Giải một bài toán khuếch tán ngược bậc phân bằng phương pháp chỉnh hóa phổ

✶

▼Ö❈ ▲Ö❈
❚r❛♥❣

▼Ö❈ ▲Ö❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶
▲❮■ ◆➶■ ✣❺❯ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷
❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❜ê trñ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺
✶✳✶ ❍➔♠ ●❛♠♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺
✶✳✷ P❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r tr♦♥❣ ❝→❝ ❦❤æ♥❣ ❣✐❛♥ L1 (R) ✈➔ L2 (R) ✳ ✶✵
❈❤÷ì♥❣ ✷✳ ●✐↔✐ ♠ët ❜➔✐ t♦→♥ ❦❤✉➳❝❤ t→♥ ♥❣÷ñ❝ ❜➟❝ ♣❤➙♥ ❜➡♥❣
♣❤÷ì♥❣ ♣❤→♣ ❝❤➾♥❤ ❤â❛ ♣❤ê ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
✷✳✶ ✣↕♦ ❤➔♠ ❜➟❝ ♣❤➙♥ ❈❛♣✉t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
✷✳✷ ●✐î✐ t❤✐➺✉ ❜➔✐ t♦→♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✷✳✸ ❈❤➾♥❤ ❤â❛ ❜➔✐ t♦→♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
❑➌❚ ▲❯❾◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷


▲❮■ ◆➶■ ✣❺❯
●➛♥ ✤➙②✱ ♣❤➨♣ t➼♥❤ ✈✐ ♣❤➙♥ ❜➟❝ ♣❤➙♥ ✈➔ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠
r✐➯♥❣ ❜➟❝ ♣❤➙♥ ✤➣ ✤÷ñ❝ sû ❞ö♥❣ ✤➸ ❣✐↔✐ ♠ët sè ❜➔✐ t♦→♥ tr♦♥❣ ❝→❝ ❧➽♥❤
✈ü❝ ✈➟t ❧þ✱ ❤â❛ ❤å❝✱ s✐♥❤ ❤å❝✱ ❝ì ❦❤➼✱ ①û ❧þ t➼♥ ❤✐➺✉✱ ✤✐➺♥ tû✱ ✤✐➲✉ ❦❤✐➸♥
tè✐ ÷✉ ✈➔ t➔✐ ❝❤➼♥❤ ✭①❡♠ ❬✻❪✮✳
P❤÷ì♥❣ tr➻♥❤ ❦❤✉➳❝❤ t→♥ ❜➟❝ ♣❤➙♥ ①✉➜t ❤✐➺♥ ❦❤✐ t❛ t❤❛② ✤↕♦ ❤➔♠
❜➟❝ ♥❣✉②➯♥ tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❦❤✉➳❝❤ t→♥ ❜➡♥❣ ♠ët ✤↕♦ ❤➔♠ ❜➟❝ ♣❤➙♥✳
❑✐➸✉ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ✤÷ñ❝ sû ❞ö♥❣ ✤➸ ♠æ t↔ ❝→❝ q✉→ tr➻♥❤ ❦❤✉➳❝❤ t→♥
❜➜t t❤÷í♥❣ ✭❛♥♦♠❛❧♦✉s ❞✐❢❢✉s✐♦♥✮ ♥❤÷ ❦❤✉➳❝❤ t→♥ tr➯♥ ✭s✉♣❡r❞✐❢❢✉s✐♦♥✮✱
❦❤✉➳❝❤ t→♥ ❞÷î✐ ✭s✉❜❞✐❢❢✉s✐♦♥✮✳ ❈→❝ q✉→ tr➻♥❤ ❦❤✉➳❝❤ t→♥ ♥➔② ❦❤æ♥❣ t✉➙♥
t❤❡♦ ✤à♥❤ ❧✉➟t ❋✐❝❦ ❝ê ✤✐➸♥✳
❇➔✐ t♦→♥ ❦❤✉➳❝❤ t→♥ ♥❣÷ñ❝ ❜➟❝ ♣❤➙♥ t❤÷í♥❣ ✤➦t ❦❤æ♥❣ ❝❤➾♥❤ t❤❡♦

♥❣❤➽❛ ❍❛❞❛♠❛r❞✳ ▼ët s❛✐ sè ♥❤ä tr♦♥❣ ✤♦ ✤↕❝ ❝ô♥❣ ❝â t❤➸ ❞➝♥ ✤➳♥ ♠ët
s❛✐ ❧➺❝❤ ❧î♥ ✈➲ ♥❣❤✐➺♠✳ ❈❤➼♥❤ ✈➻ ✈➟② ✤➸ ❣✐↔✐ q✉②➳t ❜➔✐ t♦→♥ t❛ ❝➛♥ ✤➲
①✉➜t ❝→❝ ♣❤÷ì♥❣ ♣❤→♣ ❝❤➾♥❤ ❤â❛✳ ❈❤♦ ✤➳♥ ♥❛② ✤➣ ❝â ♥❤✐➲✉ ♣❤÷ì♥❣ ♣❤→♣
❝❤➾♥❤ ❤â❛ ❞➔♥❤ ❝❤♦ ❜➔✐ t♦→♥ ❦❤✉➳❝❤ t→♥ ♥❣÷ñ❝ ❜➟❝ ♥❣✉②➯♥✳ ❚✉② ♥❤✐➯♥
❝→❝ ❦➳t qõ❛ ❝❤➾♥❤ ❤â❛ ❜➔✐ t♦→♥ ❦❤✉➳❝❤ t→♥ ♥❣÷ñ❝ ❜➟❝ ♣❤➙♥ ✈➝♥ ❝á♥ ❤↕♥
❝❤➳✳
✣➸ t➟♣ ❞÷ñt ♥❣❤✐➯♥ ❝ù✉ ❝ô♥❣ ♥❤÷ ✤➸ ❧➔♠ ♣❤♦♥❣ ♣❤ó t❤➯♠ ❝→❝ t➔✐
❧✐➺✉ ✈➲ ✈✐➺❝ ❝❤➾♥❤ ❤â❛ ❜➔✐ t♦→♥ ❦❤✉➳❝❤ t→♥ ♥❣÷ñ❝ ❜➟❝ ♣❤➙♥✱ tr➯♥ ❝ì sð
❜➔✐ ❜→♦ ✧❙♣❡❝tr❛❧ r❡❣✉❧❛r✐③❛t✐♦♥ ♠❡t❤♦❞ ❢♦r s♦❧✈✐♥❣ ❛ t✐♠❡ ✲ ❢r❛❝t✐♦♥❛❧
✐♥✈❡rs❡ ❞✐❢❢✉s✐♦♥ ♣r♦❜❧❡♠✧ ❝õ❛ ❝→❝ t→❝ ❣✐↔ ●✳ ❍✳ ❩❤❡♥❣✱ ❚✳ ❲❡✐ ✤➠♥❣ tr➯♥
t↕♣ ❝❤➼ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ♥➠♠ ✷✵✶✶✱ ❝❤ó♥❣ tæ✐ ❧ü❛

✧●✐↔✐ ♠ët ❜➔✐ t♦→♥ ❦❤✉➳❝❤
t→♥ ♥❣÷ñ❝ ❜➟❝ ♣❤➙♥ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❝❤➾♥❤ ❤â❛ ♣❤ê✧✳
❝❤å♥ ✤➲ t➔✐ ❝❤♦ ▲✉➟♥ ✈➠♥ ❝õ❛ ♠➻♥❤ ❧➔ ✿

✷




t t t




0 Dt u(x, t) = aux (x, t), x > 0, t > 0, (0, 1),
u(1, t) = f (t), t 0,

u(x, 0) = lim u(x, t) = 0,

tr õ 0 Dt u t ợ (0 <

1) ữủ

x


t
1
g (s)
=
ds, 0
(1 ) 0 (t s)
dg(t)

, = 1.
0 Dt g(t) =
dt

0 Dt g(t)



<1

r ú tổ q t tợ u ux tr
ữỡ tr tứ ỳ ữủ t x = 1 ừ u(x, t) u(1, t) = f (t)
ú ỵ r ởt t t ổ
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




ử tr t t ỡ ừ

P ờ rr tr ổ

L1 (R)



L2 (R)
ử tr t t ỡ ừ ờ
rr tr ổ L1 (R) L2 (R)

ữỡ ởt t t ữủ
ữỡ õ ờ
r ữỡ t ú tổ tr
t ởt số t t ỡ ừ õ tứ t t

õ ú tổ ợ t t t ữủ
tr t qừ õ t tr ụ ữ
t ự ởt t qừ ợ

t

ử tr t ởt số t t ỡ
ừ õ

ợ t

ử ợ t t t ữủ

õ t

ử tr ữỡ õ t t
ữủ t qừ tố ở ở tử ừ ữỡ
tr ụ ữ t ởt t qừ ợ
ữủ tỹ t rữớ ồ ữợ sỹ ữợ
ừ t ự tọ ỏ t
ỡ s s ừ t t
ỡ Pỏ t ồ ừ ữ P
ồ ỡ t ổ tr ở ổ t ữ P
ồ t t ú ù t tr sốt tớ
ồ t t 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
2

◆❤➙♥ ❤❛✐ ✈➳ ❝õ❛ ✭✶✳✸✮ ✈î✐ e−u ✈➔ ❧➜② t➼❝❤ ♣❤➙♥ tø 0 ✤➳♥ ∞ t❛ ✤÷ñ❝
∞
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 trð
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)
1
nz+m n!
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)

✶✳✷ P❤➨♣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r tr♦♥❣ ❝→❝ ❦❤æ♥❣ ❣✐❛♥
L1(R) ✈➔ L2(R)
✶✳✷✳✶ ✣à♥❤ ♥❣❤➽❛

✳

✭✣à♥❤ ♥❣❤➽❛ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r tr♦♥❣ L1 (R) ✮ ◆➳✉

f ∈ L1 (R)✱ t❛ ✤à♥❤ ♥❣❤➽❛ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ f ❧➔
1
f (ξ) := √
2π

+∞

e−ix.ξ f (x)dx (ξ ∈ R)

✭✶✳✶✻✮

−∞

✈➔ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ♥❣÷ñ❝ ❝õ❛ f ❧➔

1

f (ξ) := √
2π
∨

+∞

eix.ξ f (x)dx (ξ ∈ R).
−∞

✭✶✳✶✼✮


✶✶

❱➻ e±ixξ = 1 ✈➔ f ∈ L1 (R) ♥➯♥ ❝→❝ t➼❝❤ ♣❤➙♥ tr➯♥ ❤ë✐ tö ✈î✐ ♠é✐ ξ ∈ R.
❙❛✉ ✤➙② ❧➔ ♠ët ✈➔✐ t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ f (ξ)
✶✳ f (ξ) ❧➔ ❤➔♠ ❜à ❝❤➦♥✱ ✈➻
+∞
1
e−ix.ξ f (x)dx
f (ξ) = √
2π −∞
+∞
1
√
e−ix.ξ |f (x)| dx
2π −∞
+∞
1
=√

|f (x)| dx
2π −∞
f 1 < ∞.

✷✳ f (ξ) ❧✐➯♥ tö❝ ✤➲✉ ✈î✐ −∞ < ξ < +∞✳
◆➳✉ y > 0 t❤➻

1
f (ξ + y) − f (ξ) = √
2π
1
√
2π
1
√
2π
1
√
2π

+∞
−∞
+∞

f (x)e−ixξ (e−iyx − 1)dx

|f (x)| e−iyx − 1 dx

−∞
+∞

|f (x)| 2 sin(
−∞
+∞

xy
) dx
2

+∞

|f (x)| dx.

|f (x)| dx + yR

+
−∞

+R

−∞

−R

❱î✐ ε > 0 ✱ t❛ ❝â t❤➸ ❝❤å♥ R ✤õ ❧î♥ ✈➔ y ✤õ ❜➨ ✤➸ ❜✐➸✉ t❤ù❝ ❝ë♥❣ ❧↕✐ ❧➜②
tê♥❣ ❝✉è✐ ❝ò♥❣ ♥❤ä ❤ì♥ ε✳
✸✳ ◆➳✉ c1 ✈➔ c2 ❧➔ ❝→❝ sè t❤ü❝ t❤➻

(c1 f1 + c2 f2 ) = c1 f1 + c2 f2 .
✹✳ Df (x) = −iξ f (x) ✈î✐ Df (x) ∈ L1 (R)✳

❑þ ❤✐➺✉ t➼❝❤ ♣❤➙♥ ❦❤æ♥❣ ①→❝ ✤à♥❤ ❝õ❛ f (x) ❧➔

Df (x) =

g(y)dy


✶✷

✈➔

A

Df (A) − Df (a) =

g(y)dy.
a

◆➳✉ t❛ ❣✐ú a ❝è ✤à♥❤✱ ❝❤♦ A → ∞✱ ❞♦ g(x) ∈ L1 (R) ♥➯♥ t❛ ❝â
A

g(x)dx → c.
a

❱➻ f (A) → l✱ f (−A) → −m ♠➔ f (x) ∈ L1 (R) ♥➯♥ t❛ ♣❤↔✐ ❝â l = −m = 0.
◆➳✉ Df (x) = ψ(ξ) t❤➻
+A
1
ψ(ξ) = lim √
e−ixξ df (x)

A→∞ 2π −A
1
= lim √
(e−ixξ f (x))A
−A − iξ
A→∞ 2π

A

e−ixξ f (x)dx

a

= −iξ f (ξ).
✭❞♦ ❣✐î✐ ❤↕♥ ❜à tr✐➺t t✐➯✉✱ ❧ ❂ ✲♠ ❂ ✵ ✮✳

✶✳✷✳✷ ✣à♥❤ ♥❣❤➽❛ ✭❚➼❝❤ ❝❤➟♣✮✳ ❚➼❝❤ ❝❤➟♣ ❝õ❛ ❝→❝ ❤➔♠ f, g ❦þ ❤✐➺✉ ❧➔
f ∗ g ✈➔ ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛
+∞

(f ∗ g)(x) =

f (y)g(x − y)dy.
−∞

✶✳✷✳✸ ✣à♥❤ ❧þ✳ ❚➼❝❤ ❝❤➟♣ ❝â ❝→❝ t➼♥❤ ❝❤➜t s❛✉ ✤➙②
✶✮ ∀f, g ∈ L1 (R), f ∗ g ∈ L1 (R) ✈➔ f ∗ g

1

≤ f

1.

g 1,

✷✮ ∀f, g ∈ L1 (R), f ∗ g = g ∗ f,
✸✮ ∀f, g, h ∈ L1 (R), λ ∈ R, (λf + g) ∗ h = λf ∗ h + g ∗ h.
❈❤ù♥❣ ♠✐♥❤✳ ✶✮ ❉♦ ❤➔♠ g ❦❤↔ t➼❝❤ t✉②➺t ✤è✐ ♥➯♥ ❜à ❝❤➦♥ tr➯♥ R

∀(x, y) ∈ R, |f (y)g(x − y)| ≤ g

∞ |f (y)|.

❉♦ f ❦❤↔ t➼❝❤ t✉②➺t ✤è✐ ♥➯♥ t➼❝❤ ♣❤➙♥ s✉② rë♥❣ ❝õ❛ (f ∗ g)(x) ❤ë✐ tö


✶✸

t✉②➺t ✤è✐ ✈➔ ❜à ❝❤➦♥ ✤➲✉
+∞

f ∗g

1

+∞

f (y)g(x − y)dy dx

=
−∞
+∞

−∞
+∞

≤

|f (y)|
−∞

= f

1

|g(x − y)|dx dy
−∞

g 1.

✷✮ ❱î✐ x ∈ R t❛ ❝â
+∞

(f ∗ g)(x) =

f (y)g(x − y)dy
−∞
+∞

f (x − t)g(t)dt (✤➦t t = x − y)

=
−∞

= (g ∗ f )(x).
✸✮ ❙✉② r❛ tø t➼♥❤ t✉②➳♥ t➼♥❤ ❝õ❛ t➼❝❤ ♣❤➙♥
+∞

(λf + g)(y)h(x − y)dy

((λf + g) ∗ h)(x) =
−∞
+∞

[(λf )(y)h(x − y) + g(y)h(x − y)] dy

=
−∞

= λf ∗ h + g ∗ h.

❇➙② ❣✐í t❛ s➩ ♠ð rë♥❣ ❝→❝ ✤à♥❤ ♥❣❤➽❛ ✭✶✳✶✻✮ ✈➔ ✭✶✳✶✼✮ ❝❤♦ ❝→❝ ❤➔♠

f (x) ∈ L2 (R) ❜ð✐ ✤à♥❤ ❧þ s❛✉

✶✳✷✳✹ ✣à♥❤ ❧þ ✭✣➥♥❣ t❤ù❝ P❛rs❡✈❛❧✮✳ ●✐↔ t❤✐➳t f (x) ∈ L1(R) ∩ L2(R)✳
❑❤✐ ✤â f , f ∨ ∈ L2 (R) ✈➔

f = f∨ = f .

✭✶✳✶✽✮

❈❤ù♥❣ ♠✐♥❤✳ ❳➨t ❝→❝ ❤➔♠ g, h ∈ L1 (R)✳ ❑❤✐ ✤â g, h ∈ L∞ (R) ✭❞♦ t➼♥❤


✶✹

❝❤➜t ✶✮✳ ❚❛ ❝â
+∞

+∞

g(x)h(x)dx =
−∞

g(x)

1
√
2π

+∞

+∞

−∞

1
=√

2π
+∞

−∞

+∞

g(ξ)h(ξ)dξ =
−∞

−∞

−∞

e−ixξ g(x)h(ξ)dxdξ.

+∞

1
√
2π
−∞

e−ixξ h(ξ)dξ dx

−∞

e−ixξ g(x)dx h(ξ)dξ

−∞

+∞

+∞

1
=√
2π

+∞

e−ixξ g(x)h(ξ)dxdξ.

−∞

❙✉② r❛
+∞

+∞

g(x)h(x)dx =
−∞

✭✶✳✶✾✮

g(ξ)h(ξ)dξ.
−∞

❚❛ ❧↕✐ ❝â
+∞

e

−ixξ−tx2

+∞ −

dx =

−∞

√

e

iξ
tx+ √
2 t

2

ξ2
−
4t dx

−∞
2

ξ 2 +∞ − √tx+ iξ
√
√

1 −
2
t
4t
√
e
=
e
d( tx)
t
−∞
ξ2
ξ2
1 − √
π −
= √ e 4t π =
e 4t . (t > 0)
t
t
❙✉② r❛
+∞

ξ2
π −
e 4t (t > 0).
t

2

e−ixξ−tx dx =

−∞
2

❉♦ ✤â✱ ♥➳✉ ε > 0 ✈➔ ✤➦t gε (x) := e−ε.x t❛ ❝â

ξ2
e 4ε
gε (ξ) = √ .
2ε
−


✶✺

❱➻ t❤➳ ✈î✐ ♠é✐ ε > 0✱ tø ✭✶✳✶✾✮ s✉② r❛
+∞
−∞

2
1
h(x)e−εx dx = √
2ε

+∞

ξ2
−
h(ξ)e 4ε dξ.

✭✶✳✷✵✮

−∞

▲➜② f (x) ∈ L1 (R) ∩ L2 (R) ✈➔ ✤➦t g(x) := f (−x). ❳➨t

h := f ∗ g ∈ L1 (R) ∩ L2 (R).
❚❛ ❝â
+∞
1
e−ix.ξ (f ∗ g)(x)dx
h(ξ) = f ∗ g(ξ) = √
2π −∞
+∞
+∞
1
−ixξ
e
f (y)g(x − y)dy dx
=√
2π −∞
−∞
+∞
+∞
1
−ixξ
=√
e
f (y)
e−iξ(x−y) g(x − y)d(x − y) dy

2π −∞
−∞
+∞
√
=
e−ixξ f (y) g(ξ) = 2π f (ξ)g(ξ).
−∞

❙✉② r❛

h=

√

2π f g.

▼➦t ❦❤→❝✱ t❛ ❝â

1
g(ξ) = √
2π
❉♦ ✤â h =

√

+∞

e−ixξ f (−x)dx.

−∞

2

2π f

.

❱➻ h ❧✐➯♥ tö❝ ♥➯♥

1
lim √
ε→0 2ε
❉♦ h =

√

2

2π f

+∞

ξ2
√
−
h(ξ)e 4ε dξ = 2πh(0).

−∞

≥ 0 ♥➯♥ ❦❤✐ ❝❤♦ ε → 0+ tr♦♥❣ ✭✶✳✷✵✮ t❛ ❝â

+∞

h(x)dx =
−∞

√

2πh(0)


✶✻

✳ ❙✉② r❛

+∞ √

2

2π f (x) dx =

√

2πh(0),

−∞

❤❛②
+∞

+∞

2

f (x) dx = h(0) =

+∞

f (y)g(−y)dy =
−∞

−∞

✣✐➲✉ ♥➔② t÷ì♥❣ ✤÷ì♥❣ ✈î✐ f

2

= f

|f (y)|2 dy.

−∞
2.

❉♦ ✤â f = f . ❚÷ì♥❣ tü t❛

❝ô♥❣ ❝❤ù♥❣ ♠✐♥❤ ✤÷ñ❝ f ∨ = f .

✶✳✷✳✺ ✣à♥❤ ♥❣❤➽❛ ✭✣à♥❤ ♥❣❤➽❛ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r tr♦♥❣ L2(R) ✮✳ ❚❛ ✤à♥❤
♥❣❤➽❛ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r f ❝õ❛ f ∈ L2 (R) ♥❤÷ s❛✉
❈❤♦ ♠ët ❞➣② {fk }∞

k=1 ⊂ L1 (R) ∩ L2 (R) ✈î✐ fk → f tr♦♥❣ L2 (R)✳ ❚❤❡♦
✣➥♥❣ t❤ù❝ P❛rs❡✈❛❧✱ fk − fj = fk − fj = fk − fj ✈➔ ✈➻ t❤➳ {fk }∞
k=1
❧➔ ♠ët ❞➣② ❈❛✉❝❤② tr♦♥❣ L2 (R)✳ ❉♦ ✤â fk → f tr♦♥❣ L2 (R)✱ t❛ ❣å✐ f ❧➔
❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ f tr♦♥❣ L2 (R)✳ ❚÷ì♥❣ tü✱ t❛ ❝ô♥❣ ❝â ✤à♥❤ ♥❣❤➽❛ f ∨ ✳
✣à♥❤ ♥❣❤➽❛ f ❦❤æ♥❣ ♣❤ö t❤✉ë❝ ✈➔♦ ✈✐➺❝ ❝❤å♥ ❞➣② {fk }∞
k=1 t÷ì♥❣ ù♥❣✳
❚❤➟t ✈➟②✱ ❣✐↔ sû ❝â ❞➣② {gk }∞
k=1 ⊂ L1 (R)∩L2 (R) ✈➔ gk → f tr♦♥❣ L2 (R)✳
❱➻ {gk }k ❧➔ ❞➣② ❈❛✉❝❤② tr♦♥❣ L2 (R) ♥➯♥ gk → g tr♦♥❣ L2 (R)✳ ❚❛ ❝â

g − f = g − gk + gk − fk + fk − f
≤ g − gk + gk − f k + f k − f .
❉♦ g − gk → 0, gk − fk → 0, fk − f → 0 ♥➯♥ g − f → 0✳ ❱➟②

g ≡ f ✱ ❞♦ ✤â f ❧➔ ❞✉② ♥❤➜t✳
❙❛✉ ✤➙② ❧➔ ♠ët ✈➔✐ t➼♥❤ ❝❤➜t ❝õ❛ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥

L2 (R)

✶✳✷✳✻ ✣à♥❤ ❧þ✳ ●✐↔ t❤✐➳t f, g ∈ L2(R)✳ ❑❤✐ ✤â
✐✮

+∞

+∞

f g¯dx =
−∞

✐✐✮ Dα f =

L2 (R)✱

fˆgˆdξ ✱

−∞
(iξ)α f

✈î✐ ♠é✐ ❝❤➾ sè α ♥❣✉②➯♥ ❞÷ì♥❣ s❛♦ ❝❤♦ Dα f ∈


✶✼

✐✐✐✮ f ∗ g =

√

2π f g ✱

✐✈✮ f = (f )∨ .
❈❤ù♥❣ ♠✐♥❤✳ ✐✮ ❈❤♦ f, g ∈ L2 (R) ✈➔ α ∈ C ✳ ❑❤✐ ✤â✱ t❤❡♦ ✣à♥❤ ❧þ ✶✳✷✳✹✱
t❛ ❝â f + αg

2

= f + αg 2 . ❑❤❛✐ tr✐➸♥ t❛ ✤÷ñ❝
+∞

+∞

2

|f + αg|2 dx =
−∞

f + αg dx,
−∞

❤❛② ❧➔
+∞

+∞

(f + αg) f¯ + α
¯ g¯ dx =
−∞

f + αg

¯
f +α
¯ g¯ dξ.

−∞

❘ót ❣å♥ ✤➥♥❣ t❤ù❝ tr➯♥ t❛ ✤÷ñ❝
+∞

+∞

¯
αf g + α
¯ f g¯ dξ.

αf¯g + α
¯ f g¯ dx =
−∞

−∞

❱î✐ α = 1 t❤➻
+∞

+∞

f¯g + f g¯ dx =

¯
f g + f g¯ dξ.

✭✶✳✷✶✮

¯
−if g + if g¯ dξ.

✭✶✳✷✷✮

−∞

−∞

❱î✐ α = −i t❤➻
+∞

+∞

−if¯g + if g¯ dx =
−∞

−∞

❈ë♥❣ ✈➳ ✈î✐ ✈➳ ❝õ❛ ✭✶✳✷✶✮ ✈➔ ✭✶✳✷✷✮ t❛ ✤÷ñ❝
+∞

+∞

fˆgˆdξ.

f g¯dx =
−∞

−∞


✶✽

✐✐✮ ◆➳✉ f ❧➔ trì♥ ✈➔ ❝â ❣✐→ ❝♦♠♣❛❝t✱ t❛ ❝â

Dα f

+∞
1
=√
e−ixξ Dα f (x)dx
2π −∞
(−1)α +∞ α −ixξ
= √
Dx e
f (x)dx
2π −∞
+∞
1
e−ix.ξ (iξ)α f (x)dx
=√
2π −∞
= (iξ)α f (ξ).

❇➡♥❣ ❝→❝❤ t✐➳♥ tî✐ ❣✐î✐ ❤↕♥✱ ❝æ♥❣ t❤ù❝ tr➯♥ s➩ ✤ó♥❣ ♥➳✉ Dα f ∈ L2 (R)✳
✐✐✐✮ ❱î✐ f (x), g(x) ∈ L1 (R) ∩ L2 (R) ✈➔ ξ ∈ R t❛ ❝â

1
f ∗ g(ξ) = √
2π
1
=√
2π
1
=√
2π

e−ixξ (f ∗ g)(x)dx

−∞
+∞

e

√

f (y)g(x − y)dy dx
−∞

−∞
+∞

e
e

+∞

−ix.ξ

−ix.ξ

−∞
+∞
−iyξ

=
=

+∞

+∞

f (y)

e−i(x−y).ξ g(x − y)dx dy

−∞

f (y)dy g(ξ)

−∞

2π f (ξ)g(ξ).
2

✐✈✮ ❈è ✤à♥❤ y ∈ R, ε > 0 ✈➔ ✤➦t gε (ξ) := eiξy−εξ . ❚❛ ❝â

1
gε (ξ) = √
2π
1
=√
2π

+∞

2

e−ixξ eiyx−εx dx

−∞
+∞

2
1
e−i(ξ−y)x−εx dx = √
2π
−∞
2
−(ξ − y)
1 −
4ε
=√ e
.
2ε

−(ξ − y)2
π −
4ε
e
t

❙û ❞ö♥❣ ❝æ♥❣ t❤ù❝ ✭✶✳✶✾✮ t❛ ❦➳t ❧✉➟♥ r➡♥❣ ✈î✐ f (x) ∈ L1 (R) ∩ L2 (R) t❤➻
+∞
−∞

2

1
f eiyξ−εξ dξ = √
2ε

+∞
−∞

−(x − y)2
−
4ε
f (x)e
dx.

✭✶✳✷✸✮


✶✾

❱➳ ♣❤↔✐ ❝õ❛ ✭✶✳✷✸✮ ❞➛♥ tî✐

1
√
2π
❱➟② (f )∨ = f ✳

√

2πf (y) ❦❤✐ ε → 0+ ✳ ❙✉② r❛

+∞

f (ξ)eiξy dξ = f (y).
−∞


❈❍×❒◆● ✷

●■❷■ ▼❐❚ ❇⑨■ ❚❖⑩◆ ❑❍❯➌❈❍ ❚⑩◆ ◆●×Ñ❈ ❇❾❈
P❍❹◆ ❇➀◆● P❍×❒◆● P❍⑩P ❈❍➓◆❍ ❍➶❆ P❍✃
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ✤➛✉ t✐➯♥ ❝❤ó♥❣ tæ✐ tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ ✤↕♦ ❤➔♠ ❜➟❝
♣❤➙♥ ❈❛♣✉t♦ ✈➔ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ♥â tø t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦
❬✻❪✳ ❙❛✉ ✤â ❝❤ó♥❣ tæ✐ ❣✐î✐ t❤✐➺✉ ❜➔✐ t♦→♥ ❦❤✉➳❝❤ t→♥ ♥❣÷ñ❝ ❜➟❝ ♣❤➙♥ ✈➔
tr➻♥❤ ❜➔② ❝→❝ ❦➳t qõ❛ ❝❤➾♥❤ ❤â❛ ❜➔✐ t♦→♥ ♥➔② tr♦♥❣ ❜➔✐ ❜→♦ ❬✺❪ ❝ô♥❣ ♥❤÷
✤➲ ①✉➜t ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët ✈➔✐ ❦➳t qõ❛ ♠î✐✳

✷✳✶ ✣↕♦ ❤➔♠ ❜➟❝ ♣❤➙♥ ❈❛♣✉t♦
✷✳✶✳✶ ✣à♥❤ ♥❣❤➽❛✳ ❈❤♦ f ❧➔ ♠ët ❤➔♠ sè ❦❤↔ ✈✐ ❧✐➯♥ tö❝ ❝➜♣ n ∈ N∗
tr➯♥ [a, T ] (T > a)✳ ✣↕♦ ❤➔♠ ❜➟❝ ♣❤➙♥ ❈❛♣✉t♦ ✈î✐ ❜➟❝ α > 0 ❝õ❛ ♠ët
❤➔♠ f tr➯♥ ✤♦↕♥ [a, T ] ✤÷ñ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉
C (α)
a Dt f (t)

1
=
Γ(n − α)

C (α)
a Dt f (t)

= f (n) (t), a

t
a

f (n) (s)
ds, a
(t − s)α+1−n

t

T, n − 1 < α < n,

t

T, α = n.

✷✳✶✳✷ ◆❤➟♥ ①➨t✳ ✶✮ ❚r♦♥❣ tr÷í♥❣ ❤ñ♣ n = 1✱ t❛ ❝â
C (α)
a Dt f (t)

1
=
Γ(1 − α)

t
a

f (s)
ds, a
(t − s)α

t

T, 0 < α < 1.

✭✷✳✶✮

✷✮ ❱î✐ n ∈ N∗ ✱ α ∈ R t❤ä❛ ♠➣♥ n − 1 < α < n✱ m ∈ N ✈➔ f ❧➔ ❤➔♠ ❦❤↔
✈✐ ❧✐➯♥ tö❝ ❝➜♣ n + m t❤➻ t❛ ❝â
C (α) C m
a Dt
a Dt f (t)

✷✵

(α+m)

=C
a Dt

f (t).


✷✶

✷✳✶✳✸ ◆❤➟♥ ①➨t✳ ●✐↔ sû n ∈ N∗ ✈➔ α ❧➔ sè t❤ü❝ t❤ä❛ ♠➣♥
0

n − 1 < α < n.

❍ì♥ ♥ú❛ f ❧➔ ❤➔♠ ❦❤↔ ✈✐ ❧✐➯♥ tö❝ ❝➜♣ n + 1 tr➯♥ ✤♦↕♥ [a, T ] ✈î✐ T > a
t❤➻
(α)

n
lim C
a Dt f (t) = f (t), ∀t ∈ [a, T ].

α→n

✭✷✳✷✮

❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â

f (n) (a)(t − a)n−α
α→n
Γ(n − α + 1)
t
1
(t − τ )n−α f (n+1) (τ )dτ
+ lim
α→n Γ(n − α + 1) a

(α)

lim C
a Dt f (t) = lim

α→n

t

=f

(n)

f (n+1) (τ )dτ

(a) +
a

n

= f (t), ∀t ∈ [a, T ].

✷✳✶✳✹ ✣à♥❤ ❧þ✳ ❈❤♦ α > 0 ✈➔ λ ∈ R✳ ✣➦t f (t) = Eα,1(λtα), t ≥ 0. ❑❤✐
✤â
C (α)
0 Dt f (t)

= λEα,1 (λtα ).

❈❤ù♥❣ ♠✐♥❤✳ ◆➳✉ α = n ∈ N t❤➻
C (α)
0 Dt f (t)

dn
= n En,1 (λtn ) =
dt
∞

=
k=0

∞

k=0

dn (λtn )k
dtn Γ(kn + 1)

nk(nk − 1).....(nk − n + 1)λk tn(k−1)
.
Γ(kn + 1)

❈❤ó þ r➡♥❣

Γ(kn + 1) = knΓ(kn) = kn(kn − 1)Γ(kn − 1)
= ... = kn(kn − 1)...(kn − n + 1)Γ(kn − n + 1).


✷✷

❚❛ ❝â
∞
C (α)
0 Dt f (t)

=λ
k=1

(λtn )k−1
= λEn,1 (λtn ) = λf (t).
Γ(n(k − 1) + 1)

❱î✐ α ∈ N✱ t❤➻ tç♥ t↕✐ n ∈ N s❛♦ ❝❤♦ n − 1 < α < n✳ ❚❛ ❝â
C (α)
0 Dt f (t)

t

1
=
Γ(n − α)
1
Γ(n − α)

=

0
∞

1
f (n) (s)
ds
=
(t − s)α+1−n
Γ(n − α)

k=1

dn
α k
dsn (λs )

t
0

Γ(αk + 1)(t − s)α+1−n

0

t dn E (λsα )
dsn α,1
ds
(t − s)α+1−n

ds.

❚÷ì♥❣ tü ♥❤÷ tr÷í♥❣ ❤ñ♣ α ❧➔ sè ♥❣✉②➯♥ t❛ ❝â
C (α)
0 Dt f (t)

λ
=
Γ(n − α)
=

λ
Γ(n − α)

∞

=λ
k=1

∞

k=1
∞

t
0

λk−1 sαk−n
ds
Γ(αk − n + 1)(t − s)α+1−n

λk−1 tα(k−1) Γ(n − α)Γ(αk − n + 1)
Γ(αk − n + 1)Γ(αk − α + 1)

k=1
(λtα )k−1

Γ(α(k − 1) + 1)

= λEα,1 (λtα ) = λf (t).

✷✳✷ ●✐î✐ t❤✐➺✉ ❜➔✐ t♦→♥
❳➨t ❜➔✐ t♦→♥ ❦❤✉➳❝❤ t→♥ ❜➟❝ ♣❤➙♥
α

0 Dt u(x, t)

= −aux (x, t), x > 0, t > 0, α ∈ (0, 1),

u(1, t) = f (t), t

✭✷✳✸✮
✭✷✳✹✮

0,

✭✷✳✺✮

u(x, 0) = lim u(x, t) = 0,
x→∞

tr♦♥❣ ✤â 0 Dtα u ❧➔ ✤↕♦ ❤➔♠ ❜➟❝ ♣❤➙♥ ❈❛♣✉t♦ ✈î✐ ❜➟❝ α (0 < α

1) ✤÷ñ❝

①→❝ ✤à♥❤ ❜ð✐✿
t
1
g (s)
=
ds, 0
Γ(1 − α) 0 (t − s)α
dg(t)
α
, α = 1.

0 Dt g(t) =
dt
α
0 Dt g(t)

α<1

✭✷✳✻✮
✭✷✳✼✮


✷✸

❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ ❝❤ó♥❣ tæ✐ q✉❛♥ t➙♠ tî✐ ✈✐➺❝ ①→❝ ✤à♥❤ u ✈➔ ux
tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✸✮✕✭✷✳✺✮ tø ❞ú ❦✐➺♥ ✤÷ñ❝ ✤♦ t↕✐ x = 1 ❝õ❛ u(x, t)✿

u(1, t) = f (t)✳
❈èt ✤➸ →♣ ❞ö♥❣ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r✱ ❝❤ó♥❣ t❛ ♠ð rë♥❣ t➜t ❝↔ ❝→❝ ❤➔♠
❧➯♥ t♦➔♥ ❜ë trö❝ t❤ü❝ −∞ < t < ∞ ❜➡♥❣ ❝→❝❤ ❝❤♦ t➜t ❝↔ ❝→❝ ❤➔♠ ✤â
❜➡♥❣ ✵ ♥➳✉ t < 0✳ Ð ✤➙② ✈➔ tr♦♥❣ ❝↔ ♣❤➛♥ s❛✉✱ t❛ ❦þ ❤✐➺✉

L2 (R)✱ tù❝ ❧➔
|f (t)|2 dt

f =

·

❧➔ ❝❤✉➞♥

1
2

.

R

❇✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝õ❛ ❤➔♠ f (t) ✤÷ñ❝ ✈✐➳t ❧➔
∞

1
fˆ(ω) = √
2π
✈➔

·

p

f (t)e−iωt dt,

−∞

❧➔ ❦þ ❤✐➺✉ ❝❤✉➞♥ Hp ✱ tù❝ ❧➔

f

p

(1 + ω 2 )p |fˆ(ω)|2 dω

=

1
2

.

R

❇➙② ❣✐í t❛ ❜✐➳♥ ✤ê✐ ❋♦✉r✐❡r ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✸✮ ✈➔ ✭✷✳✹✮ ✤➸ ✤↕t ✤÷ñ❝
✭①❡♠ ❬✻❪✮

1
uˆx (x, ω) = − (iω)α uˆ(x, ω),
a
ˆ
uˆx (1, ω) = f (ω),

✭✷✳✽✮
✭✷✳✾✮

tr♦♥❣ ✤â

(iω)α =

απ
| ω |α (cos απ
2 + i sin 2 ),

ω

απ
| ω |α (cos απ
2 − i sin 2 ),

ω < 0.

0,

✭✷✳✶✵✮

❚ø ✭✷✳✽✮ ✈➔ ✭✷✳✾✮ t❛ ❞➵ ❞➔♥❣ t➼♥❤ ✤÷ñ❝
1

α

uˆ(x, ω) = e a (iω) (1−x) fˆ(ω),
1
α
1
uˆ(x, ω) = − (iω)α e a (iω) (1−x) fˆ(ω).
a

✭✷✳✶✶✮
✭✷✳✶✷✮

t r
1



u(0, ) = e a (i) f(),

1

1
u(0, ) = (i) e a (i) f().

a
ú ỵ r ữủ (i) õ tỹ ữỡ ởt s số ọ
tr t t số ừ f ụ õ t t r ởt s
ợ ữủ u
(x, ) u(0, ) ợ 0

e

1
|| cos
(1x)
a
2

x < 1 t t

+ t ử ỗ u

ux tứ u(1, t) = f (t) ởt t t ổ õ
qt t t r t t ữỡ õ

õ t
ởt tỹ ờ t tr ỷ tt t
số tr ừ t ử t t õ
s ờ rr

uc (x, ) = u(x, )max ,



uc,x (x, ) = u(x, )max ,



ừ õ

tr õ max trữ tr [max , max ] tự

max () =

1 [max , max ]
0
/ [max , max ].

õ õ õ t t ữủ sỷ ử
ờ rr ữủ

1

uc (x, t) =
2



uc (x, )eit d,



uc,x (x, )eit d.





ừ õ

1
uc,x (x, t) =
2





✷✺

❑➼ ❤✐➺✉ ♥❣❤✐➺♠ ❝❤➾♥❤ ❤â❛ ✤è✐ ✈î✐ ❞ú ❦✐➺♥ ❜à ♥❤✐➵✉ f δ ❧➔ uδc (x, t)✳ ❑❤✐ ✤â
t❛ ❝â ❝→❝ ✤→♥❤ ❣✐→ s❛✐ sè ❝õ❛ ♥❣❤✐➺♠

u(x, .) − uδc (x, .)

u(x, .) − uc (x, .) + u(x, .) − uδc (x, .) ,

✭✷✳✶✾✮

✈➔ s❛✐ sè ❝õ❛ ✤↕♦ ❤➔♠ ❝õ❛ ♥â ❧➔

ux (x, .) − uδc,x (x, .)

ux (x, .) − uc.x (x, .)
+ uc,x (x, .) − uδc,x (x, .) .

✭✷✳✷✵✮

❚✐➳♣ t❤❡♦ ❝❤ó♥❣ t❛ s➩ ✤÷❛ r❛ ✤→♥❤ ❣✐→ tè❝ ✤ë ❤ë✐ tö ❝❤♦ u(x, .)−uc (x, .)
✈➔ ux (x, .) − uc.x (x, .) ♥❤í ✈✐➺❝ ❝❤å♥ t❤➼❝❤ ❤ñ♣ t➛♥ sè ❝❤➦t ❝öt ωmax ✈➔
♠ët ❣✐↔ t❤✐➳t ✈➲ t➼♥❤ ❜à ❝❤➦♥ ❝õ❛ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ u ✈➔ ✤↕♦ ❤➔♠ ux ❝õ❛
♥â✳

✷✳✸✳✶ ✣à♥❤ ❧þ✳ ✭❬✺❪✮ ●✐↔ sû u ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✸✮✕✭✷✳✺✮✱ uδc ❧➔
♥❣❤✐➺♠ ❝❤➾♥❤ ❤â❛ ✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐ ✭✷✳✶✼✮ ✈î✐ ❞ú ❦✐➺♥ ❜à ♥❤✐➵✉ f δ ✈➔ f δ
t❤ä❛ ♠➣♥ f δ − f

δ✳
1

✭✶✮ ◆➳✉ u(0, .)

E 2

E ✤ó♥❣ ✈➔ ωmax ✤÷ñ❝ ❝❤å♥ ❧➔ ωmax = (a sec απ
2 ln δ )

t❤➻ ✈î✐ ♠å✐ x ∈ (0, 1) t❛ ❝â ✤→♥❤ ❣✐→

uδc (x, .) − u(x, .)
✭✷✮ ◆➳✉ u(0, ·)

2E 1−x δ x .

✭✷✳✷✶✮
1

E α
E ✤ó♥❣ ωmax = (a sec απ
2 ln(ln δ )) t❤➻ ✈î✐ p > 0✱

p

x = 0 t❛ ❝â ✤→♥❤ ❣✐→
uδc (0, ·) − u(0, ·)

απ
E
E
ln(ln )
δ ln + E a sec
δ
2
δ

−p
α

.

✭✷✳✷✷✮

✷✳✸✳✷ ✣à♥❤ ❧þ✳ ✭❬✺❪✮ ●✐↔ sû u ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✸✮✕✭✷✳✺✮✱ uδc,x
❧➔ ✤↕♦ ❤➔♠ ❝õ❛ ♥❣❤✐➺♠ ❝❤➾♥❤ ❤â❛ ①→❝ ✤à♥❤ t❤❡♦ ❝æ♥❣ t❤ù❝ ✭✷✳✶✽✮ ✈î✐ ❞ú
❦✐➺♥ ❜à ♥❤✐➵✉ f δ ✈➔ f δ − f
✭✶✮ ◆➳✉ u(0, ·)

δ✳
1

E 2
E ✤ó♥❣ ✈➔ ωmax ✤÷ñ❝ ❝❤å♥ ❧➔ ωmax = (a sec απ
2 ln δ )

t❤➻ ✈î✐ ♠å✐ x ∈ (0, 1) t❛ ❝â ✤→♥❤ ❣✐→

uδc,x (x, ·) − ux (x, ·)

1 + sec

απ E
ln
2
δ

E 1−x δ x .

✭✷✳✷✸✮