤➺ ✭✷✳✸✾✮✱ t❛ t❤✉ ✤÷đ❝ ❤➺ s❛✉ ✤÷đ❝ ❣å✐ ❧➔ ❤➺ ✈ỉ ❤↕♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè t✉②➳♥ t➼♥❤   −α (b − a) π (1)   An+1 +     −α (b − a) π (2)  An+1 +       n = 0, 1, 2, ∞ (1) (11) (2) (12) = Fn(1) , (1) (21) (2) (22) = Fn(2) , Aj Cnj − Aj Cnj j=1 ∞ Aj Cnj + Aj Cnj ✭✷✳✹✵✮ j=1 ❚r♦♥❣ ✤â b (22) Cnj = b Tj [η (ξ)] m22 (x − ξ) dξ, ρ (ξ) ρ (x) Un [η (x)] dx a a b (11) Cnj = b a (12) Cnj (21) Cnj  (12)   Cnj + Tj [η (ξ)] m11 (x − ξ) dξ, ρ (ξ) ρ (x) Un [η (x)] dx a β(b−a)(n+1) , (n+1) −j (n = 0, 2, 4, ; j = 2, 4, 6, = or n = 1, 3, 5, ; j = 1, 3, 5, ),   (12) Cnj , (n = 0, 2, 4, ; j = 1, 3, 5, or n = 1, 3, 5, ; j = 2, 4, 6, ) ,  (21) β(b−a)(n+1)   Cnj + (n+1)2 −j , (n = 0, 2, 4, ; j = 2, 4, 6, = or n = 1, 3, 5, ; j = 1, 3, 5, ),   (21) Cnj , (n = 0, 2, 4, ; j = 1, 3, 5, or n = 1, 3, 5, ; j = 2, 4, 6, ) , b (21) Cnj = b Tj [η (ξ)] m21 (x − ξ) dξ, ρ (ξ) ρ (x) Un [η (x)] dx a a b (12) Cnj = b Tj [η (ξ)] m12 (x − ξ) dξ, ρ (ξ) ρ (x) Un [η (x)] dx a a b Fn(1) = ρ (x) Un [η (x)] ϕ1 (x) dx, a b Fn(2) = ρ (x) Un [η (x)] (x) dx a ỵ ữỡ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦➻ ❞à ✭✷✳✸✵✮ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè t✉②➳♥ t➼♥❤ ✭✷✳✹✵✮ ❧➔ t÷ì♥❣ ✤÷ì♥❣✳ ✈➔ ❤➺ ✈ỉ ❤↕♥ ❝→❝ ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû vm (ξ) ∈ L2ρ (a, b) , ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦➻ ❞à ✭✷✳✸✵✮✳ ❚❛ ❝â ψ1 (ξ) v1 (ξ) = = ρ (ξ) ρ (ξ) v2 (ξ) = ψ2 (ξ) = ρ (ξ) ρ (ξ) ∞ A1j Tj [η (ξ)] , j=1 ∞ A2j Tj [η (ξ)] j=1 ❙û ❞ư♥❣ ❝→❝ ❜✐➳♥ ✤ê✐ ð tr➯♥ t❛ ✤÷❛ ữỡ tr t ố ợ ✈ỵ✐ v1 (ξ) , v2 (ξ) ✈➲ ❤➺ ✈ỉ ❤↕♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè t✉②➳♥ t➼♥❤ ✭✷✳✹✵✮ ✤è✐ ∞ Am j j=1 ◆❣÷đ❝ ❧↕✐✱ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ tø ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✹✵✮ s✉② r❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✸✵✮✳ ●✐↔ sû Am j ∞ j=1 ∈ l2 ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ✈ỉ ❤↕♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè t✉②➳♥ t➼♥❤ ✭✷✳✹✵✮✱ t❛ ❜✐➳♥ ✤ê✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✹✵✮ ✈➲ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✸✾✮✳ ❙û ❞ư♥❣ ❜✐➸✉ t❤ù❝ ♣❤ê ✭✷✳✸✹✮✳ ❚✐➳♣ t❤❡♦ t❛ ❤♦→♥ ✈à t❤ù tü ❧➜② tê♥❣ ✈➔ t➼❝❤ ♣❤➙♥✱ ✈ỵ✐ ❦➼ ❤✐➺✉ ✭✷✳✸✼✮✱ ✭✷✳✸✽✮ t❛ t❤✉ ữủ ữỡ tr t ợ m () = vm (ξ) ρ (ξ) , v1 (ξ) , v2 (ξ)✳ t❛ t❤✉ ✤÷đ❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❦➻ ❞à ✭✷✳✸✵✮ ✤è✐ ✣à♥❤ ❧➼ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❚❛ ❦➼ ❤✐➺✉ (1) (2) X2k−1 = Ak , X2k = Ak , (k = 1, 2, 3, ) , 4 (1) (2) E2l+1 = − Fl , E2l+2 = − Fl , (l = 0, 1, 2, ) , α (b − a) π α (b − a) π 4 (11) (12) C2j+1,2n−1 = − Cjn , C2j+1,2n = + Cjn , α (b − a) π α (b − a) π 4 (21) (22) C2j+2,2n−1 = − Cjn , C2j+2,2n = − Cjn α (b − a) π α (b − a) π ✷✽ ✭✷✳✹✶✮ ✭✷✳✹✷✮ ✭✷✳✹✸✮ ✭✷✳✹✹✮ õ õ t t ữợ Xn + Cnj Xj = En (n = 1, 2, ) ✭✷✳✹✺✮ j=1 ❇ê ✤➲ ✷✳✷✳✾✳ ❇➜t ✤➥♥❣ t❤ù❝ s❛✉ ✤➙② ❧✉ỉ♥ ❧➔ ❜➜t ✤➥♥❣ t❤ù❝ ✤ó♥❣ |Cnj | ≤ L (n ≥ 1, j ≥ 2) nj ✭✷✳✹✻✮ ❚r♦♥❣ ✤â ▲ ❧➔ ♠ët ❤➡♥❣ sè ❞÷ì♥❣ ♥❤➜t ✤à♥❤✳ ❇ê ✤➲ ✷✳✷✳✶✵✳ ◆➳✉ ❝→❝ ✤↕♦ ❤➔♠ ϕ(k) m (x) , m = 1, ❧➔ ❝→❝ ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ ❬❛✱ ❜❪ t❤➻ ❜➜t ✤➥♥❣ t❤ù❝ s❛✉ ❧✉ỉ♥ ✤ó♥❣ |En | ≤ L (n = 1, 2, ; k = 0, 1, ) nk ỵ ✷✳✷✳✶✶✳ ●✐↔ sû ϕ1 (x) ✈➔ ϕ2 (x) ❧➔ ❝→❝ ❤➔♠ sè ✤➣ ❝❤♦✱ ❣✐↔ t❤✐➳t r➡♥❣ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✹✷✮ t❤✉ë❝ ❦❤æ♥❣ ❣✐❛♥ l2✳ ❑❤✐ ✤â ❤➺ ✈æ ❤↕♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè t✉②➳♥ t➼♥❤ ✭✷✳✹✵✮ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t {Xn}∞n=1 ∈ l2✳ ❍➺ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ❧➔ ❤➺ tü❛ ❤♦➔♥ t♦➔♥ ❝❤➼♥❤ q✉②✳ {En }∞ n=1 ❈❤ù♥❣ ♠✐♥❤✳ ❑➼ ❤✐➺✉ L ❧➔ ♠❛ tr➟♥ ✈æ ❤↕♥ ð ✈➳ tr→✐ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✹✵✮✳ ❚ø ❜➜t ✤➥♥❣ t❤ù❝ ✭✷✳✹✻✮ t❛ s✉② r❛ ❤➺ ❝➦♣ ❝❤✉é✐ tr♦♥❣ t❤➔♥❤ ♣❤➛♥ ❝õ❛ ❤ë✐ tö✱ ❞♦ ✤â L L ❧➔ t♦→♥ tû ❤♦➔♥ t♦➔♥ ❧✐➯♥ tư❝ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ❧➔ l2 ✳ ❉♦ ✈➟②✱ ❤➺ ✈æ ❤↕♥ ✭✷✳✹✵✮ ❧➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❋r❡❞❤♦❧♠ tr♦♥❣✳ ❚➼♥❤ ❞✉② ♥❤➜t ♥❣❤✐➺♠ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ✤÷đ❝ s✉② r❛ tø t➼♥❤ ❞✉② ♥❤➜t ♥❣❤✐➺♠ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✺✮✳ ◆❤÷ ✈➟②✱ ❤➺ ✈ỉ ❤↕♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè t✉②➳♥ t➼♥❤ ✭✷✳✹✵✮ ❝â ❞✉② ♥❤➜t ♥❣❤✐➺♠ t❤✉ë❝ ❦❤ỉ♥❣ ❣✐❛♥✳ ❱ỵ✐ ♠é✐ ♥ ❂ ◆ ✤õ ❧ỵ♥✱ t❛ ❝â ∞ L |Cnj | ≤ n j=1 ∞ j=1 ≤ − θ < (n = N + 1, N + 2, ) j2 ❉♦ ✤â ❤➺ ✈ỉ ❤↕♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè t✉②➳♥ t➼♥❤ ✭✷✳✹✵✮ ❧➔ ❤➺ tü❛ ❤♦➔♥ t♦➔♥ ❝❤➼♥❤ q✉②✳ ✣à♥❤ ❧➼ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ✷✾ ❑➳t ❧✉➟♥ ❝õ❛ ❧✉➟♥ ✈➠♥ ▲✉➟♥ ✈➠♥ ♥➔② ✤➣ tr➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ◆❣✉②➵♥ ❱➠♥ ◆❣å❝ ✈➔ ❝ë♥❣ sü ❬✹❪ ✈➲ t➼♥❤ ❣✐↔✐ ✤÷đ❝ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ✈ỵ✐ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ①✉➜t ❤✐➺♥ ❦❤✐ ❣✐↔✐ ❜➔✐ t♦→♥ ❜✐➯♥ ❤é♥ ❤đ♣ ❝õ❛ ♣❤÷ì♥❣ tr s ỏ tr ỗ ỗ tự ỡ ❜↔♥ ✈➲ ❤➺ ✈ỉ ❤↕♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè t✉②➳♥ t➼♥❤✱ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❝→❝ ❤➔♠ ❝ì ❜↔♥ ❣✐↔♠ ♥❤❛♥❤✱ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❝→❝ ❤➔♠ s✉② rë♥❣ t➠♥❣ ❝❤➟♠✱ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈✱ ❦❤ỉ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ✈❡❝tì✱ t♦→♥ tû ❣✐↔ ✈✐ ♣❤➙♥✳ ✷✳ ❈→❝❤ ✤➦t ❜➔✐ t♦→♥ ❜✐➯♥✱ ✤÷❛ ❜➔✐ t♦→♥ ❜✐➯♥ ✈➲ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r✳ ✲ ❚➼♥❤ ❣✐↔✐ ✤÷đ❝ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r ①✉➜t ❤✐➺♥ ❦❤✐ ❣✐↔✐ ❜➔✐ t ộ ủ ố ợ ỗ ❦➳t q✉↔ ❝❤➼♥❤ ❧➔ ❝→❝ ✣à♥❤ ❧➼ ✭✷✳✷✳✷✮✱ ✣à♥❤ ❧➼ ✭✷✳✷✳✹✮✳ ✲ ✣÷❛ ❝→❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ❝➦♣ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r ✈➲ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦ý ❞à✳ ✲ ✣÷❛ t✐➳♣ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❦ý ❞à ✈➲ ❤➺ ✈ỉ ❤↕♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè t✉②➳♥ t➼♥❤✳ ❍➺ ✈ỉ ❤↕♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕✐ sè t✉②➳♥ t➼♥❤ ♥➔② ❧➔ ❤➺ tü❛ ❤♦➔♥ t♦➔♥ ❝❤➼♥❤ q✉②✳ ✸✵ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❉✉❞✉❝❤❛✈❛✳ ❘ ✭✶✾✼✾✮✱ ■♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ✇✐t❤ ❢✐①❡❞ s✐♥❣✉❧❛r✐t❡s✱ ❚❡✉❜♥❡r ❱❡r❧❛❣s❣❡s❡❧❧s❝❤❛❢t✳ ▲❡✐♣③✐❣✳ ❬✷❪ ◆✳❱✳ ◆❣♦❝ ❛♥❞ P♦♣♦✈✳ ●✳ ❨❛✳ ✭✶✾✽✻✮✱ ❉✉❛❧ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❋♦✉r✐❡r tr❛♥s❢♦r♠s✱ ❯❦r❛✐♥s❦✐✐ ♠❛t❡♠t✐❝❤❡s❦✐✐ ❩❤✉r♥❛❧ ✸✽ ✭✷✮✱ ✶✽✽✲ ✶✾✺✳ ❖♥ t❤❡ s♦❧✈❛❜✐❧✐t② ♦❢ ❞✉❛❧ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ✐♥✈♦❧✈✐♥❣ ❬✸❪ ◆✳❱✳ ◆❣♦❝ ✭✶✾✽✽✮✱ ❬✹❪ ◆❣✉②❡♥ ❱❛♥ ◆❣♦❝ ❛♥❞ ◆❣✉②❡♥ ❚❤✐ ◆❣❛♥✱ ✭✷✵✶✵✮✱ ❋♦✉r✐❡r tr❛♥s❢♦r♠s✱ ❆❝t❛ ▼❛t❤✳ ❱✐❡t◆❛♠✱ ✶✸ ✭✷✮✱ ✷✶✲✸✵✳ ❙♦❧✈❛❜✐❧✐t② ♦❢ ❛ s②st❡♠ ♦❢ ❞✉❛❧ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s ✐♥✈♦❧✈✐♥❣ ❋♦✉r✐❡r tr❛♥s❢♦r♠s✱ ❱✐❡t♥❛♠ ❏♦✉♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝s ✸✽✿✹✱ ✹✻✼✲✹✽✸✳ ■♠♣r❡ss✐♦♥ ♦❢ ❛ s❡♠✐✐♥❢✐♥✐t❡ st❛♠♣ ✐♥ ❛♥ ❛❧❛st✐❝ str✐♣ ✇✐t❤ r❡❣❛r❞ ❢♦r ❢r✐❝t✐♦♥ ❛♥❞ ❛❞❤❡s✐♦♥✱ ❏✳ ▼❛t❤✳ ❙❝✐✳ ✶✻✵ ✭✹✮✱ ✹✺✸✲✹✻✾✳ ❬✺❪ ❖str②❦✳ ❱✳ ■ ✭✷✵✵✾✮✱ ❬✻❪ ❙♦❧❞❛t❡♥❦♦✈✳ ■ ✳❆ ✭✷✵✵✸✮✱ ❚❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❝♦♥t❛❝t ♣r♦❜❧❡♠ ♦❢ t❤❡ t❤❡♦r② ♦❢ ❡❧❛st✐❝✐t② ❢♦r ❛ t❤✐❝❦ str✐♣ ✇✐t❤ ❛❞❤❡s✐♦♥✱ ❏✳ ❆♣♣❧✳ ▼❛t❤✳ ▼❡❝❤✳ ✻✼ ✭✺✮✱ ✼✼✺✲✼✽✷✳ ✸✶