Phương pháp phần tử hữu hạn cho phương trình vi phân tuyến tính cấp 2

✣❸■ ❍➴❈ ✣⑨ ◆➂◆●
❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼
❑❍❖❆ ❚❖⑩◆
✯✯✯✯✯

▲➊ ❚❍➚ ❍❯❨➋◆

P❍×❒◆● P❍⑩P P❍❺◆ ❚Û ❍Ú❯ ❍❸◆
❈❍❖ P❍×❒◆● ❚❘➐◆❍ ❱■ P❍❹◆ ❚❯❨➌◆ ❚➑◆❍
❈❻P ✷
◆●⑨◆❍ ✣⑨❖ ❚❸❖✿ ❚❖⑩◆ Ù◆● ❉Ö◆●
❑❍➶❆ ▲❯❾◆ ❚➮❚ P
ữợ P ỵ ữớ

✹ ♥➠♠ ✷✵✶✾


▲❮■ ❈❷▼ ❒◆
▲í✐ ✤➛✉ t✐➯♥✱ tỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ỡ s s tợ t ữợ
P ỵ ữớ t t ữợ tổ tr sốt q tr➻♥❤ t❤ü❝
❤✐➺♥✳
❚ỉ✐ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ ✤➳♥ t➜t ❝↔ ❝→❝ t❤➛② ❝æ ❣✐→♦ ✤➣
t➟♥ t➻♥❤ ❣✐↔♥❣ ❞↕②✱ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❤å❝ t➟♣ t↕✐ ❑❤♦❛ ❚♦→♥
✲ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠✱ ✣↕✐ ❤å❝ ✣➔ ◆➤♥❣✳
❈✉è✐ ❝ị♥❣✱ tỉ✐ ❝ơ♥❣ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ✤➳♥ ❝→❝ t❤➔♥❤ ✈✐➯♥ tr♦♥❣ ▲ỵ♣
✶✺❈❚❯❉❊ ✤➣ ♥❤✐➺t t➻♥❤ ❣✐ó♣ ✤ï tỉ✐ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ t↕✐ ❚r÷í♥❣✳

❙✐♥❤ ✈✐➯♥
▲➯ ❚❤à ❍✉②➲♥

▼Ư❈ ▲Ư❈
▲❮■ ◆➶■ ✣❺❯
❈❍×❒◆● ✶✳ ▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❈❍❯❽◆ ❇➚

✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳
✳✳✳✳✳✳✳✳✳✳

✶
✸

✶✳✶✳ ▼ët sè ❦➳t q✉↔ q✉❛♥ trå♥❣ tø ❣✐↔✐ t➼❝❤ ❤➔♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸
✶✳✷✳ ❈→❝ ❦❤æ♥❣ ❣✐❛♥ ❤➔♠ ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻
✶✳✷✳✶✳ ❈→❝ ❦❤æ♥❣ ❣✐❛♥ ❤➔♠ ❝ê ✤✐➸♥

✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✻

✶✳✷✳✷✳ ▼ët sè ❦❤ỉ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻

❈❍×❒◆● ✷✳ ❚✃◆● ◗❯❆◆ ❱➋ P❍×❒◆● ❚❘➐◆❍ ❱■ P❍❹◆
❚❯❨➌◆ ❚➑◆❍ ❈❻P ✷ ❱⑨ P❍×❒◆● P❍⑩P P❍❺◆ ❚Û
❍Ú❯ ❍❸◆
✽
✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳

✷✳✶✳ ❚ê♥❣ q✉❛♥ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✽
✷✳✶✳✶✳ ❈➜✉ tró❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤
❝➜♣ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾
✷✳✶✳✷✳ P❤÷ì♥❣ ♣❤→♣ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥
t➼♥❤ ❝➜♣ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾
✷✳✷✳ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ✷ ❤➺ sè ❤➡♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶

✷✳✷✳✶✳ P❤÷ì♥❣ ♣❤→♣ t➻♠ ♥❣❤✐➺♠ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥
t➼♥❤ ❝➜♣ ✷ ❤➺ sè ❤➡♥❣ t❤✉➛♥ ♥❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶
✷✳✷✳✷✳ P❤÷ì♥❣ ♣❤→♣ t➻♠ ♥❣❤✐➺♠ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥
t➼♥❤ ❝➜♣ ✷ ❤➺ sè ❤➡♥❣ ❦❤æ♥❣ t❤✉➛♥ ♥❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷
✷✳✸✳ ❚ê♥❣ q✉❛♥ ✈➲ ♣❤÷ì♥❣ ♣❤→♣ ♣❤➛♥ tû ❤ú✉ ❤↕♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸


✷✳✸✳✶✳ ◆❣❤✐➺♠ ②➳✉ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸
✷✳✸✳✷✳ ❑❤ỉ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ♠ơ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
ị tữ ừ ♣❤÷ì♥❣ ♣❤→♣ ♣❤➛♥ tû ❤ú✉ ❤↕♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺
✷✳✸✳✹✳ ▼❛ tr➟♥ ♣❤➛♥ tû ✈➔ ✈➨❝tì ♣❤➛♥ tû ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼
✷✳✸✳✺✳ ❚➼♥❤ ❣➛♥ ✤ó♥❣ ♠❛ tr➟♥ ♣❤➛♥ tû ✈➔ ✈➨❝tì ♣❤➛♥ tû ✳ ✳ ✳ ✷✷
✷✳✹✳ ❚è❝ ✤ë ❤ë✐ tö ♣❤➛♥ tû ❤ú✉ ❤↕♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻

❈❍×❒◆● ✸✳ ▲❾P ❚❘➐◆❍ P❍❺◆ ❚Û ❍Ú❯ ❍❸◆

✳✳✳✳✳✳✳

✷✽

✸✳✶✳ ❚ê♥❣ q✉❛♥ ✈➲ ❧➟♣ tr➻♥❤ ♣❤➛♥ tû ❤ú✉ ❤↕♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽
✸✳✷✳ ❈❤÷ì♥❣ tr➻♥❤ ▼❆❚▲❆❇ ❝❤♦ ❝→❝ ✈➼ ❞ö sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾

❑➌❚ ▲❯❾◆
❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖

✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳
✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳✳

✹✸

✹✹


✶

▲❮■ ◆➶■ ✣❺❯
P❤÷ì♥❣ ♣❤→♣ ♣❤➛♥ tû ❤ú✉ ❤↕♥ ❧➔ ♠ët ♣❤÷ì♥❣ ♣❤→♣ sè ❞ị♥❣ ✤➸ ❣✐↔✐
❣➛♥ ✤ó♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣✳ ◆❣➔② ♥❛②✱
♣❤÷ì♥❣ ♣❤→♣ ♣❤➛♥ tû ❤ú✉ ❤↕♥ ❝á♥ ✤÷đ❝ ♣❤→t tr✐➸♥ ✈➔ ù♥❣ ❞ư♥❣ ✤➸ ❣✐↔✐
♥❤✐➲✉ ❜➔✐ t♦→♥ ❦❤♦❛ ❤å❝ ❦➽ t❤✉➟t✳ P❤÷ì♥❣ ♣❤→♣ ♣❤➛♥ tû ❤ú✉ ❤↕♥ ✤➣ trð
t❤➔♥❤ ♠ët tr♦♥❣ ♥❤ú♥❣ ❝ỉ♥❣ ❝ư q✉❛♥ trå♥❣ ✈➔ ❤✐➺✉ q✉↔ tr♦♥❣ ❦➽ t❤✉➟t
✈➔ ❦❤♦❛ ❤å❝✳ ❇➯♥ ❝↕♥❤ ✤â✱ ▼❆❚▲❆❇ ❧➔ ♣❤➛♥ ♠➲♠ ❝✉♥❣ ❝➜♣ ♠ỉ✐ tr÷í♥❣
t➼♥❤ t♦→♥ sè ✈➔ ❧➟♣ tr➻♥❤✳ ▼❆❚▲❆❇ ❝❤♦ t t số ợ tr
ỗ t số ỗ tổ t tỹ tt t t
ữớ ũ t ợ ỳ ❝❤÷ì♥❣ tr➻♥❤ ♠→② t➼♥❤ ✈✐➳t tr➯♥
♥❤✐➲✉ ♥❣ỉ♥ ♥❣ú ❧➟♣ tr➻♥❤ t ỏ ú ỷ ỵ t
t ✤ó♥❣ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ✷ ❜➡♥❣
♣❤÷ì♥❣ ♣❤→♣ ♣❤➛♥ tû ❤ú✉ ❤↕♥✳ ❚ø ♥❤✉ ❝➛✉ ❝õ❛ ❜↔♥ t❤➙♥ ✈➲ ❤å❝ t➟♣✱ t➻♠
❤✐➸✉ ♣❤÷ì♥❣ ♣❤→♣ ♣❤➛♥ tû ❤ú✉ ❤↕♥ ✈➔ ▼❆❚▲❆❇ ❝ơ♥❣ ♥❤÷ ♠♦♥❣ ♠✉è♥
♥❣❤✐➯♥ ❝ù✉ s➙✉ ❤ì♥ ✈➲ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ sè ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✱ tæ✐
✤➣ ♥❣❤✐➯♥ ❝ù✉ ✈➔ t❤❛♠ ❦❤↔♦ ♥❤✐➲✉ t➔✐ tỹ õ
ỳ ỵ ữ tr ũ ợ sỹ ữợ ừ P ỵ

Pữỡ tỷ ỳ
ữỡ tr t✉②➳♥ t➼♥❤ ❝➜♣ ✷ ✑ ❧➔♠ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣✳ ▼ư❝
▼÷í✐✱ tỉ✐ ✤➣ q✉②➳t ✤à♥❤ ❝❤å♥ ✤➲ t➔✐✿ ✏

t✐➯✉ ❝õ❛ ❦❤â❛ ❧✉➟♥ ♥❤➡♠ t❤➜✉ ❤✐➸✉ ❜↔♥ ❝❤➜t ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ♣❤➛♥ tû
❤ú✉ ❤↕♥ ✈➔ ❜✐➳t ❝→❝❤ →♣ ❞ö♥❣ ❧➟♣ tr➻♥❤ ❜➡♥❣ ▼❆❚▲❆❇ ✤➸ ❣✐↔✐ ❝→❝ ❜➔✐
t♦→♥ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ✷✳

❈❤ó♥❣ tỉ✐ sû ❞ư♥❣ ữỡ ự ỵ tt tr q tr
tỹ t rữợ t ú tổ t t t t
ừ ỳ t trữợ q✉❛♥ ✤➳♥ ♣❤÷ì♥❣ ♣❤→♣ ♣❤➛♥ tû ❤ú✉ ❤↕♥✱
♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ✷ ✈➔ ❧➟♣ tr➻♥❤ ▼❆❚▲❆❇✳ ❙❛✉ ✤â✱


✷
❜➡♥❣ ❝→❝❤ t÷ì♥❣ tü ❤â❛✱ ❦❤→✐ q✉→t ❤â❛ ♥❤ú♥❣ ❦➳t q õ ú tổ s ữ
r ỳ t q ợ ❝❤♦ ✤➲ t➔✐✳
◆ë✐ ❞✉♥❣ ❦❤â❛ ❧✉➟♥ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❜❛ ❝❤÷ì♥❣✳ ◆❣♦➔✐ r❛✱ ❦❤â❛
❧✉➟♥ ▲í✐ ❝↔♠ ì♥✱ ▼ư❝ ❧ư❝✱ ▲í✐ ♥â✐ ✤➛✉✱ ❑➳t ❧✉➟♥ ✈➔ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳

❈❤÷ì♥❣ ✶✱ tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➔ t➼♥❤ ❝❤➜t ❧✐➯♥ q✉❛♥ ❝õ❛ ❣✐↔✐
t➼❝❤ ❤➔♠✱ ♠ët sè ❦❤æ♥❣ ❣✐❛♥ ❤➔♠ ❝ì ❜↔♥ ♥❤➡♠ ♣❤ư❝ ✈ư ❝❤♦ ✈✐➺❝ ♥❣❤✐➯♥
❝ù✉ ❈❤÷ì♥❣ ✷✳
❈❤÷ì♥❣ ✷✱ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥
t✉②➳♥ t➼♥❤ ❝➜♣ ✷✱ ♠ët sè ❦❤→✐ ♥✐➺♠ ✈➔ t➼♥❤ ❝❤➜t ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ♣❤➛♥ tû
❤ú✉ ❤↕♥✱ tè❝ ✤ë ❤ë✐ tư ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ♣❤➛♥ tû ❤ú✉ ❤↕♥✳
❈❤÷ì♥❣ ✸✱ tr➻♥❤ ❜➔② ❝→❝❤ ❣✐↔✐ ❝→❝ ✈➼ ❞ư sè ❝ư t❤➸ ❜➔✐ t♦→♥ ♣❤÷ì♥❣ tr➻♥❤
✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ✷ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ♣❤➛♥ tû ❤ú✉ ❤↕♥ sû ❞ư♥❣ ❧➟♣
tr➻♥❤ ▼❆❚▲❆❇✳




ì


r ữỡ ú tæ✐ tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t q✉❛♥
trå♥❣ tø ❣✐↔✐ t➼❝❤ ❤➔♠ ✈➔ ♠ët sè ❦❤æ♥❣ ❣✐❛♥ ❤➔♠ ❝ì ❜↔♥ ♥❤➡♠ ♣❤ư❝ ✈ư

❝❤♦ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ♣❤÷ì♥❣ ♣❤→♣ ♣❤➛♥ tû ❤ú✉ ❤↕♥✳ ✣➸ ❝â ♠ët ❦✐➳♥ t❤ù❝
✤➛② ✤õ ✈➲ ❣✐↔✐ t➼❝❤ ❤➔♠ ✈➔ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❤➔♠ ❝ì ❜↔♥✱ ♥❣÷í✐ ✤å❝ ❝â t❤➸
t❤❛♠ ❦❤↔♦ ð ❝→❝ t➔✐ ❧✐➺✉ ❬✷❪✱ ❬✶✵❪ ✳

✶✳✶✳ ▼ët sè ❦➳t q✉↔ q✉❛♥ trå♥❣ tø t

X


t ổ ữợ

(., .)X

ởt ổ tỡ tr trữớ

ợ

ừ ố ợ

.

X t❤➻ ♥â ✤÷đ❝ ❣å✐

P❤➛♥ ❜ị trü❝ ❣✐❛♦ ❝õ❛ t➟♣ ❝♦♥

U

tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥

◆➳✉

X

R

❧➔ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✳
❍✐❧❜❡rt

X✱

U ⊥ ✱ ❧➔ ❦❤ỉ♥❣
U ⊥ = {v ∈ X |(v, u)X = 0,

✤÷đ❝ ❦➼ ❤✐➺✉ ❜ð✐

❣✐❛♥ ❝♦♥ ❜à ✤â♥❣ s❛♦ ❝❤♦✿
✈ỵ✐ ♠å✐

u ∈ U} .

✣à♥❤ ❧➼ ✶✳✶✳✸✳ ❈❤♦ U ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ õ ừ ổ rt

X

õ f


X

tỗ t↕✐ ❝→❝ ❤➔♠ ❞✉② ♥❤➜t u ∈ U ✈➔ v ∈ U ⊥ s❛♦
f = u + v,

✈➔ t❛ ✈✐➳t X = U

U ⊥✳

❈❤ó♥❣ t❛ s➩ tr➻♥❤ ❜➔② ♠ët sè ❦➳t q✉↔ ❝ì ❜↔♥ ❧✐➯♥ q✉❛♥ ✤➳♥ ❜➔✐ t♦→♥
❜✐➳♥ ♣❤➙♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✹✳
a(., .) : X × Y → R

❈❤♦

X

✈➔

Y

❧➔ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✳ ▼ët →♥❤ ①↕

✤÷đ❝ ❣å✐ ❧➔ ❞↕♥❣ s♦♥❣ t✉②➳♥ t➼♥❤ ♥➳✉✿

a(α1 u + α2 v, φ) = α1 a(u, φ) + α2 a(v, φ),


✹

✈ỵ✐ ♠å✐

α1 , α2 ∈ R, u, v ∈ X, φ ∈ Y.
a(u, β1 φ + β2 ϑ) = β1 a(u, φ) + β2 a(u, ϑ),

✈ỵ✐ ♠å✐

β1 , β2 ∈ R, u ∈ X, ϑ, φ ∈ Y.

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✺✳

▼ët s t t

X ìY



tr õ

X

Y

a(., .) ữủ tr➯♥

❧➔ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✱ ✤÷đ❝ ❣å✐ ❧➔ ❜à ❝❤➦♥

C ✤ë❝ ❧➟♣ ❝õ❛ u ∈ X ✈➔ φ ∈ Y s❛♦ ❝❤♦✿
|a(u, φ)| ≤ C u X φ Y ✱ ✈ỵ✐ ♠å✐ u ∈ X ✈➔ φ ∈ Y.

♥➳✉ ❝â ♠ët ❤➡♥❣ sè

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✻✳
X

❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt✱ ✤÷đ❝ ❣å✐ ❧➔ ❝÷ï♥❣

♣❤ư t❤✉ë❝ ✈➔♦

u ∈ X s❛♦ ❝❤♦✿
|a(u, u)| ≥ α u

❈❤♦ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt
❜à ❝❤➦♥
❚➻♠

a(., .)

tr➯♥

u∈X

a : X × X → R✱ tr♦♥❣ ✤â
❜ù❝ ♥➳✉ ❝â ♠ët α > 0 ổ

ởt s t t

X ì X

X

2
X

,

ợ ồ

u ∈ X.

✈➔ ♠ët ❞↕♥❣ s♦♥❣ t✉②➳♥ t➼♥❤ ❝÷ï♥❣ ❜ù❝✱

t❛ ①➨t ❜➔✐ t♦→♥ ❜✐➳♥ ♣❤➙♥✿

s❛♦ ❝❤♦✿

a(u, φ) = f (φ), ∀φ ∈ X,
tr♦♥❣ ✤â

f ∈X

❇ê ✤➲ ✶✳✶✳✼

✭✶✳✶✮

❧➔ ♠ët ♣❤✐➳♠ ❤➔♠ t✉②➳♥ t trữợ

r

sỷ a : X × X → R ❧➔ ♠ët ❞↕♥❣ s♦♥❣

t✉②➳♥ t ữù ự õ ợ ộ f X tỗ t ởt
tỷ t u ∈ X t❤ä❛ ♠➣♥ a(u, φ) = f (φ) ✈ỵ✐ ♠å✐ φ ✈➔✿
u

X

≤

C
f
α

X

,

tr♦♥❣ ✤â C ✈➔ α ❧➔ ❝→❝ ❤➡♥❣ sè tr♦♥❣ ❝→❝ ✤à♥❤ ♥❣❤➽❛ t➼♥❤ ❝÷ï♥❣ ❜ù❝ ✈➔ ❜à
❝❤➦♥ ð tr➯♥✳
●✐↔ sû r➡♥❣ ❝❤ó♥❣ t❛ ❝â ♠ët ❞➣② ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❝♦♥ ❤ú✉ ❤↕♥ ❝❤✐➲✉

Xh , h > 0 ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt X ✳ Ð ✤➙②✱ t❛ ❣✐↔ sû
Xh ⊂ X ✱ h > 0✱ ✈➔ ❝❤ó♥❣ ❧➔ ỳ ứ Xh X ợ

ữủ ❜ð✐
❦❤ỉ♥❣ ❣✐❛♥
♠é✐

h✱

t❛ ♥â✐ r➡♥❣ sü ①➜♣ ①➾ ❧➔ ♣❤ị ❤đ♣✳

❇ê ✤➲ ✶✳✶✳✽

✳ ●✐↔ sû Xh ⊂ X ✱h > 0✱ ❧➔ ♠ët ❤å ❝→❝ ❦❤æ♥❣ ❣✐❛♥

✭❈❡❛✮

❝♦♥ ❤ú✉ ❤↕♥ ❝❤✐➲✉ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❍✐❧❜❡rt X ✳ ●✐↔ sû a : X × X → R ❧➔
❞↕♥❣ s♦♥❣ t✉②➳♥ t➼♥❤ ❝÷ï♥❣ ❜ù❝✱ ❜à ❝❤➦♥ ✈➔ f ∈ X ✳ ❑❤✐ ✤â✱ ❜➔✐ t♦→♥ t➻♠


✺

uh ∈ Xh

s❛♦ ❝❤♦✿
a(uh , φh ) = f (φh ), ∀φh ∈ Xh

✭✶✳✷✮

❝â ❞✉② ♥❤➜t ♠ët ♥❣❤✐➺♠✳ ◆➳✉ u ∈ X ❧➔ ♠ët ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ❝õ❛
a(u, φ) = f (φ) ✈ỵ✐ ♠å✐ φ ∈ X,
t❤➻ ❝â ♠ët ❤➡♥❣ sè C ✤ë❝ ❧➟♣ ✈ỵ✐ u, uh ✈➔ h s❛♦ ❝❤♦✿
u − uh

❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â Xh

X

⊂ X✱

≤ C inf

xh ∈Xh

u − xh

tø ✤â ❝â t❤➸ t❤➜②

X

.

✭✶✳✸✮

a : Xh × X h → R

❦➳

t❤ø❛ t➼♥❤ ❜à ❝❤➦♥ ✈➔ ❝→❝ t➼♥❤ t ữù ự tứ s t t tr

X ìX

ợ ❝ị♥❣ ❝→❝ ❤➡♥❣ sè✳ ❉♦ ✤â✱ ù♥❣ ❞ư♥❣ ❜ê ✤➲ r

t t r

uh Xh

tỗ t↕✐✳ ❇➙② ❣✐í t❛ ❧➜②

φ = φh

tr♦♥❣

✭✶✳✶✮ ✈➔ trø ✭✶✳✷✮ ❜ð✐ ✭✶✳✶✮ ❝❤♦ t❛ q✉❛♥ ❤➺ ❝â t➼♥❤ trü❝ ❣✐❛♦ ●❛❧❡r❦✐♥✱

a(u − uh , φh ) = 0, ∀ φh Xh .
ữ ợ t ý

x h Xh

a(u − uh , u − uh ) = a(u − uh , u − xh ) + a(u − uh , xh − uh )
= a(u − uh , u − xh ).
❙û ❞ö♥❣ ❜✐➸✉ t❤ù❝ ♥➔② ✈➔ ❝→❝ t➼♥❤ ❝❤➜t t➼♥❤ ❝÷ï♥❣ ❜ù❝ ✈➔ t➼♥❤ ❜à ❝❤➦♥ ❝õ❛

a(., .)✱

t❛ ❝â✿

α u − uh

2
X

≤ |a(u − uh , u − uh )|
= |a(u − uh , u − xh )|
≤ C u − uh

❉♦ ✤â✱

u − uh

X

◆❤➟♥ ①➨t ✶✳✶✳✾✳

≤

C
α

inf

u − xh

X ợ ồ

u xh

X

.

x h Xh .

ìợ ữủ ữủ ồ ởt ữợ ữủ s số ❣➛♥

♥❤÷ tè✐ ÷✉✱ s❛✐ sè t❤ü❝

xh ∈Xh

u − xh

X

u − uh

X ❜à ❝❤➦♥ ❜ð✐ s❛✐ sè ①➜♣ ①➾ tèt ♥❤➜t

X ởt ữợ ữủ tố ữ õ

C = 1


✻

✶✳✷✳ ❈→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❤➔♠ ❝ì ❜↔♥
❚r♦♥❣ ♣❤➛♥ ♥➔② ❝❤ó♥❣ t❛ tâ♠ ❧÷đ❝ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❤➔♠ ❝ì ❜↔♥ ❧➔♠
❝ì sð ❝❤♦ ✈✐➺❝ tr➻♥❤ ❜➔② ❜✐➸✉ t❤ù❝ ❜✐➳♥ ♣❤➙♥✱ ❝æ♥❣ t❤ù❝ ♥❣❤✐➺♠ ②➳✉ ❝õ❛
♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✈➔ ♣❤÷ì♥❣ ♣❤→♣ ♣❤➛♥ tû ❤ú✉ ❤↕♥✳

✶✳✷✳✶✳ ❈→❝ ❦❤æ♥❣ ❣✐❛♥ ❤➔♠ ❝ê ✤✐➸♥
(a, b) R ợ a < b ú t ỵ ❤✐➺✉ C(a, b) ❧➔ t➟♣ t➜t ❝↔ ❝→❝
k
❤➔♠ ❧✐➯♥ tö❝ f : (a, b) → R. ▼ët ❝→❝❤ tê♥❣ q✉→t✱ C (a, b) ❧➔ t➟♣ t➜t ❝↔
❝→❝ ❤➔♠ sè f : (a, b) → R ❦❤↔ ✈✐ ❧✐➯♥ tö❝ ✤➳♥ ❝➜♣ k ✳
❈❤♦

◆❣❤➽❛ ❧➔✿

C k [a, b] = {f : [a, b] → R|f

k tr➯♥ C[a, b]} .
C 0 (a, b) = C(a, b)✳ Ck0 (a, b) ❧➔ t➟♣

❦❤↔ ✈✐ ❧✐➯♥ tư❝ ✤➳♥ ❝➜♣

✣➸ t❤✉➟♥ t✐➺♥✱ ❝❤ó♥❣ t❛ ❝ơ♥❣ q✉② ữợ
ừ

C k (a, b) ỗ số õ ❣✐→ ❧➔ ♠ët t➟♣ ❝♦♠♣❛❝t tr♦♥❣ ❦❤♦↔♥❣

(a, b)✳

✶✳✷✳✷✳ ▼ët sè ổ ỡ
t ú t ỵ t➟♣ ❤ñ♣

b
2

L [a, b] :=

f (x)2 dx < +∞ ,

f : (a, b) → R|
a

❧➔ t➟♣ t➜t ❝↔ ❝→❝ ❤➔♠ sè

f

✤♦ ✤÷đ❝ s❛♦ ❝❤♦

f2

❦❤↔ t➼❝❤ ▲❡❜❡s❣✉❡ tr➯♥

(a, b)✳
f ✈➔ g tở L2 (a, b) ữủ ồ ỵ ❤✐➺✉
f = g ✱ ♥➳✉ f (x) = g(x) ❤➛✉ ❦❤➢♣ ♥ì✐ tr➯♥ (a, b)✳ ❉➵ t❤➜② r➡♥❣ ❝→❝ ❤➔♠
sè ❧✐➯♥ tư❝ ❤♦➦❝ ❧✐➯♥ tư❝ tø♥❣ ❦❤ó❝ tr➯♥ ✤♦↕♥ [a, b] ❧➔ ❝→❝ ♣❤➛♥ tû t❤✉ë❝
L2 (a, b)✳ ❚➟♣ ❤ñ♣ ũ ợ t tổ tữớ ở ❤➔♠
❍❛✐ ❤➔♠ sè

sè ✈➔ ♥❤➙♥ ♠ët sè ✈ỵ✐ ♠ët ❤➔♠✱ t↕♦ t❤➔♥❤ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì ✈ỉ ❤↕♥ ❝❤✐➲✉
✈➔ ✤÷đ❝ ❣å✐ ❧➔
♥ú❛✱

❦ ❤ỉ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ❜➻♥❤ ♣❤÷ì♥❣ ❦❤↔ t➼❝❤ tr➯♥ (a, b)✳ ❍ì♥

L2 (a, b) ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❍✐❧❜❡rt ợ t ổ ữợ ữủ



b

(f, g)

L2 (a,b)

:=

f (x)g(x)dx.
a

rữợ ❦❤✐ ❣✐ỵ✐ t❤✐➺✉ ♠ët sè ❦❤ỉ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ ❦❤→❝✱ ❝❤ó♥❣ t❛ ❝➛♥ ❜✐➳t


✼
✈➲ ❦❤→✐ ♥✐➺♠ ✤↕♦ ❤➔♠ ②➳✉✳ ❑❤→✐ ♥✐➺♠ ✤↕♦ ❤➔♠ ②➳✉ ❝❤♦ ❤➔♠ sè ♠ët ❜✐➳♥
✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ s❛✉✿

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳

f ∈ L2 (a, b)✱ ❤➔♠ sè g ∈ L2 (a, b)
f tr♦♥❣ ❦❤♦↔♥❣ (a, b) ♥➳✉✿

❈❤♦ ❤➔♠ sè

❣å✐ ❧➔ ♠ët ✤↕♦ ❤➔♠ ②➳✉ ❝õ❛

b

b

f (x)ϕ (x)dx, ∀ϕ ∈ C 1 (a, b).

g(x)(x)dx =

a

a

õ số

g

ữủ ỵ ❤✐➺✉ ❧➔✿

g=f.
f

❚ø ✤à♥❤ ♥❣❤➽❛✱ ❝❤ó♥❣ t❛ t❤➜② r➡♥❣✱ ♥➳✉
tư❝ ✭t❤❡♦ ♥❣❤➽❛ ❝ê ✤✐➸♥✮ tr➯♥

(a, b)

✈➔ ✤↕♦ ❤➔♠ ❝õ❛

❝õ❛ ❤➔♠ sè
◆➳✉

f

f

[a, b] t❤➻ f

❧➔ ❤➔♠ sè ❝â ✤↕♦ ❤➔♠ ❧✐➯♥

❝ô♥❣ ❝â ✤↕♦ ❤➔♠ ②➳✉ tr➯♥ ❦❤♦↔♥❣

✭t❤❡♦ ♥❣❤➽❛ ❝ê ✤✐➸♥✮ ❝ô♥❣ ❝❤➼♥❤ ❧➔ ✤↕♦ ❤➔♠ ②➳✉

tr➯♥ ❦❤♦↔♥❣ ✤â✳

f ∈ C 1 [a, b]

t ợ

C01 [a, b]

b



ữủ

t õ

b

b

f (x) (x)dx =

f (x)ϕ(x)|ba

−

a

f (x)ϕ(x)dx.

f (x)ϕ(x)dx =
a

a

✣↕♦ ❤➔♠ ❝ê ✤✐➸♥

f

❧➔ ✤↕♦ ❤➔♠ ②➳✉ ❝õ❛

f

tr➯♥

(a, b)

❤❛② ✤↕♦ ❤➔♠ ②➳✉

❧➔ ♠ët ♠ð rë♥❣ ❝õ❛ ❦❤→✐ ♥✐➺♠ ✤↕♦ ❤➔♠ ❝ê ✤✐➸♥✳
❚ø ✤â✱ t❛ ❝â ✤à♥❤ ♥❣❤➽❛ ❦❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✷✳

❈❤♦

a, b ∈ R, a < b✳

H 1 (a, b)✿

❑❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈

H 1 (a, b)

❧➔

t➟♣ t➜t ❝↔ ❝→❝ ❤➔♠ ❝â ✤↕♦ ❤➔♠ ②➳✉ ❝ị♥❣ ✈ỵ✐ ❤❛✐ ♣❤➨♣ t♦→♥✿ ❝ë♥❣ ❤❛✐ ❤➔♠
sè ✈➔ ♥❤➙♥ ♠ët sè t❤ü❝ ✈ỵ✐ ♠ët ❤➔♠ sè✳ ❚❛ ❝â✿

H 1 (a, b) = f L2 (a, b)|

ú ỵ
H 1 (a, b)


❝â ✤↕♦ ❤➔♠ ②➳✉ tr➯♥

(a, b)} .

◆❣÷í✐ t❛ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ r➡♥❣ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈

❝ỉ♥❣ t❤ù❝ t➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥ ✈➝♥ ✤ó♥❣✱ tù❝ ❧➔✿

u, v ∈ H 1 (a, b)✱

t❛ ❝â✿

b

b

udv =
a

uv|ba

−

vdu.
a


✽

❈❍×❒◆● ✷

❚✃◆● ◗❯❆◆ ❱➋ P❍×❒◆● ❚❘➐◆❍ ❱■ P❍❹◆
❚❯❨➌◆ ❚➑◆❍ ❈❻P Pì PP
P
ữỡ tr ♠ët sè ❦✐➳♥ t❤ù❝ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥
t➼♥❤ ❝➜♣ ✷✱ ♠ët sè ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t q✉❛♥ trå♥❣ ừ ữỡ
tỷ ỳ ố ợ ữỡ tr ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ✷✳ Ð ❝❤÷ì♥❣ ♥➔②✱
t❛ ❦❤ỉ♥❣ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ t➼♥❤ ❝❤➜t✱ ❦➳t q✉↔✳ ❈→❝ ❝❤ù♥❣ ♠✐♥❤ ❝â t❤➸ t❤❛♠
❦❤↔♦ ð ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪✱ ❬✹❪✱ ❬✶✵❪✱✳✳✳ ✣➙② ❧➔ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥✱ ❝➛♥ t❤✐➳t
✤➸ t✐➳♣ ❝➟♥ ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ✷ ❜➡♥❣

♣❤÷ì♥❣ ♣❤→♣ ♣❤➛♥ tû ❤ú✉ ❤↕♥ sû ❞ư♥❣ ❧➟♣ tr➻♥❤ ▼❆❚▲❆❇ ð ❝❤÷ì♥❣ s❛✉✳

✷✳✶✳ ❚ê♥❣ q✉❛♥ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ✷
✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✶✳

P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ❤❛✐ ❧➔ ♣❤÷ì♥❣

tr➻♥❤ ❝â ❞↕♥❣✿

u + p(x)u + q(x)u = f (x), x ∈ (a, b),

✭✷✳✶✮

tr♦♥❣ ✤â

♣✭①✮✱ q✭①✮✱ ❢✭①✮ ❧➔ ❝→❝ ❤➔♠ ❧✐➯♥ tö❝ trữợ tr [a, b] u

t

u ,u



❧÷đt ❧➔ ✤↕♦ ❤➔♠ ❝➜♣ ✶✱ ❝➜♣ ✷ ❝õ❛

f (x) = 0

ợ ồ

x

u

t ữỡ tr ữủ ồ ữỡ tr ✈✐

♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ✷ t❤✉➛♥ ♥❤➜t✳ ◆❣÷đ❝ ❧↕✐✱ ♣❤÷ì♥❣ tr➻♥❤ ✤÷đ❝ ❣å✐ ❧➔
♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ✷ ❦❤æ♥❣ t❤✉➛♥ ♥❤➜t✳


✾

✷✳✶✳✶✳ ❈➜✉ tró❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤
❝➜♣ ✷
✣à♥❤ ❧➼ ✷✳✶✳✷✳ ❈❤♦ u1(x), u2(x) ❧➔ ❤❛✐ ♥❣❤✐➺♠ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤ tr♦♥❣
(a, b)

❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t ❝➜♣ ✷✿

u + p(x)u + q(x)u = 0.

✭✷✳✷✮

❑❤✐ ✤â✱ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮ ❝â ❞↕♥❣✿
u0 = C1 u1 (x) + C2 u2 (x),

✈ỵ✐ C1, C2 ❧➔ ❝→❝ số tũ ỵ
tờ qt ừ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❦❤ỉ♥❣ t❤✉➛♥
♥❤➜t ✭✷✳✶✮ ❝â ❞↕♥❣✿
utq = u0 + ur ,

tr♦♥❣ ✤â u0 ❧➔ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t ✭✷✳✷✮ ✈➔ ur
❧➔ ♥❣❤✐➺♠ r✐➯♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❦❤ỉ♥❣ t❤✉➛♥ ♥❤➜t ✭✷✳✶✮✳

✷✳✶✳✷✳ P❤÷ì♥❣ ♣❤→♣ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥
t➼♥❤ ❝➜♣ ✷
❈❤ó♥❣ t❛ ❦❤æ♥❣ ❝â ❝→❝❤ ❣✐↔✐ tê♥❣ q✉→t ✤➸ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ✷✳ ❚✉② ♥❤✐➯♥✱ t❛ ❝â t❤➸ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣
tr➻♥❤ ♥❤÷ s❛✉✿
●✐↔ sû

u1 (x)

❧➔ ♥❣❤✐➺♠ r✐➯♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t ✭✷✳✷✮✳

u2 =
−p(x)dx

❚➻♠ ♥❣❤✐➺♠ t❤ù ❤❛✐ ð ❞↕♥❣

❚❤➳ ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮✱

e
dx.
u21 (x)
e−p(x)dx
u2 (x) = u1 (x).
dx.
u21 (x)

t❛ s➩ s✉② r❛ ✤÷đ❝

❱➻ ✈➟②✱

u1 (x).u(x)✳

u=

◆❣❤✐➺♠ r✐➯♥❣ ❝õ❛ ✭✷✳✶✮ ✤÷đ❝ t➻♠ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ t❤✐➯♥ ❤➡♥❣ sè✱
t❛ ❣✐↔ sû ♥❣❤✐➺♠ r✐➯♥❣ ❝â ❞↕♥❣✿

ur = C1 (x)u1 (x) + C2 (x)u2 (x).


✶✵
❙✉② r❛✱ t❛ ❝â✿

ur = C1 (x)u1 (x) + C1 (x)u1 (x) + C2 (x)u2 (x) + C2 (x)u2 (x),
ur = C1 (x)u1 (x) + C1 (x)u1 (x) + C1 (x)u1 (x) + C1 (x)u1 (x)
+ C2 (x)u2 (x) + C2 (x)u2 (x) + C2 (x)u2 (x) + C2 (x)u2 (x).
❚❤❛② ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮✱ t❛ s✉② r❛✿


C u + C u = 0
1 1
2 2
C u + C u = f (x).
2 2
1 1

C1 , C2 ✳

❚❛ ❣✐↔✐ ❤➺ t➻♠

❚ø ✤â s✉② r❛

❑❤✐ ✤â✱ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ✭✷✳✶✮

✣à♥❤ ❧➼ ✷✳✶✳✹

C1 , C2 .
❧➔ ✿ utq = u0 + ur .

ỵ tỗ t t

❝→❝ ❤➔♠ sè

p(x), q(x), f (x) ❧✐➯♥ tö❝ tr➯♥ ❦❤♦↔♥❣ (a, b) t❤➻ ✈ỵ✐ ♠å✐ x0 ∈ (a, b) ✈➔ ợ
ồ tr y0, y0 t ữỡ tr õ ♥❣❤✐➺♠ ❞✉② ♥❤➜t t❤ä❛ ✤✐➲✉ ❦✐➺♥

✤➛✉✿

❱➼ ❞ö ✷✳✶✳✺✳

y(x0 ) = y0 , y (x0 ) = y0 .
●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤

x2 u − xu + u = 4x3 .

●✐↔✐✳

✭✷✳✸✮

1
1
u − u + 2 u = 4x
x
x
1
1
u − u + 2 u = 0.
x
x

❚❛ ❝â ✭✷✳✸✮ ✤÷❛ ✈➲ ❞↕♥❣ ❝❤✉➞♥✿
✈➔ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t✿

❚❛ ❞ü ✤♦→♥ ♠ët ♥❣❤✐➺♠ r✐➯♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t✿

u1 (x) = x.

❙❛✉ ✤➙②✱ t❛ t✐➳♥ ❤➔♥❤ t➻♠ ♥❣❤✐➺♠ r✐➯♥❣ t❤ù ❤❛✐ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥
♥❤➜t✿

u2 (x) = u1 (x).

e−p(x)dx
dx = x.
u21 (x)

−1

e x dx
dx = xln|x|.
x2

❚➻♠ ♥❣❤✐➺♠ r✐➯♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✸✮ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❜✐➳♥ t❤✐➯♥
❤➡♥❣ sè✳
❚r♦♥❣ ❜➔✐ ♥➔②✱

u = x3 .

❱➟② ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ✭✷✳✸✮ ❧➔✿

utq = C1 x + xln|x| + x3 .


✶✶

✷✳✷✳ P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ✷ ❤➺ sè ❤➡♥❣
✣à♥❤ ♥❣❤➽❛ ✷✳✷✳✶✳

P❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ❤❛✐ ❤➺ sè ❤➡♥❣

❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❝â ❞↕♥❣✿

u + pu + qu = f (x),
tr♦♥❣ ✤â
◆➳✉

f (x)

❧➔ ❤➔♠ ❧✐➯♥ tö❝✱

f (x) = 0

ợ ồ

x

p, q



số

t ữỡ tr➻♥❤ ✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐

♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ✷ ❤➺ sè ❤➡♥❣ t❤✉➛♥ ♥❤➜t✳ ◆❣÷đ❝ ❧↕✐✱ ♣❤÷ì♥❣ tr➻♥❤
✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ✷ ❤➺ sè ❤➡♥❣ ❦❤ỉ♥❣ t❤✉➛♥
♥❤➜t✳

✷✳✷✳✶✳ P❤÷ì♥❣ ♣❤→♣ t➻♠ ♥❣❤✐➺♠ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥
t➼♥❤ ❝➜♣ ✷ ❤➺ sè ❤➡♥❣ t❤✉➛♥ ♥❤➜t
❈❤♦ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t ❝â ❞↕♥❣✿

u + pu + qu = 0.

✭✷✳✺✮

✣➸ t➻♠ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t ✭✷✳✺✮✱ t❛ ❧➔♠ ♥❤÷

s❛✉✿
❚❛ ❝â ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ♥❤÷ s❛✉✿

k 2 + pk + q = 0.
❙❛✉ ✤â✱ t❛ t✐➳♥ ❤➔♥❤ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣✳ ❑❤✐ ✤â✱ s➩ ①↔② r❛ ❝→❝
tr÷í♥❣ ❤đ♣ s❛✉✿

✭✶✮

P❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❝â ✷ ♥❣❤✐➺♠ t❤ü❝ ♣❤➙♥ ❜✐➺t✳
◆❣❤✐➺♠ tê♥❣ q✉→t

✭✷✮

P❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❝â ✶ ♥❣❤✐➺♠ ❦➨♣
◆❣❤✐➺♠ tê♥❣ q✉→t

✭✸✮

u0 = C1 ek1 x + C2 ek2 x .
k0 .

u0 = ek0 x (C1 + C2 x).
k1 = a + bi.
u0 = eax (C1 cosbx + C2 sinbx).

P❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❝â ✶ ♥❣❤✐➺♠ ♣❤ù❝
◆❣❤✐➺♠ tê♥❣ q✉→t

✶✷

✷✳✷✳✷✳ P❤÷ì♥❣ ♣❤→♣ t➻♠ ♥❣❤✐➺♠ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥
t➼♥❤ ❝➜♣ ✷ ❤➺ sè ❤➡♥❣ ❦❤ỉ♥❣ t❤✉➛♥ ♥❤➜t
❈❤♦ ♣❤÷ì♥❣ tr➻♥❤ ❦❤æ♥❣ t❤✉➛♥ ♥❤➜t ❝â ❞↕♥❣✿

u + pu + qu = f.

✭✷✳✻✮

✣➸ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❦❤ỉ♥❣ t❤✉➛♥ ♥❤➜t ✭✷✳✻✮✱ t ữ s

ữợ

tờ qt

u0

ừ ữỡ tr t t ữ

tr

ữợ

r ừ ♣❤÷ì♥❣ tr➻♥❤ ❦❤ỉ♥❣ t❤✉➛♥ ♥❤➜t ✭✷✳✻✮✳ ❑❤✐

✤â✱ s➩ ①↔② r❛ ❝→❝ tr÷í♥❣ ❤đ♣ s❛✉✿

✭✶✮ f (x) = Pn(x)eax
❚➻♠

ur

ð ❞↕♥❣

✱

Pn (x)

❧➔ ✤❛ t❤ù❝ ❜➟❝ ♥✳

ur = xs eax Qn (x).

✭✐✮ s❂✵ ♥➳✉

a

❦❤ỉ♥❣ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣✳

✭✐✐✮ s❂✶ ♥➳✉

a

❧➔ ♥❣❤✐➺♠ ✤ì♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣✳

✭✐✐✐✮ s❂✷ ♥➳✉

a

❧➔ ♥❣❤✐➺♠ ❦➨♣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣✳

✭✷✮ f (x) = eax(Pn(x)cosbx + Qm(x)sinbx).
❚➻♠

ur

ð ❞↕♥❣

ur = xs eax (Hk (x)cosbx + Tk (x)sinbx).

✭✐✮ s❂✵ ♥➳✉

a + ib

❦❤ỉ♥❣ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣✳

✭✐✐✮ s❂✶ ♥➳✉

a + ib

❧➔ ♥❣❤✐➺♠ ✤ì♥ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣✳

Hk , Tk

k = max {m, n} .
t❤❛② ur ✈➔♦ ♣❤÷ì♥❣

❧➔ ✤❛ t❤ù❝ ❜➟❝ tè✐ ✤❛ ❧➔

✣➸ t➻♠ ❝→❝ ❤➺ sè ❝õ❛

Hk , Tk ✱

tr➻♥❤ ❦❤ỉ♥❣ t❤✉➛♥

♥❤➜t✿

❱➼ ❞ư ✷✳✷✳✷✳
●✐↔✐✳

u + pur + qur = f (x).
●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤

u − 5u + 6u = e−x .

P❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✿

k 2 − 5k + 6 = 0 ⇔ k1 = 2 ∨ k2 = 3.

✭✷✳✼✮


✶✸

❚❛ ❝â ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t

f (x) = e−x = Pn (x)eax ✱

s✉② r❛

a = −1, Pn (x)

u0 = C1 e2x + C1 e3x .

❜➟❝ ✵✳

❙✉② r❛✿

ur = x0 e−x A = Ae−x ,
ur = −Ae−x ,
ur = Ae−x .
❚❛ s✉② r❛✿

Ae−x + 5Ae−x + 6Ae−x = e−x ⇔ A =

1
.
12

❱➟② ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❧➔✿

utq = u0 + ur = C1 e2x + C1 e3x +

1 −x
e .
12

✷✳✸✳ ❚ê♥❣ q✉❛♥ ✈➲ ♣❤÷ì♥❣ ♣❤→♣ tỷ ỳ

ị tữ ỡ t ừ ữỡ ♣❤→♣ ♣❤➛♥ tû ❤ú✉ ❤↕♥ ❧➔ ❝❤✐❛ ♠✐➲♥

❜➔✐ t♦→♥ t❤➔♥❤ ❝→❝ ♣❤➛♥ tû ❤ú✉ ❤↕♥✱ s❛✉ ✤â t➼♥❤ t♦→♥ ♠❛ tr➟♥ ✈➔ ✈➨❝tì
tr➯♥ ♠é✐ ♣❤➛♥ tû ✈➔ ❧➢♣ ❣❤➨♣ ❝❤ó♥❣ ❧↕✐ ✤➸ ♥❤➟♥ ✤÷đ❝ ♠❛ tr➟♥ t♦➔♥ ❝ư❝

A

✈➔ ✈➨❝tì t♦➔♥ ❝ư❝

F✳

❈✉è✐ ❝ị♥❣ ✤➸ ♥❤➟♥ ✤÷đ❝ ♥❣❤✐➺♠ ①➜♣ ①➾✱ t❛ t➼❝❤

❤đ♣ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ✈➔♦ ♠❛ tr➟♥

A

✈➔ ✈➨❝tì

F

❞➝♥ ✤➳♥ ♠ët ❤➺ ♣❤÷ì♥❣

tr➻♥❤ t✉②➳♥ t➼♥❤✳ ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ♥➔②✱ ❝❤ó♥❣ t❛ t➻♠ ✤÷đ❝
♥❣❤✐➺♠ ①➜♣ ①➾ ❝õ❛ ❜➔✐ t♦→♥✳

✷✳✸✳✶✳ ◆❣❤✐➺♠ ②➳✉ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥
❳➨t ❜➔✐ t♦→♥ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥

− (p(x)u ) + q(x)u = f (x) trong (a, b),
p ∈ C 1 , q, f C[a, b]

trữợ

u



ữ t

ũ ợ ởt tr♦♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿

•

✣✐➲✉ ❦✐➺♥ ❜✐➯♥ ❉✐r❡❝❧❡t✿

•

✣✐➲✉ ❦✐➺♥ ❜✐➯♥ ◆❡✉♠❛♥♥✿

•

✣✐➲✉ ❦✐➺♥ ❜✐➯♥ ❤é♥ ❤đ♣✿

u(b) = k2 ✮✳

u(a) = k1

u(a) = k1

u(a) = k1

u(b) = k2 ✳

✈➔

✈➔

✈➔

u(b) = k2 ✳

u(b) = k2

✭❤♦➦❝

u(a) = k1

✈➔



ừ ữỡ tr ợ

v L2 (a, b)

b

✈➔ ❧➜② t➼❝❤ ♣❤➙♥✿

b

f (x)vdx, ∀v ∈ L2 (a, b).

[−(p(x)u ) + q(x)u]vdx =
a

✭✷✳✾✮

a

❚❛ s✉② r❛ ✤÷đ❝✿

b

b

[pu v + quv]dx =

b

f vdx + pu v

a

, ∀v ∈ H 1 (a, b).

a

a

✭✷✳✶✵✮

❑❤✐ ✤â✱ ✭✷✳✶✵✮ ✤÷đ❝ ❣å✐ ❧➔ ❝ỉ♥❣ t❤ù❝ ♥❣❤✐➺♠ ②➳✉ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✽✮✳
◆❤÷ s➩ t❤➜② ð ♣❤➛♥ t✐➳♣ t❤❡♦ ✈➲ ♣❤÷ì♥❣ ♣❤→♣ ♣❤➛♥ tû ❤ú✉ ❤↕♥✱ ✈✐➺❝
❞ị♥❣ ❝ỉ♥❣ t❤ù❝ ❞↕♥❣ ②➳✉ ✈➔ t➻♠ ♥❣❤✐➺♠ ②➳✉ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ❝â
♥❤✐➲✉ ❧đ✐ t❤➳ ✈÷đt trë✐✳ ❚❤ù ♥❤➜t✱ ♥❣❤✐➺♠ ②➳✉

u

❝❤➾ ②➯✉ ❝➛✉ ❝â ✤↕♦ ❤➔♠

②➳✉ ❝➜♣ ✶✳ ❚❤ù ✷✱ ❝ỉ♥❣ t❤ù❝ ♥❣❤✐➺♠ ②➳✉ ❝ị♥❣ ✈ỵ✐ ✈✐➺❝ ❝❤å♥ ❝→❝ ❤➔♠ ❦✐➸♠
tr❛

V

❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ●❛❧❡r❦✐♥ ❞➝♥ ✤➳♥ ✈✐➺❝ ❣✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥

t➼♥❤ ❞↕♥❣

Ax = b✱

A

tr♦♥❣ ✤â

❧➔ ♠❛ tr➟♥ ✤è✐ ①ù♥❣ ✈➔ ❝❤➾ ❝â ❝→❝ ♣❤➛♥ tû

♥➡♠ tr➯♥ ✤÷í♥❣ ❝❤➨♦ ❝❤➼♥❤ ữớ s s ợ ữớ
ổ ◆❤ú♥❣ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ♥❤÷ t❤➳ ❝â t❤➸ ✤÷đ❝ ❣✐↔✐
❜ð✐ ♥❤✐➲✉ t❤✉➟t t♦→♥ ♥❤❛♥❤ ✈➔ ❤ú✉ ❤✐➺✉✳

✷✳✸✳✷✳ ❑❤ỉ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ♠ô
❳➨t ✤♦↕♥

[a, b]✱

❝❤✐❛

[a, b]

t❤➔♥❤

N −1

✤♦↕♥ ❜ð✐ ❝→❝ ✤✐➸♠✿

a = x1 < x2 < ... < xN = b.
õ ỡ s ổ ụ ỗ ❝→❝ ❤➔♠

ψi , i = 1, N

✤à♥❤ ❜ð✐✿

• ψi (xi ) = 1, ∀i = j,
• ψi (xj ) = 0, ∀i = j,
• ψi |[xk ,xk+1 ] ❧➔ ✤❛ t❤ù❝

❜➟❝

≤ 1✳

❈ö t❤➸✱ t❛ ❝â✿

x −x
 2
, x ∈ [x1 , x2 )
x
−
x
2
1
ψ1 (x) =

0,
x1 ≥ x2 .

✤÷đ❝ ①→❝


✶✺
❱ỵ✐

i = 2, N − 1✱

t❛ ❝â✿



0,





x − xi−1


,

x
−
x
i
i−1
ψi (x) = xi+1 − x


,


x
−
x

i+1
i


0,
❱ỵ✐

i = N✱

x ≤ xi−1
x ∈ [xi−1 , xi )
x ∈ [xi , xi+1 )
x ≥ xi+1 .

t❛ ❝â✿



0,
x ≤ xN −1
ψN (x) =
x − xN −1

, x ∈ [xN −1 , xN ).

xN − xN −1

✷✳✸✳✸✳ Þ t÷ð♥❣ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ♣❤➛♥ tû ❤ú✉ ❤↕♥
❳➨t ❜➔✐ t➟♣ ♠➝✉✿


−(pu ) + qu = f trong (a, b)
u(a) = α, u(b) = β.

✭✷✳✶✶✮

❈ỉ♥❣ t❤ù❝ ♥❣❤✐➺♠ ②➳✉ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✿

b

b

[pu v + quv]dx =
a

b

f vdx + pu v

, v H 1 (a, b).

a

a



ữợ t

ữợ



[a, b]

t

N 1

❝❤✐❛✿

a = x1 < x2 < ... < xN = b.
{ψ1 , ψ2 , ..., ψN }✳
U = u1 ψ1 + u2 ψ2 + ... + uN ψN .

❳→❝ ✤à♥❤ ỡ s ỗ ụ
õ

ữợ

ỡ s ổ tr ữủ ồ ữỡ

r

i =

ữợ

u
= i , i = 1, N .
∂ui

◆❣❤✐➺♠ ①➜♣ ①➾ t❤ä❛✿

b

b

pu ϕi + quϕi dx =
a
❚❤➳

U=

b

f ϕi dx + pu ϕi
a

n
j=1 uj ϕj ✈➔♦ ✭✷✳✶✸✮ t❛ ✤÷đ❝✿

a

, ∀i = 1, N .

✭✷✳✶✸✮


✶✻
n

b

uj
j=1

b

pϕj ϕi + qϕj ϕi dx =
a

b

f ϕi dx + pu ϕi
a

, ∀i = 1, N .

a

❚❛ ✤➦t ♠❛ tr➟♥ ❆✱ ✈➨❝tì ❝ët ❋ ✈➔ ❇ ♥❤÷ s❛✉✿

b

Aij =

pϕj ϕi + qϕj ϕi dx,
a
b

Fi =

✭✷✳✶✹✮

f ϕi dx,
a

b

Bi = pu ϕi .
a

❑❤✐ ✤â✱ ❤➺ ✭✷✳✶✸✮ t÷ì♥❣ ✤÷ì♥❣✿




u1
 
 u2 

A
 ... = F + B.

uN

ú ỵ B1 = pu ϕ1 a = p(b)u (b)ϕ1(b) − p(a)u (a)ϕ1(a) = −p(a)u (a).
b

i = 2, N − 1✱ t❛ ❝â✿
Bi = 0 ✈➔ BN = p(b)u (b).


−p(a)u (a)




0



 ♥➯♥ ❤➺ ✭✷✳✶✸✮ t÷ì♥❣ ữỡ ợ
B =
...




0


p(b)u (b)



F1 p(a)u (a)
u1


F2

 

 u2  
.
=
A

...

 ...  

  
FN 1


uN
FN + p(b)u (b)
ợ

ữợ



ỷ

❝â

N +2

➞♥ ♥❤÷♥❣ ❝❤➾ ❝â

✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ✤➸ ❣✐↔✐ ❤➺ tr➯♥✳

N

♣❤÷ì♥❣ tr➻♥❤✱ ✈➟② t❛ s➩ ❦➳t ❤đ♣

✶✼

u(a) = α
u(b) = β

❚ø ✤✐➲✉ ❦✐➺♥ ❜✐➯♥

u1 = α
.
uN = β

✱ s✉② r❛

❑➳t ❤ñ♣ ❤➺ ✭✷✳✶✺✮ ✈➔ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥✱ t❛ t➻♠ ✤÷đ❝

u1 , u2 , ..., uN .

❚❛ t❤÷í♥❣ ❦➳t ❤đ♣ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ✈➔♦ ❤➺ ✭✷✳✶✺✮ ♥❤÷ s❛✉✿



1

...

...

0

✣à♥❤ ♥❣❤➽❛ ✷✳✸✳✶✳




α
0
u1


    F2 
  

...
  u2  =  ...  .
  

...

  ...  
FN −1 
1
uN
β


...
...
...
...

▼❛ tr➟♥

A✱



✈➨❝tì

F

✈➔ ✈➨❝tì

B

tr♦♥❣ ✭✷✳✶✹✮ ❧➛♥ ❧÷đt

✤÷đ❝ ❣å✐ ❧➔ ♠❛ tr➟♥ t♦➔♥ ❝ư❝✱ ✈➨❝tì t♦➔♥ ❝ư❝ ✈➔ ✈➨❝tì ✤✐➲✉ ❦✐➺♥ ❜✐➯♥✳
❚ø ✈✐➺❝ t➼♥❤ ✤÷đ❝ ♠❛ tr➟♥

A✱ ✈➨❝tì F

✈➔ ✈➨❝tì

B ✱ t❛ ❝â t❤➸ t➻♠ ♥❣❤✐➺♠

❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t✉②➳♥ t➼♥❤ ❝➜♣ ✷ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ♣❤➛♥ tû ❤ú✉
❤↕♥✳ ❚✉② ♥❤✐➯♥✱ ✈✐➺❝ t➼♥❤ ♠❛ tr➟♥ t♦➔♥ ❝ư❝

A

✈➔ ✈➨❝tì t♦➔♥ ❝ư❝

F

t❤ỉ♥❣

q✉❛ ❦❤ỉ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ ♠ơ ❧➔ rt t tớ ỗ t
số ❝❤✐❛ ❧ỵ♥ ✈➔ ❞➵ s❛✐ sât✳ ❱➻ ✈➟②✱ ✤➸ ❦❤➢❝ ử ữủ ữớ
t ợ t tr➟♥ ♣❤➛♥ tû ✈➔ ✈➨❝tì ♣❤➛♥ tû✳

✷✳✸✳✹✳ ▼❛ tr➟♥ ♣❤➛♥ tû ✈➔ ✈➨❝tì ♣❤➛♥ tû
❳➨t ✷ t➼❝❤ ♣❤➙♥✿

xi+1

(pu v + quv)dx,
xi
xi+1

f vdx.
xi
❚r➯♥ ✤♦↕♥

xi , xi+1

t❛ ❝â✿

u = ui ψi + ui+1 ψi+1 = ui H1 + ui+1 H2 .
❚r♦♥❣ ✤â✿

H1 = ϕi

xi ,xi+1

H2 = ϕi+1

xi+1 − x
,
hi
x − xi
⇒ H2 (x) =
,
hi

⇒ H1 (x) =

xi ,xi+1


✶✽

❉♦ ✤â✱
❚❤➳

u

ui
.
ui+1

U = H1 H2
❜ð✐

U

✈➔ ❝❤♦

v

❧➛♥ ❧÷đt ❜➡♥❣

xi+1
xi

H1 , H2

t❛ ❝â✿

(p H1 H2

ui
H1 + q H1 H2
ui+1

ui
H1 )dx
ui+1

(p H1 H2

ui
H2 + q H1 H2
ui+1

ui
H2 )dx.
ui+1

✈➔

xi+1
xi

●ë♣ ❧↕✐ t❛ s➩ ❝â✿

xi+1

i

A =

(p

H1
H2

f

H1

dx.
H2

xi
xi+1

Fi =

xi
i

A
Fi

H1 H2 + q

✤÷đ❝ ❣å✐ ❧➔ ♠❛ tr➟♥ ♣❤➛♥ tû t❤ù
✤÷đ❝ ❣å✐ ❧➔ ✈➨❝tì ♣❤➛♥ tû t❤ù

◆❤➟♥ ①➨t ✷✳✸✳✷✳
t♦➔♥ ❝ư❝

A✱

H1
H2

H1 H2 )dx,

i✱

i✳

✧❈ë♥❣✧ t➜t ❝↔ ❝→❝ ♠❛ tr➟♥ ♣❤➛♥ tû t❛ ✤÷đ❝ ♠❛ tr➟♥

✧❝ë♥❣✧ t➜t ❝↔ ❝→❝ ✈➨❝tì ♣❤➛♥ tû t❛ ✤÷đ❝ ✈➨❝tì t♦➔♥ ❝ư❝

F✳

✣➸ ❤✐➸✉ rã ❝→❝❤ t➼♥❤ ♠❛ tr➟♥ ♣❤➛♥ tû ✈➔ ✈➨❝tì ♣❤➛♥ tû✱ t❛ ①➨t ❝→❝ ✈➼
❞ư s❛✉ ✤➙②✿

❱➼ ❞ö ✷✳✸✳✸✳

❚➻♠ ♥❣❤✐➺♣ ①➜♣ ①➾ ❝õ❛ ❜➔✐ t♦→♥ s❛✉✿


−xu = 1 trong (1, 3)
u(1) = 0, u (3) = 1.

●✐↔✐✳

❈ỉ♥❣ t❤ù❝ ♥❣❤✐➺♠ ②➳✉ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✿

3

3

1

3

vdx + xvu , ∀v ∈ H1 (1, 3).

u (v + xv )dx =
1

1

❚❛ s✉② r❛ ✤÷đ❝ ♠❛ tr➟♥ ♣❤➛♥ tû✿

Ai =

xi+1
xi

H1
H1
+x
H2
H2

H1 H2

dx,


✶✾
✈➔ ✈➨❝tì ♣❤➛♥ tû✿

xi+1

i

F =
xi

H1
hi /2
dx =
.
H2
hi /2

❙✉② r❛✿

xi+1

i

H1
H2

A =
xi
xi+1

=

dx

xi

=

−1/hi 1/hi + x

H1
H2

−1/hi
−1/hi
xi+1

−1/hi 1/hi +

xdx
xi

−1/hi 1/hi

1/h2i −1/h2i
−1/h2i 1/h2i

xi + x1+1 1/hi −1/hi
.
2
−1/hi 1/hi





1 −1 0
0 0 0




⇒ A1 = −2 2 0 , A2 = 0 2 −2 .
0 0 0
0 −3 3


1 −1 0


⇒ A = A1 + A2 = −2 4 −2 .
0 −3 3

hi /2
hi /2

−1/hi 1/hi +

ữỡ tỹ ố ợ

F

t õ

1/2
0




F 1 = 1/2 , F 2 = 1/2 .
0
1/2


1/2


⇒ F = F1 + F2 =  1 .
1/2
❚❛ t✐➳♥ ❤➔♥❤ ①û ❧➼ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥✿

3

dx

Bi = xu ϕi = 3u (3)ϕi (3) − u (1)ϕi (1).
1



−u (1)


⇒ B =  0 .
3u (3)


✷✵



❚ø

u(1) = 0
⇒ u1 = 0
u (3) = 1

❚➼❝❤ ❤ñ♣ ✤✐➲✉ ❦✐➺♥

✈➔

u1 = 0✱


u (1)


B =  0 .
3

t❛ ✤÷đ❝✿



⇒

❱➟② ♥❣❤✐➺♠ ①➜♣ ①➾

❱➼ ❞ö ✷✳✸✳✹✳

  

−1 −1 0
u1
0

  

4 −2 u2  =  1  .
 −2
0−3 3
u3
7/2



u =0

 1

u2 = 5/3



u = 17/6.
3


0


u =  5/3  .
17/6

❚➻♠ ♥❣❤✐➺♠ ①➜♣ ①➾ ❝õ❛ ❜➔✐ t♦→♥✿


−u = 1 trong (1, 4)
u (1) = −1, u(4) = 0.

●✐↔✐✳

❈æ♥❣ t❤ù❝ ♥❣❤✐➺♠ ②➳✉ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✿

4

4

vdx + u v , ∀v ∈ H 1 (1, 4).

u v dx =
1

4
1

1

❚❛ s✉② r❛ ✤÷đ❝ ♠❛ tr➟♥ ♣❤➛♥ tû✿

xi+1

i

A =
xi
xi+1

=
xi

=
❱➔ ✈➨❝tì ♣❤➛♥ tû✿

H1
H2

H1 H2

−1/hi

1/hi

1/hi −1/hi
,
−1/hi 1/hi

dx

−1/hi 1/hi

dx


✷✶

xi+1

i

F =
xi

H1
dx
H2


1
 hi 
=  21  .

hi
2


[1, 4] t❤➔♥❤ ✹ ✤♦↕♥✿



1 −1 0 0
0 0 0



−1 1 0 0

 , A2 = 0 1 −1
A1 = 
0
0 −1 1
0 0 0



0 0 0 0
0 0 0


1 −1 0 0



−1 2 −1 0 

⇒A=
 0 −1 2 −1 .


0 0 −1 1






1/2
0
0






1/2 2 1/2 3  0 
1





F =

 0  , F = 1/2 , F = 1/2 .






0
0
1/2


1/2


1/2

⇒F =
1/2 .


1/2

❚❛ ❝❤✐❛



0
0




0
 , A3 = 0
0
0


0
0

❙❛✉ ✤â t✐➳♥ ❤➔♥❤ ①û ❧➼ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥✿

4

Bi = u ϕi

1

 
1
 
0

⇒B=
0 .

0

ủ ợ t ữủ

Au = F + B.


0 0 0

0 0 0
.
0 1 −1

0 −1 1