ĐẠIăH CăĐÀăN NG TR NGăĐẠIăH CăS ăPHẠM PHẠMăPH CăHUY CHUỖI FOURIER VÀ ỨNG DỤNG CHO BÀI TOÁNăDAOăĐỘNG CỦA SỢI DÂY LU NăV NăTHẠCăSĨăTOÁN H C ĐàăN ng, N mă2018 ĐẠIăH CăĐÀăN NG TR NGăĐẠIăH CăS ăPHẠM PHẠMăPH CăHUY CHUỖI FOURIER VÀ ỨNG DỤNG CHO BÀI TOÁNăDAOăĐỘNG CỦA SỢI DÂY Chuyên ngành: Tốnăgi iătích Mưăsố:ă846.01.02 LU NăV NăTHẠCăSĨăTỐN H C Ng iăh ngăd năkhoaăh c:ăTS.ăLÊăH IăTRUNG ĐàăN ng, N mă2018 ▲❮■ ❈❆▼ ✣❖❆◆ ❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ✤➙② ❧➔ ❝æ♥❣ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ r✐➯♥❣ tæ✐✳ ❈→❝ sè ❧✐➺✉✱ ❦➳t q✉↔ ♥➯✉ tr♦♥❣ ❧✉➟♥ ✈➠♥ ❧➔ tr✉♥❣ t❤ü❝ ✈➔ ❝❤÷❛ tø♥❣ ✤÷đ❝ ❛✐ ❝ỉ♥❣ ❜è tr♦♥❣ ❜➜t ❦➻ ❝ỉ♥❣ tr➻♥❤ ♥➔♦ P Pữợ ợ t t t ữủ tä ❧á♥❣ ❜✐➳t ì♥ ✤➳♥ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ P❤↕♠ ✕ ✣↕✐ ❤å❝ ✣➔ ◆➤♥❣✱ P❤á♥❣ ✣➔♦ t↕♦ ❙❛✉ ✣↕✐ qỵ ổ ợ t t ữợ ❞➝♥✱ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ ❝❤♦ t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ✣➦❝ ❜✐➺t✱ t→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ ❚❙✳ ▲➯ ❍↔✐ ❚r✉♥❣✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✕ ✣↕✐ ❤å❝ ✣➔ ◆➤♥❣✱ ♥❣÷í✐ ❚❤➛② trỹ t ữợ ợ ỳ tự qỵ ❝❤➾ ❜↔♦ ❣✐ó♣ ✤ï t→❝ ❣✐↔ tü t✐♥✱ ✈÷đt q✉❛ ♥❤ú♥❣ ❦❤â ❦❤➠♥✱ trð ♥❣↕✐ tr♦♥❣ q✉→ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ✤➸ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❳✐♥ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ ❝õ❛ t→❝ ❣✐↔ ✤➳♥ ▲➣♥❤ ✤↕♦ tr÷í♥❣ ❚❍❈❙ P ỗ ữớ t ✤➣ ✤ë♥❣ ✈✐➯♥✱ ❣✐ó♣ ✤ï✱ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ✤➸ t→❝ ❣✐↔ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❤å❝✳ ❉ò t→❝ ❣✐↔ ✤➣ r➜t ❝è ❣➢♥❣✱ s♦♥❣ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ t❤➸ tr→❝❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳t sât✱ ❦➼♥❤ ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ sü ❣â♣ þ✱ ❝❤➾ ❞➝♥ ❝õ❛ q✉þ t❤➛②✱ ❝æ ❣✐→♦✱ ❝→❝ ❜↕♥ ỗ ỳ ữớ q t t ♥❣❤✐➯♥ ❝ù✉✳ ❳✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ✦ ❚→❝ ❣✐↔ P Pữợ ệ ệ é ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ❈❍×❒◆● ✶✳ ❑■➌◆ ❚❍Ù❈ ❈❒ ❙Ð ❱➋ ❈❍❯➱■ ❋❖❯❘■❊❘✸ ✶✳✶✳ ❈❍❯➱■ ▲×Đ◆● ●■⑩❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷✳ ❈❍❯➱■ ❋❖❯❘■❊❘ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✸✳ ✣■➋❯ ❑■➏◆ ✣Õ ✣➎ ❍⑨▼ ❙➮ ❑❍❆■ ❚❘■➎◆ ✣×Đ❈ ❚❍⑨◆❍ ❈❍❯➱■ ❋❖❯❘■❊❘ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✼ ✶✳✹✳ ❑❍❆■ ❚❘■➎◆ ▼❐❚ ❍⑨▼ ❙➮ ❇❻❚ ❑Ý ❚❍⑨◆❍ ❈❍❯➱■ ❋❖❯❘■❊❘ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✶✺ ✶✳✺✳ ✣➃◆● ❚❍Ù❈ P❆❘❙❊❱❆▲ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✻✳ ❉❸◆● P❍Ù❈ ❈Õ❆ ❈❍❯➱■ ❋❖❯❘■❊❘ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✷✸ ❈❍×❒◆● ✷✳ P❍×❒◆● ❚❘➐◆❍ ❉❆❖ ✣❐◆● ❈Õ❆ ❉❹❨✷✽ ✷✳✶✳ P❍×❒◆● ❚❘➐◆❍ ❉❆❖ ✣❐◆● ❈Õ❆ ❉❹❨ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✷✽ ✷✳✷✳ ❈⑩❈ ✣■➋❯ ❑■➏◆ ❇■➊◆ ❱⑨ ✣■➋❯ ❑■➏◆ ❇❆◆ ✣❺❯ ❈❍❖ P❍×❒◆● ❚❘➐◆❍ ❉❆❖ ✣❐◆● ❈Õ❆ ❉❹❨ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✸✳ ❇⑨■ ❚❖⑩◆ ❚❍Ù ◆❍❻❚ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✸✷ ✷✳✹✳ ❇⑨■ ❚❖⑩◆ ❚❍Ù ❍❆■ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✺✳ ❇⑨■ ❚❖⑩◆ ❚❍Ù ❇❆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✸✽ ❑➌❚ ▲❯❾◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✶ ▼Ð ỵ t ự ộ rr t ỗ tứ ỳ t tr t ỵ t t q ❞❛♦ ✤ë♥❣ ✈➔ ❝→❝ ❜➔✐ t♦→♥ tr✉②➲♥ ♥❤✐➺t✳ ❏✳ ❋♦✉r✐❡r ❧➔ ♥❣÷í✐ ✤➛✉ t✐➯♥ ♥❣❤✐➯♥ ❝ù✉ ❝❤✉é✐ ❧÷đ♥❣ ❣✐→❝ t❤❡♦ ổ tr trữợ õ r rt ❇❡r♥♦✉❧❧✐✳ ❏✳ ❋♦✉r✐❡r ✤➣ →♣ ❞ö♥❣ ❝❤✉é✐ ❋♦✉r✐❡r ✤➸ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ tr✉②➲♥ ♥❤✐➺t tr♦♥❣ ❝→❝ ❝ỉ♥❣ tr➻♥❤ ✤➛✉ t✐➯♥ ❝õ❛ ỉ♥❣ ✤÷đ❝ ❝ỉ♥❣ ❜è ✈➔♦ ♥➠♠ ✶✽✵✼ ✈➔ ✶✽✶✶✳ ❚r♦♥❣ ❝✉è♥ ▲➼ t❤✉②➳t ❣✐↔✐ t➼❝❤ ✈➲ ♥❤✐➺t ❤å❝ ✭❚❤➨♦r✐❡ ❛♥❛❧②t✐q✉❡ ❞❡ ❧❛ ❝❤❛❧❡✉r✮ ❝õ❛ ỉ♥❣ ✤÷đ❝ ❝ỉ♥❣ ❜è ✈➔♦ ♥➠♠ ✶✽✷✷ ✤➣ tr➻♥❤ ❜➔② ♠ët ❝→❝❤ ✤➛② ✤õ ✈✐➺❝ ❣✐↔✐ q✉②➳t ❜➔✐ t♦→♥ tr✉②➲♥ ♥❤✐➺t ✈➔ ❞❛♦ ✤ë♥❣ ❜➡♥❣ ❝❤✉é✐ ❋♦✉r✐❡r✱ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ ♥ê✐ t✐➳♥❣✱ tr♦♥❣ ✤â ❝â ❘✐❡♠❛♥♥✱ ❈❛♥t♦r ✈➔ ▲❡❜❡s❣✉❡ ❝ơ♥❣ ✤➣ ❣➢♥ ❧✐➲♥ ✈ỵ✐ ữợ ự t õ t õ r tr♦♥❣ t❤í✐ ✤↕✐ ❝õ❛ ❝❤ó♥❣ t❛✱ ✈ỵ✐ sù❝ ❤➜♣ ❞➝♥ ✈➔ sü ♣❤→t tr✐➸♥ ❝õ❛ ♠➻♥❤✱ ❝❤✉é✐ ❋♦✉r✐❡r ✤❛♥❣ ❝❤✐➳♠ ♠ët ✈à tr➼ q✉❛♥ trå♥❣ tr♦♥❣ ❣✐↔✐ t➼❝❤✳ ✷✳ ▼ö❝ t✐➯✉ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ✤➲ t➔✐ ✰ ❍➺ t❤è♥❣ ❝→❝ ❦✐➳♥ t❤ù❝ ✈➲ ❝❤✉é✐ ❋♦✉r✐❡r✳ ✰ ❙û ❞ö♥❣ ❦✐➳♥ t❤ù❝ ✈➲ ❝❤✉é✐ ❋♦✉r✐❡r ✤➸ ❣✐↔✐ q✉②➳t ❜➔✐ t♦→♥ ❞❛♦ ✤ë♥❣ sủ ợ ố tữủ ♥❣❤✐➯♥ ❝ù✉ ◆❣❤✐➯♥ ❝ù✉ ❝→❝❤ ❣✐↔✐ q✉②➳t ❜➔✐ t♦→♥ ❞❛♦ ✤ë♥❣ sđ✐ ❞➙② ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✉é✐ ❋♦✉r✐❡r✳ ✹✳ P❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉ ●✐↔✐ q✉②➳t ❜➔✐ t♦→♥ ❞❛♦ ✤ë♥❣ sñ✐ ❞➙② ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ ✈➔ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❝❤✉é✐ ❋♦✉r✐❡r✳ ✷ ✺✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ sû ❞ö♥❣ ❝→❝ ❦✐➳♥ t❤ù❝ ♥➡♠ tr♦♥❣ ❝→❝ ❧➽♥❤ ✈ü❝ s❛✉ ✤➙②✿ ●✐↔✐ t➼❝❤✱ ●✐↔✐ t➼❝❤ ❤➔♠✱ ●✐↔✐ t➼❝❤ ❋♦✉r✐❡r✱ P❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣✱ ✻✳ ✣â♥❣ ❣â♣ ❝õ❛ ✤➲ t➔✐ ✣➲ t➔✐ ❝â þ ♥❣❤➽❛ ✈➲ ♠➦t ❧þ t❤✉②➳t✱ ❝â t❤➸ sû ❞ö♥❣ ♥❤÷ ❧➔ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❞➔♥❤ ❝❤♦ ❝→❝ ✤è✐ t÷đ♥❣ q✉❛♥ t➙♠ ✤➳♥ ❜➔✐ t♦→♥ ❞❛♦ ✤ë♥❣ ❝õ❛ sđ✐ ❞➙②✳ ✼✳ ❈➜✉ tró❝ ❧✉➟♥ ✈➠♥ ❇è ❝ư❝ ❝õ❛ ❧✉➟♥ ỗ ử ❦➳t ❧✉➟♥ ✈➔ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❝❤✐❛ t❤➔♥❤ ✷ ❝❤÷ì♥❣✿ ❈❤÷ì♥❣ ✶✱ tr➻♥❤ ❜➔② ❝→❝ ❦✐➳♥ t❤ù❝ ❝ì sð ❝õ❛ ❝❤✉é✐ ❋♦✉r✐❡r✳ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② t→❝ ❣✐↔ ♥❤➢❝ ❧↕✐ ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ❝❤✉é✐ ❋♦✉r✐❡r✱ ❝❤✉é✐ ❧÷đ♥❣ ❣✐→❝✱ ✤➥♥❣ t❤ù❝ P❛rs❡✈❛❧✱ ❞↕♥❣ ♣❤ù❝ ❝õ❛ ❝❤✉é✐ ❋♦✉r✐❡r✳ ❈❤÷ì♥❣ ✷✱ tr➻♥❤ ❜➔② ù♥❣ ❞ö♥❣ ❝❤✉é✐ ❋♦✉r✐❡r tr♦♥❣ ❜❛ ❜➔✐ t♦→♥ ❞❛♦ ✤ë♥❣ ❝õ❛ sđ✐ ❞➙②✳ ✸ ❈❍×❒◆● ✶ ❑■➌◆ ❚❍Ù❈ ❈❒ ❙Ð ❱➋ ❈❍❯➱■ ❋❖❯❘■❊❘ ❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ❝→❝ ❦✐➳♥ t❤ù❝ ✤÷đ❝ tr➼❝❤ ❞➝♥ ❝❤õ ②➳✉ tø ❝→❝ t➔✐ ❧✐➺✉ ❬✷❪✱ ị ữỡ ❣✐ỵ✐ t❤✐➺✉ ❝→❝ ❦✐➳♥ t❤ù❝ ✈➲ ❝❤✉é✐ ❋♦✉r✐❡r✱ ❝→❝❤ ❦❤❛✐ tr✐➵♥ ♠ët sè ❤➔♠ t❤➔♥❤ ❝❤✉é✐ ❋♦✉r✐❡r tr♦♥❣ ❝→❝ tr÷í♥❣ ❤đ♣ ❝ư t❤➸ ✈➔ tê♥❣ q✉→t✳ ❚r♦♥❣ t❤➳ ❣✐ỵ✐ q✉❛♥❤ t tữớ ỳ tữủ õ ữợ ✤✐ ❧➦♣ ❧↕✐ t✉➛♥ ❤♦➔♥✳ ❈❤ó♥❣ ✤÷đ❝ ♠ỉ t↔ ❜ð✐ ♥❤ú♥❣ ❤➔♠ sè t✉➛♥ ❤♦➔♥✳ ◆❤ú♥❣ ❤➔♠ sè t✉➛♥ ❤♦➔♥ ✤ì♥ ❣✐↔♥ ♥❤➜t ❧➔ ♥❤ú♥❣ ❤➔♠ sè An sin(nωt + ϕn ), n = 1, 2, , ❜✐➸✉ ❞✐➵♥ ♥❤ú♥❣ ❞❛♦ ✤ë♥❣ ✤✐➲✉ ❤♦➔ ✈ỵ✐ ❜✐➯♥ ✤ë An ✱ ❝❤✉ ❦➻ T = 2π ✳ nω 2π ✱ t❛ ❝â t❤➸ ❦❤❛✐ tr✐➸♥ ♥â ω An sin(nωt + ϕn ) ❜➡♥❣ ❝→❝❤ ✤➦t ωt = x✱ t❛ ❝â ❑❤✐ ❤➔♠ sè gt t✉➛♥ ❤♦➔♥ ✈ỵ✐ ❝❤✉ ❦➻ T = ữợ g(t) = A0 + n=1 x g(t) = g( ) := f (x)✳ ❑❤✐ ✤â f (x) ❧➔ ♠ët ❤➔♠ sè t✉➛♥ ❤♦➔♥ ❝❤✉ ❦➻ 2π ✱ ❦❤✐ ω ✤â ❦❤❛✐ tr✐➸♥ tr➯♥ ✤÷❛ ✈➲ ❞↕♥❣✿ ∞ a0 (an cos nx + bn sin nx), + n=1 tr♦♥❣ ✤â a0 = 2A0 , an = An sin ϕn , bn = Bn cos ϕn ìẹ ộ ữủ ❣✐→❝ ❧➔ ❝❤✉é✐ ❤➔♠ ✶ sè ❝â ❞↕♥❣ a0 + ∞ (an cos nx + bn sin nx) n=1 tr♦♥❣ ✤â a0 , an , bn ❧➔ ♥❤ú♥❣ ❤➡♥❣ sè✳ ✶ ❈❤✉é✐ ❤➔♠ ❧➔ ❝❤✉é✐ un (x)✱ tr♦♥❣ ✤â ❝→❝ un (x) ❧➔ ❝→❝ ❤➔♠ ❝õ❛ x✳ ✭✶✳✶✮ ✸✻ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ✈➔ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ ♥❤÷ s❛✉ u(0, t) = u(l, t) = 2hx khi0 ≤ x ≤ 2l l , 2h u(x, 0) = f (x) = (l − x), 2l ≤ x ≤ l, l ut (x, 0) = ✭✷✳✷✵✮ ❇➔✐ t♦→♥ ✤➣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t tr♦♥❣ ❧➼ t❤✉②➳t ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣✳ Ð ✤➙② t❛ ❞ị♥❣ ♣❤÷ì♥❣ ♣❤→♣ t→❝❤ rr t rữợ t t t r u ổ ỗ t õ t u(x, t) = X(x)T (t) ✭✷✳✷✶✮ ❚❛ ❝â utt = a2uxx ⇒ X ′′(x)T (t) = a2X(x)T ′′(t)✳ ❙✉② r❛ X ′′ (x) a2 T ′′ (t) = X(x) T (t) tr→✐ ♣❤ö t❤✉ë❝ x✱ ♥❣❤➽❛ ❱➳ ♣❤↔✐ ♣❤ö t❤✉ë❝ t✱ ✈➳ ❧➔ ❝❤♦ ❞ò ❝→❝ ❜✐➳♥ sè ❝â t❤❛② ✤ê✐✱ ♥❤÷♥❣ t➾ sè ❧✉ỉ♥ ❧✉ỉ♥ ❜➡♥❣ ♥❤❛✉✳ ◆â ❝❤➾ ❝â t❤➸ t❤ä❛ ♠➣♥ ♥➳✉ ❜➡♥❣ ♠ët ❤➡♥❣ sè ữủ ợ số ữ ✈➟② X ′′ (x) a2 T ′′ (t) = = −λ X(x) T (t) ✭✷✳✷✷✮ ❚❛ ♥❤➟♥ ✤÷đ❝ ❤❛✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ X ′′ (x) = −λX(x), T ′′ = −a2 λT, X(x) = ✭✷✳✷✸✮ T (t) = ❈→❝ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ✭✷✳✷✵✮ ❝❤♦ t❛ u(0, t) = X(0)T (t) = u(l, t) = X(l)T (t) = ⇒ X(0) = X(l) = 0do T (t) = ✣➸ ✤✐➲✉ ❦✐➺♥ ✤➛✉ ✤÷đ❝ t❤ä❛ ♠➣♥ t❤➻ ✭✷✳✷✹✮ ●✐↔✐ ❜➔✐ t♦→♥ ✤ì♥ ❣✐↔♥ ♥❤➜t ✈➲ trà r✐➯♥❣ ✿ ❚➻♠ ❣✐→ trà ❝õ❛ t❤❛♠ sè λ ✤➸ ♣❤÷ì♥❣ tr➻♥❤ X ′′ + λX = 0, X(0) = X(l) = ✭✷✳✷✺✮ X(0) = X(l) = ✸✼ ❝â ♥❣❤✐➺♠ ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣✳ ▲➼ t❤✉②➳t ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝❤♦ t❤➜② ✤➸ ✭✷✳✷✺✮ ❝â ♥❣❤✐➺♠ ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣ t❤➻ λ ♣❤↔✐ ❞÷ì♥❣✳ ❑❤✐ ✤â t❛ ❝â X(x) = C1 cos √ λx + C2 sin √ λx √ X(0) = C1 = 0, X(l) = C2 sin λx = ✭C1 , C2 ❧➔ ❤➡♥❣ sè✮✳ ❙✉② r❛ ♣❤÷ì♥❣ tr➻♥❤ t➻♠ trà r✐➯♥❣ sin √ λl = ⇒ √ nπ l ✭✷✳✷✻✮ (n = 1, 2, 3, ) ✭✷✳✷✼✮ λ= ❉♦ ✤â ❜➔✐ t♦→♥ ❝❤➾ ❝â ♥❣❤✐➺♠ ❦❤ỉ♥❣ t➛♠ t❤÷í♥❣ ❦❤✐ ❣✐→ trà r✐➯♥❣ nπ l λ = λn = ❚÷ì♥❣ ù♥❣ t❛ ❝â ❝→❝ ❤➔♠ r✐➯♥❣ nπx l ✭✷✳✷✽✮ nπat nπat + Bn sin l l ✭✷✳✷✾✮ Xn (x) = sin ợ tr r ữỡ tr t❤❡♦ ❜✐➳♥ t ❝â ❞↕♥❣ Tn = An cos tr♦♥❣ õ An , Bn số tũ ỵ ❙✉② r❛ ♥❣❤✐➺♠ r✐➯♥❣ ✭✷✳✷✶✮ ❝â ❞↕♥❣ un (x, t) = Xn (x)Tn (t) = An cos nπat nπat nπx + Bn sin ✭✷✳✸✵✮ sin l l l ❚❛ s t tờ qt u ữợ ộ s u(x, t) = ∞ nπat nπx nπat + Bn sin sin l l l An cos n=1 ✭✷✳✸✶✮ ❑❤✐ ✤â ✤✐➲✉ ❦✐➺♥ ✤➛✉ ✭✷✳✸✵✮ ①→❝ ✤à♥❤ ❝❤♦ t❛ ❝→❝ số tũ ỵ An , Bn õ u(x, 0) = ut (x, 0) = ∞ An sin n=1 ∞ Bn n=1 nπx = f (x) l nπx nπa sin = g(x) = l l ❇➡♥❣ ❝→❝❤ ❦❤❛✐ tr✐➸♥ f ✈➔ g t❤❡♦ ❝❤✉é✐ s✐♥ t❛ ✤÷đ❝ An = l l f (x) sin nπx dx l ✸✽ = l l nπx 2hx sin dx + l l l l l nπx 2h(l − x) sin dx l l l 4h nπx 4h l nπx (l − x) sin = x sin dx + dx l l l 2l l 8h nπ = 2 sin , n = 1, 2, π n l nπx Bn = dx, n = 1, 2, g(x) sin anπ l ❱➟② t ữủ ợ ❤➺ sè An , Bn ✳ ♥❤÷ tr➯♥✳ ✷✳✺✳ ❇⑨■ ❚❖⑩◆ ❚❍Ù ❇❆ ❚❛ ①➨t ❜➔✐ t♦→♥ t➻♠ ♥❣❤✐➺♠ u ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❜✐➸✉ ❞✐➵♥ ❞❛♦ ✤ë♥❣ ❝÷ï♥❣ ❜ù❝ ❝õ❛ ♠ët sđ✐ ❞➙② ❝â ❤❛✐ ✤➛✉ ♠ót ❝è ✤à♥❤ ∂ 2u ∂ 2u = + x(x − 1) ∂ 2t ∂x (0 ≤ x ≤ 1, t ≥ 0) ✭✷✳✸✷✮ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ✈➔ ✤✐➲✉ ❦✐➺♥ ✤➛✉ ♥❤÷ s❛✉ u(0, t) = u(1, t) = 0, u(x, 0) = u′ t (x, 0) = 0, ✭✷✳✸✸✮ t≥0 (0 ≤ x ≤ 1) ✭✷✳✸✹✮ ❇➔✐ t♦→♥ ✤➣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t tr♦♥❣ ❧➼ t❤✉②➳t ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣✳ Ð ✤➙② t❛ ❞ị♥❣ ❝ỉ♥❣ ❝ö ❝❤✉é✐ ❋♦✉r✐❡r ✤➸ ❣✐↔✐ ♥❣❤✐➺♠ ❜➔✐ t♦→♥✳ ❚❛ ❦❤❛✐ tr✐➸♥ ❤➔♠ f (x, t) = x(x − 1) t❤➔♥❤ ❝❤✉é✐ s✐♥ f (x, t) = ∞ k=1 bk (t) sin kπx , l ✈ỵ✐ l = ❚➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥ tr♦♥❣ ❝æ♥❣ t❤ù❝ ❤➺ sè ❋♦✉r✐❡r bk (t)✱ t❛ ❝â l kπx (x2 − x) sin kπxdx f (x, t) sin dx = l 0 1 = −(x2 − x) cos(kπx) + (2x − 1) sin(kπx) − cos(kπx) kπ (kπ)2 (kπ)3 (1 − cos(kπ)) = (kπ)3 bk = l l ✸✾ = 0, k = 2m −8 , (2m−1) π ✭✷✳✸✺✮ (m ∈ Z) k = 2m 1, s t t ữợ ❞↕♥❣ ∞ ∞ kπx = ck (t) sin u(x, t) = l k=1 ck (t) sin k=1 kπx l ✭✷✳✸✻✮ ❚❤❛② ✭✷✳✸✻✮ ✈➔♦ ✭✷✳✸✷✮ t❛ ✤÷đ❝ ∞ k=1 [c′′k (t) + (kπ)2 ck (t) − bk (t)] sin(kπx) = ◆❤➙♥ ❤❛✐ ✈➳ ❝õ❛ ✤➥♥❣ t❤ù❝ tr➯♥ ✈ỵ✐ sin(kπx) t❛ ✤÷đ❝ ∞ k=1 [c′′k (t) + (kπ)2 ck (t) − bk (t)]sin2 (kπx) = ▲➜② t➼❝❤ ♣❤➙♥ t❤❡♦ x✱ t❛ ✤÷đ❝ c′′k (t) + (kπ)2 ck (t) − bk (t) = 0, k = 1, 2, ❚❤❛② bk ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ð tr➯♥ t❛ ✤÷đ❝ c′′ 2m (t) + (2mπ)2 c2m (t) = c′′ 2m+1 (t) + ((2m − 1)π)2 c2m−1 (t) + = (2m − 1)3 π ✭✷✳✸✼✮ ✭✷✳✸✽✮ ❙û ❞ö♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤➛✉ ✭✷✳✸✺✮ t❛ ❝â u(x, 0) = ∞ ck (0) sin kπx = k=1 ✈➔ u′t (x, 0) = ∞ c′ k (0) sin(kπx) = k=1 ❚❛ s✉② r❛ ck (0) = c′k (0) = 0, k = 1, 2, ✭✷✳✸✾✮ ❚❤❡♦ ❧➼ t❤✉②➳t ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✱ t❛ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❝õ❛ ✭✷✳✸✼✮ ❧➔ c2m (t) = 0, m = 1, 2, ✹✵ ✈➔ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ✭✷✳✸✽✮ ❧➔ c2m−1 (t) = β2m−1 cos(2m − 1)πt + γ2m−1 sin(2m − 1)πt − ✣è✐ ❝❤✐➳✉ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥ ✭✷✳✸✾✮✱ t❛ s✉② r❛ β2m−1 = ❱➟② ∞ , (2m − 1)5 π (2m − 1)5 π γ2m−1 = − cos(2m − 1)πt sin(2m − 1)πx u(x, t) = − π m=1 (2m − 1)5 ✹✶ ❑➌❚ ▲❯❾◆ ❙❛✉ t❤í✐ ❣✐❛♥ ❞➔✐ t➻♠ ❤✐➸✉ ✈➔ ♥❣❤✐➯♥ ❝ù✉✱ ❧✉➟♥ ✈➠♥ ✈ỵ✐ ✤➲ t➔✐ ✏ ❈❤✉é✐ ❋♦✉r✐❡r ✈➔ ❜➔✐ t♦→♥ ❞❛♦ ✤ë♥❣ sđ✐ ❞➙②✑ ✤➣ t❤ü❝ ❤✐➺♥ ✤÷đ❝ ♥❤ú♥❣ ♥ë✐ ❞✉♥❣ s❛✉✿ ❚r♦♥❣ ❝❤÷ì♥❣ ✶✱ ❧✉➟♥ ✈➠♥ ✤➣ tr➻♥❤ ❜➔② ♠ët ❝→❝❤ ❤➺ t❤è♥❣ ✈➔ ❝❤✐ t✐➳t ✈➲ ❝❤✉é✐ rr ự tt ỵ ✶✳✸✳✼ ①➙② ❞ü♥❣ ❝æ♥❣ t❤ù❝ t➼♥❤ ❤➺ sè ❝❤✉é✐ ❋♦✉r✐❡r✱ ✤✐➲✉ ❦✐➺♥ ✤õ ✤➸ tr✐➸♥ ❦❤❛✐ t❤➔♥❤ ❝❤✉é✐ ❋♦✉r✐❡r✱ ❞↕♥❣ ♣❤ù❝ ❝õ❛ ❝❤✉é✐ ❋♦✉r✐❡r✳ ▲✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ❝❤✐ t✐➳t ❝→❝ ✈➼ ❞ö ♠✐♥❤ ❤å❛ ✈➲ ❦❤❛✐ tr✐➸♥ ❤➔♠ sè t❤➔♥❤ ❝❤✉é✐ ❋♦✉r✐❡r✳ ❍➛✉ ❤➳t ❝→❝ ✈➼ ❞ö t→❝ ♥➔② ❣✐↔ tü ✤÷❛ r❛ ✈➔ ❧➜② ✤➲ ❜➔✐ tø ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✷❪✱ ❬✺❪ s❛✉ ✤â ❣✐↔✐ ❝❤✐ t✐➳t tr♦♥❣ ❧✉➟♥ ✈➠♥✱ ♠é✐ ✈➼ ❞ö ✤✐➸♥ ❤➻♥❤ ❝❤♦ ♠é✐ ♥ë✐ ❞✉♥❣ ❦❤→❝ ♥❤❛✉✳ Ð ♣❤➛♥ ✶✳✹ ✈➲ ✈✐➺❝ ❦❤❛✐ tr✐➸♥ ❤➔♠ sè ❜➜t ❦➻ t❤➔♥❤ ❝❤✉é✐ ❋♦✉r✐❡r✱ ❧✉➟♥ r ữợ tỹ t ✈➔ ♠ët sè ✈➼ ❞ö ♠✐♥❤ ❤å❛ rã r➔♥❣✳ ❈✉è✐ ❝❤÷ì♥❣ ■✱ t→❝ ❣✐↔ ✤➣ ✤➲ ❝➟♣ ✤➳♥ ❞↕♥❣ ♣❤ù❝ ❝õ❛ ❝❤✉é✐ ❋♦✉r✐❡r✳ ❚ò② t❤❡♦ ❤➔♠ sè ✤➲ ❜➔✐ ❝❤♦ ♠➔ t❛ ❝â t❤➸ →♣ ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ t➼♥❤ ❤➺ sè ❋♦✉r✐❡r ❞↕♥❣ ♣❤ù❝ ❤♦➦❝ ❞↕♥❣ t❤ü❝ ✤➸ t➼♥❤ t♦→♥ ✤ì♥ ❣✐↔♥ ❤ì♥✳ ❚r♦♥❣ ❝❤÷ì♥❣ ✷✱ ❧✉➟♥ ✈➠♥ ✤➣ tr➻♥❤ ❜➔② ❜➔✐ t♦→♥ ❞❛♦ ✤ë♥❣ sđ✐ ❞➙②✱ ♣❤÷ì♥❣ tr➻♥❤ ❞❛♦ ✤ë♥❣ sñ✐ ❞➙②✱ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ✈➔ ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ❞❛♦ ✤ỉ♥❣ sđ✐ ❞➙②✱ ❜❛ ❜➔✐ t♦→♥ ❞❛♦ ✤ë♥❣ sđ✐ ❞➙②✳ ❈→❝ ✈➼ ❞ư ð ❝❤÷ì♥❣ ♥➔② t→❝ ❣✐↔ s÷✉ t➛♠ tø t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✹❪✱ s❛✉ ✤â ❣✐↔✐ ❝❤✐ t✐➳t✳ ◗✉❛ ♥ë✐ ❞✉♥❣ tr➯♥✱ ❝❤ó♥❣ t❛ t❤➜② ❝❤✉é✐ ❋♦✉r✐❡r ❧➔ ❝ỉ♥❣ ❝ư tèt ✤➸ ❣✐↔✐ q✉②➳t ❝→❝ ❜➔✐ t♦→♥ ❞❛♦ ✤ë♥❣ sñ✐ ❞➙② ♠➔ tr♦♥❣ ❧✉➟♥ ✈➠♥ ✤➣ ✤➲ ❝➟♣✳ ❚r♦♥❣ q✉→ tr➻♥❤ t❤ü❝ ❤✐➺♥ ✤➲ t➔✐✱ t→❝ ❣✐↔ ✤➣ ❝è ❣➢♥❣ ♥❣❤✐➯♥ ❝ù✉✱ t✉② ♥❤✐➯♥ ❞♦ t❤í✐ ❣✐❛♥ ❝â ❤↕♥ ✈➔ ❦✐➳♥ t❤ù❝ ❝á♥ ❤↕♥ ❝❤➳ ♥➯♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ữủ sỹ õ ỵ t qỵ t ổ ỗ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤✐➺♥ ❤ì♥✳ ✹✷ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ❚r➛♥ ❆♥❤ ❇↔♦✱ ◆❣✉②➵♥ ❱➠♥ ❑❤↔✐✱ P❤↕♠ ❱➠♥ ❑✐➲✉✱ ◆❣ỉ ❳✉➙♥ ❙ì♥ ✭✷✵✵✼✮✱ ●✐↔✐ t➼❝❤ sè✱ ◆❤➔ ①✉➜t ❜↔♥ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠✳ ❬✷❪ ◆❣✉②➵♥ ❱✐➳t ✣ỉ♥❣✱ ▲➯ ❚❤à ❚❤✐➯♥ ❍÷ì♥❣✱ ◆❣✉②➵♥ ❆♥❤ ❚✉➜♥✱ ▲➯ ❆♥❤ ❱ơ ✭✶✾✾✽✮✱ ❚♦→♥ ❝❛♦ ❝➜♣ ❚➟♣ ✶✱ ◆❤➔ ①✉➜t ❜↔♥ ●✐→♦ ❞ư❝✳ ❬✸❪ ◆❣✉②➵♥ ❚❤ø❛ ❍đ♣ ✭✷✵✵✻✮✱ ●✐→♦ tr➻♥❤ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ t➟♣ ■ ✈➔ ■■✱ ◆❤➔ ①✉➜t ❜↔♥ ✣❍◗● ❍➔ ◆ë✐✳ ❬✹❪ ✣✐♥❤ ❳✉➙♥ ❑❤♦❛✱ ◆❣✉②➵♥ ❍✉② ❇➡♥❣ tr t t ỵ t ❍➔ ◆ë✐✳ ❬✺❪ ◆❣✉②➵♥ ✣➻♥❤ ❚r➼ ✭✷✵✶✶✮✱ ❚♦→♥ ❤å❝ ❝❛♦ ❝➜♣✱ ◆❤➔ ①✉➜t ❜↔♥ ●✐→♦ ❞ư❝✳ ❬✻❪ ◆❣✉②➵♥ ❈ỉ♥❣ ❚➙♠ ổ ữỡ tr t ỵ t t ố P ỗ ❚✐➳♥❣ ❆♥❤ ❬✼❪ ❊❞✇❛r❞s ❈✳❍❡♥❞r②✱ ❉❛✈✐❞ ❊✳P❡♥♥❡② ✭✷✵✵✼✮✱ ❊❧❡♠❡♥t❛r② ❞✐❢❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✇✐t❤ ❜♦✉♥❞❛r② ✈❛❧✉❡ ♣r♦❜❧❡♠✱ Pr❡♥t✐❝❡ ❍❛❧❧✳ ❬✽❪ ▼❛tt❤❡✇ ❏✳❍ ✭✷✵✵✺✮✱ ▲✐♥❡❛r P❛rt✐❛❧ ❉✐❢❢❡r❡♥t✐❛❧ ❊q✉❛t✐♦♥s✱ ❋❛❧❧✳ ❬✾❪ ❆♥❞❡r ❱r❡t❜❧❛❞ ✭✷✵✵✸✮✱ ❋♦✉r✐❡r ❆♥❛❧②s✐s ❛♥❞ ■ts ❆♣♣❧✐❝❛t✐♦♥s✱ ❙♣r✐♥❣❡r✳ ...ĐẠIăH CăĐÀăN NG TR NGăĐẠIăH CăS ăPHẠM PHẠMăPH CăHUY CHUỖI FOURIER VÀ ỨNG DỤNG CHO BÀI TOÁN? ?DAO? ?ĐỘNG CỦA SỢI DÂY Chun ngành: Tốnăgi iătích Mưăsố:ă846.01.02 LU NăV NăTHẠCăSĨăTỐN