Về phương pháp lồi Lôgarit và một vài ứng dụng

ĐẠI HỌC THÁI NGUYÊN
TRƢỜNG ĐẠI HỌC KHOA HỌC
---------------------------

PHẠM LỆ QUYÊN

VỀ PHƢƠNG PHÁP LỒI LÔGARIT
VÀ MỘT VÀI ỨNG DỤNG

LUẬN VĂN THẠC SĨ TOÁN HỌC

THÁI NGUYÊN - 2019


ĐẠI HỌC THÁI NGUYÊN
TRƢỜNG ĐẠI HỌC KHOA HỌC
---------------------------

PHẠM LỆ QUYÊN

VỀ PHƢƠNG PHÁP LỒI LƠGARIT
VÀ MỘT VÀI ỨNG DỤNG
Chun ngành: Tốn Ứng Dụng
Mã số: 8 46 01 12

LUẬN VĂN THẠC SĨ TOÁN HỌC

NGƯỜI HƯỚNG DẪN KHOA HỌC
TS. Bùi Việt Hƣơng

THÁI NGUYÊN - 2019

ử ử








ỗ ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
ỗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺
✶✳✷✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝ì sð ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽
✶✳✷✳✶✳ P❤➙♥ ❧♦↕✐ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ❝➜♣ ❤❛✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾
✶✳✷✳✷✳ ▼➦t ✤➦❝ tr÷♥❣✳ ❇➔✐ t♦→♥ ợ ỳ tr t
trữ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶
✶✳✷✳✸✳ ❙ü ♣❤ö t❤✉ë❝ ❧✐➯♥ tö❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
Pữỡ ỗ ổrt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹

✷ ▼❐❚ ❱⑨■ Ù◆● ❉Ư◆● ❈Õ❆ P❍×❒◆● P❍⑩P ▲➬■ ▲➷●❆❘■❚

✷✳✶✳ Ù♥❣ ❞ư♥❣ tr♦♥❣ ❜➔✐ t♦→♥ ❈❛✉❝❤② ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ♣❛r❛❜♦❧✐❝ ♥❣÷đ❝ t❤í✐
❣✐❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✶✳✶✳ P❤÷ì♥❣ tr➻♥❤ ♣❛r❛❜♦❧✐❝ ♥❣÷đ❝ t❤í✐ ❣✐❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✶✳✷✳ ✣→♥❤ ❣✐→ ê♥ ✤à♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✷✳ Ù♥❣ ❞ư♥❣ tr♦♥❣ ❜➔✐ t♦→♥ ❈❛✉❝❤② ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✷✳✶✳ P❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✷✳✷✳ ✣→♥❤ ❣✐→ ê♥ ✤à♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✷✵

✷✵
✷✵
✷✹
✷✽
✷✽
✷✾
✹✵


▼Ð ✣❺❯
❇➔✐ t♦→♥ ✤➦t ❦❤æ♥❣ ❝❤➾♥❤ ①✉➜t ❤✐➺♥ tr♦♥❣ ♥❤✐➲✉ ❧➽♥❤ ✈ü❝ ù♥❣ ❞ö♥❣✳ ❇➔✐ t♦→♥
♥➔② ❝â ❧✐➯♥ q✉❛♥ ✤➳♥ t ỵ t ỵ s t ❧➽♥❤ ✈ü❝ ✤✐➺♥ s✐♥❤
❤å❝✳✳✳ ❚r♦♥❣ ♠ët ❜➔✐ ❜→♦ ♥ê✐ t✐➳♥❣ ừ r t t
ữủ ợ t ♥❤÷ ❧➔ ♠ët ✈➼ ❞ư ❦✐♥❤ ✤✐➸♥ ✈➲ ❜➔✐ t♦→♥ ✤➦t ❦❤æ♥❣ ❝❤➾♥❤✳ ✣➦❝
✤✐➸♠ ♥ê✐ ❜➟t ❝õ❛ ❜➔✐ t♦→♥ ♥➔② ❧➔ ♠ët t❤❛② ✤ê✐ ♥❤ä tr♦♥❣ ❞ú ❦✐➺♥ ❝ô♥❣ ❝â t❤➸
❞➝♥ ✤➳♥ ♠ët s❛✐ ❧➺❝❤ ❧ỵ♥ ✈➲ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥✳ ❍❛❞❛♠❛r❞ ❝❤♦ r➡♥❣ ❝→❝ ❜➔✐
t♦→♥ ✤➦t ❦❤æ♥❣ ❝❤➾♥❤ ❦❤æ♥❣ õ ỵ t ❝ù✉ ❝→❝
❜➔✐ t♦→♥ ✤➦t ❦❤æ♥❣ ❝❤➾♥❤ ✤➸ t➻♠ r❛ ❝→❝ ✤→♥❤ ❣✐→ ê♥ ✤à♥❤ ✈➔ ❝→❝ ♣❤÷ì♥❣ ♣❤→♣
❝❤➾♥❤ ❤â❛ ❧➔ ởt q trồ
Pữỡ ỗ ổrt ởt tr ♥❤ú♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❞ị♥❣ ✤➸ ê♥ ✤à♥❤ ❤â❛
❝→❝ ❜➔✐ t♦→♥ ✤➦t ❦❤ỉ♥❣ ❝❤➾♥❤ tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣✳ P❤÷ì♥❣ ♣❤→♣
♥➔② ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ❜ð✐ P✉❝❝✐ ✭✶✾✺✺✮✱ ❏♦❤♥ ✭✶✾✺✺✱ ✶✾✻✵✮✱ ▲❛✈r❡♥t✐❡✈ ✭✶✾✺✻✮ ❛♥❞
P❛②♥❡ ✭✶✾✻✵✮✱ ✣✐♥❤ ◆❤♦ ❍➔♦ ✈➔ ◆❣✉②➵♥ ❱➠♥ ✣ù❝ ✭✷✵✵✾✱ ✷✵✶✵✱ ✷✵✶✶✮✳ ✣➙② ❧➔ ❦➽
t❤✉➟t ✤→♥❤ ❣✐→ ❞ü❛ tr➯♥ ❝→❝ ❜➜t ✤➥♥❣ t❤ù❝ ❜➟❝ ❤❛✐ ✈➲ ✤↕♦ ❤➔♠ ữ r ợ
tr ợ ữợ ởt ỗ ổrt ởt ừ

✤→♥❤ ❣✐→ ✤â ✤÷đ❝ ❞ị♥❣ ✤➸ t❤✐➳t ❧➟♣ t➼♥❤ ❞✉② ♥❤➜t ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✈➔
t❛ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ sü ♣❤ư t❤✉ë❝ ❧✐➯♥ tư❝ ❝õ❛ ♥❣❤✐➺♠ ✈➔♦ ❞ú ❦✐➺♥ ✤➣ ❝❤♦
t❤❡♦ ♠ët ♥❣❤➽❛ ♥➔♦ ✤â✳
▲✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ữỡ ỗ ổrt ởt số ự ử ❝õ❛
♣❤÷ì♥❣ ♣❤→♣ ✤➸ ê♥ ✤à♥❤ ❤â❛ ❜➔✐ t♦→♥ ✤➦t ❦❤ỉ♥❣ ❝❤➾♥❤ tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦
❤➔♠ r✐➯♥❣✳ ❈ư t❤➸✱ ❧✉➟♥ ✈➠♥ ỗ ữỡ ữỡ t tr
ỗ ởt tự ỡ ừ ữỡ tr r ữỡ
ỗ ổrt ữỡ t ❣✐↔ tr➻♥❤ ❜➔② ❤❛✐ ❜➔✐ t♦→♥ ♠✐♥❤ ❤å❛ ❝❤♦ ♣❤÷ì♥❣

✶


♣❤→♣ ♥➔②✱ ✤â ❧➔ ❜➔✐ t♦→♥ ❈❛✉❝❤② ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ♣❛r❛❜♦❧✐❝ ♥❣÷đ❝ t❤í✐ ❣✐❛♥ ✈➔
❜➔✐ t♦→♥ ❈❛✉❝❤② ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ▲❛♣❧❛❝❡✳ ✣➙② ❧➔ ❝→❝ ❜➔✐ t♦→♥ ✤➦t ❦❤æ♥❣ ❝❤➾♥❤
✈➔ t→❝ sỷ ử ữỡ ỗ ổrt ữ r❛ ✤→♥❤ ❣✐→ ê♥ ✤à♥❤ ❝❤♦
♥❣❤✐➺♠ ❝õ❛ ❝→❝ ❜➔✐ t♦→♥ ợ ữủ ờ s P ố ữỡ ✷✱
t→❝ ❣✐↔ ❝â tr➻♥❤ ❜➔② t❤➯♠ ♠ët ❜➔✐ t♦→♥ ❝â t❤➸ ①❡♠ ♥❤÷ ♠ð rë♥❣ ❝õ❛ ❜➔✐ t♦→♥
❈❛✉❝❤② ❝❤♦ ♣❤÷ì♥❣ tr
ữủ t ữợ sỹ ữợ ừ ũ t ữỡ ổ
t t ữợ ❝❤➾ ❜↔♦ ❡♠ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉✳
❊♠ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ ❈ỉ✳
❊♠ ❝ơ♥❣ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ tr➙♥ t tợ ổ
trữớ ❤å❝ ❑❤♦❛ ❤å❝✱ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ t➟♥ t➻♥❤ ❣✐↔♥❣ ❞↕② ✈➔
t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ tr♦♥❣ q✉→ tr➻♥❤ ❡♠ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ t↕✐ tr÷í♥❣✳
❊♠ ①✐♥ tr➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❚❙✳ ▼❛✐ ❱✐➳t ❚❤✉➟♥ ✈➔ ❚❙✳ ❚r÷ì♥❣ ▼✐♥❤ ❚✉②➯♥ ✤➣
❞➔♥❤ sü q✉❛♥ t➙♠ ✈➔ ❝â ♥❤ú♥❣ ❧í✐ ✤ë♥❣ ✈✐➯♥ ❦à♣ t❤í✐ ✤➸ ❡♠ ❝è ❣➢♥❣ ❤♦➔♥ t❤➔♥❤
❧✉➟♥ ✈➠♥ ♥➔②✳
❈✉è✐ ❝ị♥❣ ❡♠ ①✐♥ ❝↔♠ ì♥ ❣✐❛ ✤➻♥❤✱ ỗ ổ ở
t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ ❡♠ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥✳

✷


ữỡ


ỗ ỗ
ử tr➻♥❤ ❜➔② ♠ët sè ❦❤→✐ ♥✐➺♠✱ ✤à♥❤ ♥❣❤➽❛ ✈➔ ❦➳t q tt
q ỗ t ỗ ◆ë✐ ❞✉♥❣ ❝õ❛ ♠ư❝ ✤÷đ❝ t❤❛♠ ❦❤↔♦ tø ❬✷❪✳

✶✳✶✳✶✳ ❚➟♣ ỗ
a, b Rn
✣÷í♥❣ t❤➥♥❣ ✤✐ q✉❛ ❤❛✐ ✤✐➸♠ a ✈➔ b ❧➔ t➟♣ ❤ñ♣ ❝â ❞↕♥❣

{x ∈ Rn |x = λa + (1 − λ)b, λ ∈ R}.
✐✐✮ ✣♦↕♥ t❤➥♥❣ ✤✐ q✉❛ ❤❛✐ ✤✐➸♠ a ✈➔ b ❧➔ t➟♣ ❤ñ♣ ❝â ❞↕♥❣

{x ∈ Rn |x = λa + (1 − λ)b, λ ∈ [0, 1]}.

✣à♥❤ ♥❣❤➽❛ ✶✳✷ ❚➟♣ C ⊂ Rn ✤÷đ❝ ồ t ỗ C ự ồ t
ố ❤❛✐ ✤✐➸♠ ❜➜t ❦ý ❝õ❛ ♥â✱ tù❝ ❧➔

∀x, y ∈ C, ∀λ ∈ [0, 1], t❛ ❝â λx + (1 − λ)y ∈ C.

✣à♥❤ ♥❣❤➽❛ ✶✳✸ ✐✮ ❚❛ ♥â✐ x tờ ủ ỗ ừ tỡ x1, x2, · · · , xk
♥➳✉

k

k

λj x ✈ỵ✐ λj > 0, ∀j = 1, 2, · · · , k ✈➔
j

x=
j=1

λj = 1.
j=1

✸


✐✐✮ ❚❛ ♥â✐ x ❧➔ tê ❤ñ♣ ❛❢❢✐♥❡ ❝õ❛ ❝→❝ ✤✐➸♠ ✭✈❡❝tì✮ x1 , x2 , · · · , xk ♥➳✉
k

k

λj x ✈ỵ✐
j

x=
j=1

λj = 1.
j=1

▼➺♥❤ ✤➲ ✶✳✶ ❚➟♣ ❤đ♣ C ỗ õ ự ồ tờ ủ ỗ ừ
ừ õ tự ợ ♠å✐ k ∈ N✱ ✈ỵ✐ ♠å✐ λ1 , λ2 , · · · , λk > 0 s❛♦ ❝❤♦

k

λj = 1
j=1

✈➔ ✈ỵ✐ ♠å✐ x1 , x2 , · · · , xk ∈ C t❛ ❝â
k

λj xj ∈ C.
j=1

✣à♥❤ ♥❣❤➽❛ ởt t C ữủ ồ õ ợ ♠å✐ λ > 0✱ ✈ỵ✐ ♠å✐ x ∈ C
t❛ ❝â λx ∈ C ✳
✐✮ ▼ët ♥â♥ ✤÷đ❝ ❣å✐ ❧➔ ♥â♥ ỗ õ t ỗ
ởt õ ỗ ữủ ❣å✐ ❧➔ ♥â♥ ♥❤å♥ ♥➳✉ ♥â ❦❤ỉ♥❣ ❝❤ù❛ ✤÷í♥❣ t❤➥♥❣✱ ❦❤✐ ✤â
t❛ ♥â✐ 0 ❧➔ ✤➾♥❤ ❝õ❛ ♥â♥✳ ◆➳✉ ♥â♥ ởt t ỗ t t õ õ
õ ỗ

C Rn ởt t ỗ x C
❚➟♣

NC (x) = {w : w, y − x ≤ 0, ∀y ∈ C},
✤÷đ❝ ❣å✐ ❧➔ ♥â♥ ♣❤→♣ t✉②➳♥ ✭♥❣♦➔✐✮ ❝õ❛ C t↕✐ x✳
✐✐✮ ❚➟♣

−NC (x) = {w : w, y − x ≥ 0, ∀y ∈ C},
✤÷đ❝ ❣å✐ ❧➔ õ t tr ừ C t x

ỵ ỵ t t t ỗ ồ t ỗ õ
rộ ổ trũ ợ t ở ổ ❣✐❛♥ ✤➲✉ ❧➔ ❣✐❛♦ ❝õ❛ t➜t ❝↔ ❝→❝ ♥û❛ ❦❤æ♥❣

❣✐❛♥ tü❛ ❝õ❛ ♥â✳

✹


✣à♥❤ ♥❣❤➽❛ ✶✳✻ ❈❤♦ ❤❛✐ t➟♣ C ✈➔ D ❦❤→❝ ré♥❣✱ t❛ ♥â✐ s✐➯✉ ♣❤➥♥❣ aT x = α
t→❝❤ C ✈➔ D ♥➳✉

aT x ≤ α ≤ aT y, ∀a ∈ C, y ∈ D.
❚❛ ♥â✐ s✐➯✉ ♣❤➥♥❣ aT x = α t→❝❤ ❝❤➦t C ✈➔ D ♥➳✉

aT x < α < aT y, ∀a ∈ C, y ∈ D.
❚❛ ♥â✐ s✐➯✉ ♣❤➥♥❣ aT x = α t→❝❤ ♠↕♥❤ C ✈➔ D ♥➳✉

sup aT x < α < inf aT y, a C, y D.
xC

yD

ỵ ỵ t C D t ỗ rộ tr Rn
s C D = ∅✳ ❑❤✐ ✤â ❝â ♠ët s✐➯✉ ♣❤➥♥❣ t→❝❤ C ✈➔ D

ỵ ỵ t C

D t ỗ õ rộ

tr Rn s C ∩ D = ∅✳ ●✐↔ sû ➼t ♥❤➜t ♠ët tr♦♥❣ ❤❛✐ t➟♣ ❧➔ t➟♣ ❝♦♠♣❛❝t✳
❑❤✐ ✤â✱ ❤❛✐ t➟♣ ♥➔② ❝â t❤➸ t→❝❤ ♠↕♥❤ ✤÷đ❝ ❜ð✐ ♠ët s✐➯✉ ♣❤➥♥❣✳

✶✳✶✳✷✳ ❍➔♠ ỗ

C Rn t ỗ f : C → R✳ ❚❛ ❦➼ ❤✐➺✉
❞♦♠f = {x ∈ C : f (x) < +∞},
❡♣✐f = {(x, α) ∈ C × R : f (x) ≤ α}.

✣à♥❤ ♥❣❤➽❛ ✶✳✼ ❚➟♣ ❞♦♠f ✤÷đ❝ ❣å✐ ❧➔ ♠✐➲♥ ❤ú✉ ❤✐➺✉ ❝õ❛ f ✳ f ữủ
ồ tr ỗ t ừ f
❝→❝❤ ✤➦t f (x) = +∞ ♥➳✉ x ∈
/ C ✱ t❛ ❝â t❤➸ ❝♦✐ f ①→❝ ✤à♥❤ tr➯♥ t♦➔♥
❦❤æ♥❣ ❣✐❛♥✳ ❑❤✐ ✤â✱ t❛ ❝â
❞♦♠f = {x ∈ Rn : f (x) ≤ +∞},
❡♣✐f = {(x, α) ∈ Rn × R : f (x) ≤ α}.

✺


✣à♥❤ ♥❣❤➽❛ ✶✳✽ ❈❤♦ C ⊂ Rn✱ C = ∅ t ỗ f : C [, +] õ
f ỗ tr C f t ỗ tr Rn+1
tr tữỡ ữỡ ợ ∀x, y ∈ C, ∀λ ∈ (0, 1) t❛ ❝â

f [λx + (1 − λ)y] ≤ λf (x) + (1 − λ)f (y).

◆❤➟♥ ①➨t ✶✳✶ ❱➲ ♠➦t ❤➻♥❤ ❤å❝✱ ✤÷í♥❣ ởt ỗ tọ
t ❝❤➜t s❛✉
✐✮ ❦❤æ♥❣ ♥➡♠ tr➯♥ ✤♦↕♥ t❤➥♥❣ ♥è✐ ❜➜t ❦ý tở ữớ
ổ ữợ t t✉②➳♥ t↕✐ ❜➜t ❦ý ✤✐➸♠ ♥➔♦ t❤✉ë❝ ✤÷í♥❣ ❝♦♥❣✳

❱➲ ♠➦t t t tr õ t ữợ ❞↕♥❣ ❜➜t ✤➥♥❣ t❤ù❝ s❛✉

f (a) + f (a)(x − a) ≤ f (x) ≤ f (a) +

f (b) − f (a)
(x − a).
b−a

✭✶✳✶✮

✣à♥❤ ♥❣❤➽❛ ✶✳✾ ❈❤♦ C ⊂ Rn✱ C = t ỗ
f : Rn [, +] ữủ ồ ỗ t tr C ♥➳✉

f [λx + (1 − λ)y] < λf (x) + (1 − λ)f (y),

∀x, y ∈ C, ∀λ ∈ (0, 1).

✐✐✮ ❍➔♠ f : Rn → [−∞, +∞] ✤÷đ❝ ❣å✐ ỗ tr C ợ số > 0 ♥➳✉
✈ỵ✐ ♠å✐ x, y ∈ C, ✈ỵ✐ ♠å✐ λ ∈ (0, 1)

1
f [λx + (1 − λ)y] ≤ λf (x) + (1 − λ)f (y) − ηλ(1 − λ) x − y 2 .
2

✻


✐✐✐✮ ❍➔♠ f : Rn → [−∞, +∞] ✤÷đ❝ ❣å✐ ó tr C f ỗ
tr C ✳

▼➺♥❤ ✤➲ ✶✳✷ ▼ët ❤➔♠ f : C → R ỗ tr C ✈ỵ✐ ♠å✐
x, y ∈ C ✱ ✈ỵ✐ ♠å✐ α, β t❤ä❛ ♠➣♥ f (x) < α, f (y) < β ✱ ✈ỵ✐ ♠å✐ sè λ ∈ [0, 1] t❛ ❝â
f [λx + (1 − λ)y] ≤ λα + (1 − ).

ử ởt số ử ỗ
||x|| ởt ỗ tr Rn tr♦♥❣ ✤â x ∈ Rn ✳
✐✐✮ ❈❤♦ C ⊂ Rn t ỗ rộ ừ C ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛


0
♥➳✉ x ∈ C
δC (x) :=

+∞ ♥➳✉ x
/C
ởt ỗ
C Rn t ỗ rộ tỹ ừ C ữủ

SC (x) := sup y, x
yC

ởt ỗ
C Rn t ỗ rộ ❝→❝❤ ✤➳♥ t➟♣ C ✱ ✤÷đ❝ ✤à♥❤
♥❣❤➽❛

dC (x) := min x y
yC

ởt ỗ

f

ữủ ❣å✐ ❧➔ ❤➔♠ ❝❤➼♥❤ t❤÷í♥❣ ♥➳✉ ❞♦♠f = ∅ ✈➔

f (x) > ợ ồ x

f

ữủ ❣å✐ ❧➔ ❤➔♠ ✤â♥❣ ♥➳✉ ❡♣✐f ❧➔ t➟♣ ✤â♥❣ tr♦♥❣

❦❤æ♥❣ ❣✐❛♥ Rn+1 ✳

✼


t f ởt ỗ t f t ỗ
f ữủ ❣å✐ ❧➔ t❤✉➛♥ ♥❤➜t ❞÷ì♥❣ ✭❜➟❝ ✶✮ tr➯♥ Rn ♥➳✉
f (λx) = λf (x), ∀x ∈ Rn , ∀λ > 0.

▼➺♥❤ ✤➲ ✶✳✸ ❈❤♦ f

❧➔ ❤➔♠ t❤✉➛♥ ♥❤➜t ❞÷ì♥❣ tr➯♥ Rn õ f ỗ

❦❤✐

f (x + y) ≤ f (x) + f (y), ∀x, y ∈ Rn .

▼➺♥❤ ✤➲ ✶✳✹ ◆➳✉ f1, f2 ỗ tữớ t f1 + f2 ụ
ỗ

q ờ qt f1, f2, à à à , fm ỗ t❤÷í♥❣
✈➔ λ1 , λ2 , · · · , λm ❧➔ ❝→❝ sè ❞÷ì♥❣ t❤➻ ❤➔♠ λ1 f1 + λ2 f2 + · · · + λm fm ❧➔ ❤➔♠ ỗ

ởt số tự ỡ s ữỡ tr ✤↕♦ ❤➔♠

r✐➯♥❣
❚r♦♥❣ ♠ư❝ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ✤➲ ❝➟♣ ✤➳♥ ♠ët ✈➔✐ ✈➜♥ ✤➲ ❝ì ❜↔♥ ✈➲ ♣❤÷ì♥❣ tr➻♥❤
✤↕♦ ❤➔♠ r✐➯♥❣✳ ❈→❝ ❦✐➳♥ t❤ù❝ ♥➔② ✤÷đ❝ t❤❛♠ ❦❤↔♦ tø ❬✶❪✳
▼ët ♣❤÷ì♥❣ tr➻♥❤ ❧✐➯♥ ❤➺ ❣✐ú❛ ➞♥ ❤➔♠ u(x1 , x2 , . . . , xn )✱ ❝→❝ ❜✐➳♥ ✤ë❝ ❧➟♣

x1 , x2 , . . . , xn ✈➔ ❝→❝ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝õ❛ ♥â ✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ ✤↕♦
❤➔♠ r✐➯♥❣✳ ◆â ❝â ❞↕♥❣

F x1 , x2 , . . . , xn , u,

∂u
∂u
∂ku
,...,
, . . . , k1
,...
∂x1
∂xn
∂x1 ...∂xknn

= 0,

tr♦♥❣ ✤â F ❧➔ ♠ët ❤➔♠ ♥➔♦ ✤â ❝õ❛ ❝→❝ ✤è✐ sè ❝õ❛ ♥â❀ k = (k1 , k2 , . . . , kn ) ởt
ở ỗ số ổ tọ |k| = k1 + k2 + . . . + kn ✈➔ ✤÷đ❝ ❣å✐
❧➔ ♠ët ✤❛ ❝❤➾ sè✳
❈➜♣ ❝❛♦ ♥❤➜t ❝õ❛ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝õ❛ ❤➔♠ u ❝â ♠➦t tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✤÷đ❝
❣å✐ ❧➔ ❝➜♣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✳ ❈❤➥♥❣ ❤↕♥ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝➜♣ ❤❛✐ ❝õ❛

✽

❤➔♠ ❤❛✐ ❜✐➳♥ ❝â ❞↕♥❣

∂u ∂u ∂ 2 u ∂ 2 u ∂ 2 u
F x, y, , , 2 ,
,
= 0.
∂x ∂y ∂x ∂x∂y ∂y 2
P❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ✤÷đ❝ ❣å✐ ❧➔ t✉②➳♥ t➼♥❤ ♥➳✉ ♥❤÷ ♥â t✉②➳♥ t➼♥❤
✤è✐ ✈ỵ✐ ➞♥ ❤➔♠ ✈➔ t➜t ❝↔ ❝→❝ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝õ❛ ♥â✳ ❈❤➥♥❣ ❤↕♥ ♣❤÷ì♥❣ tr➻♥❤
t✉②➳♥ t➼♥❤ ❝➜♣ ❤❛✐ ❝â ❞↕♥❣

∂ 2u
∂ 2u
∂ 2u
∂u
∂u
+ c(x, y) 2 + d(x, y)
+ e(x, y)
+ f (x, y)u
a(x, y) 2 + b(x, y)
∂x
∂x∂y
∂y
∂x
∂y
= g(x, y).
❚r♦♥❣ ♥ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥✱ t❛ ❝❤➾ ①➨t ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ t✉②➳♥
t➼♥❤ ❝➜♣ ❤❛✐✳ ❱➔ ✤➸ ✤ì♥ ❣✐↔♥✱ t❛ ✈✐➳t ux , uy , uxx , uxy , uyy t ỵ

u ∂u ∂ 2 u ∂ 2 u ∂ 2 u
, ,
,
,
.
∂x ∂y ∂x2 ∂x∂y ∂y 2
◆❤÷ t❛ ✤➣ ❜✐➳t✱ ❤❛✐ ❜➔✐ t♦→♥ q✉❛♥ trå♥❣ tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣
❧➔
✐✮ ❇➔✐ t♦→♥ ❣✐→ trà ❜❛♥ ✤➛✉ ✭■✳❱✳P✮
✐✐✮ ❇➔✐ t♦→♥ ❣✐→ trà ❜✐➯♥ ✭❇✳❱✳P✮
❇➔✐ t♦→♥ ❣✐→ trà ❜❛♥ ✤➛✉ t❤÷í♥❣ ✤÷đ❝ ❣å✐ ❧➔ ❜➔✐ t♦→♥ ❈❛✉❝❤②✳ ❱ỵ✐ ❜➔✐ t♦→♥ ❣✐→
trà ❜✐➯♥✿ ♥➳✉ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ❝â ❞↕♥❣ B(u) = u tr➯♥ ∂Ω t❤➻ ❜➔✐ t♦→♥ ✤÷đ❝ ❣å✐
❧➔ ❜➔✐ t♦→♥ ❜✐➯♥ ❉✐r✐❝❤❧❡t❀ ♥➳✉ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ❝â ❞↕♥❣ B(u) = ∇u · n ✈ỵ✐ n ❧➔
✈➨❝ tì ♣❤→♣ t✉②➳♥ ✤ì♥ ✈à ♥❣♦➔✐ tr➯♥ ∂Ω t❤➻ ❜➔✐ t♦→♥ ✤÷đ❝ ❣å✐ ❧➔ ❜➔✐ t♦→♥ ❜✐➯♥
◆❡✉♠❛♥♥❀ ♥➳✉ ✤✐➲✉ ❦✐➺♥ ❝â ❞↕♥❣ B(u) = λu + µ∇ · n ✈ỵ✐ λ, µ ❧➔ ❝→❝ ❤➡♥❣ sè t❤➻
❜➔✐ t♦→♥ ✤÷đ❝ ❣å✐ ❧➔ ❜➔✐ t♦→♥ ❜✐➯♥ ❘♦❜✐♥ ❤❛② ❜➔✐ t♦→♥ ❜✐➯♥ ❤é♥ ❤đ♣✳

✶✳✷✳✶✳ P❤➙♥ ❧♦↕✐ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ❝➜♣ ❤❛✐
❳➨t ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ❝➜♣ ❤❛✐
n

aij (x1 , x2 , . . . , xn )uxi xj + F (x1 , x2 , . . . , xn , u, ux1 , ux2 , . . . , uxn ) = 0,
i,j=1

✾

✭✶✳✷✮


tr♦♥❣ ✤â aij = aij ✈➔ ❧➔ ❝→❝ ❤➔♠ ❝õ❛ ❜✐➳♥ x1 , x2 , . . . , xn ✳

●✐↔ sû x = (x1 , x2 , . . . , xn ) ∈ Rn ✳ ❳➨t ♠❛ tr➟♥

A(x) = aij (x) .
❚❛ ❝â t❤➸ ❝♦✐ A(x) ❧➔ ♠ët ♠❛ tr➟♥ ✤è✐ ①ù♥❣✳ ❚❛ ❝è ✤à♥❤ ✤✐➸♠ x0 = (x01 , x02 , . . . , x0n )
♥➔♦ ✤â✳ ❑❤✐ ✤â✱ A(x) ❧➔ ♠ët ♠❛ tr➟♥ ❤➡♥❣ A(x0 )✳ P❤÷ì♥❣ tr➻♥❤
❞❡t (A(x0 ) − λE) = 0,

✭✶✳✸✮

tr♦♥❣ ✤â E ❧➔ ♠❛ tr➟♥ ✤ì♥ ✈à✱ λ ❧➔ ♠ët sè t❤ü❝✱ ✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝
tr÷♥❣ ❝õ❛ ✭✶✳✷✮ t↕✐ ✤✐➸♠ x0 ✳ ❱➔ t❛ ❝â ✤à♥❤ ♥❣❤➽❛ s❛✉

✣à♥❤ ♥❣❤➽❛ ✶✳✶✸

✐✮ P❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✮ ✤÷đ❝ ❣å✐ ❧➔ t❤✉ë❝ ❧♦↕✐ ❡❧✐♣t✐❝ t↕✐

✤✐➸♠ x0 ♥➳✉ t↕✐ ✤✐➸♠ ✤â ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ✭✶✳✸✮ ❝â n ♥❣❤✐➺♠ ❦❤→❝

0 ✈➔ ❝ò♥❣ ♠ët ❞➜✉✳ ✭❚r♦♥❣ trữớ ủ t ữỡ tữỡ ự
ợ õ

n

aij (x0 )ti tj
i,j=1

❧➔ ♠ët ❞↕♥❣ ①→❝ ✤à♥❤ ❞÷ì♥❣ ❤♦➦❝ ①→❝ ✤à♥❤ ➙♠✳✮
✐✐✮ P❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✮ ✤÷đ❝ ❣å✐ ❧➔ t❤✉ë❝ ❧♦↕✐ ❤②♣❡❝❜♦❧✐❝ t↕✐ ✤✐➸♠ x0 ♥➳✉ t↕✐
✤✐➸♠ ✤â ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ✭✶✳✸✮ ❝â n ♥❣❤✐➺♠ ❦❤→❝ 0 ✈➔ t❤ä❛ ♠➣♥ ❝â

(n − 1) ♥❣❤✐➺♠ ❝ò♥❣ ❞➜✉✱ ♠ët ♥❣❤✐➺♠ ❝á♥ ❧↕✐ ❦❤→❝ ❞➜✉✳
✐✐✐✮ P❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✮ ✤÷đ❝ ❣å✐ ❧➔ t❤✉ë❝ ❧♦↕✐ ♣❛r❛❜♦❧✐❝ t↕✐ ✤✐➸♠ x0 ♥➳✉ t↕✐ ✤✐➸♠
✤â ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ✭✶✳✸✮ ❝â n ♥❣❤✐➺♠✱ tr♦♥❣ ✤â ❝â ♠ët ♥❣❤✐➺♠ ❜➡♥❣

0 ✈➔ (n − 1) ♥❣❤✐➺♠ ❝á♥ ❧↕✐ ❦❤→❝ 0 ✈➔ ❝ị♥❣ ❞➜✉✳
◆➳✉ ♥❤÷ t↕✐ ♠å✐ ✤✐➸♠ tr♦♥❣ ♠✐➲♥ Ω ⊂ Rn ✱ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✮ t❤✉ë❝ ❝ị♥❣
♠ët ❧♦↕✐ t❤➻ t❛ ♥â✐ r➡♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✮ t❤✉ë❝ ❧♦↕✐ ✤â tr♦♥❣ Ω✳
❑❤✐ n = 2✱ t❛ ①➨t ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ❝➜♣ ❤❛✐ s❛✉

a(x, y)uxx + 2b(x, y)uxy + c(x, y)uyy + F (x, y, u, ux , uy ) = 0.

✶✵

✭✶✳✹✮


❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ ♠❛ tr➟♥ A ❝â ❞↕♥❣


a(x, y) b(x, y)
.
A(x, y) = 
b(x, y) c(x, y)
❳➨t ✤✐➸♠ (x0 , y0 ) ∈ R2 ❝è ✤à♥❤✱ ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❝â ❞↕♥❣
❞❡t (A − λE) = (a − λ)(c − λ) − b2 = λ2 − (a + c)λ + ac − b2 = 0.
❉♦ ✤â✱ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✹✮ t↕✐ ✤✐➸♠ (x0 , y0 ) ✤÷đ❝ ❣å✐ ❧➔
✐✮ t❤✉ë❝ ❧♦↕✐ ❡❧❧✐♣t✐❝ ♥➳✉ t↕✐ ✤✐➸♠ ✤â

b2 − ac < 0,
✐✐✮ t❤✉ë❝ ❧♦↕✐ ❤②♣❡❝❜♦❧✐❝ ♥➳✉ t↕✐ ✤✐➸♠ ✤â

b2 − ac > 0,
✐✐✐✮ t❤✉ë❝ ❧♦↕✐ ♣❛r❛❜♦❧✐❝ ♥➳✉ t↕✐ ✤✐➸♠ ✤â

b2 − ac = 0.
ú ỵ r ờ = ξ(x, y), η = η(x, y) t❛ ❝â t❤➸ ✤÷❛ ♣❤÷ì♥❣
tr➻♥❤ t❤✉ë❝ tø♥❣ ❧♦↕✐ ✈➲ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❝â ❞↕♥❣ ✤➦❝ ❜✐➺t ♥➔♦ ✤â ♠➔ t❛ ❣å✐ ❧➔
❝→❝ ❞↕♥❣ ❝❤➼♥❤ t

t trữ t ợ ỳ ❝❤♦ tr➯♥ ♠➦t ✤➦❝
tr÷♥❣
❳➨t ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ❝➜♣ ❤❛✐
n

aij (x)uxi xj + F (x1 , x2 , . . . , xn , u, ux1 , ux2 , . . . , uxn ) = 0.



i,j=1

ữỡ ự ợ õ t❛ t❤✐➳t ❧➟♣ ♣❤÷ì♥❣ tr➻♥❤
n

aij (x)ωxi ωxj = 0.
i,j=1

✶✶

✭✶✳✻✮

P❤÷ì♥❣ tr➻♥❤ ✭✶✳✻✮ ✤÷đ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❝→❝ ♠➦t ✤➦❝ tr÷♥❣ ✭❤❛② ♣❤÷ì♥❣

tr➻♥❤ ❝→❝ ✤÷í♥❣ ✤➦❝ tr÷♥❣ ❦❤✐ n = 2✮✳
▼➦t S ✤÷đ❝ ❣å✐ ❧➔ ♠➦t ✤➦❝ tr÷♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✺✮ ♥➳✉ ♣❤÷ì♥❣ tr➻♥❤
❝õ❛ ♥â ❝â t❤➸ ✈✐➳t ữủ ữợ

(x1 , x2 , . . . , xn ) = 0,

✭✶✳✼✮

tr♦♥❣ ✤â ❤➔♠ ω(x1 , x2 , . . . , xn ) tr➯♥ ♠➦t S t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✻✮ ✈➔
n
2
i=1 ωxi

= 0✳ ❑❤✐ ✤â✱ t❛ õ t t s

ỵ t trữ ❜➜t ❜✐➳♥ q✉❛ ❝→❝ ♣❤➨♣ ✤ê✐ ❜✐➳♥ sè✳
✣➸ t➻♠ ❤✐➸✉ ✈➲ ❜➔✐ t♦→♥ ❈❛✉❝❤② ✈ỵ✐ ❞ú ❦✐➺♥ ❝❤♦ tr➯♥ ♠➦t ✤➦❝ tr÷♥❣✱ t❛ ①➨t

S ❧➔ ♠ët ♠➦t trì♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ Rn ✳ ❚↕✐ ♠é✐ ✤✐➸♠ x ∈ S ✱ t ởt ữợ
ổ t ú ợ t S ✳ ❇➔✐ t♦→♥ ❈❛✉❝❤② ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✺✮
❧➔ ❜➔✐ t♦→♥ s❛✉✿ ❚r♦♥❣ ❧➙♥ ❝➟♥ ♠➦t S ✱ t➻♠ ♠ët ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✺✮
t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥

u

S

∂u
∂λ

S

= ϕ(x)

✭✶✳✽✮

= (x),



tr õ (x), (x) trữợ tr ♠➦t S t❤ä❛ ♠➣♥ ❣✐↔ t❤✐➳t✿ ϕ(x) ❧➔
♠ët ❤➔♠ ❦❤↔ ✈✐ ❧✐➯♥ tö❝✱ ψ(x) ❧➔ ♠ët ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ S ✳ ❈→❝ ❤➔♠ ϕ(x), ψ(x)
✤÷đ❝ ❣å✐ ❧➔ ❝→❝ ❞ú ❦✐➺♥ ❈❛✉❝❤②✱ ♠➦t S ❣å✐ ❧➔ ♠➦t ♠❛♥❣ ❞ú ❦✐➺♥ ❈❛✉❝❤② ❤❛② ❣å✐
t➢t ❧➔ ♠➦t ❈❛✉❝❤②✳
❚❛ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ t➼♥❤ ❝❤➜t s❛✉ ✭①❡♠ ❬✶❪✮
✶✳ ❇✐➳t ❝→❝ ❞ú ❦✐➺♥ ❈❛✉❝❤②✱ ❝â t❤➸ t➻♠ t➜t ❝↔ ❝→❝ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝➜♣ ♠ët ❝õ❛
♥❣❤✐➺♠ ð tr➯♥ ♠➦t ❈❛✉❝❤②✳
✷✳ ❚r➯♥ ♠➦t ✤➦❝ tr÷♥❣✱ ❝→❝ ❞ú ❦✐➺♥ ❈❛✉❝❤② ♣❤ư t❤✉ë❝ ❧➝♥ ♥❤❛✉✱ tù❝ ố
ợ t tr t trữ ỳ ổ
t ởt tũ ỵ




✸✳ ▼➦t ✤➦❝ tr÷♥❣ ❧➔ ♠➦t ✧tr✉②➲♥ ❝→❝ ❣✐→♥ ✤♦↕♥✧ ❝õ❛ ❝→❝ ✤↕♦ ❤➔♠ ❝➜♣ ❝❛♦
❝õ❛ ♥❣❤✐➺♠✳

✶✳✷✳✸✳ ❙ü ♣❤ö t❤✉ë❝ ❧✐➯♥ tö❝
❚r➯♥ t❤ü❝ t➳✱ ❦❤✐ t❛ ✤✐ ❣✐↔✐ ❝→❝ ❜➔✐ t t ỵ tữớ s số
s số ♣❤→t s✐♥❤ tø ✈✐➺❝ ✤♦ ✤↕❝ ❝→❝ ❞ú ❦✐➺♥ ❝❤♦ trữợ ữ
ỹ t ở t ỵ õ t s số tr♦♥❣ q✉→ tr➻♥❤
t➼♥❤ t♦→♥✳ ❈➙✉ ❤ä✐ ✤÷đ❝ ✤➦t r❛ ❧➔ ❝→❝ s❛✐ sè ♥➔② ❝â ↔♥❤ ❤÷ð♥❣ ♥❤÷ t❤➳ ♥➔♦ ✤➳♥
♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥✳ ✣➙② ❧➔ sü q✉❛♥ t➙♠ ✈➲ sü ♣❤ư t❤✉ë❝ ❧✐➯♥ tư❝ ❤❛② tê♥❣
q✉→t ❤ì♥ ❧➔ sü ê♥ ✤à♥❤ ❝õ❛ ♥❣❤✐➺♠✳ ❱ỵ✐ ❝→❝ ❜➔✐ t♦→♥ ✤➦t ❦❤ỉ♥❣ ❝❤➾♥❤✱ ❝❤ó♥❣ t❛
t❤÷í♥❣ q✉❛♥ t➙♠ ✤➳♥ sü ♣❤ư t❤✉ë❝ ❧✐➯♥ tử t oăr ỳ
✤÷đ❝ ✤÷❛ r❛ ❜ð✐ ❏♦❤♥ ✈➔♦ ♥➠♠ ✶✾✻✵✳
❳➨t ❜➔✐ t♦→♥ ❝â t➟♣ ♥❣❤✐➺♠ ❧➔ U ✱ t➟♣ ❝→❝ ❞ú ❦✐➺♥ ❧➔ F ✈➔ A ❧➔ →♥❤ ①↕ tø F
✈➔♦ U ✳ ●✐↔ sû✱ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✤÷đ❝ ①→❝ ✤à♥❤ tr➯♥ t➟♣ ❝♦♥ R ⊂ U ✈➔
❝→❝ ❞ú ❦✐➺♥ ①→❝ ✤à♥❤ tr➯♥ t➟♣ G ⊂ F t❤ä❛ ♠➣♥ G, R ❧➔ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ t✉②➳♥
t➼♥❤ ✤à♥❤ ❝❤✉➞♥ ✈ỵ✐

·

G✱

·

R

❧➔ ❝→❝ tữỡ ự tr ổ G

R ợ ộ t➟♣ ❝♦♥ S ❝õ❛ R ✈ỵ✐ ❝❤✉➞♥

·

S✱

♥➳✉ u1 , u2 ∈ U ✈➔ f1 , f2 ∈ F t❤ä❛

♠➣♥ u1 = Af1 ✈➔ u2 = Af2 ✱ t❛ ♥â✐ →♥❤ A tử oăr t f1 ❝❤➾
♥➳✉

sup u1 − u2

S

< M εα ,

u2 ∈S

♥➳✉ f1 − f2

G

< ε,

tr♦♥❣ ✤â M, α ❧➔ ❝→❝ ❤➡♥❣ sè ❞÷ì♥❣ ♣❤ư t❤✉ë❝ ✈➔♦ U ✈➔ S ✳
❱ỵ✐ ❝→❝ ❜➔✐ t♦→♥ t ỵ ú t t

R = {u(t) U t ∈ [0, T )},
✈ỵ✐ [0, T ) ❝❤➾ ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥✱

S = {u(t) ∈ U t ∈ [0, T1 )},

T1 ≤ T

✈➔

G = {f ∈ F ∃u ∈ U t❤ä❛ ♠➣♥ u(0) = f }.

✶✸


u1 (Ã, t) ữủ ồ ờ oăr tr♦♥❣ ❦❤♦↔♥❣ t ∈ [0, T ) ♥➳✉ ✈➔ ❝❤➾
♥➳✉ ợ > 0 trữợ ợ ồ u(Ã, 0) ∈ F t❤ä❛ ♠➣♥ u1 (·, 0) − u2 (·, 0)

0

<ε

t❤➻

u1 (·, t) − u2 (·, t)

sup

t

< Cεα ,

0≤t≤T1
✈ỵ✐ α ∈ (0, 1]✱ ❝❤✉➞♥

·

t

✈➔

·

0

t÷ì♥❣ ù♥❣ ❧➔ ❝→❝ ❝❤✉➞♥ ❝õ❛ ♥❣❤✐➺♠ u(·, t) t↕✐

t❤í✐ ✤✐➸♠ t ✈➔ t❤í✐ ✤✐➸♠ ❜❛♥ ✤➛✉ t = 0✱ C ❧➔ ❤➡♥❣ sè ❞÷ì♥❣ ❦❤ỉ♥❣ ♣❤ö t❤✉ë❝
✈➔♦ ε✳ ❑❤✐ ✤â✱ t❛ ❝â t❤➸ ♥â✐ ♥❣❤✐➺♠ ừ t ử tở tử oăr t

ợ t [0, T ) ú ỵ r tr tr sỹ ờ
oăr ữủ ✤à♥❤ tr➯♥ ❦❤♦↔♥❣ ❝♦♥ ❝♦♠♣❛❝t ❝õ❛ ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥ ❤ú✉

Pữỡ ỗ ổrt
r ử ú tổ tr ỗ ổrt ữỡ tỹ
ữ ỗ t tổ tữớ ỗ ổrt tọ ♠➣♥ ❜➜t ✤➥♥❣
t❤ù❝ ✈➲ ✤↕♦ ❤➔♠ ❝➜♣ ❤❛✐✳ ❉ü❛ tr➯♥ t tự õ ú tổ ữ r ợ
tr ữợ ỗ ổrt rữợ t ú tổ ợ
t ỗ ổrt

ởt ữủ ồ ỗ ổrt õ ổ
ổrt ừ õ ởt ỗ
sỷ F (x) ỗ ❧ỉ❣❛r✐t✱ tù❝ ❧➔ F (x) > 0 ✈ỵ✐ ♠å✐ x ∈ [x1 , x2 ] ✈➔
t❤ä❛ ♠➣♥
✭✶✳✶✵✮

f (x) = ln F (x)
ỗ

f (x) ỗ t❛ ❝â f (x) ≥ 0✱ ✈ỵ✐ ♠å✐ x ∈ [x1 , x2 ]✳ ❉♦ ✤â✱ t❛ ❝â

d2 [ln F (x)]
dx2

≥ 0,

∀x ∈ [x1 , x2 ].

▼➔ t❛ ❧↕✐ ❝â

d2 [ln F (x)]
dx2

F (x)F (x) − [F (x)[2
=

F 2 (x)

✶✹

≥ 0.


❙✉② r❛

F (x)F (x) − [F (x)]2 ≥ 0,

✭✶✳✶✶✮

∀x ∈ [x1 , x2 ].

▼➦t ❦❤→❝✱ ✈➻ f (x) = ln F (x) ỗ t t tự ✭✶✳✶✮ t❛ ❝â

ln F (x) ≥ ln F (x1 ) + [ln F (x)] (x1 )(x − x1 )
= ln F (x1 ) +

F (x1 )
(x − x1 )
F (x1 )

= ln F (x1 ) · exp

F (x1 )
F (x1 )

(x − x1 )

.

❍❛② t❛ ❝â

F (x) ≥ F (x1 ) · exp

F (x1 )
F (x1 )

(x − x1 ) ,

∀x ∈ [x1 , x2 ].

✭✶✳✶✷✮

❱➔ t❛ ❝ô♥❣ ❝â

ln F (x) ≤ ln F (x1 ) +
1
=

x2 − x1
1

=

x2 − x1

ln F (x2 ) − ln F (x1 )
(x − x1 )
x2 − x2

[ln F (x1 )(x2 − x1 ) + ln F (x2 )(x − x1 ) − ln F (x1 )(x − x1 )]
[(x2 − x) ln F (x1 ) + (x − x1 ) ln F (x2 )]
x2 −x

x−x1

= ln F (x1 ) x2 −x1 · F (x2 ) x2 −x1 .
❉♦ ✤â
x2 −x

x−x1

F (x) ≤ F (x1 ) x2 −x1 · F (x2 ) x2 −x1 ,

∀x ∈ [x1 , x2 ].

✭✶✳✶✸✮

❚ø ✭✶✳✶✷✮ ✈➔ ✭✶✳✶✸✮✱ ✈ỵ✐ ♠å✐ x ∈ [x1 , x2 ] t❛ ✤÷đ❝

F (x1 ). exp

F (x1 )
x2 −x
x−x1
(x − x1 ) ≤ F (x) ≤ F (x1 ) x2 −x1 · F (x2 ) x2 −x1 .
F (x1 )

✭✶✳✶✹✮

❈æ♥❣ t❤ù❝ ✭✶✳✶✹✮ ❝❤♦ t❛ ợ tr ợ ữợ ừ ởt ỗ ổ
rt ổ tự q trồ ữủ sỷ ❞ư♥❣ r➜t ♥❤✐➲✉ tr♦♥❣ ❝→❝ ✤→♥❤ ❣✐→
ð ❈❤÷ì♥❣ ✷✳

✶✺


❚r♦♥❣ ♣❤➛♥ t✐➳♣ t❤❡♦✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët ♠ð rë♥❣ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝
✭✶✳✶✶✮✳ ●✐↔ sû ❤➔♠ F (x) > 0 ✈ỵ✐ ♠å✐ x ∈ [x1 , x2 ] ✈➔ t❤ä❛ ♠➣♥ ❜➜t ✤➥♥❣ t❤ù❝

F (x)F (x) − [F (x)]2 ≥ −a1 F (x)F (x) − a2 F 2 (x)

✭✶✳✶✺✮

✈ỵ✐ a1 , a2 ❧➔ ❝→❝ ❤➡♥❣ sè✳
❑❤✐ ✤â✱ ✈ỵ✐ ♠å✐ x ∈ [x1 , x2 ] t❛ ❝â

F (x)F (x) − [F (x)]2 + a1 F (x) · F (x) + a2 F 2 (x)
F 2 (x)

≥ 0.

❍❛②

F (x)F (x) − [F (x)]2

F (x)
+ a1

F 2 (x)

F (x)

+ a2 ≥ 0.

❙✉② r❛
✭✶✳✶✻✮

[ln F (x)] + a1 [ln F (x)] + a2 ≥ 0.
●✐↔ sû a1 = 0✱ t❛ ✤➦t

σ = e−a1 x ,

∀x ∈ [x1 , x2 ].

❳➨t ❤➔♠
2

ln F (σ)σ −a2 /a1 = ln F (σ) −

a2
a21

ln σ.

❚❛ ❝â
2

d ln F (σ)σ −a2 /a1
dσ

= ln F (σ) ·

−1
a1 σ

−

a2
a21 σ

.

❙✉② r❛
2

d2 F (σ)σ −a2 /a1
dσ 2

= ln F (σ)

·

1
a2
1
+
ln
F
(σ)
·
+
a21 σ 2
a1 σ 2 a21 σ 2

1
=

a21 σ 2

ln F (σ)

+ a1 ln F (σ) + a2 .

❚❤❡♦ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✶✻✮ t❛ ❝â

ln F (σ)

+ a1 ln F (σ) + a2 ≥ 0,

✶✻

∀σ ∈ [σ1 , σ2 ],


tr♦♥❣ ✤â✱ σ1 = e−a1 x1 , σ2 = e−a1 x2 ✳ ❉♦ ✤â✱ t❛ ♥❤➟♥ ✤÷đ❝
2

d2 ln F (σ) · σ −a2 /a1
≥ 0,

dσ 2

✭✶✳✶✼✮

∀σ ∈ [σ1 , σ2 ].

❚❤❡♦ ✤→♥❤ ❣✐→ tr➯♥ ❝õ❛ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✶✹✮ t❛ ❝â

F (σ)σ

−a2 /a21

≤ F (σ1 )σ1

σ2 −σ
σ2 −σ1

σ−σ1
σ2 −σ1

−a /a2
F (σ2 )σ2 2 1

·

.

❚❛ ✤➦t

δ :=

σ2 − σ
σ2 − σ1

e−a1 x2 − e−a1 x
=

e−a1 x2 − e−a1 x1

.

❑❤✐ ✤â✱ t❛ ✤÷đ❝

σ − σ1
σ2 − σ1

e−a1 x − e−a1 x1
=

e−a1 x2 − e−a1 x1

= 1 − δ.

❉♦ ✤â✱ t❛ ❝â
2

F (σ)σ −a2 /a1 ≤ F (σ1 )σ1

δ

−a2 /a21

1−δ

· F (σ2 )σ2

.

❱➻ σ = e−a1 x ♥➯♥ t❛ ❝â

F (e−a1 x )ea2 x/a1 ≤ F (e−a1 x1 )ea2 x1 /a1

δ

· F (e−a1 x2 )ea2 x2 /a1

1−δ

.

❙✉② r❛

F (x)ea2 x/a1 ≤ F (x1 )ea2 x1 /a1

δ

· F (x2 )ea2 x2 /a1

1−δ

.

❱➟②✱ t❛ t❤✉ ✤÷đ❝

F (x) ≤ e−a2 x/a1 · F (x1 )ea2 x1 /a1

δ

· F (x2 )ea2 x2 /a1

1−δ

,

∀x ∈ [x1 , x2 ].

t t ữợ ừ t t❤ù❝ ✭✶✳✶✹✮✱ t❛ ❝â




−a2 /a21


F (σ)σ
(σ1 )
−a2 /a21
−a2 /a21
(σ − σ1 ) .
F (σ)σ
≥ F (σ1 )σ1
· exp
2



 F (σ1 )σ1−a2 /a1

✶✼

❱➻
2

F (σ)σ −a2 /a1 (σ1 )
−a /a2
F (σ1 )σ1 2 1

(σ − σ1 ) =
−1
F (σ1 )

−a2 /a21

· σ1

a1 σ1

=

−

a2

−a2 /a21 −1

· F (σ1 )σ1
a21

(σ − σ1 )

−a2 /a! 12

F (σ1 )σ1
F (σ1 ) ·
=

=

1
+

a2

a1 σ1 a21 σ1
F (σ1 )

· F (σ1 )

F (σ1 ) + a2 /a1 · F (σ1 )
a1 F (σ1 )

(σ1 − σ)
1−

σ
σ1

.

◆➯♥
−a2 /a21

2

F (σ)σ −a2 /a1 ≥ F (σ1 )σ1

F (σ1 ) + a2 /a1 · F (σ1 )

· exp

a1 F (σ1 )

1−

σ
σ1

.

❙✉② r❛✱ t❛ ❝â

F (σ) ≥ F (σ1 ) · exp

F (σ1 ) + a2 /a1 · F (σ1 )
a1 F (σ1 )

1−

σ
σ1

−a2 /a21

· σ1

2

· σ a2 /a1 .

❚❤❛② ❜✐➳♥ σ = e−a1 x t❛ t❤✉ ✤÷đ❝
a2

−a2

F (e−a1 x ) ≥ F (e−a1 x1 )e a1 x1 e a1 x ×


F (e−a1 x1 ) + (a2 /a1 )F (e−a1 x1 )
e−a1 x
× exp 
1 − −a1 x1 
a1 F (e−a1 x1 )
e


F (e−a1 x1 ) + (a2 /a1 )F (e−a1 x1 )
a2
= F (e−a1 x1 ) · exp 

1 − ea1 (x−x1 ) −
(x − x1 ) .
−a
x
1
1
a1 F (e
)
a1
❱➟②✱ ✈ỵ✐ ♠å✐ x ∈ [x1 , x2 ] t❛ ❝â


F (x) ≥ F (x1 ) · exp 

F (x1 ) + (a2 /a1 )F (x1 )
a1 F (x1 )

1 − ea1 (x−x1 ) −

a2
a1


(x − x1 ) .
✭✶✳✶✾✮

✶✽


●✐è♥❣ ❝æ♥❣ t❤ù❝ ✭✶✳✶✹✮✱ ❝æ♥❣ t❤ù❝ ✭✶✳✶✾✮ ✈➔ ✭✶✳✶✽✮ ❧➔ ❝ỉ♥❣ ❝ư q✉❛♥ trå♥❣ ✤➸ ✤÷❛

r❛ ❝→❝ ✤→♥❤ ❣✐→ ð ❈❤÷ì♥❣ ✷✳
❚r♦♥❣ ♠ët tr÷í♥❣ ❤đ♣ ❦❤→❝✱ ♥➳✉ a1 = 0 t❤➻ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✶✺✮ trð t❤➔♥❤

F (x)F (x) − [F (x)]2 ≥ −a2 F 2 (x),

∀x ∈ [x1 , x2 ],

✭✶✳✷✵✮

✈ỵ✐ a2 ❧➔ ❤➡♥❣ sè✳
❑❤✐ ✤â✱ t❛ ❝â

F (x)F (x) − [F (x)]2 + a2 F 2 (x)
F 2 (x)

≥ 0.

❙✉② r❛

+ a2 ≥ 0.

ln F (x)
❉♦ ✤â✱ t❛ ❝â

d2 ln F (x) · ea2 x

2

/2+αx+β

≥ 0,

dx2

∀x ∈ [x1 , x2 ],

✭✶✳✷✶✮

tr♦♥❣ ✤â✱ α, β ❧➔ ❝→❝ ❤➡♥❣ sè tũ ỵ õ t ụ õ ố ợ tr
ữợ ừ F (x)




ữỡ

ệ ế Pì PP

❞ư♥❣ tr♦♥❣ ❜➔✐ t♦→♥ ❈❛✉❝❤② ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤
♣❛r❛❜♦❧✐❝ ♥❣÷đ❝ t❤í✐ ❣✐❛♥

✷✳✶✳✶✳ P❤÷ì♥❣ tr➻♥❤ ♣❛r❛❜♦❧✐❝ ♥❣÷đ❝ t❤í✐ ❣✐❛♥
❈❤♦ Ω = [0, 1], T > 0✳ ❳➨t ❜➔✐ t♦→♥ ♣❛r❛❜♦❧✐❝ tr♦♥❣ tr÷í♥❣ ❤ñ♣ ♠ët ❝❤✐➲✉

ut = uxx , 0 ≤ x ≤ 1, 0 < t ≤ T,

✭✷✳✶✮

u(0, t) = u(1, t) = 0, 0 < t ≤ T,

✭✷✳✷✮

u(x, 0) = u0 (x), 0 ≤ x ≤ 1.

✭✷✳✸✮

●✐↔ sû u0 (x) ❧➔ ❤➔♠ ❧✐➯♥ tư❝ tø♥❣ ❦❤ó❝ ✈➔ tr✐➺t t✐➯✉ t↕✐ x = 0, x = 1✱ tù❝ ❧➔

u0 (0) = u0 (1) = 0.
❙û ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ t→❝❤ ❜✐➳♥✱ t❛ t➻♠ ổ t tữớ ừ t
ữợ

u(x, t) = X(x) · T (t).
❚❤❛② ✈➔♦ ✭✷✳✶✮ t❛ ❝â

X(x) · T (t) = X (x) · T (t),
❤❛②

X (x)
T (t)
=
= −λ,
T (t)
X(x)

✷✵


tr♦♥❣ ✤â✱ λ ❧➔ ❤➡♥❣ sè✳ ❙✉② r❛✱ ♣❤÷ì♥❣ tr➻♥❤ tữỡ ữỡ ợ ữỡ
tr tữớ

T (t) + λT (t) = 0,

✭✷✳✹✮

X (x) + λX(x) = 0.

✭✷✳✺✮

❙û ❞ư♥❣ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ✭✷✳✷✮ t❛ ♥❤➟♥ ✤÷đ❝
✭✷✳✻✮

X(0) = X(1) = 0.

rữợ t t t ữỡ tr t t➼♥❤ ❝➜♣ ❤❛✐ ✭✷✳✺✮ ✈ỵ✐ ✤✐➲✉ ❦✐➺♥
❜❛♥ ✤➛✉ ✭✷✳✻✮



X (x) + λX(x) = 0,

X(0) = X(1) = 0.
❚❛ ①➨t ❝→❝ tr÷í♥❣ ❤đ♣ s❛✉ ❝õ❛ t❤❛♠ sè λ

✰ ❚r÷í♥❣ ❤đ♣ ✶✿ ◆➳✉ λ < 0 t❤➻ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✺✮ ❝â
❞↕♥❣

X(x) = C1 e

√

−λx

√
−λx

+ C2 e−

, ✈ỵ✐ C1 , C2 ❧➔ ❤➡♥❣ sè.

❚ø ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ ✭✷✳✻✮ t❛ t❤✉ ✤÷đ❝

X(0) = C1 + c2 = 0,
√
−λ

X(1) = C1 e

+ C2 e−

√
−λ

= 0.

❉♦ ✤â✱ t❛ ♥❤➟♥ ✤÷đ❝ C1 = C2 = 0 ❤❛② X(x) ≡ 0✳ ❱➟②✱ ✈ỵ✐ λ < 0✱ t❤➻ u(x, t) ≡ 0.

✰ ❚r÷í♥❣ ❤đ♣ ✷✿ ◆➳✉ λ = 0 t❤➻ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ữỡ tr õ


X(x) = ax + b, ợ a, b ❧➔ ❤➡♥❣ sè✳

❚ø ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ t❛ ❝ô♥❣ s✉② r❛ ✤÷đ❝ a = b = 0 ❤❛② X(x) ≡ 0✳ ❱➟②✱ ✈ỵ✐

λ = 0 t❤➻ t❛ ❝ơ♥❣ ❝â u(x, t) ≡ 0.
✰ ❚r÷í♥❣ ❤đ♣ ✸✿ ◆➳✉ λ > 0 t❤➻ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✺✮ ❝â
❞↕♥❣

√
√
X(x) = C1 cos( λx) + C2 sin( λx), ✈ỵ✐ C1 , C2 ❧➔ ❤➡♥❣ sè✳

✷✶


❚❤❛② ✤✐➲✉ ❦✐➺♥ ❜❛♥ ✤➛✉ t❛ ❝â

√
X(0) = C1 cos( λx) = 0.
√
❙✉② r❛✱ C1 = 0✳ ❚❤❛② ✤✐➲✉ ❦✐➺♥ X(1) = 0 ✈➔ C1 = 0✱ t❛ ❝â C2 sin( λ) = 0✳ ✣➸
√
♥❣❤✐➺♠ u(x, t) = 0 t❤➻ t❛ ♣❤↔✐ ❝â C2 = 0✳ ❉♦ ✤â✱ λ = kπ ❤❛②
λ = k2π2,

✭✷✳✼✮

k ∈ Z.

❑❤✐ ✤â✱ t❛ ❝â X(x) = C2 sin(kπx)✳
❱➻ ❤➔♠ X(x) ♣❤ö t❤✉ë❝ ✈➔♦ k ♥➯♥ t ỵ

Xk (x) = Ak sin(kx).
= k 2 π 2 ✈➔♦ ✭✷✳✹✮ t❤➻ ♣❤÷ì♥❣ tr➻♥❤

T (t) + k 2 T (t) = 0,
❝â ♥❣❤✐➺♠

Tk (t) = Bk e−tk

2 2

π

✭✷✳✾✮

.

❚ø ✭✷✳✽✮ ✈➔ ✭✷✳✾✮✱ t❛ ❝â ♥❣❤✐➺♠ r✐➯♥❣ ❝õ❛ ❜➔✐ t♦→♥ ✤➣ ❝❤♦ ❝â ❞↕♥❣

uk (x, t) = Ck e−tk

2 2

π

sin(kπx),

✭✷✳✶✵✮

✈ỵ✐ Ck = Ak · Bk ❧➔ ❤➡♥❣ sè tũ ỵ t ổ tự tọ

❦✐➺♥ ❜✐➯♥ ✭✷✳✷✮✳ ❚❛ ①➙② ❞ü♥❣ ♠ët ❝→❝❤ ❤➻♥❤ t❤ù❝ ❝❤✉é✐
∞

u(x, t) =

∞

Ck e−tk

uk (x, t) =
k=1

2 2

π

sin(kπx),

✭✷✳✶✶✮

k=1

✈ỵ✐ ❤➺ sè Ck ①→❝ ✤à♥❤ s❛♦ ❝❤♦ ❝❤✉é✐ ✭✷✳✶✶✮ t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮ ✈➔ ✤✐➲✉
❦✐➺♥ ❜❛♥ ✤➛✉ ✭✷✳✸✮✱ tù❝ ❧➔
∞

u(x, 0) =

Ck sin(kπx) = u0 (x).
k=1

✷✷