TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI KHOA TOÁN ************* PHÙNG THỊ HƯƠNG PHƯƠNG PHÁP NHIỄU GIẢI PHƯƠNG TRÌNH VI PHÂN KHÓA LUẬN TỐT NGHIỆP ĐẠI HỌC Chuyên ngành: Giải tích HÀ NỘI – 2018 TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI KHOA TOÁN ************* PHÙNG THỊ HƯƠNG PHƯƠNG PHÁP NHIỄU GIẢI PHƯƠNG TRÌNH VI PHÂN KHĨA LUẬN TỐT NGHIỆP ĐẠI HỌC Chuyên ngành: Giải tích Người hướng dẫn khoa học PGS.TS KHUẤT VĂN NINH HÀ NỘI – 2018 ▲❮■ ❈❷▼ ❒◆ ✣➸ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✱ ❡♠ ①✐♥ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤➳♥ P●❙✳❚❙✳ ❑❤✉➜t ❱➠♥ ◆✐♥❤ ✲ ◆❣÷í✐ trü❝ t✐➳♣ t➟♥ t ữợ ữợ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❡♠ ❧➔♠ ❜➔✐ ❦❤â❛ ❧✉➟♥ ❝õ❛ ỗ tớ ụ t ỡ ❝→❝ t❤➛② ❝æ tr♦♥❣ tê ●✐↔✐ t➼❝❤ ✈➔ ❝→❝ t❤➛② ❝ỉ tr♦♥❣ ❦❤♦❛ ❚♦→♥ ✲ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✷✱ ❇❛♥ ❝❤õ ♥❤✐➺♠ ❦❤♦❛ ❚♦→♥ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ ❡♠ ❤♦➔♥ t❤➔♥❤ tèt ❜➔✐ ❦❤â❛ ❧✉➟♥ ♥➔② ✤➸ ❝â ❦➳t q✉↔ ♥❤÷ ♥❣➔② ❤ỉ♠ ♥❛②✳ ▼➦❝ ❞ò ✤➣ ❝â r➜t ♥❤✐➲✉ ❝è ❣➢♥❣✱ s♦♥❣ t❤í✐ ❣✐❛♥ ✈➔ ❦✐♥❤ ♥❣❤✐➺♠ ❜↔♥ t❤➙♥ ❝á♥ ♥❤✐➲✉ ❤↕♥ ❝❤➳ ♥➯♥ ❦❤â❛ ❧✉➟♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât r➜t ữủ sỹ õ õ ỵ t ❝æ ❣✐→♦✱ ❝→❝ ❜↕♥ s✐♥❤ ✈✐➯♥ ✈➔ ❜↕♥ ✤å❝✳ ❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❍➔ ◆ë✐✱ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✶✽ ❚→❝ ❣✐↔ ❦❤â❛ ❧✉➟♥ P❤ò♥❣ ❚❤à ❍÷ì♥❣ ▲❮■ ❈❆▼ ✣❖❆◆ ❊♠ ①✐♥ ❝❛♠ ✤♦❛♥ ❑❤â❛ ❧✉➟♥ ♥➔② ❧➔ ❝æ♥❣ tr ự r ữợ sỹ ữợ ❝õ❛ t❤➛② P●❙✳❚❙✳ ❑❤✉➜t ❱➠♥ ◆✐♥❤ ✳ ❚r♦♥❣ ❦❤✐ ♥❣❤✐➯♥ ❝ù✉✱ ❤♦➔♥ t❤➔♥❤ ❜↔♥ ❦❤â❛ ❧✉➟♥ ♥➔② ❡♠ ✤➣ t❤❛♠ ❦❤↔♦ ♠ët sè t➔✐ ❧✐➺✉ ✤➣ ❣❤✐ tr♦♥❣ ♣❤➛♥ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳ ❊♠ ①✐♥ ❦❤➥♥❣ ✤à♥❤ ❦➳t q✉↔ ❝õ❛ ✤➲ t➔✐✿ ✏ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ P❤÷ì♥❣ ♣❤→♣ ♥❤✐➵✉ ❣✐↔✐ ✑ ❧➔ ❦➳t q✉↔ ❝õ❛ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ♥é ❧ü❝ ❤å❝ t➟♣ ❝õ❛ ❜↔♥ t❤➙♥✱ ❦❤ỉ♥❣ trò♥❣ ❧➦♣ ✈ỵ✐ ❦➳t q✉↔ ❝õ❛ ❝→❝ ✤➲ t➔✐ ❦❤→❝✳ ◆➳✉ s❛✐ ❡♠ ①✐♥ ❝❤à✉ ❤♦➔♥ t♦➔♥ tr→❝❤ ♥❤✐➺♠✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✶✽ ❚→❝ ❣✐↔ ❦❤â❛ ❧✉➟♥ P❤ò♥❣ ❚❤à ❍÷ì♥❣ ▼ö❝ ❧ö❝ ▼Ð ✣❺❯ ✶ ✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ✸ ✶✳✶ ✶✳✷ ▼ët sè ❦✐➳♥ t❤ù❝ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ✳ ✳ ✳ ✸ ✶✳✶✳✶ ▼ët sè ❦❤→✐ ♥✐➺♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✶✳✷ ▼ët sè ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✤➣ ❜✐➳t ❝→❝❤ ❣✐↔✐ ✳ ✹ ✶✳✶✳✸ ❇➔✐ t♦→♥ ❈❛✉❝❤② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ❈❤✉é✐ ❧ô② t❤ø❛ ✶✳✷✳✶ ✣à♥❤ ♥❣❤➽❛ ❝❤✉é✐ ❧ô② t❤ø❛ ✈➔ ❜→♥ ❦➼♥❤ ❤ë✐ tư ❝õ❛ ❝❤✉é✐ ❧ơ② t❤ø❛ ✶✳✷✳✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ỵ tr t ❝❤✉é✐ ❧ơ② t❤ø❛ ✳ ✳ ✾ ✷ P❍×❒◆● P❍⑩P ◆❍■➍❯ Pì P ị tữ ✤➳♥ ♣❤÷ì♥❣ ♣❤→♣ ♥❤✐➵✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✷✳✷ P❤÷ì♥❣ ♣❤→♣ ♥❤✐➵✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✷✳✸ P❤÷ì♥❣ ♣❤→♣ ♥❤✐➵✉ ❦ý ❞à ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ Pữỡ ợ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❑➌❚ ▲❯❾◆ ✹✽ ✐✐ P❍Ị◆● ❚❍➚ ❍×❒◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✺✵ ✐✐✐ P❍Ị◆● ❚❍➚ ❍×❒◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ▲❮■ ◆➶■ ✣❺❯ ✶✳ ỵ t Pữỡ tr ♠ët ❝❤✉②➯♥ ♥❣➔♥❤ q✉❛♥ trå♥❣ ❝õ❛ ❚♦→♥ ❤å❝ ✈➔ ❝â r➜t ♥❤✐➲✉ ù♥❣ ❞ö♥❣ tr♦♥❣ ❝→❝ ❧➽♥❤ ✈ü❝ ❦❤♦❛ ❤å❝✱ ❝ỉ♥❣ ♥❣❤➺✱ ♥â ❝á♥ ✤÷đ❝ ❝♦✐ ♥❤÷ ❧➔ ❝➛✉ ♥è✐ ỳ ỵ tt ự ữỡ tr ✈✐ ♣❤➙♥ ❧➔ ♠ỉ♥ ❤å❝ ✤÷đ❝ ❣✐↔♥❣ ❞↕② rë♥❣ r➣✐ trữớ tr ữợ ú t❛ ❜✐➳t r➡♥❣ ❝❤➾ ❝â ♠ët sè ➼t ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❧➔ ❝â t❤➸ t➻♠ ✤÷đ❝ ♥❣❤✐➺♠ ❝❤➼♥❤ tr õ ợ ữỡ tr ♣❤➙♥ ♥↔② s✐♥❤ tø ❝→❝ ❜➔✐ t♦→♥ t❤ü❝ t✐➵♥ ✤➲✉ ❦❤ỉ♥❣ t➻♠ ✤÷đ❝ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝✳ ❉♦ ✈➟②✱ ♠ët ✈➜♥ ✤➲ ✤➦t r❛ ❧➔ t➻♠ ❝→❝❤ ✤➸ ①→❝ ✤à♥❤ ♥❣❤✐➺♠ ❣➛♥ ✤ó♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣✳ ❳✉➜t ♣❤→t tø ♥❤✉ ❝➛✉ ✤â✱ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ ✤➣ t➻♠ r❛ ♥❤✐➲✉ ♣❤÷ì♥❣ ♣❤→♣ ✤➸ ❣✐↔✐ ❣➛♥ ✤ó♥❣ ♣❤÷ì♥❣ tr➻♥❤ tữớ ợ ố t ự s ỡ ữợ P t ◆✐♥❤ P❤÷ì♥❣ ♣❤→♣ ♥❤✐➵✉ ✤➸ ❣✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ sỹ ữợ t ❝ù✉ ✤➲ ✑ ✤➸ t❤ü❝ ❤✐➺♥ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ❝õ❛ ♠➻♥❤✳ ✷✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉ ❑❤â❛ ❧✉➟♥ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ♥❤✐➵✉ ✤➸ ❣✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✳ ✸✳ ✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉ P❤÷ì♥❣ ♣❤→♣ ữỡ ý ữỡ ợ ❜✐➯♥ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣✳ ✶ P❍Ị◆● ❚❍➚ ì õ tốt Pữỡ ự Pữỡ ự ỵ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉ tê♥❣ ❦➳t t➔✐ ❧✐➺✉✳ ✺✳ ❈➜✉ tró❝ ✤➲ t➔✐ ❑❤â❛ ❧✉➟♥ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❤❛✐ ❝❤÷ì♥❣ ❈❤÷ì♥❣ ✶✳✏❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✑✳ ❈❤÷ì♥❣ ♥➔② ♥❤➢❝ ❧↕✐ ♠ët sè ❦✐➳♥ t❤ù❝ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✱ ❦❤→✐ ♥✐➺♠ ❝❤✉é✐ ❧ơ② t❤ø❛✱ ❜→♥ ❦➼♥❤ ❤ë✐ tư ✈➔ ✤à♥❤ ỵ tr t ộ ụ tứ ữỡ ✏P❤÷ì♥❣ ♣❤→♣ ♥❤✐➵✉ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✑✳ ▼ư❝ ✤➼❝❤ ữỡ ợ t ữỡ ♥❤✐➵✉ ✤➸ ❣✐↔✐ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ♣❤✐ t✉②➳♥ ✈➔ ♠ët sè ✈➼ ❞ö →♣ ❞ö♥❣✳ ❍➔ ◆ë✐✱ t❤→♥❣ ✵✺ ♥➠♠ ✷✵✶✽ ❚→❝ ❣✐↔ ❦❤â❛ ❧✉➟♥ P❤ò♥❣ ❚❤à ❍÷ì♥❣ ✷ ❈❤÷ì♥❣ ✶ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚ ❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✱ ❦❤→✐ ♥✐➺♠ ❝❤✉é✐ ❧ô② t❤ø❛✱ ❜→♥ ❦➼♥❤ ❤ë✐ tử ỵ tr t ộ ụ t❤ø❛✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ❝❤÷ì♥❣ ✤÷đ❝ t❤❛♠ ❦❤↔♦ tr♦♥❣ ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪✱❬✷❪✳ ✶✳✶ ▼ët sè ❦✐➳♥ t❤ù❝ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ✶✳✶✳✶ ▼ët sè ❦❤→✐ ♥✐➺♠ ✲ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❜➟❝ ♠ët ❝â ❞↕♥❣ tê♥❣ q✉→t✿ F (x, y, y ) = tr♦♥❣ ✤â ❤➔♠ F ◆➳✉ tr♦♥❣ ♠✐➲♥ ①→❝ ✤à♥❤ tr♦♥❣ ♠✐➲♥ D✱ ✭✶✳✶✮ D ⊂ R3 tø ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✮ t❛ ❝â t❤➸ ❣✐↔✐ ✤÷đ❝ y = f (x, y) ✸ y✿ ✭✶✳✷✮ P❍Ị◆● ❚❍➚ ❍×❒◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ t❤➻ t❛ ✤÷đ❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ♠ët ✤➣ ❣✐↔✐ r❛ ✤↕♦ ❤➔♠✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳ ❦❤♦↔♥❣ ✐✳ I = (a, b) ①→❝ ✤à♥❤ ✈➔ ❦❤↔ ✈✐ tr➯♥ ✤÷đ❝ ❣å✐ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✮ ♥➳✉ (x, ϕ(x), ϕ (x)) ∈ D ✐✐✳F (x, ϕ(x), ϕ y = ϕ(x) ❍➔♠ sè ✈ỵ✐ ♠å✐ (x)) ≡ I✳ tr➯♥ ✲ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❜➟❝ x∈I n ❝â ❞↕♥❣ tê♥❣ q✉→t✿ F (x, y, y , y , , y (n) ) = ❍➔♠ F ①→❝ ✤à♥❤ tr♦♥❣ ♠ët ♠✐➲♥ ✭✶✳✸✮ G ♥➔♦ ✤➜② ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ Rn+2 ❚r♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✸✮ ❝â t❤➸ ✈➢♥❣ ♠➦t ♠ët sè tr♦♥❣ ❝→❝ ❜✐➳♥ x, y, y , , y (n−1) ♥❤÷♥❣ y (n) ♥❤➜t t❤✐➳t ♣❤↔✐ ❝â ♠➦t✳ ◆➳✉ tø ✭✶✳✸✮ t❛ ❣✐↔✐ r❛ ✤÷đ❝ ✤↕♦ ❤➔♠ ❝➜♣ ❝❛♦ ♥❤➜t✱ tù❝ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✸✮ ❝â ❞↕♥❣ y (n) = f (x, y, y , y , , y (n−1) ) ✭✶✳✹✮ t❤➻ t❛ ✤÷đ❝ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝➜♣ ♥ ✤➣ ❣✐↔✐ r❛ ✤è✐ ✈ỵ✐ ✤↕♦ ❤➔♠ ❝➜♣ ❝❛♦ ♥❤➜t✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ◆❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✸✮ ❧➔ ❤➔♠ y = ϕ(x) ❦❤↔ ✈✐ ♥ ❧➛♥ tr➯♥ ❦❤♦↔♥❣ ✭❛✱❜✮ s❛♦ ❝❤♦ ✐✳ (x, ϕ(x), ϕ (x), , ϕ(n) (x)) ∈ G, ∀x ∈ (a, b) ✐✐✳ ◆â ♥❣❤✐➺♠ ✤ó♥❣ ♣❤÷ì♥❣ t➻♥❤ ✭✶✳✸✮ tr➯♥ ✭❛✱❜✮✳ ✶✳✶✳✷ ▼ët sè ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✤➣ ❜✐➳t ❝→❝❤ ❣✐↔✐ ❛✳ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❜✐➳♥ sè ♣❤➙♥ ❧② ✹ P❍Ị◆● ❚❍➚ ❍×❒◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❧➔ −x x − e ε + e ε + e eε y(x) = eε ú ỵ r ợ ỳ t õ ❤ì♥✱ t❛ ❦❤ỉ♥❣ t➻♠ ✤÷đ❝ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝✳ Ð ✤➙② ❝❤ó♥❣ t❛ sû ❞ư♥❣ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ♥➔② ✤➸ ❦✐➸♠ tr❛ ❤✐➺✉ ❧ü❝ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣✳ ✣➛✉ t✐➯♥ t❛ t❤➳ ❝→❝ ❜✐➸✉ t❤ù❝ y(x) = y0 (x) + εy1 (x) + ε2 y2 (x) + ✈➔ y (x) = y0 (x) + εy1 (x) + ε2 y2 (x) + ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✽✮✳ ❙❛✉ ✤â ❝➙♥ ❜➡♥❣ ❤➺ sè ❝õ❛ ε0 ✈➔ ε1 ✱ t❛ ✤÷đ❝ ε0 : y0 (x) = 1, y0 (0) = 0.y0 (1) = ε1 : y1 (x) = 0, y1 (0) = 0, y1 (1) = ❉♦ ✤â✱ ❝❤✉é✐ ♥❣❤✐➺♠ ❧➔ y(x) = y0 (x) = ✭✷✳✶✾✮ Ð ✤➙② t❛ t❤➜② r➡♥❣ ♥❣❤✐➺♠ t❤ä❛ ♠➣♥ ❞✉② ♥❤➜t ♠ët ✤✐➲✉ ❦✐➺♥ ✤÷❛ r❛ ❜ð✐ y(1) = ♥➯♥ t❛ ♥â✐ ✤➙② ❧➔ ❧ỵ♣ ❜✐➯♥ ❣➛♥ ❇ð✐ ✈➟②✱ t❛ ❝❤✐❛ ♠✐➲♥ ❝õ❛ x ∈ [0, 1] x = t❤➔♥❤ ❤❛✐ ♣❤➛♥✳ ▼ët ♣❤➛♥ ♠✐➲♥ tr♦♥❣ ❤♦➦❝ ❧ỵ♣ ❜✐➯♥ ①✉♥❣ q✉❛♥❤ ✈➔ ♠ët ♠✐➲♥ ♥❣♦➔✐✳ ✸✻ P❍Ị◆● ❚❍➚ ❍×❒◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ▼✐➲♥ ♥❣♦➔✐ ▼✐➲♥ tr♦♥❣ ✿ ◆❣❤✐➺♠ ð ♠✐➲♥ ♥❣♦➔✐ ❧➔ Yo (x) = ✭♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✾✮✮ ✿ ❚r♦♥❣ ♠✐➲♥ tr♦♥❣ t❛ ❝❤å♥ ♠ët ❜✐➳♥ ✤ë❝ ❧➟♣ ♠ỵ✐ ✤à♥❤ ❜ð✐ ξ= x ✳ εα ❙❛✉ ✤â t❤➳ ε1−2α ξ ξ ①→❝ ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✽✮✱ t❛ ✤÷đ❝ d2 y − y = −1, dξ ❙û ❞ö♥❣ t➼♥❤ ♥❤➜t q✉→♥✱ t❛ ❝â α= y(ξ = 0) = ❚❤➳ ❜✐➸✉ t❤ù❝ y(ξ) = y0 (ξ) + εy1 (ξ) + ε2 y2 (ξ) + ✈➔♦ ữỡ tr trữợ õ tr ữủ d y0 − y0 = −1, dξ y0 (0) = ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ♥➔②✱ t❛ ✤÷đ❝ y0 (ξ) = C2 (−eξ + e−ξ ) − eξ + = Yi (ξ) ❙û ❞ư♥❣ ❦ÿ t❤✉➟t ❦➳t ❤đ♣✱ lim Yi (ξ) = lim Yo (x) x→x0 ξ→∞ t❛ ✤÷đ❝ ❉♦ ✤â✱ ♥❣❤✐➺♠ tr♦♥❣ ❧➔ Yi (ξ) = − e−ξ −x =1−e ε ✈➔ M atch = lim Yi (ξ) = lim Yo (x) x→x0 ξ→∞ M atch = lim (1 − e−ξ ) = lim = x→0 ξ→∞ ✸✼ C2 = −1 P❍Ò◆● ❚❍➚ ❍×❒◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❉♦ ✤â✱ ♥❣❤✐➺♠ ❣❤➨♣ ❧➔✿ ycomp = Yi (ξ) + Yo (x) − M atch ycomp −x = − e−ξ + − = − e ε ✈➔ ✤➙② ❧➔ ♠ët ❧ỵ♣ ❜✐➯♥ t↕✐ x = 0✳ ❱➼ ❞ư ✷✳✹✳✷✳ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ εy + y + y = 0, y(0) = 0, y(1) = ✭✷✳✷✵✮ ▲í✐ ❣✐↔✐ ❚❤➳ ❜✐➸✉ t❤ù❝ y(x) = y0 (x) + εy1 (x) + ε2 y2 + ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦✱ t❛ ✤÷đ❝ ε(y0 + εy1 + ε2 y2 + ) + (y0 + εy1 + ε2 y2 + ) + (y0 + εy1 + ε2 y2 + ) = ❈➙♥ ❜➡♥❣ ❤➺ sè ❝õ❛ ε0 t❛ ❝â y0 + y0 = s✉② r❛ y0 (x) = Ce−x tr♦♥❣ ✤â C ❧➔ ♠ët ❤➡♥❣ sè ❜➜t ❦ý✳ ❘ã r➔♥❣✱ ♥❣❤✐➺♠ ♥➔② ❦❤æ♥❣ t❤➸ t❤ä❛ ♠➣♥ ❝↔ ❤❛✐ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ❝ò♥❣ ✸✽ P❍Ị◆● ❚❍➚ ❍×❒◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ♠ët ❧ó❝✱ ♥❤÷♥❣ t❛ ❝â t❤➸ t➻♠ ♥❣❤✐➺♠ ❜➡♥❣ ❝→❝❤ ①➨t ✈ỵ✐ tø♥❣ ✤✐➲✉ ❦✐➺♥ ♠ët✳ ❱➻ ❝â ❤❛✐ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ♥➯♥ t❛ s➩ t➻♠ r❛ ❤❛✐ ♥❣❤✐➺♠ tr♦♥❣ ✤â ♠ët ♥❣❤✐➺♠ sû ❞ö♥❣ ✤✐➲✉ ❦✐➺♥ ❦✐➺♥ y(1) = y(0) = ✈➔ ♠ët ♥❣❤✐➺♠ sû ❞ö♥❣ ✤✐➲✉ ❉♦ ✤â t❛ ①➨t ❤❛✐ tr÷í♥❣ ❤đ♣ s❛✉ ❚r÷í♥❣ ❤đ♣ ✶✳ ◆➳✉ ❣✐↔ sû ❧ỵ♣ ❜✐➯♥ t↕✐ ❦❤ỉ♥❣ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ t↕✐ x = 0, x=0 ♥❣❤➽❛ ữỡ tr t ợ Yo (x) = y0 (x) = e1−x ❙û ❞ö♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ξ= x εα ❙❛✉ ✤â t❤❛② ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ t❛ ✤÷đ❝ ε1−2α Y + εY + Y = 0, Y (0) = ❚❤➳ Y (ξ) = Y0 (ξ) + εY1 (ξ) + ε2 Y2 (ξ) + ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ ✈➔ ✤➦t α = 1✱ t❛ ✤÷đ❝ (Y0 +εY1 +ε2 Y2 + )+(Y0 +εY1 +εY2 + )+ε(Y0 +εY1 +εY2 + ) = ❈➙♥ ❜➡♥❣ ❤➺ sè ❝õ❛ ε0 t❛ ❝â Y0 + Y0 = ❙✉② r❛ Y0 (ξ) = C(1 − e−ξ ) ✸✾ P❍Ị◆● ❚❍➚ ❍×❒◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❑➳t ❤ñ♣ lim Y0 (ξ) = lim (Y0 (x) = y0 (x)) x→x0 ξ→∞ lim C(1 − e−ξ ) = lim e1−x x→0 ξ→∞ ❙✉② r❛ C=e ❚❤➳ C=e ✈➔♦ Y0 (ξ) t❛ ✤÷đ❝ Y0 (ξ) = e(1 − e−ξ ) = e − e1−ξ ◆❣❤✐➺♠ ❦➳t ❤ñ♣ lim Y0 (ξ) = lim y0 (x) x→x0 ξ→∞ lim (e − e1−ξ ) = lim e1−x = e x→0 ξ→∞ ❉♦ ✤â✱ ycomp = Y0 (x) + Y0 (ξ) − M atch ycomp = e1−x + e − e1−ξ − e ∼ e1−x − e1−ξ x = e1−x − e ε 1− ✣➙② ❝❤➼♥❤ ❧➔ sè ❤↕♥❣ ①➜♣ ①➾ ❝õ❛ ❝↔ ❧ỵ♣ tr♦♥❣ ✈➔ ❧ỵ♣ ♥❣♦➔✐ ❝õ❛ ♥❣❤✐➺♠ ①➜♣ ①➾ t✐➺♠ ❝➟♥✳ rữớ ủ t sỷ ợ t x=1 ❧➔ Yo (x) = y0 (x) = ❚❛ sû ❞ö♥❣ ♣❤➳♣ ❜✐➳♥ ✤ê✐ ξ= x−1 εα ✹✵ t❤➻ ợ Pề ì õ tốt ✣↕✐ ❤å❝ ❚❤➳ ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ❜❛♥ ✤➛✉ t❛ ✤÷đ❝ ε1−2α Y + εY + Y = 0, Y (1) = ❚✐➳♣ tö❝ sû ❞ö♥❣ t➼♥❤ ♥❤➜t q✉→♥ ❦✐➸♠ tr❛ α = ✈➔ ❧➔♠ t÷ì♥❣ tü tr÷í♥❣ ❤đ♣ ✶✱ t❛ ✤÷đ❝ Y0 (ξ) = e−ξ + C(1 − e1−ξ ) tr♦♥❣ ✤â C ❧➔ ❤➡♥❣ sè tò② ②✳ ❑➳t ❤ñ♣ lim Y0 (ξ) = lim (Yo (x) = y0 (x)) x→x0 ξ→∞ ❚❛ t❤➜② ✤✐➲✉ ♥➔② ❧➔ ❦❤æ♥❣ t❤➸ ❝❤♦ ❜➜t ❦ý ❣✐→ trà ♥➔♦ ❝õ❛ ❱➻ ✈➟②✱ ✤➙② ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ❧ỵ♣ ❜✐➯♥ t↕✐ C x = ❱➼ ❞ö ✷✳✹✳✸✳ ●✐↔✐ ❜➔✐ t♦→♥ ❜✐➯♥ εy − y + y = 0, y(0) = 0, y(1) = ✭✷✳✷✶✮ ▲í✐ ❣✐↔✐ ❚❤➳ y(x) = y0 (x) + εy1 (x) + ε2 y2 + ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦✱ t❛ ✤÷đ❝ ε(y0 + εy1 + ε2 y2 + ) − (y0 + εy1 + ε2 y2 + ) + (y0 + εy1 + ε2 y2 + ) = ❈➙♥ ❜➡♥❣ ❤➺ sè ❝õ❛ ε0 t❛ ❝â −y0 + y0 = ✹✶ P❍Ị◆● ❚❍➚ ❍×❒◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ s✉② r❛ y0 (x) = Cex tr♦♥❣ ✤â C ❧➔ ♠ët ❤➡♥❣ sè ❜➜t ❦ý✳ ❘ã r➔♥❣✱ ♥❣❤✐➺♠ ♥➔② ❦❤æ♥❣ t❤➸ t❤ä❛ ♠➣♥ ❝↔ ❤❛✐ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ❝ò♥❣ ♠ët ❧ó❝✱ ♥❤÷♥❣ t❛ ❝â t❤➸ t➻♠ ♥❣❤✐➺♠ ❜➡♥❣ ❝→❝❤ ①➨t ✈ỵ✐ tø♥❣ ✤✐➲✉ ❦✐➺♥ ♠ët✳ ❱➻ ❝â ❤❛✐ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ♥➯♥ t❛ s➩ t➻♠ r❛ ❤❛✐ ♥❣❤✐➺♠ tr♦♥❣ ✤â ♠ët ♥❣❤✐➺♠ sû ❞ö♥❣ ✤✐➲✉ ❦✐➺♥ ❦✐➺♥ y(1) = y(0) = ✈➔ ♠ët ♥❣❤✐➺♠ sû ❞ư♥❣ ✤✐➲✉ ❉♦ ✤â t❛ ①➨t ❤❛✐ tr÷í♥❣ ủ s rữớ ủ sỷ ợ t↕✐ ❦❤æ♥❣ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ t↕✐ x = 0, x=0 ữỡ tr t ợ ❧➔ Yo (x) = y0 (x) = ex−1 ❙û ❞ö♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ξ= x εα ❙❛✉ ✤â t❤❛② ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ t❛ ✤÷đ❝ ε1−2α Y − ε−α Y + Y = ❚❤➳ ❜✐➸✉ t❤ù❝ Y (ξ) = Y0 (ξ) + εY1 (ξ) + ε2 Y2 (ξ) + ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ ✈➔ ✤➦t α = 1✱ t❛ ✤÷đ❝ (Y0 +εY1 +ε2 Y2 + )−(Y0 +εY1 +ε2 Y2 + )+ε(Y0 +εY1 +ε2 Y2 + ) = ✹✷ P❍Ị◆● ❚❍➚ ❍×❒◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❈➙♥ ❜➡♥❣ ❤➺ sè ❝õ❛ ε0 t❛ ❝â Y0 − Y0 = ❙✉② r❛ Y0 (ξ) = C(eξ − 1) tr♦♥❣ ✤â ❈ ❧➔ ❤➡♥❣ sè ❜➜t ❦ý✳ ❑➳t ❤ñ♣ lim Y0 (ξ) = lim (Yo (x) = y0 (x)) x→x0 ξ→∞ lim C(eξ − 1) = lim ex−1 = e−1 x→0 ξ→∞ ✣➳♥ ✤➙② t❛ ❦❤æ♥❣ t❤➸ t➻♠ ✤÷đ❝ ♠ët ❣✐→ trà ♥➔♦ ❝õ❛ ❦✐➺♥ ❦➳t ❤đ♣ tr õ ổ ợ t rữớ ❤đ♣ ✷✳ ◆➳✉ t❛ ❣✐↔ sû ❧ỵ♣ ♥❣♦➔✐ t↕✐ x=1 C t❤ä❛ ♠➣♥ ✤✐➲✉ x = t❤➻ ♥❣❤✐➺♠ ❧ỵ♣ ♥❣♦➔✐ ❧➔ Yo (x) = y0 (x) = ❚❛ sû ❞ö♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ξ= x−1 εα ❚❤➳ ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ❜❛♥ ✤➛✉ t❛ ✤÷đ❝ ε1−2α Y − ε−α Y + Y = 0, ❚✐➳♣ tö❝ sû ❞ö♥❣ t➼♥❤ ♥❤➜t q✉→♥ ❦✐➸♠ tr❛ Y (1) = α = ✈➔ ❧➔♠ t÷ì♥❣ tü tr÷í♥❣ ❤đ♣ ✶✱ t❛ ✤÷đ❝ Y0 (ξ) = + C(eξ − 1) ✹✸ P❍Ò◆● ❚❍➚ ❍×❒◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ tr♦♥❣ ✤â C ❧➔ ❤➡♥❣ sè tü ❞♦✳ ❑➳t ❤ñ♣ lim Y0 (ξ) = lim (Yo (x) = y0 (x)) x→x0 ξ→∞ lim [1 + C(eξ − 1)] = ξ→∞ s✉② r❛ C=1 ❚❤➳ C ✈➔♦ Y0 (ξ) t❛ ✤÷đ❝ Y0 (ξ) = eξ ◆❣❤✐➺♠ ❦➳t ❤ñ♣ lim Y0 (ξ) = lim (Yo (x) = y0 (x)) ξ→∞ x→x0 lim eξ = ξ→∞ ❉♦ ✤â✱ ycomp (x) ∼ Y0 (ξ) + Yo (x) − M atch ❱➟② x−1 ycomp (x) ∼ + eξ − = eξ = e ε ✣➙② ❝❤➼♥❤ ❧➔ sè ❤↕♥❣ ①➜♣ ①➾ ✤➛✉ t✐➯♥ ❝➛♥ t➻♠ ❝õ❛ ❝↔ ❧ỵ♣ tr♦♥❣ ✈➔ ❧ỵ♣ ♥❣♦➔✐ ❝õ❛ ♥❣❤✐➺♠ ①➜♣ ①➾ t✐➺♠ ❝➟♥✳ ❱➼ ❞ư ✷✳✹✳✹✳ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ εy + (1 + ε)y + y = 0, ▲í✐ ❣✐↔✐ ✹✹ y(0) = 0, y(1) = ✭✷✳✷✷✮ P❍Ị◆● ❚❍➚ ❍×❒◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❚❤➳ y(x) = y0 (x) + εy1 (x) + ε2 y2 + ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦✱ t❛ ✤÷đ❝ ε(y0 +εy1 +ε2 y2 + )+(1+ε)(y0 +εy1 +ε2 y2 + )+(y0 +εy1 +ε2 y2 + ) = ❈➙♥ ❜➡♥❣ ❤➺ sè ❝õ❛ ε0 t❛ ❝â y0 + y0 = s✉② r❛ y0 (x) = Ce−x tr♦♥❣ ✤â C ❧➔ ♠ët ❤➡♥❣ sè ❜➜t ❦ý✳ ❘ã r➔♥❣✱ ♥❣❤✐➺♠ ♥➔② ❦❤æ♥❣ t❤➸ t❤ä❛ ♠➣♥ ❝↔ ❤❛✐ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ❝ò♥❣ ♠ët ❧ó❝✱ ♥❤÷♥❣ t❛ ❝â t❤➸ t➻♠ ♥❣❤✐➺♠ ❜➡♥❣ ❝→❝❤ ①➨t ✈ỵ✐ tø♥❣ ✤✐➲✉ ❦✐➺♥ ♠ët✳ ❱➻ ❝â ❤❛✐ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ♥➯♥ t❛ s➩ t➻♠ r❛ ❤❛✐ ♥❣❤✐➺♠ tr♦♥❣ ✤â ♠ët ♥❣❤✐➺♠ sû ❞ö♥❣ ✤✐➲✉ ❦✐➺♥ ❦✐➺♥ y(1) = y(0) = ✈➔ ♠ët ♥❣❤✐➺♠ sû ❞ö♥❣ ✤✐➲✉ ❉♦ ✤â t❛ ①➨t ❤❛✐ tr÷í♥❣ ❤đ♣ s❛✉ ❚r÷í♥❣ ❤đ♣ ✶✳ ◆➳✉ ❣✐↔ sû ❧ỵ♣ ❜✐➯♥ t↕✐ ❦❤ỉ♥❣ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ t↕✐ x = 0, x=0 t❤➻ ♥❣❤✐➺♠ ❧ỵ♣ ♥❣♦➔✐ ❧➔ Yo (x) = y0 (x) = e1−x ❙û ❞ö♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ξ= ✹✺ x εα ♥❣❤➽❛ ❧➔ ữỡ tr Pề ì õ tốt ❤å❝ ❙❛✉ ✤â t❤❛② ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ t❛ ✤÷đ❝ ε1−2α Y + ε−α (1 + ε)Y + Y = ❚❤➳ Y (ξ) = Y0 (ξ) + εY1 (ξ) + ε2 Y2 (ξ) + ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ ✈➔ ✤➦t α = 1✱ t❛ ✤÷đ❝ (Y0 +εY1 +ε2 Y2 + )+(1+ε)(Y0 +εY1 +ε2 Y2 + )+ε(Y0 +εY1 +ε2 Y2 + ) = ❈➙♥ ❜➡♥❣ ❤➺ sè ❝õ❛ ε0 t❛ ❝â Y0 + Y0 = 0, Y0 (0) = ❙✉② r❛ Y0 (ξ) = C(e−ξ − 1) ❑➳t ❤ñ♣ lim Y0 (ξ) = lim (Y0 (x) = y0 (x)) ξ→∞ x→x0 lim C(e−ξ − 1) = lim e1−x x→x0 ξ→∞ ❙✉② r❛ C = −e ❚❤➳ C=e ✈➔♦ Y0 (ξ) t❛ ✤÷đ❝ Y0 (ξ) = −e(e−ξ − 1) = e − e1−ξ ✹✻ P❍Ị◆● ❚❍➚ ❍×❒◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ◆❣❤✐➺♠ ❦➳t ❤ñ♣ lim Y0 (ξ) = lim Yo (x) x→x0 ξ→∞ lim (e − e1−ξ ) = lim e1−x = e x→x0 ξ→∞ ❉♦ ✤â✱ ycomp ∼ Yo (x) + Y0 (ξ) − M atch ycomp ∼ e1−x + e − e1−ξ − e = e1−x − e1−ξ ❚r÷í♥❣ ❤đ♣ ✷✳ ◆➳✉ t❛ ❣✐↔ sû ❧ỵ♣ ♥❣♦➔✐ t↕✐ −x = e(e−x − e ε ) x=1 t❤➻ ♥❣❤✐➺♠ ❧ỵ♣ ♥❣♦➔✐ ❧➔ Yo (x) = y0 (x) = ❚❛ sû ❞ö♥❣ ♣❤➨♣ ❜✐➳♥ ✤ê✐ ξ= x−1 εα ❚❤➳ ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ❜❛♥ ✤➛✉ t❛ ✤÷đ❝ ε1−2α Y + ε−α (1 + ε)Y + Y = 0, ❚✐➳♣ tö❝ sû ❞ö♥❣ t➼♥❤ ♥❤➜t q✉→♥ ❦✐➸♠ tr❛ Y (1) = α = ✈➔ ❧➔♠ t÷ì♥❣ tü tr÷í♥❣ ❤đ♣ ✶✱ t❛ ✤÷đ❝ Y0 (ξ) = Ce−ξ − Ce−1 + tr♦♥❣ ✤â C ❧➔ số tũ ỵ t ủ lim Y0 () = lim (Yo (x) = y0 (x)) ξ→∞ x→x0 ❚❛ t❤➜② ✤✐➲✉ ♥➔② ❧➔ ❦❤æ♥❣ t❤➸ ❝❤♦ ❜➜t ❦ý ❣✐→ trà ♥➔♦ ❝õ❛ ✹✼ C P❍Ị◆● ❚❍➚ ❍×❒◆● ❑❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ ✣↕✐ ❤å❝ ❱➻ ✈➟②✱ ✤➙② ❦❤æ♥❣ ♣❤↔✐ ❧➔ ❧ỵ♣ ❜✐➯♥ t↕✐ ✹✽ x = ❑➌❚ ▲❯❾◆ ❉ü❛ tr➯♥ ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❧✉➟♥ ✈➠♥ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ✈➜♥ ✤➲ s❛✉ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✱ ✤à♥❤ ♥❣❤➽❛ ❝❤✉é✐ ụ tứ tử ỵ tr✐➸♥ ❤➔♠ t❤➔♥❤ ❝❤✉é✐ ❧ơ② t❤ø❛✳ ✷✳ P❤÷ì♥❣ ♣❤→♣ ♥❤✐➵✉ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✈➔ ♠ët sè ✈➼ ❞ư →♣ ❞ö♥❣✳ ❚r➯♥ ✤➙② ❧➔ t♦➔♥ ❜ë ♥ë✐ ❞✉♥❣ ❦❤♦→ ❧✉➟♥ ✈➲ ✤➲ t➔✐ ✏ ♥❤✐➵✉ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ P❤÷ì♥❣ ♣❤→♣ ✑✳ ▼➦❝ ❞ò ✤➣ ❤➳t sù❝ ❝è ❣➢♥❣ s♦♥❣ ❞♦ ❤↕♥ ❝❤➳ ✈➲ t❤í✐ ❣✐❛♥✱ ❦✐➳♥ t❤ù❝ ✈➔ ❦✐♥❤ ♥❣❤✐➺♠ ♥➯♥ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✳ ❊♠ r➜t ♠♦♥❣ ♥❤➟♥ ✤÷đ❝ sü q✉❛♥ t➙♠ õ õ ỵ ổ ❜↕♥ ✤➸ ❦❤â❛ ❧✉➟♥ ♥➔② ✤÷ì❝ ✤➛② ✤õ ✈➔ ❤♦➔♥ t ỡ rữợ t tú õ ởt ❧➛♥ ♥ú❛ ❡♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ ✤è✐ ✈ỵ✐ ❝→❝ ❚❤➛②✱ ❈ỉ ❣✐→♦ tr♦♥❣ ❑❤♦❛ ❚♦→♥✱ ✤➦❝ ❜✐➺t ❧➔ t❤➛② ❣✐→♦ P●❙✳❚❙✳ ❑❤✉➜t ❱➠♥ ◆✐♥❤ ❣✐ó♣ ✤ï tæ✐ ❤♦➔♥ t❤➔♥❤ ❦❤â❛ ❧✉➟♥ ♥➔②✳ ❊♠ ①✐♥ ❝❤➙♥ t ỡ t t ữợ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬❆❪ ❚➔✐ ❧✐➺✉ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ◆❣✉②➵♥ ❚❤➳ ❍♦➔♥✱ P❤↕♠ P❤✉ ✭✷✵✵✺✮✱ ❈ì sð ♣❤÷ì♥❣ tr➻♥❤ ỵ tt ❱✐➺t ◆❛♠✳ ❬✷❪ ❚r➛♥ ✣ù❝ ▲♦♥❣✱ ◆❣✉②➵♥ ✣➻♥❤ ❙❛♥❣✱ ❍♦➔♥❣ ◗✉è❝ ❚♦➔♥ ✭✷✵✵✻✮✱ ●✐→♦ tr➻♥❤ ❣✐↔✐ t➼❝❤ t➟♣ ✷✱ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ●✐❛ ❍➔ ◆ë✐✳ ❬❇❪ ❚➔✐ ❧✐➺✉ ❚✐➳♥❣ ❆♥❤ ❬✸❪ ❘✳❇❡❧❧♠❛♥ ✭✶✾✻✻✮✱ P❡rt✉r❜❛t✐♦♥ ❚❡❝❤♥✐q✉❡s ✐♥ ▼❛t❤❡♠❛t✐❝s✱ ❊♥✲ ❣✐♥❡❡r✐♥❣ ❛♥❞ P❤②s✐s✱ ◆❡✇ ❨♦r❦✿ ❉♦✈❡r✳ ❬✹❪ ❚✳❙✳▲✳ ❘❛❞❤✐❦❛✱ ❚✳❑✳❱✳■②❡♥❣❛r✱ ❚✳❘❛❥❛ ❘❛♥✐ ✭✷✵✶✺✮✱ ❆♣♣♦①✐♠❛t❡ ❛♥❛❧②t✐❝❛❧ ♠❡t❤♦❞s ❢♦r s♦❧✈✐♥❣ ♦r❞✐♥❛r② ❞✐❢❢❡♥t✐❛❧ ❡q✉❛t✐♦♥s✱ ❈❘❈ Pr❡ss ❚❛②❧♦r ✫ ❋r❛♥❝✐s ●r♦✉♣✱ ❇♦❝❛ ❘❛t♦♥ ▲♦♥❞♦♥ ◆❡✇ ❨♦r❦✳ ✺✵ ... PHẠM HÀ NỘI KHOA TOÁN ************* PHÙNG THỊ HƯƠNG PHƯƠNG PHÁP NHIỄU GIẢI PHƯƠNG TRÌNH VI PHÂN KHÓA LUẬN TỐT NGHIỆP ĐẠI HỌC Chuyên ngành: Giải tích Người hướng dẫn khoa học PGS.TS KHUẤT VĂN