Giải một số bài toán truyền nhiệt bậc phân ngược thời gian bằng chỉnh hóa điều kiện biên

✶

▼Ö❈ ▲Ö❈
❚r❛♥❣

▼Ö❈ ▲Ö❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶
▲❮■ ◆➶■ ✣❺❯ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷
❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❜ê trñ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻
✶✳✶ ❍➔♠ ●❛♠♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻
✶✳✷ ❍➔♠ ▼✐tt❛❣✲▲❡❢❢❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶
✶✳✸ ▼ët sè ❦❤æ♥❣ ❣✐❛♥ ❤➔♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻
❈❤÷ì♥❣ ✷✳ ●✐↔✐ ♠ët ❜➔✐ t♦→♥ tr✉②➲♥ ♥❤✐➺t ❜➟❝ ♣❤➙♥ ♥❣÷ñ❝ t❤í✐
❣✐❛♥ ❜➡♥❣ ❝❤➾♥❤ ❤â❛ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✷✳✶ ●✐î✐ t❤✐➺✉ ❜➔✐ t♦→♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✷✳✷ ❈❤➾♥❤ ❤â❛ ❜➔✐ t♦→♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
✷✳✸ ❚è❝ ✤ë ❤ë✐ tö ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❝❤➾♥❤ ❤â❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
❑➌❚ ▲❯❾◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺
❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻



ừ ồ ổ ữ t t t
tr ổ tr q ổ trữớ ữợ t ỷ ỵ
t ợ ữ t ữủ ữỡ tr r
ởt tr ỳ ữợ ự tr ỹ ữỡ tr
r ữủ tớ
Pữỡ tr r ữủ tớ t tr ỵ tt
tr t t t ở t ởt tớ õ
tr q ự q t ở ữủ t tớ t t
ụ tữớ t tr t ỵ ồ t
từ ở ồ

t ữủ tr tữớ t ổ t r
ởt t ữủ ồ t õ tọ õ
õ t ử tở tử t ởt
tổổ õ t ỳ ừ t ữ t t ởt tr
ổ tọ t t õ r t t ổ
r r t t ổ ổ õ ỵ t
ỵ ữ õ tr t tỹ t ừ ồ
ổ t t ổ ỳ
ỵ tứ t ừ t trữợ ổ tr
ự tợ t t ổ t ồ
rt P
ỳ ữớ t tr ỹ
õ rt ổ tr t ữỡ tr r
ữủ tớ t ổ tr õ ữỡ





tr r t qừ ữỡ tr r
ỏ ữỡ tr r
ữủ tớ ữủ t ồ tr ữợ q
t ự ổ ố ởt số t qừ tr t õ t
tr tớ
t ữủt ự ụ ữ ú t t
õ t tr t ữủ tớ
tr ỡ s rt s r
r t rrt ừ t
tr t r tts ú
tổ ỹ ồ t ừ ởt t

tr t ữủ tớ õ

t t tr t ữủ tớ


u

= u, (x, t) ì (0, T ],
t
u(x,
t) = 0, x , t [0, T ],


u(x, T ) = uT (x), x ,



tr õ (0, 1) ú t tt r t õ ởt t
u(x, t) r ú tổ q t tợ
u(x, t) ợ t [0, T ) tứ ỳ uT (x) ừ tr ố uT (x) tọ


uT (ã) uT (ã)

L2 ()

,

tr õ ự s số > 0 ữủ t
ợ ử ữ t t
ỗ õ ữỡ ỹ tr t ố ử s

ữỡ ởt số tự ờ trủ

ữỡ ử tr ởt số tự q
ở ữỡ ừ ữủ ú tổ t tr t


✹

✈➔ ❬✺❪✳

✶✳✶✳ ❍➔♠ ●❛♠♠❛

▼ö❝ ♥➔② tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❤➔♠ ●❛♠♠❛✳

✶✳✷✳ ❍➔♠ ▼✐tt❛❣✲▲❡❢❢❧❡r

▼ö❝ ♥➔② tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ✈➔ ❝→❝ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❤➔♠ ▼✐tt❛❣✲
▲❡❢❢❡r✳

✶✳✸✳ ▼ët sè ❦❤æ♥❣ ❣✐❛♥ ❤➔♠

▼ö❝ ♥➔② tr➻♥❤ ❜➔② ♠ët sè ❦❤æ♥❣ ❣✐❛♥ ❤➔♠ ❧✐➯♥ q✉❛♥ ✤➳♥ ♥ë✐ ❞✉♥❣
❝❤÷ì♥❣ ✷ ♥❤÷ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ H 1 (Ω)✱ H 2 (Ω)✱ H01 (Ω)✱ C([0, T ], L2 (Ω))✱ ✳✳✳

❈❤÷ì♥❣ ✷✳ ●✐↔✐ ♠ët ❜➔✐ t♦→♥ tr✉②➲♥ ♥❤✐➺t ❜➟❝ ♣❤➙♥ ♥❣÷ñ❝

t❤í✐ ❣✐❛♥ ❜➡♥❣ ❝❤➾♥❤ ❤â❛ ✤✐➲✉ ❦✐➺♥ ❜✐➯♥

❈❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tæ✐ tr➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣ ❝❤➾♥❤ ❤â❛ ❜➔✐ t♦→♥

❦❤✉➳❝❤ t→♥ ❜➟❝ ♣❤➙♥ ♥❣÷ñ❝ t❤í✐ ❣✐❛♥ tr♦♥❣ ❜➔✐ ❜→♦ ❬✸❪ ❝ô♥❣ ♥❤÷ ✤➲ ①✉➜t
✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët ✈➔✐ ❦➳t qõ❛ ♠î✐✳

✷✳✶✳ ●✐î✐ t❤✐➺✉ ❜➔✐ t♦→♥

▼ö❝ ♥➔② ♥❤➡♠ ❣✐î✐ t❤✐➺✉ ❜➔✐ t♦→♥ tr✉②➲♥ ♥❤✐➺t ❜➟❝ ♣❤➙♥ ♥❣÷ñ❝ t❤í✐
❣✐❛♥✳

✷✳✷✳ ❈❤➾♥❤ ❤â❛ ❜➔✐ t♦→♥

▼ö❝ ♥➔② ♥❤➡♠ tr➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣ ❝❤➾♥❤ ❤â❛ ❝❤♦ ❜➔✐ t♦→♥ tr✉②➲♥
♥❤✐➺t ❜➟❝ ♣❤➙♥ ♥❣÷ñ❝ t❤í✐ ❣✐❛♥ tr♦♥❣ ❜➔✐ ❜→♦ ❬✸❪✳

✷✳✸✳ ❚è❝ ✤ë ❤ë✐ tö ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ❝❤➾♥❤ ❤â❛

▼ö❝ ♥➔② ♥❤➡♠ tr➻♥❤ ❜➔② ❝→❝ ❦➳t qõ❛ ✤→♥❤ ❣✐→ tè❝ ✤ë ❤ë✐ tö ❝õ❛
♣❤÷ì♥❣ ♣❤→♣ tr♦♥❣ ❜➔✐ ❜→♦ ❬✸❪ ❝ô♥❣ ♥❤÷ ✤➲ ①✉➜t ♠ët ✈➔✐ ❦➳t qõ❛ ♠î✐✳
▲✉➟♥ ✈➠♥ ✤÷ñ❝ t❤ü❝ ❤✐➺♥ t↕✐ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❱✐♥❤ ❞÷î✐ sü ❤÷î♥❣
❞➝♥ ❝õ❛ t❤➛② ❣✐→♦✱ ❚❙✳ ◆❣✉②➵♥ ❱➠♥ ✣ù❝✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥
s➙✉ s➢❝ ❝õ❛ ♠➻♥❤ ✤➳♥ ❚❤➛②✳ ◆❤➙♥ ❞à♣ ♥➔②✱ t→❝ ❣✐↔ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠
ì♥ ❇❛♥ ❝❤õ ♥❤✐➺♠ ♣❤á♥❣ ❙❛✉ ✤↕✐ ❤å❝✱ ❇❛♥ ❝❤õ ♥❤✐➺♠ ❦❤♦❛ ❚♦→♥ ❤å❝
✈➔ ❝↔♠ ì♥ ❝→❝ t❤➛②✱ ❝æ ❣✐→♦ tr♦♥❣ ❜ë ♠æ♥ ●✐↔✐ t➼❝❤✱ ❦❤♦❛ ❚♦→♥ ❤å❝ ✤➣
♥❤✐➺t t➻♥❤ ❣✐↔♥❣ ❞↕② ✈➔ ❣✐ó♣ ✤ï t→❝ ❣✐↔ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❤å❝ t➟♣ ✈➔

t ữỡ ố ũ t ỡ
ỗ t tr ợ ồ t
ở t ú ù ở t tr sốt q tr ồ t
ự
ũ õ ố ữ ổ tr ọ
ỳ t sõt ú tổ rt ữủ ỳ ỵ
õ õ ừ t ổ ữủ
t ỡ
t




❈❍×❒◆● ✶

▼❐❚ ❙➮ ❑■➌◆ ❚❍Ù❈ ❇✃ ❚❘Ñ
❈❤÷ì♥❣ ♥➔② ♥❤➡♠ ♠ö❝ ✤➼❝❤ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❧✐➯♥ q✉❛♥ ✤➳♥
♥ë✐ ❞✉♥❣ ❝❤÷ì♥❣ ✷✱ ❝❤õ ②➳✉ ✤÷ñ❝ ❝❤ó♥❣ tæ✐ t❤❛♠ ❦❤↔♦ tr♦♥❣ t➔✐ ❧✐➺✉ ❬✹❪
✈➔ ❬✺❪✳

✶✳✶ ❍➔♠ ●❛♠♠❛

✶✳✶✳✶ ✣à♥❤ ♥❣❤➽❛✳ ❍➔♠ ❣❛♠♠❛ Γ ❤❛② t➼❝❤ ♣❤➙♥ ❊✉❧❡r ❧♦↕✐ ✷ ❧➔ t➼❝❤

♣❤➙♥

∞

Γ(z) =

e−t tz−1 dt

✭✶✳✶✮

0

✈î✐ z t❤✉ë❝ ♥û❛ ♠➦t ♣❤➥♥❣ ❜➯♥ ♣❤↔✐ ❝õ❛ ♠➦t ♣❤➥♥❣ ♣❤ù❝ ❘❡z > 0✳

✶✳✶✳✷ ◆❤➟♥ ①➨t✳ ❚➼❝❤ ♣❤➙♥ ✭✶✳✶✮ ❤ë✐ tö ✈î✐ ♠å✐ z ∈ C t❤ä❛ ♠➣♥ ❘❡z >
0✳ ❚❤➟t ✈➟②✱ ✈î✐ z ∈ C t❤ä❛ ♠➣♥ ❘❡z > 0✱ t❛ ❝â t❤➸ ❜✐➸✉ ❞✐➵♥ z = x + iy

✈î✐ x, y ∈ R ✈➔ x > 0✳ ❑❤✐ ✤â t❛ ❝â
∞

Γ(z) = Γ(x + iy) =
∞

=
0

=

∞

e−t tx−1+iy dt

0

e−t tx−1 eiy ln t dt

e−t tx−1 (cos(y ln t) + i sin(y ln t)) dt.

✭✶✳✷✮

0

❱➻ ✤↕✐ ❧÷ñ♥❣ (cos(y ln t) + i sin(y ln t)) ❜à ❝❤➦♥ ♥➯♥ ❞➵ ♥❤➟♥ t❤➜② r➡♥❣ t➼❝❤
♣❤➙♥ ✭✶✳✷✮ ❤ë✐ tö ✈î✐ ♠å✐ x > 0 ✈➔ ♠å✐ y ∈ R✳
✻


✼

✶✳✶✳✸ ✣à♥❤ ❧þ✳ ❍➔♠ ❣❛♠♠❛ Γ ❝â ❝→❝ t➼♥❤ ❝❤➜t s❛✉
✶✮ Γ(z + 1) = zΓ(z), ∀z ∈ C, ❘❡z > 0✱
✷✮ Γ(1) = 1✱
✸✮ Γ(n + 1) = n!, ∀n ∈ N∗ ✱
√
1
= π✱
✹✮ Γ
2
(2n)! √
1
= 2n
π✳
✺✮ Γ n +
2
2 n!
❈❤ù♥❣ ♠✐♥❤✳ 1) ❱î✐ ♠å✐ z ∈ C, ❘❡z > 0 t❛ ❝â

∞

Γ(z + 1) =

e−t tz dt

0
−t z

= −e t

∞

t=∞

+z
t=0

e−t tx−1+iy dt

0

= zΓ(z).
2) ❚❛ ❝â
∞

−t 0

−t

e t dt = −e

Γ(1) =

t=∞

= 1.
t=0

0

3) ❙û ❞ö♥❣ ❝→❝ t➼♥❤ ❝❤➜t 1) ✈➔ 2)✱ t❛ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ ❝❤➜t 3)

❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ q✉② ♥↕♣✳
∞ −x2
dx✳
0 e

4) ❚r÷î❝ ❤➳t t❛ t➼♥❤ t➼❝❤ ♣❤➙♥ I =

∞

I=u

✣➦t x = ut, u > 0✱ t❛ ❝â

2 2

e−u t dt.

✭✶✳✸✮

0

◆❤➙♥ ❤❛✐ ✈➳ ❝õ❛ ✭✶✳✸✮ ✈î✐ e−u ✈➔ ❧➜② t➼❝❤ ♣❤➙♥ tø 0 ✤➳♥ ∞ t❛ ✤÷ñ❝
2

∞
2

I =

e

−u2

0

u

2 2

e−u t dt du

0
∞

∞

=

0

1
=
2

∞

∞
0

2

e−u

(1+t2 )

0

dt
π
=
.
1 + t2
4

udu dt

ớ ổ tự tỹ
2
ờ t = u2 t s t ữủ

õ I =

x2
dx
0 e

=



(z) = 2

2

eu u2z1 du.



0

t z =

1
t ữủ
2


1
2



=2

2

eu du = 2I =



.

0

5) ứ t t 1) t t 4) t õ
n+

1
2

1
1

n
2
2
3
3
1
n
n
= n
2
2
2
3
3 1
1
1
n
ã ã ã . .
= n
2
2
2 2
2
(2n)!
.
= 2n
2 n!
=

n

ỵ ợ ồ z C tọ z > 0 t õ
n!nz
n z(z + 1) ã ã ã (z + n)

(z) = lim



ự ự ổ tự trữợ t ú t t
n

fn (z) =
0

ờ =

t
n

t
1
n

n

tz1 dt.

s õ sỷ ử t tứ ú t




t ữủ
1
z

(1 )n z1 d

fn (z) = n

0
1

nz

=

z
= ããã

(1 )n1 z d

n
0

n!nz
=

z(z + 1) ã ã ã (z + n 1)
n!nz
=
.
z(z + 1) ã ã ã (z + n)

1

z+n1 d
0



ú ỵ r
lim

n

n

t
1
n

= et .

õ ử t t ừ ú t ự tự
n

t

1
lim fn (z) = lim
n
n 0
n

t
=
lim 1
n
0 n

n

tz1 dt
n


z1

t

dt =

et tz1 dt = (z).

0

t ữủ ử ú t ữủ


= (z) fn (z) =
0
n

=

t

e
0

t
1
n

et tz1 dt fn (z)
n


z1

t

dt +

et tz1 dt.

n

> 0 tũ ỵ ứ sỹ ở tử ừ t ợ ồ z C tọ
z > 0 t s r tỗ t số tỹ n0 s ợ ồ n N
n

n0 t õ


t z1

e t
n



dt
n


et tx1 dt < , (x = z).
3




✶✵

❱î✐ ♠å✐ n ∈ N∗ ♠➔ n > n0 ✱ t❛ ✈✐➳t ∆ t❤➔♥❤ tê♥❣ ❝õ❛ ❜❛ t➼❝❤ ♣❤➙♥ s❛✉
N

∆=

e

−t

0
n

tz−1 dt
n

t
− 1−
n

−t

+

n

t
− 1−
n

e

N

∞
z−1

t

dt +

e−t tz−1 dt.

✭✶✳✶✶✮

n

❚❛ ❝â
n

e
N

−t

n

t
− 1−
n

n

n

t

z−1

t
dt
e − 1−
tx−1 dt
n
N
∞
ε
<
e−t tx−1 dt < , (x = ❘❡z).
3
n
✭✶✳✶✷✮
−t

✣➸ ✤→♥❤ ❣✐→ t➼❝❤ ♣❤➙♥ t❤ù ♥❤➜t ð ✭✶✳✶✶✮✱ t❛ ❝➛♥ ❜➜t ✤➥♥❣ t❤ù❝ ❜ê trñ
s❛✉
−t

0
t
− 1−

n

n

t2
<
, 0 < t < n.
2n

✭✶✳✶✸✮

❚❤➟t ✈➟②✱ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✶✸✮ ✤÷ñ❝ s✉② r❛ tø ♠è✐ q✉❛♥ ❤➺
−t

e

t
− 1−
n

n

t

eτ 1 −

=
0

τ

n

n

τ
dτ
n

✈➔ ❜➜t ✤➥♥❣ t❤ù❝
t

τ
1−
n

τ

0<

e
0

n

τ
dτ <
n

t
0

t2
e dτ = e
.
n
2n
ττ

t

❙û ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ✭✶✳✶✸✮ t❛ ❝â ✤→♥❤ ❣✐→ s❛✉ ✈î✐ n ✤õ ❧î♥
N

e
0

−t

t
− 1−
n

n

t

z−1

1
dt <

2n

N
0

ε
tx+1 dt < , (x = ❘❡z). ✭✶✳✶✹✮
3

✣➥♥❣ t❤ù❝ ✭✶✳✽✮ ❜➙② ❣✐í ✤÷ñ❝ s✉② r❛ tø ✭✶✳✶✵✮✱ ✭✶✳✶✶✮✱ ✭✶✳✶✷✮ ✈➔ ✭✶✳✶✹✮✳
✣à♥❤ ❧þ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤✳

✶✳✶✳✺ ◆❤➟♥ ①➨t✳ ◆❤í t➼♥❤ ❝❤➜t 1) tr♦♥❣ ✣à♥❤ ❧þ ✶✳✶✳✸✱ ♥❣÷í✐ t❛ ❝â t❤➸

✤à♥❤ ♥❣❤➽❛ ❤➔♠ ❣❛♠♠❛ Γ(z) ❝❤♦ ♠å✐ z ∈ C ♠➔ z = 0, −1, −2, · · · ❈ö


✶✶

t❤➸✱ ♥➳✉ z ∈ C ♠➔ −m < ❘❡z

−m + 1 ✈î✐ m ❧➔ ♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣

♥➔♦ ✤â t❤➻ t❛ ①→❝ ✤à♥❤ Γ(z) t❤❡♦ ❝æ♥❣ t❤ù❝
Γ(z) =

Γ(z + m)
.
z(z + 1) · · · (z + m − 1)

✭✶✳✶✺✮

✶✳✶✳✻ ◆❤➟♥ ①➨t✳ ❈æ♥❣ t❤ù❝ ✭✶✳✺✮ ❦❤æ♥❣ ❝❤➾ ✤ó♥❣ ✈î✐ ♠å✐ z ∈ C t❤ä❛

♠➣♥ ❘❡z > 0 ♠➔ ❝á♥ ✤ó♥❣ ✈î✐ ♠å✐ z ∈ C ♠➔ z = 0, −1, −2, · · · ❚❤➟t
✈➟②✱ tø ❝æ♥❣ t❤ù❝ ✭✶✳✶✺✮ ✈➔ ✣à♥❤ ❧þ ✶✳✶✳✹ t❛ ❝â
Γ(z + m)
z(z + 1) · · · (z + m − 1)
nz+m n!
1
lim
=
z(z + 1) · · · (z + m − 1) n→∞ (z + m) · · · (z + m + n)
1
nz n!
=
lim
z(z + 1) · · · (z + m − 1) n→∞ (z + m)(z + m + 1) · · · (z + n)
nm
× lim
n→∞ (z + n)(z + n + 1) · · · (z + n + m)

Γ(z) =

1
nz n!
lim
z(z + 1) · · · (z + m − 1) n→∞ (z + m)(z + m + 1) · · · (z + n)
1
nz n!

= lim
n→∞ z(z + 1) · · · (z + m − 1) (z + m)(z + m + 1) · · · (z + n)
n!nz
= lim
.
n→∞ z(z + 1) · · · (z + n)

=

✶✳✷ ❍➔♠ ▼✐tt❛❣✲▲❡❢❢❧❡r

✶✳✷✳✶ ✣à♥❤ ♥❣❤➽❛✳ ❍➔♠ ▼✐tt❛❣✲▲❡❢❢❧❡r ♠ët t❤❛♠ sè ❧➔ ❤➔♠ ❝â ❞↕♥❣
∞

Eα (z) =
k=0

zk
, α > 0, z ∈ C.
Γ(αk + 1)

✭✶✳✶✻✮

❍➔♠ ▼✐tt❛❣✲▲❡❢❢❧❡r ❤❛✐ t❤❛♠ sè ❧➔ ❤➔♠ ❝â ❞↕♥❣
∞

Eα,β (z) =
k=0

zk

, α > 0, β > 0, z ∈ C.
Γ(αk + β)

✭✶✳✶✼✮




t ợ ồ z C z = 0 t õ


zk
= E (z), > 0,
(k + 1)

1) E,1 (z) =
k=0


2) E1,1 (z) =
k=0


3) E1,2 (z) =
k=0


zk
=
(k + 1)

zk
=
(k + 2)

2

4) E2,1 (z ) =
k=0

2

5) E2,2 (z ) =
k=0




k=0


zk
= ez ,
k!

k=0


z 2k
=
(2k + 1)

k=0

z 2k+1
sinh(z)
=
,
(2k + 1)
z


zk

6) E1/2,1 (z) =
k=0

k=0

z (k+1)
ez 1
=
,
(k + 1)!
z

z 2k
= cosh(z),
(2k)!

k=0

z 2k
1
=
(2k + 2) z



zk
1
=
(k + 1)! z

2 z2

=
e

( k2 + 1)

2

2

et dt = ez r(z).

z

ú t (, ) ( > 0, 0 <

) ữớ ỗ t

s
arg = | |


arg



| | =

arg = | |



ữớ (, ) ( > 0, 0 <

) t ự t

t G (, ) G+ (, ) ữủt tr
ừ ữớ (, )
0 < < t G (, ) G+ (, )
ổ = t G (, ) tr t trỏ
| | <

ỵ 0 < < 2 số ự tũ ỵ à số tỹ tọ

< à < min{, }.
2




✶✸

❑❤✐ ✤â ✈î✐ ε > 0 tò② þ t❛ ❝â

exp(ζ 1/α )ζ (1−β)/α
1
dζ, z ∈ G− (ε, µ)
2απi γ(ε,µ)
ζ −z
1
Eα,β (z) = z (1−β)/α exp z 1/α
α
1
exp(ζ 1/α )ζ (1−β)/α
dζ, z ∈ G+ (ε, µ).
+
2απi γ(ε,µ)
ζ −z
Eα,β (z) =

✭✶✳✶✾✮

✭✶✳✷✵✮

❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ |z| < ε t❤➻

z
< 1, ζ ∈ γ(ε, µ).
ζ

✭✶✳✷✶✮

❉♦ ✤â ✈î✐ 0 < α < 2 ✈➔ |z| < ε t❛ ❝â
∞

Eα,β (z) =
k=0

1
2απi

1
=
2απi
=

1
2απi

exp(ζ 1/α )ζ (1−β)/α−k−1 dζ
γ(ε,µ)
∞

exp(ζ

1/α

)ζ

(1−β)/α−1

γ(ε,µ)

k=0

exp(ζ 1/α )ζ (1−β)/α
γ(ε,µ)

zk

ζ −z

z
ζ

dζ.

dζ

✭✶✳✷✷✮

◆❤í ✤✐➲✉ ❦✐➺♥ ✭✶✳✶✽✮✱ t➼❝❤ ♣❤➙♥ ð ✭✶✳✷✷✮ ❤ë✐ tö t✉②➺t ✤è✐ ✈➔ ①→❝ ✤à♥❤ ♠ët
❤➔♠ ❜✐➳♥ z ❣✐↔✐ t➼❝❤ tr♦♥❣ ❝→❝ ♠✐➲♥ G− (ε, µ) ✈➔ G+ (ε, µ)✳ ▼➦t ❦❤→❝✱ ✈î✐

πα
, min{π, πα} ✱ ❤➻♥❤ trá♥ |z| < ε ♥➡♠ tr♦♥❣ ♠✐➲♥ G− (ε, µ)✳
♠é✐ µ ∈
2
❉♦ ✤â✱ t➼❝❤ ♣❤➙♥ ✭✶✳✷✷✮ ❜➡♥❣ Eα,β (z) ❦❤æ♥❣ ❝❤➾ tr♦♥❣ ♠✐➲♥ ❤➻♥❤ trá♥
|z| < ε ♠➔ tr♦♥❣ t♦➔♥ ❜ë ♠✐➲♥ G− (ε, µ) ♥➯♥ t❛ ❝â ❝æ♥❣ t❤ù❝ ✭✶✳✶✾✮✳

❇➙② ❣✐í ❝❤ó♥❣ t❛ ❧➜② z ∈ G+ (ε, µ)✳ ❑❤✐ ✤â ✈î✐ ❜➜t ❦ý ε1 > |z| t❛ ❝â
z ∈ G− (ε1 , µ)✳ ❙û ❞ö♥❣ ❝æ♥❣ t❤ù❝ ✭✶✳✶✾✮ t❛ ❝â
Eα,β (z) =

1
2απi

exp(ζ 1/α )ζ (1−β)/α
dζ, z ∈ G− (ε1 , µ).
ζ −z
γ(ε1 ,µ)

✭✶✳✷✸✮

▼➦t ❦❤→❝✱ ♥➳✉ ε < |z| < ε1 ✈➔ −µ < arg(z) < µ t❤➻ t❤❡♦ ✤à♥❤ ❧þ ❈❛✉❝❤②


✶✹

t❛ ❝â
1
2απi

exp(ζ 1/α )ζ (1−β)/α
1
dζ = z (1−β)/α exp z 1/α .
ζ −z
α
γ(ε1 ,µ)−γ(ε,µ)
✭✶✳✷✹✮

❚ê ❤ñ♣ ✭✶✳✷✸✮ ✈➔ ✭✶✳✷✹✮ t❛ ✤↕t ✤÷ñ❝ ❝æ♥❣ t❤ù❝ ✭✶✳✷✵✮✳

✶✳✷✳✹ ✣à♥❤ ❧þ✳ ◆➳✉ 0 < α < 2✱ β ❧➔ sè ♣❤ù❝ tò② þ ✈➔ µ ❧➔ sè t❤ü❝ t❤ä❛
♠➣♥

πα
< µ < min{π, πα}
2
t❤➻ ✈î✐ sè ♥❣✉②➯♥ ❞÷ì♥❣ p tò② þ t❛ ❝â ✤→♥❤ ❣✐→

1
Eα,β (z) = z (1−β)/α exp z 1/α −
α
|z| → ∞, | arg(z)|

p

k=1

z −k
+ O |z|−1−p
Γ(β − αk)

✭✶✳✷✺✮

µ.

❈❤ù♥❣ ♠✐♥❤✳ ❈❤ó♥❣ t❛ ❜➢t ✤➛✉ ❝❤ù♥❣ ♠✐♥❤ ❝æ♥❣ t❤ù❝ ✭✶✳✷✺✮ ❜➡♥❣ ❝→❝❤

❝❤å♥ ϕ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥
πα
< µ < ϕ < min{π, πα}.
2
❚✐➳♣ t❤❡♦ ❝❤å♥ ε = 1 ✈➔ t❤❛② t❤➳
1
=−
ζ −z

p

k=1

ζp
ζ k−1
+ p
z (ζ − z)
zk

✭✶✳✷✻✮

✭✶✳✷✼✮

tr♦♥❣ ❝æ♥❣ t❤ù❝ ✭✶✳✷✵✮ ❝õ❛ ✣à♥❤ ❧þ ✶✳✷✳✸ ❝❤ó♥❣ t❛ ❝â ❝æ♥❣ t❤ù❝ ❜✐➸✉ ❞✐➵♥
s❛✉ ✤➙② ❝❤♦ ❤➔♠ ▼✐tt❛❣✲▲❡❢❢❧❡r Eα,β (z) tr♦♥❣ ♠✐➲♥ G+ (1, ϕ) ✭♥❣❤➽❛ ❧➔
♠✐➲♥ ❜➯♥ ♣❤↔✐ ❝õ❛ ✤÷í♥❣ γ(1, ϕ)✮✳
1
Eα,β (z) = z (1−β)/α exp z 1/α
α
p
1
−
exp(ζ 1/α )ζ (1−β)/α+k−1 dζ z −k
2απi γ(1,ϕ)
k=1

+

1
2απiz p

exp(ζ 1/α )ζ (1−β)/α+p ζ.
γ(1,ϕ)

✭✶✳✷✽✮


✶✺

❈❤ó þ r➡♥❣
1
2απi

exp(ζ 1/α )ζ (1−β)/α+k−1 dζ =
γ(1,ϕ)

1
, k ∈ N∗ .
Γ(β − αk)

✭✶✳✷✾✮

❚❤❛② t❤➳ ✭✶✳✷✾✮ ✈➔♦ ✭✶✳✷✽✮ t❛ ✤↕t ✤÷ñ❝
1
Eα,β (z) = z (1−β)/α exp z 1/α −
α
1
2απiz p
(| arg(z)|

p

k=1

z −k
Γ(β − αk)

exp(ζ 1/α )ζ (1−β)/α+p ζ,

+

✭✶✳✸✵✮

γ(1,ϕ)

µ, |z| > 1).

❈❤ó♥❣ t❛ ❤➣② ✤→♥❤ ❣✐→ t➼❝❤ ♣❤➙♥
Ip (z) =

1
2απiz p

✈î✐ |z| ✤õ ❧î♥ ✈➔ | arg(z)|

✭✶✳✸✶✮

exp(ζ 1/α )ζ (1−β)/α+p ζ,
γ(1,ϕ)

µ✳ ❱î✐ |z| ✤õ ❧î♥ ✈➔ | arg(z)|

µ t❛ ❝â

min |ζ − z| = |z| sin(ϕ − µ).
ζ∈γ(1,ϕ)

❉♦ ✤â ✈î✐ |z| ✤õ ❧î♥ ✈➔ | arg(z)|
|Ip (z)|

|z|−1−p
2πα sin(ϕ − µ)

µ t❛ ✤↕t ✤÷ñ❝
exp(ζ 1/α ) ζ (1−β)+p ζ.

✭✶✳✸✷✮

γ(1,ϕ)

❚➼❝❤ ♣❤➙♥ ð ✈➳ ♣❤↔✐ ❝õ❛ ✭✶✳✸✷✮ ❤ë✐ tö✱ ❜ð✐ ✈➻ ✈î✐ ζ t❤ä❛ ♠➣♥ arg(ζ) =
±ϕ ✈➔ |ζ|

1 t❛ ❝â
exp(ζ 1/α ) = exp |ζ|1/α cos

ϕ
α

ϕ
< 0 ❞♦ ✤✐➲✉ ❦✐➺♥ ✭✶✳✷✻✮✳ ❚ê ❤ñ♣ ✭✶✳✸✵✮ ✈➔ ✭✶✳✸✷✮ t❛ ✤↕t ✤÷ñ❝
α
❝æ♥❣ t❤ù❝ ✭✶✳✷✺✮✳

✈➔ cos

✶✳✷✳✺ ✣à♥❤ ❧þ✳ ◆➳✉ 0 < α < 2✱ β ❧➔ sè ♣❤ù❝ tò② þ ✈➔ µ ❧➔ sè t❤ü❝ t❤ä❛
♠➣♥

πα
< µ < min{π, πα}
2

✶✻

t❤➻ ✈î✐ sè ♥❣✉②➯♥ ❞÷ì♥❣ p tò② þ t❛ ❝â ✤→♥❤ ❣✐→
p

Eα,β (z) = −
k=1

z −k
+ O |z|−1−p
Γ(β − αk)

|z| → ∞, µ

| arg(z)|

✭✶✳✸✸✮

π.

❈❤ù♥❣ ♠✐♥❤✳ ❚r÷î❝ ❤➳t✱ ❝❤ó♥❣ t❛ ❧➜② ϕ t❤ä❛ ♠➣♥

πα
< ϕ < µ < min{π, πα}.
2

✭✶✳✸✹✮

❚✐➳♣ t❤❡♦ ❝❤å♥ ε = 1 tr♦♥❣ ❝æ♥❣ t❤ù❝ ✭✶✳✷✵✮ ❝õ❛ ✣à♥❤ ❧þ ✶✳✷✳✸ ✈➔ sû ❞ö♥❣

❝æ♥❣ t❤ù❝ ✭✶✳✷✼✮ t❛ ✤↕t ✤÷ñ❝
p

Eα,β (z) = −
k=1

z −k
+ Ip (z), z ∈ G− (1, ϕ)
Γ(β − αk)

✭✶✳✸✺✮

✈î✐ Ip (z) ✤➣ ✤÷ñ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ✣à♥❤ ❧þ ✶✳✷✳✹✳ ❱î✐ |z|
✤õ ❧î♥ ✈➔ µ

| arg(z)|

π t❛ ❝â

min |ζ − z| = |z| sin(ϕ − µ).
ζ∈γ(1,ϕ)

❍ì♥ ♥ú❛✱ ♠✐➲♥ µ

π ♥➡♠ tr♦♥❣ ♠✐➲♥ G− (1, ϕ) ♥➯♥ ✤➥♥❣

| arg(z)|

t❤ù❝ ✭✶✳✸✺✮ ✤ó♥❣ ✈î✐ z t❤ä❛ ♠➣♥ µ
✈➔ µ

| arg(z)|
|Ip (z)|

| arg(z)|

π ✳ ❉♦ ✤â ✈î✐ |z| ✤õ ❧î♥

π t❛ ✤↕t ✤÷ñ❝

|z|−1−p
2πα sin(ϕ − µ)

exp(ζ 1/α ) ζ (1−β)+p ζ.

✭✶✳✸✻✮

γ(1,ϕ)

❚ê ❤ñ♣ ✭✶✳✸✺✮ ✈➔ ✭✶✳✸✻✮ t❛ ✤↕t ✤÷ñ❝ ❝æ♥❣ t❤ù❝ ✭✶✳✸✸✮✳

✶✳✸ ▼ët sè ❦❤æ♥❣ ❣✐❛♥ ❤➔♠

✶✳✸✳✶ ✣à♥❤ ♥❣❤➽❛✳ ●✐↔ sû u, v ∈ L1❧♦❝ (U ) ✈➔ α ❧➔ ♠ët ✤❛ ❝❤➾ sè✳ ❚❛ ♥â✐
r➡♥❣ v ❧➔ ✤↕♦ ❤➔♠ ②➳✉ ❝➜♣ α ❝õ❛ u✱ ❦þ ❤✐➺✉ ❧➔
Dα u = v,


✶✼

♥➳✉
uDα φdx = (−1)|α|

✈î✐ ♠å✐ ❤➔♠ t❤û φ ∈

vφdx

U
Cc∞ (U )✳

U

✶✳✸✳✷ ❇ê ✤➲✳ ✭❚➼♥❤ ❞✉② ♥❤➜t ❝õ❛ ✤↕♦ ❤➔♠ ②➳✉✮ ▼ët ✤↕♦ ❤➔♠ ②➳✉ ❝➜♣
α ❝õ❛ u ♥➳✉ tç♥ t↕✐ t❤➻ ✤÷ñ❝ ①→❝ ✤à♥❤ ♠ët ❝→❝❤ ❞✉② ♥❤➜t✱ s❛✐ ❦❤→❝ tr➯♥
t➟♣ ❝â ✤ë ✤♦ ❦❤æ♥❣✳
❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû r➡♥❣ u, v ∈ L1❧♦❝ (U ) t❤ä❛ ♠➣♥

uDα φdx = (−1)|α|
U

vφdx = (−1)|α|
U

vφdx
U

✈î✐ ♠å✐ φ ∈ Cc∞ (U )✳ ❑❤✐ ✤â
(v − v) φdx = 0
U

✈î✐ ♠å✐ φ ∈ Cc∞ (U )✳ ❙✉② r❛ v − v = 0 ❤➛✉ ❦❤➢♣ ♥ì✐ ✭❤✳❦✳♥✮✳

✶✳✸✳✸ ❈→❝ ✈➼ ❞ö✳ ✶✳ ❈❤♦ n = 1✱ U = (0, 2) ✈➔
u (x) =

❳→❝ ✤à♥❤
v (x) =

x ♥➳✉ 0 < x
1 ♥➳✉ 1

1

x < 2.

1 ♥➳✉ 0 < x

1

0 ♥➳✉ 1 < x < 2.

❚❛ ❝❤ù♥❣ tä u = v t❤❡♦ ♥❣❤➽❛ ②➳✉✳ ❚❤➟t ✈➟②✱ ❝❤å♥ ❜➜t ❦ý φ ∈ Cc∞ (U )✳
❚❛ ❝➛♥ ❝❤➾ r❛

2

2

uφ dx = −
0

vφdx.
0

❚❤➟t ✈➟②✱
2

1

uφ dx =
0

2

xφ dx +
0

φ dx
1

1

=−

2

φdx + φ (1) − φ (1) = −
0

vφdx.

0


✶✽

✷✳ ❱î✐ n = 1✱ U = (0, 2) ✈➔
x ♥➳✉ 0 < x

u (x) =

1

2 ♥➳✉ 1 < x < 2.

❚❛ ❦❤➥♥❣ ✤à♥❤ ❦❤æ♥❣ tç♥ t↕✐ ✤↕♦ ❤➔♠ ②➳✉ u ✳ ▼✉è♥ t❤➳✱ t❛ ❝➛♥ ❝❤➾ r❛
r➡♥❣ ❦❤æ♥❣ tç♥ t↕✐ ❜➜t ❦ý ❤➔♠ v ∈ L✶❧♦❝ (U ) ♥➔♦ t❤ä❛ ♠➣♥
2

2

uφ dx = −

✭✶✳✸✼✮

vφdx

0

0

✈î✐ ♠å✐ φ ∈ Cc∞ (U )✳
●✐↔ sû ♥❣÷ñ❝ ❧↕✐✱ ❝â ❤➔♠ v ♥➔♦ ✤â t❤ä❛ ♠➣♥ ✭✶✳✸✼✮✳ ❑❤✐ ✤â
2

−

2

vφdx =
0

1

uφ dx =

2

xφ dx + 2

0

φ dx

0

✭✶✳✸✽✮

1
1

=−

φdx − φ (1) .
0

▲➜② ♠ët ❞➣② ❤➔♠ trì♥ {φm }∞
m=1 t❤ä❛ ♠➣♥
0

φm

1, φm (1) = 1, φm (x) → 0 ✈î✐ ♠å✐ x = ✶.

❚r♦♥❣ ✭✶✳✸✽✮ t❤❛② φ ❜ð✐ φm ✈➔ ❝❤♦ m → ∞✱ t❛ ❝â
1

2

vφm dx −

1 = lim φm (1) = lim
m→∞

m→∞

φm dx = 0,
0

0

♠➙✉ t❤✉➝♥✳

✶✳✸✳✹ ✣à♥❤ ♥❣❤➽❛✳ ❑❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ W k,p (U ) ❜❛♦ ❣ç♠ t➜t ❝↔ ♥❤ú♥❣
❤➔♠ ❦❤↔ tê♥❣ ✤à❛ ♣❤÷ì♥❣ u : U → R s❛♦ ❝❤♦ ✈î✐ ♠é✐ ✤❛ ❝❤➾ sè α, |α|

✤↕♦ ❤➔♠ ②➳✉ Dα u ❧➔ tç♥ t↕✐ ✈➔ t❤✉ë❝ Lp (U )✳

✶✳✸✳✺ ✣à♥❤ ♥❣❤➽❛✳ ◆➳✉ u ∈ W k,p (U )✱ t❛ ✤à♥❤ ♥❣❤➽❛ ❝❤✉➞♥ ❝õ❛ ♥â ❧➔
u

W k,p (U )

:=

 p1









|Dα u|p dx
|α| k









ess supU |Dα u|
|α| k

(1

p < ∞)

U

(p = ∞) .

k✱


✶✾

✶✳✸✳✻ ✣à♥❤ ♥❣❤➽❛✳ ✐✮ ❈❤♦ {um}∞m=1 ⊂ W k,p (U )✳ ❚❛ ♥â✐ um ❤ë✐ tö ✤➳♥

u tr♦♥❣ W k,p (U )✱ ❦þ ❤✐➺✉ ❧➔ um → u tr♦♥❣ W k,p (U ) ♥➳✉
lim

m→∞

um − u

W k,p (U )

= 0.

k,p
✐✐✮ ❚❛ ♥â✐ um → u tr♦♥❣ W❧♦❝
(U ) ♥➳✉ um → u tr♦♥❣ W k,p (V ) ✈î✐ ♠é✐

V ⊂⊂ U ✳

✶✳✸✳✼ ✣à♥❤ ♥❣❤➽❛✳ ❚❛ ♥â✐ W0k,p (U ) ❧➔ ❜❛♦ ✤â♥❣ ❝õ❛ Cc∞ (U ) tr♦♥❣

W k,p (U )✳ ◆❤÷ ✈➟②✱ u ∈ W0k,p (U ) ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ tç♥ t↕✐ ❝→❝ ❤➔♠ um ∈

Cc∞ (U ) s❛♦ ❝❤♦ um → u tr♦♥❣ W k,p (U )✳ ❚❛ ❝♦✐ W0k,p (U ) ♥❤÷ ❧➔ t➟♣ ❤ñ♣

♥❤ú♥❣ ❤➔♠ u ∈ W k,p (U ) s❛♦ ❝❤♦
Dα u = 0 tr➯♥ ∂u ✈î✐ ♠å✐ |α|

k − 1.

❑þ ❤✐➺✉✿ H0k (U ) = W0k,2 (U )

✶✳✸✳✽ ✣à♥❤ ❧þ✳ ❱î✐ ♠é✐ k = 1, 2, ... ✈➔ 1

∞✱ ❦❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈

p

W k,p (U ) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳
❈❤ù♥❣ ♠✐♥❤✳ ✶✳ ❚r÷î❝ ❤➳t t❛ ❦✐➸♠ tr❛ r➡♥❣ u

r➔♥❣ λu

W k,p (U )

= |λ| u

W k,p (U )

✈➔ u

W k,p (U )

❧➔ ♠ët ❝❤✉➞♥✳ ❘ã

= 0 ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ u =

W k,p (U )

0 ❤➛✉ ❦❤➢♣ ♥ì✐✳

❚✐➳♣ t❤❡♦ ❣✐↔ sû u, v ∈ W k,p (U )✳ ❑❤✐ ✤â ✈î✐ 1
t❤ù❝ ▼✐♥❦♦✇s❦✐ t❛ ❝â

u+v

W k,p (U )

p < ∞✱ t❤❡♦ ❜➜t ✤➥♥❣

 p1
Dα u + Dα v

=

p

Lp (U )

|α| k

 p1


Dα u



p
Lp (U )

+ Dα v

p

Lp (U )

|α| k

 p1


Dα u



p

Lp (U )

W k,p (U )

Dα v

+

|α| k

= u

 p1


|α| k

+ v

W k,p (U ) .

p


Lp (U )


✷✵

✷✳ ❈❤ù♥❣ tä r➡♥❣ W k,p (U ) ❧➔ ✤➛② ✤õ✳ ●✐↔ sû {um }∞
m=1 ❧➔ ♠ët ❞➣② ❈❛✉❝❤②
k, {Dα um }∞
m=1 ❧➔ ♠ët ❞➣② ❈❛✉❝❤②

tr♦♥❣ W k,p (U )✳ ❑❤✐ ✤â ✈î✐ ♠é✐ |α|

tr♦♥❣ Lp (U )✳ ❱➻ Lp (U ) ❧➔ ✤➛② ✤õ ♥➯♥ tç♥ t↕✐ ❝→❝ ❤➔♠ uα ∈ Lp (U ) s❛♦
❝❤♦
Dα um → uα tr♦♥❣ Lp (U )

✈î✐ ♠é✐ |α|

k ✳ ❚r♦♥❣ tr÷í♥❣ ❤ñ♣ ♥➔②
um → u(0,0,...0) =: u tr♦♥❣ Lp (U ) .

✸✳ ❇➙② ❣✐í t❛ ❦❤➥♥❣ ✤à♥❤
u ∈ W k,p (U ) , Dα u = uα

(|α|

k) .

✭✶✳✸✾✮

✣➸ t❤û ❧↕✐ ❦❤➥♥❣ ✤à♥❤ ♥➔②✱ t❛ ❝è ✤à♥❤ φ ∈ Cc∞ (U )✳ ❑❤✐ ✤â
uDα φdx = lim

m→∞ U

U

um Dα φdx

= lim (−1)|α|
m→∞

Dα um φdx
U

= (−1)|α|

uα φdx.
U

◆❤÷ ✈➟② ❝æ♥❣ t❤ù❝ ✭✶✳✸✾✮ ✤ó♥❣✳ ❇ð✐ ✈➟② Dα um → Dα u tr♦♥❣ Lp (U ) ✈î✐
♠å✐ |α|

k ✱ t❛ t❤➜② r➡♥❣ um → u tr♦♥❣ W k,p (U ) .

✶✳✸✳✾ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱ ❦❤æ♥❣ ❣✐❛♥ Lp (0, T ; X)

❣ç♠ t➜t ❝↔ ❝→❝ ❤➔♠ ✤♦ ✤÷ñ❝ u : [0, T ] → X ✈î✐
✐✮ u

✐✐✮ u

Lp (0,T ;X)

T
0

=

L∞ (0,T ;X)

p

u (t) dt

1
p

< ∞ ♥➳✉ 1

p < ∞ ✈➔

= ❡ss sup u (t) < ∞✳
0 t T

✶✳✸✳✶✵ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ X

❧➔ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✱ ❦❤æ♥❣ ❣✐❛♥

C ([0, T ] ; X) ❜❛♦ ❣ç♠ t➜t ❝↔ ❝→❝ ❤➔♠ ❧✐➯♥ tö❝ u : [0, T ] → X ✈î✐

u

C([0,T ];X)

= sup
0 t T

u (t) < ∞.


✷✶

✶✳✸✳✶✶ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ u ∈ L1 (0, T ; X)✳ ❚❛ ♥â✐ v ∈ L1 (0, T ; X) ❧➔
✤↕♦ ❤➔♠ ②➳✉ ❝õ❛ u✱ ❦þ ❤✐➺✉ u = v ♥➳✉
T

T

φ (t) u (t) dt = −
0

φ (t) v (t) dt
0

✈î✐ ♠å✐ ❤➔♠ t❤û ✈æ ❤÷î♥❣ φ ∈ Cc∞ (0, T ) .

✶✳✸✳✶✷ ✣à♥❤ ♥❣❤➽❛✳ ✐✮ ❑❤æ♥❣ ❣✐❛♥ ❙♦❜♦❧❡✈ W 1,p (0, T ; X) ❜❛♦ ❣ç♠ t➜t
❝↔ ❝→❝ ❤➔♠ u ∈ Lp (0, T ; X) s❛♦ ❝❤♦ tç♥ t↕✐ ✤↕♦ ❤➔♠ ②➳✉ u ♥➡♠ tr♦♥❣
Lp (0, T ; X)✳ ❍ì♥ ♥ú❛✱





u

W 1,p (0,T ;X)

T

u (t)

=

p

p

1
p

+ u (t) dt

(1

p < ∞)

0




 ❡ss sup

(p = ∞) .

u (t) + u (t)

0 t T

✐✐✮ ❚❛ ❦þ ❤✐➺✉ H −1 (0, T ; X) = W 1,2 (0, T ; X)✳

✶✳✸✳✶✸ ✣à♥❤ ❧þ✳ ❈❤♦ u ∈ W 1,p (0, T ; X) ✈î✐ 1

p < ∞✳ ❑❤✐ ✤â

✐✮ u ∈ C (0, T ; X) ✭❈â t❤➸ ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ ❧↕✐ tr➯♥ ♠ët t➟♣ ❝â ✤ë ✤♦
❦❤æ♥❣✮✱
✐✐✮ u (t) = u (s) +
✐✐✐✮ sup
0 t T

tr➯♥ T ✳

u (t)

t
s u

(T ) dT ✈î✐ 0

C u

W 1,p (0,T ;X)

s

t

T✱

✈î✐ C ❧➔ ❤➡♥❣ sè ❝❤➾ ♣❤ö t❤✉ë❝


❈❍×❒◆● ✷

●■❷■ ▼❐❚ ❇⑨■ ❚❖⑩◆ ❚❘❯❨➋◆ ◆❍■➏❚ ❇❾❈ P❍❹◆
◆●×Ñ❈ ❚❍❮■ ●■❆◆ ❇➀◆● ❈❍➓◆❍ ❍➶❆ ✣■➋❯ ❑■➏◆
❇■➊◆
❈❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tæ✐ tr➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣ ❝❤➾♥❤ ❤â❛ ❜➔✐ t♦→♥
❦❤✉➳❝❤ t→♥ ❜➟❝ ♣❤➙♥ ♥❣÷ñ❝ t❤í✐ ❣✐❛♥ tr♦♥❣ ❜➔✐ ❜→♦ ❬✸❪ ❝ô♥❣ ♥❤÷ ✤➲ ①✉➜t
✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët ✈➔✐ ❦➳t q✉↔ ♠î✐✳

✷✳✶ ●✐î✐ t❤✐➺✉ ❜➔✐ t♦→♥
P❤÷ì♥❣ tr➻♥❤ tr✉②➲♥ ♥❤✐➺t
∂u
= ∆u, (x, t) ∈ Ω × (0, T )
∂t

✭✷✳✶✮

2

∂
N
✈î✐ ∆ = ΣN
j=1 ∂x2 ✱ Ω ⊂ R ❧➔ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ❦✐♥❤ ✤✐➸♥ tr♦♥❣ ❧þ t❤✉②➳t
i

♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ✈➔ ✤➣ ✤÷ñ❝ ♥❣❤✐➯♥ ❝ù✉ ❜ð✐ r➜t ♥❤✐➲✉ ♥❤➔ t♦→♥
❤å❝✳ ❚r♦♥❣ ♥❤ú♥❣ ♥➠♠ ❣➛♥ ✤➙②✱ ♥❣÷í✐ t❛ q✉❛♥ t➙♠ ♥❤✐➲✉ ✤➳♥ ♣❤÷ì♥❣
tr➻♥❤ tr✉②➲♥ ♥❤✐➺t ❜➟❝ ♣❤➙♥ t❤❡♦ ❜✐➳♥ t❤í✐ ❣✐❛♥
∂γ u
= ∆u, (x, t) ∈ Ω × (0, T )
∂tγ

✭✷✳✷✮

✈î✐ γ ∈ (0, 1] ✈➔ ✤➣ ❝❤➾ r❛ r➡♥❣ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ❝ô♥❣ ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣
tr♦♥❣ t❤ü❝ t✐➵♥✳ Ð ✤➙②✱ ✤↕♦ ❤➔♠ ❜➟❝ ♣❤➙♥ ✤÷ñ❝ ❤✐➸✉ t❤❡♦ ♥❣❤➽❛ ❈❛♣✉t♦✱
tù❝ ❧➔
dγ u
1
:=
dtγ
Γ(1 − γ)

✈î✐ Γ(·) ❧➔ ❤➔♠ ❣❛♠♠❛✳
✷✷

t
0

f (s)
ds,
(t − s)γ

✭✷✳✸✮




r ữỡ ú tổ q t tợ u(x, t)
ừ ữỡ tr tr t ữủ tớ

u

= u, (x, t) ì (0, T ],
t
u(x, t) = 0, x , t [0, T ],


u(x, T ) = uT (x), x ,



tr õ (0, 1) tứ ỳ uT (x) ừ tr ố uT (x) tọ

uT (ã) uT (ã)

L2 ()

,

tr õ ự s số > 0 ữủ t

õ t

ờ sỷ r (0, 1) õ
1
1 x1/
1

e
+O
, x +,

x(1 )
x2
1
1
E, 1 (x) =
+O
, x .
x(1 )
x2

E, 1 (x) =

sỷ r 0 < 0 < 1 < 1 õ tỗ t số C1 C2

ử tở 0 1 s
C1
1
E, 1 (x)
(1 ) 1 x

C2
1
, ợ ồ x
(1 ) 1 x

0.

tr ợ ồ [0 , 1 ]
ự P q trỹ t ừ ỵ

ỵ tr tr õ


1
1
=
+O
x 1x

1
x2

=

1
+o
1x

1
1x

, x .

õ tỗ t M > 0 s
o

1
1x

1
1
, x (, M ]
2(1 ) 1 x




ợ ồ [0 , 1 ] (0, 1) õ
E, 1 (x) =

1
1
1
+O

(1 ) 1 x
1x
1
1
, x (, M ].
2(1 ) 1 x

ợ x [M, 0] ú ỵ r E,1 (x) > 0 ợ ồ x

0 t t ừ

tử tr ởt t õ t ồ số C1 = C1 (0 , 1 ) > 0
ừ ọ s
E, 1 (x)

min
(x,)[M,0]ì[0 ,1 ]

E, 1 (x)

C1
C1
(1 1 )
(1 )
C1
1
.
(1 ) 1 x

tr ợ ồ [0 , 1 ]

ờ ữủ ự
sỷ g L2 () t t
w
(x, t) ì (0, T ],
t = w,
w(x, t) = 0,
x , t [0, T ],

w(x, 0) + w(x, T ) = g(x), x ,



tr õ > 0 t số õ

ỵ ợ ồ > 0 ồ g L2 tỗ t ởt

t [g](x, t) C((0, T ]; H01 ()) ừ t

ự ữợ t ú t sỷ ử ữỡ t

t ữủ ừ t (x, t) = (x)(t)
t t ữủ
+ (x) = 0, x ,
(x) = 0,
x ,




✷✺

✈➔
dγ
ψ(t) + λψ(t) = 0,
dtγ

✭✷✳✽✮

✈î✐ λ ❧➔ ❤➡♥❣ sè✳ ❚ø ❧þ t❤✉②➳t ✈➲ ❜➔✐ t♦→♥ ❣✐→ trà r✐➯♥❣✱ ✭✷✳✼✮ ❝â ❝→❝ ❣✐→
trà r✐➯♥❣ λn > 0 t÷ì♥❣ ù♥❣ ✈î✐ ❝→❝ ❤➔♠ r✐➯♥❣ φn (x), n = 1, 2, ..., s❛♦ ❝❤♦
2
λn → ∞ ❦❤✐ n → ∞ ✈➔ {φn }∞
n=1 ❧➔ ♠ët ❝ì sð trü❝ ❝❤✉➞♥ ❝õ❛ L (Ω)✳

❇➡♥❣ ❝→❝❤ t➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥✱ t❛ t❤➜② r➡♥❣
λn = ∇φn

2
L2 (Ω) ,

λ2n = ∇φn

2
L2 (Ω)

✈➔
∇φn .∇φm dx = 0,
Ω

∆φn ∆φm dx = 0, n = m.

Ω

❇➡♥❣ ❝→❝❤ sû ❞ö♥❣ ❜✐➳♥ ✤ê✐ ▲❛♣❧❛❝❡✱ ❝❤ó♥❣ t❛ ❝â t❤➸ ❣✐↔✐ ✭✷✳✽✮ ✈î✐ λ = λn
❝è ✤à♥❤ ✤➸ ✤↕t ✤÷ñ❝ ♥❣❤✐➺♠
ψn (t) = Dn Eγ,1 (−λn tγ ).

❇➙② ❣✐í ❝❤ó♥❣ t❛ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✻✮ ❝â ❞↕♥❣
∞
α

Dn Eγ,1 (−λn tγ )φn (x).

ω (x, t) =

✭✷✳✾✮

n=1

❚ø ✤✐➲✉ ❦✐➺♥ αω(x, 0) + ω(x, T ) = g(x)✱ t❛ ❝â
1
g(x)φn (x)dx
α + Eγ,1 (−λn T γ ) Ω
gn
:=
.
α + Eγ,1 (−λn T γ )

Dn =

✭✷✳✶✵✮

❚✐➳♣ t❤❡♦✱ ❝❤ó♥❣ t❛ ❦✐➸♠ tr❛ r➡♥❣ ω α (x, t) ✤÷ñ❝ ①➙② ❞ü♥❣ ð tr➯♥ ❧➔
♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✻✮✳ ❚❤➟t ✈➟②✱ t❛ ❝â
dγ
Eγ,1 (−λn tγ ) = −λn Eγ,1 (−λn tγ ),
γ
dt

∆φn = −λn φn (x).