Xác định nguồn nhiệt trong phương trình truyền nhiệt bậc phân bằng phương pháp chặt cụt
Bạn đang xem bản rút gọn của tài liệu. Xem và tải ngay bản đầy đủ của tài liệu tại đây (412.67 KB, 30 trang )
✶
▼Ö❈ ▲Ö❈
❚r❛♥❣
▼Ö❈ ▲Ö❈ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶
▲❮■ ◆➶■ ✣❺❯ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷
❈❤÷ì♥❣ ✶✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❜ê trñ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹
✶✳✶ ❍➔♠ ●❛♠♠❛ ✈➔ ❤➔♠ ▼✐tt❛❣✲▲❡❢❢❧❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹
✶✳✷ ✣↕♦ ❤➔♠ ❜➟❝ ♣❤➙♥ ❈❛♣✉t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹
❈❤÷ì♥❣ ✷✳ ❳→❝ ✤à♥❤ ♥❣✉ç♥ ♥❤✐➺t tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ tr✉②➲♥
♥❤✐➺t ❜➟❝ ♣❤➙♥ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ❝❤➦t ❝öt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻
✷✳✶ ●✐î✐ t❤✐➺✉ ❜➔✐ t♦→♥ ✈➔ ✤→♥❤ ❣✐→ ê♥ ✤à♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻
✷✳✷ ❈❤➾♥❤ ❤â❛ ❝❤➦t ❝öt ✈➔ ✤→♥❤ ❣✐→ s❛✐ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
❑➌❚ ▲❯❾◆ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾
❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵
r ỳ ữỡ tr r
t út ữủ sỹ q t ự ừ ồ
ổ sỷ ử ữỡ tr r ữủ ự
ử t ổ ổ t t tr s t t ỵ õ ồ
õ s t t tr tr
ữỡ tr t ữủ ự rở r ợ
ở ữ ỵ ỹ sỹ tỗ t t t
ữỡ số ữ ữỡ tỷ ỳ ữỡ
s ỳ ụ ữủ ử t tr
tr ởt số t ố ử t ởt tr tổ t
ỳ ỳ số t ỗ ổ
ữủ t ú t ú tứ ỳ ờ s
s ởt số t t ữủ
ởt số ổ tr t ỹ t ữủ ổ
tr ừ r ổ ố tr õ ởt số
ổ tr ừ t ụ ữủ t
ở sỹ t ởt t ữủ ợ ử ừ
số t tr ữỡ tr t
ữ r ởt t qừ t t
t t ữủ tớ ữỡ tr
t t tớ õ ụ
t ữỡ tr t t tớ
ữỡ õ rr ữỡ
ữỡ tr
ữợ ự t ữủt ự ụ ữ
ú t t t ữủ tr ữỡ tr
t tr ỡ s t sr
trt s qt trt t ừ
t tr t
tt
tts
ú tổ ỹ ồ t
ỗ t tr ữỡ tr tr
t ữỡ t ửt
ừ
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 ồ ỡ
t ổ tr ở ổ t ồ t t
ú ù t tr sốt tớ ồ 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➔✐ ❧✐➺✉ ❬✺❪✳
✶✳✶ ❍➔♠ ●❛♠♠❛ ✈➔ ❤➔♠ ▼✐tt❛❣✲▲❡❢❢❧❡r
✶✳✶✳✶ ✣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∗✱
✹✮ Γ 12 = √π✱
√
✺✮ Γ n + 12 = 2(2n)!
π✱
2n n!
✻✮ ❱î✐ ♠å✐ sè t❤ü❝ x > 0 t❛ ❝â
∞
Γ(x) = 2
2
e−t t2x−1 dt.
0
❈❤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
∞
Γ(1) =
e−t t0 dt = −e−t
t=∞
= 1.
t=0
0
3) ❙û ❞ö♥❣ ❝→❝ t➼♥❤ ❝❤➜t 1) ✈➔ 2)✱ t❛ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ t➼♥❤ ❝❤➜t 3)
❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ q✉② ♥↕♣✳
4) ❚r÷î❝ ❤➳t t❛ t➼♥❤ t➼❝❤ ♣❤➙♥ I =
I=u
∞ −x2
dx✳
0 e
∞
−u2 t2
e
✣➦t x = ut, u > 0✱ t❛ ❝â
dt.
✭✶✳✸✮
0
2
◆❤➙♥ ❤❛✐ ✈➳ ❝õ❛ ✭✶✳✸✮ ✈î✐ e−u ✈➔ ❧➜② t➼❝❤ ♣❤➙♥ tø 0 ✤➳♥ ∞ t❛ ✤÷ñ❝
∞
2
I =
e
−u2
0
∞
=
0
1
2
u
2 2
e−u t dt du
0
∞
=
∞
∞
0
2
e−u
(1+t2 )
0
dt
π
= .
2
1+t
4
udu dt
✻
√
π
✳ ❇➙② ❣✐í ð ❝æ♥❣ t❤ù❝ ✭✶✳✶✮✱ ❜➡♥❣ ❝→❝❤ t❤ü❝
2
❤✐➺♥ ♣❤➨♣ ✤ê✐ ❜✐➳♥ t = u2 t❛ s➩ t❤✉ ✤÷ñ❝
❉♦ ✤â I =
∞ −x2
dx
0 e
=
∞
Γ(z) = 2
2
e−u u2z−1 du.
✭✶✳✹✮
0
❇➡♥❣ ❝→❝❤ t❤❛② z =
1
✈➔♦ ✭✶✳✹✮ t❛ ✤÷ñ❝
2
∞
1
2
Γ
=2
2
e−u du = 2I =
√
π.
0
5) ❚ø t➼♥❤ ❝❤➜t 1) ✈➔ t➼♥❤ ❝❤➜t 4) t❛ ❝â
Γ n+
1
2
1
1
Γ n−
2
2
1
3
3
= n−
n−
Γ n−
2
2
2
1
3
3 1
1
n−
· · · . .Γ
= n−
2
2
2 2
2
(2n)! √
π.
= 2n
2 n!
=
n−
6) ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ❤➔♠ ❣❛♠♠❛ t❛ ❝â
∞
Γ(x) =
e−t tx−1 dt.
0
❚❤ü❝ ❤✐➺♥ ♣❤➨♣ ✤ê✐ ❜✐➳♥ t = u2 ✈î✐ u ❜✐➳♥ t❤✐➯♥ tø ✵ ✤➳♥ ∞ t❛ ❝â
∞
Γ(x) =
0
∞
=
0
e−t tx−1 dt
∞
=2
0
=2
0
2
e−u (u2 )x−1 .2udu
∞
2
e−u u2x−1 du
2
e−t t2x−1 dt.
✼
✶✳✶✳✹ ✣à♥❤ ❧þ✳ ❱î✐ ♠å✐ z ∈ C t❤ä❛ ♠➣♥ ❘❡z > 0 t❛ ❝â
n!nz
.
n→∞ z(z + 1) · · · (z + n)
Γ(z) = lim
✭✶✳✺✮
❈❤ù♥❣ ♠✐♥❤✳ ✣➸ ❝❤ù♥❣ ♠✐♥❤ ❝æ♥❣ t❤ù❝ ✭✶✳✺✮✱ tr÷î❝ ❤➳t ❝❤ó♥❣ t❛ ①➨t ❤➔♠
n
fn (z) =
0
❇➡♥❣ ❝→❝❤ ✤ê✐ ❜✐➳♥ τ =
n
t
1−
n
tz−1 dt.
✭✶✳✻✮
s❛✉ ✤â sû ❞ö♥❣ t➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥ ❝❤ó♥❣ t❛
t
n
✤➦t ✤÷ñ❝
1
(1 − τ )n τ z−1 dτ
z
fn (z) = n
0
1
nz
=
z
= ···
(1 − τ )n−1 τ z dτ
n
0
n!nz
=
z(z + 1) · · · (z + n − 1)
n!nz
=
.
z(z + 1) · · · (z + n)
1
τ z+n−1 dτ
0
✭✶✳✼✮
❈❤ó þ r➡♥❣
lim
n→∞
n
t
1−
n
= e−t .
❉♦ ✤â ♠ö❝ ✤➼❝❤ t✐➳♣ t❤❡♦ ❝õ❛ ❝❤ó♥❣ t❛ ❧➔ ❝❤ù♥❣ ♠✐♥❤ ✤➥♥❣ t❤ù❝
n
t
1−
lim fn (z) = lim
n→∞
n→∞ 0
n
∞
t
=
lim 1 −
n
0 n→∞
n
tz−1 dt
n
∞
z−1
t
dt =
e−t tz−1 dt = Γ(z).
0
✭✶✳✽✮
✣➸ ✤↕t ✤÷ñ❝ ♠ö❝ ✤➼❝❤ ♥➔②✱ ❝❤ó♥❣ t❛ ❤➣② ✤→♥❤ ❣✐→ ✤↕✐ ❧÷ñ♥❣
∞
∆ = Γ(z) − fn (z) =
0
n
=
−t
e
0
t
− 1−
n
e−t tz−1 dt − fn (z)
n
∞
z−1
t
dt +
n
e−t tz−1 dt.
✭✶✳✾✮
> 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
n0 t õ
n
t z1
e t
et tx1 dt < , (x = z).
3
dt
n
n
ợ ồ n N n > n0 t t t tờ ừ t s
N
=
e
t
0
n
e
N
õ
n
e
N
t
tz1 dt
n
t
1
n
t
+
n
t
1
n
n
t
1
n
z1
t
dt +
t
n
n
n
z1
et tz1 dt.
t
e 1
dt
tx1 dt
n
N
<
et tx1 dt < , (x = z).
3
n
t
t tự t t t tự ờ tr
s
t
0
t
1
n
n
t2
<
, 0 < t < n.
2n
t t tự ữủ s r tứ ố q
t
e
t
1
n
n
t
e 1
=
0
n
n
d
n
t tự
t
1
n
0<
e
0
n
d <
n
t
0
t2
e d = e
.
n
2n
t
ỷ ử t tự t õ s ợ n ừ ợ
N
e
0
t
t
1
n
n
t
z1
1
dt <
2n
N
0
tx+1 dt < , (x = z).
3
tự ớ ữủ s r tứ
ỵ ữủ ự
✾
✶✳✶✳✺ ◆❤➟♥ 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) =
nz n!
1
=
lim
z(z + 1) · · · (z + m − 1) n→∞ (z + m)(z + m + 1) · · · (z + n)
nz n!
1
= lim
n→∞ z(z + 1) · · · (z + m − 1) (z + m)(z + m + 1) · · · (z + n)
n!nz
= lim
.
n→∞ z(z + 1) · · · (z + n)
✶✳✶✳✼ ✣à♥❤ ♥❣❤➽❛✳ ❍➔♠ ▼✐tt❛❣✲▲❡❢❢❧❡r ♠ët t❤❛♠ sè
∞
Eα (z) =
k=0
zk
, α > 0, z ∈ C.
Γ(αk + 1)
❍➔♠ ▼✐tt❛❣✲▲❡❢❢❧❡r ❤❛✐ t❤❛♠ sè
∞
Eα,β (z) =
k=0
❧➔ ❤➔♠ ❝â ❞↕♥❣
✭✶✳✶✻✮
❧➔ ❤➔♠ ❝â ❞↕♥❣
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)✮✳
Eα,β (z) =
1 (1−β)/α
z
exp z 1/α
α
p
1
−
exp(ζ 1/α )ζ (1−β)/α+k−1 dζ
2απi γ(1,ϕ)
z −k
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❝ ✭✶✳✸✸✮✳
✶✳✷ ✣↕♦ ❤➔♠ ❜➟❝ ♣❤➙♥ ❈❛♣✉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 (n)
a Dt f (t)
= f (n) (t), a
t
a
f (n) (s)
ds, a
(t − s)α+1−n
t
T, α = n.
t
T, n − 1 < 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 f (t)
a Dt
(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
(α)
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
lim C
a Dt f (t) = lim
α→n
t
=f
(n)
n
f (n+1) (τ )dτ
(a) +
a
= f (t), ∀t ∈ [a, T ].
❈❍×❒◆● ✷
❳⑩❈ ✣➚◆❍ ◆●❯➬◆ ◆❍■➏❚ ❚❘❖◆● P❍×❒◆● ❚❘➐◆❍
❚❘❯❨➋◆ ◆❍■➏❚ ❇❾❈ P❍❹◆ ❇➀◆● P❍×❒◆● P❍⑩P
❈❍➄❚ ❈Ö❚
❈❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tæ✐ tr➻♥❤ ❜➔② ❝→❝ ❦➳t qõ❛ tr♦♥❣ ❜➔✐ ❜→♦ ❬✹❪ ❝ô♥❣
♥❤÷ ✤➲ ①✉➜t ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët ✈➔✐ ❦➳t qõ❛ ♠î✐✳
✷✳✶ ●✐î✐ t❤✐➺✉ ❜➔✐ t♦→♥ ✈➔ ✤→♥❤ ❣✐→ ê♥ ✤à♥❤
❳➨t ❜➔✐ t♦→♥ tr✉②➲♥ ♥❤✐➺t ❜➟❝ ♣❤➙♥
α
0 ∂t u = uxx + f (x), 0 < x < 1, 0 < t < T,
u(0, t) = u(1, t) = 0, 0 t T,
u(x, 0) = 0, 0 x 1,
✭✷✳✶✮
γ
tr♦♥❣ ✤â 0 ∂t u ❧➔ ✤↕♦ ❤➔♠ ❜➟❝ ♣❤➙♥ ❈❛♣✉t♦ ❜➟❝ α ✤è✐ ✈î✐ t ✈➔ ✤÷ñ❝ ①→❝
✤à♥❤ ❜ð✐ ✭①❡♠ ❬✺❪✮
α
0 ∂t u
1
=
Γ(1 − α)
t
0
∂u(x, s) ds
, 0 < α < 1,
∂s (t − s)α
✈î✐ Γ(·) ❧➔ ❤➔♠ ●❛♠♠❛✳
❇➔✐ t♦→♥ ✭✷✳✶✮ ❧➔ ❜➔✐ t♦→♥ t❤✉➟♥ ❦❤✐ f (x) ✤➣ ✤÷ñ❝ ❝❤♦ t❤➼❝❤ ❤ñ♣✳
❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❝❤ó♥❣ tæ✐ q✉❛♥ t➙♠ tî✐ ✧❜➔✐ t♦→♥ ♥❣✉ç♥ ♥❣÷ñ❝✧
❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮✳ ❚ù❝ ❧➔ ❜➔✐ t♦→♥ ①→❝ ✤à♥❤ ♥❣✉ç♥ ♥❤✐➺t ❝❤÷❛ ✤÷ñ❝
❜✐➳t f (x) ❞ü❛ ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮ ✈➔ ♠ët t❤æ♥❣ t✐♥ ❜ê s✉♥❣ ✈➲ ♥❣❤✐➺♠
❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮ t↕✐ t❤í✐ ✤✐➸♠ t = T
u(x, T ) = g(x), 0 < x < 1.
✶✻
✭✷✳✷✮
✶✼
❚r♦♥❣ ❝→❝ ù♥❣ ❞ö♥❣ ❝ö t❤➸✱ ❞ú ❦✐➺♥ ✤➛✉ ✈➔♦ g(x) ✤÷ñ❝ ❝✉♥❣ ❝➜♣ ❜ð✐
✤♦ ✤↕❝ ♥➯♥ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ s❛✐ sè✳ ❉♦ ✤â✱ t❤❛② ✈➻ ❜✐➳t ❝❤➼♥❤ ①→❝ g(x)
t❛ ❝❤➾ ❜✐➳t ❞ú ❦✐➺♥ ✤♦ ✤↕❝ g δ ∈ L2 (0, 1) t❤ä❛ ♠➣♥
gδ − g
✈î✐
·
✭✷✳✸✮
δ,
❧➔ ❝❤✉➞♥ L2 ✈➔ ♠ù❝ s❛✐ sè δ > 0 ✤➣ ✤÷ñ❝ ❜✐➳t✳
✷✳✶✳✶ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✹❪✮✣↕♦ ❤➔♠ ❈❛♣✉t♦ ❜➟❝ α ✤÷ñ❝ ①→❝ ✤à♥❤ t❤❡♦ ❝æ♥❣
t❤ù❝
α
σ ∂t z(t)
✈➔
1
=
Γ(1 − α)
t
σ
z (s)
ds,
(t − s)α
0<α<1
✭✷✳✹✮
✤↕♦ ❤➔♠ ❘✐❡♠❛♥♥✲▲✐♦✉✈✐❧❧❡ ❜➟❝ α ✤÷ñ❝ ①→❝ ✤à♥❤ t❤❡♦ ❝æ♥❣ t❤ù❝
α
σ Dt z(t)
∂
1
=
Γ(1 − α) ∂t
t
σ
z(s)
ds, 0 < α < 1.
(t − s)α
✭✷✳✺✮
✷✳✶✳✷ ▼➺♥❤ ✤➲✳ ✭❬✺❪✮ ●✐ú❛ ✤↕♦ ❤➔♠ ❜➟❝ ♣❤➙♥ ❈❛♣✉t♦ ✈➔ ✤↕♦ ❤➔♠ ❜➟❝
♣❤➙♥ ❘✐❡♠❛♥♥✲▲✐♦✉✈✐❧❧❡ ❝â ♠è✐ q✉❛♥ ❤➺ s❛✉
α
σ Dt z(t)
=
z(σ)
1
+σ ∂tα z(t).
α
Γ(1 − α) (t − σ)
✷✳✶✳✸ ✣à♥❤ ♥❣❤➽❛✳ ✭❬✹❪✮❍➔♠ ▼✐tt❛❣✲▲❡❢❢❧❡r ❤❛✐ t❤❛♠ sè
✭✷✳✻✮
✤÷ñ❝ ①→❝ ✤à♥❤
❜ð✐
∞
Eα,β (z) =
k=0
zk
, ∀α > 0, β > 0.
Γ(αk + β)
✭✷✳✼✮
✷✳✶✳✹ ▼➺♥❤ ✤➲✳ ✭❬✹❪✮ ❱î✐ λ > 0 t❛ ❝â ❝→❝ ✤➥♥❣ t❤ù❝ s❛✉
α
α
0 ∂t Eα,1 (−λt )
= −λEα,1 (−λtα ), t > 0, 0 < α < 1,
d
Eα,1 (−λtα ) = −λtα−1 Eα,α (−λtα ), t > 0, α > 0.
dt
✭❬✹❪✮ ❍➔♠ Eα,1 (−t) ❧➔ ♠ët ❤➔♠ t❤ä❛ ♠➣♥
✭✷✳✽✮
✭✷✳✾✮
✷✳✶✳✺ ▼➺♥❤ ✤➲✳
dn
(−1) n Eα,1 (−t) 0
dt
n = 0, 1, 2, · · · ❉♦ ✤â t❛ ❝â
n
✈î✐ ♠å✐ t
0
✈➔
1 = Eα,1 (0) > Eα,1 (−t) > 0, t > 0.
✭✷✳✶✵✮
✶✽
❈❤ó þ r➡♥❣ ♥❣❤✐➺♠ u(x, t) ❝õ❛ ❜➔✐ t♦→♥ ✭✷✳✶✮ ❝â t❤➸ ❜✐➸✉ ❞✐➵♥ ❞÷î✐
❞↕♥❣
t
u(x, t) =
v(x, t; τ )dτ,
✭✷✳✶✶✮
0
tr♦♥❣ ✤â v(x, t; τ ) ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ s❛✉
α
τ ∂t v(x, t; τ ) = vxx (x, t; τ ), (x, t) ∈ (0, 1) × (τ, T ),
v(x, t; τ ) |t=τ =0 Dτ1−α f (x), 0 x 1,
v(1, t; τ ) = v(0, t; τ ) = 0, τ t T
✭✷✳✶✷✮
✈î✐ 0 Dτ1−α ❧➔ ✤↕♦ ❤➔♠ ❘✐❡♠❛♥♥✲▲✐♦✉✈✐❧❧❡ ✤÷ñ❝ ✤à♥❤ ♥❣❤➽❛ t❤❡♦ ❝æ♥❣ t❤ù❝
✭✷✳✺✮✳ ❇➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ t→❝❤ ❜✐➳♥ ✈➔ sû ❞ö♥❣ ❝æ♥❣ t❤ù❝ ✭✷✳✽✮ t❛ ❝â
∞
1−α
1Eα,1 (−k 2 π 2 (t − τ )α )fk Xk
0 Dτ
v(x, t; τ ) =
k=1
tr♦♥❣ ✤â {Xk =
√
2 sin(kπx), k = 1, 2, ...} ❧➔ ♠ët ❝ì sð trü❝ ❝❤✉➞♥ tr♦♥❣
1
√
2
L (0, 1) ✈➔ fk := (f, Xk ) = 2
f (x) sin(kπx)dx✳ ❑❤✐ ✤â ♥❣❤✐➺♠ ❝õ❛
0
❜➔✐ t♦→♥ ✭✷✳✶✮ ✤÷ñ❝ ✈✐➳t ❧↕✐ t❤➔♥❤
t
v(x, t; τ )dτ
u(x, t) =
0
∞
t
1−α
1Eα,1 (−k 2 π 2 (t − τ )α )dτ fk Xk .
0 Dτ
=
✭✷✳✶✸✮
0
k=1
▼➦t ❦❤→❝✱ t❛ ❜✐➳t r➡♥❣ ✭①❡♠ ❬✸❪✮
t
1−α
1Eα,1 (−k 2 π 2 (t − τ )α )dτ fk Xk
0 Dτ
0
t
(t − τ )α−1 Eα.α (−k 2 π 2 (t − τ )α )dτ fk Xk .
=
✭✷✳✶✹✮
0
❙û ❞ö♥❣ ❝æ♥❣ t❤ù❝ ✭✷✳✾✮ t❛ ✤↕t ✤÷ñ❝
∞
u(x, t) =
k=1
1 − Eα,1 (−k 2 π 2 τ α )
(f, Xk )Xk .
k2π2
✭✷✳✶✺✮
✶✾
✣à♥❤ ♥❣❤➽❛ t♦→♥ tû A : f → g ✳ ❑❤✐ ✤â ❝❤ó♥❣ t❛ ❝â
∞
Af (x) =
k=1
1 − Eα,1 (−k 2 π 2 τ α )
(f, Xk )Xk = g(x).
k2π2
❉➵ t❤➜② r➡♥❣ A ❧➔ t♦→♥ tû ❝♦♠♣❛❝t ✈î✐ ❝→❝ ❣✐→ trà ❦ý ❞à {σk }∞
k=1 ✤÷ñ❝
①→❝ ✤à♥❤ ❜ð✐
1 − Eα,1 (−k 2 π 2 τ α )
σk =
k2π2
✈➔
1 − Eα,1 (−k 2 π 2 τ α )
(g, Xk ) =
(f, Xk ).
k2π2
1
❉♦ ✤â t❛ ❝â (f, Xk ) = (g, Xk ) ✈➔
σk
∞
−1
f (x) = A
g(x) =
k=1
k2π2
(g, Xk )Xk .
1 − Eα,1 (−k 2 π 2 τ α )
✭✷✳✶✻✮
1
= O(k 2 ) ❧➔ ✤↕✐ ❧÷ñ♥❣ ❦❤æ♥❣ ❜à ❝❤➦♥ ❦❤✐ k → ∞ ✈➔ ♥❣❤✐➺♠ f
σk
✤÷ñ❝ ❣✐↔ t❤✐➳t t❤✉ë❝ ❦❤æ♥❣ ❣✐❛♥ L2 (0, 1) ♥➯♥ ❞ú ❦✐➺♥ ❝❤➼♥❤ ①→❝ g ♣❤↔✐
❱➻
❣✐↔♠ ♥❤❛♥❤ ❝ï O(k −2 )✳ ❚✉② ♥❤✐➯♥ ❞ú ❦✐➺♥ g ♥❤➻♥ ❝❤✉♥❣ ❝â ✤÷ñ❝ ❞♦ q✉❛♥
s→t✱ ✤♦ ✤↕❝ ♥➯♥ ❝❤ó♥❣ t❛ ❝❤➾ ❝â ✤÷ñ❝ ❞ú ❦✐➺♥ ❜à ♥❤✐➵✉ g δ ∈ L2 (0, 1) ✈î✐
g − gδ
δ ✳ ❱➻ ❝❤ó♥❣ t❛ ❦❤æ♥❣ ❤✐ ✈å♥❣ r➡♥❣ ❞ú ❦✐➺♥ ❜à ♥❤✐➵✉ g δ ❣✐↔♠
♥❤❛♥❤ ♥❤÷ ❞ú ❦✐➺♥ ❝❤➼♥❤ ①→❝ g ♥➯♥ ♥❣❤✐➺♠ t÷ì♥❣ ù♥❣ ♥❤➻♥ ❝❤✉♥❣ s➩
❦❤æ♥❣ t❤✉ë❝ ✈➔♦ ❦❤æ♥❣ ❣✐❛♥ L2 (0, 1)✳ ❉♦ ✤â ❜➔✐ t♦→♥ ♥➔② ❧➔ ❜➔✐ t♦→♥ ✤➦t
❦❤æ♥❣ ❝❤➾♥❤✱ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❦❤æ♥❣ ♣❤ö t❤✉ë❝ ❧✐➯♥ tö❝ ✈➔♦ ❞ú ❦✐➺♥✳
◆❤÷ ❝❤ó♥❣ t❛ ✤➣ ❜✐➳t ✤è✐ ✈î✐ ❜➜t ❦ý ♠ët ❜➔✐ t♦→♥ ✤➦t ❦❤æ♥❣ ❝❤➾♥❤✱
♠ët ❣✐↔ t❤✐➳t ✈➲ ❣✐î✐ ❤↕♥ t✐➯♥ ♥❣❤✐➺♠ ✤è✐ ✈î✐ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ❝➛♥ ✤÷ñ❝
✤➦t r❛✳ ◆➳✉ ❦❤æ♥❣ ❝â ❣✐↔ t❤✐➳t ♥➔②✱ sü ❤ë✐ tö ❝õ❛ ♥❣❤✐➺♠ ❝❤➾♥❤ ❤â❛ s➩
❦❤æ♥❣ ✤↕t ✤÷ñ❝ ❤♦➦❝ tè❝ ✤ë ❤ë✐ tö ❝â t❤➸ ❝❤➟♠ tò② þ✳ ❚r♦♥❣ ❧✉➟♥ ✈➠♥
♥➔②✱ ❝❤ó♥❣ t❛ ❣✐↔ t❤✐➳t ❣✐î✐ ❤↕♥ t✐➺♥ ♥❣❤✐➺♠ ♥❤÷ s❛✉
f
H p (0,1)
E, p > 0,
✭✷✳✶✼✮
tr õ E số ữỡ
ã
ỵ tr ổ
H p (0,1)
H p (0, 1) ữủ
1
2
f
H p (0,1)
(1 + k 2 )p | fk |2
=
.
k=1
ớ ú t t ỵ ờ õ
t
ỵ sỷ r f (x) ừ t ỗ
ữủ ữủ tọ t s ú
p
p+2
2
1 E,1 ( 2 T )
f
2
E p+2
p
p+2
g
.
ự ỷ ử t tự r t
õ
f
2
=
k=1
=
k=1
=
k=1
k=1
k=1
k=1
2
k22
1 E,1 (k 2 2 T )
2
| gk |
2
1 E,1 ( 2 T )
4
p+2
2
p+2
p
p+2
(| gk |
2
p+2
| gk |2
gk
2p
p+2
2
p+2
p
1 E,1 (k 2 2 T )
2p
4
| gk | p+2 | gk | p+2
k=1
1 E,1 (k 2 2 T )
k22
2
| gk |2
p+2
2
2
p+2
k22
=
k22
1 E,1 (k 2 2 T )
k22
1 E,1 (k 2 2 T )
=
k22
(g, Xk )Xk
1 E,1 (k 2 2 T )
| fk |2
gk
2
p+2
(1 + k 2 )p | fk |2
k=1
2p
p+2
gk
2p
p+2
2p
p+2
)
p+2
p
p
p+2
2p
p+2
2
1 E,1 ( 2 T )
õ t õ
f
4
E p+2
g
2p
p+2
2
1 E,1 ( 2 T )
.
p
p+2
2
E p+2
g
p
p+2
.
t f1(x) f2(x) ừ t ỗ
ữủ ợ ỳ g1 (x) g2 (x) tữỡ ự t s
ú
f1 (ã) f2 (ã)
2
1 E,1 ( 2 T )
p
p+2
f1 (ã) f2 (ã)
2
p+2
H p (0,1)
g1 (ã) g2 (ã)
p
p+2
.
ó r r g1 (ã) g2 (ã) 0 t f1 (ã) f2 (ã) 0
t qừ ờ õ ổ sỹ ờ ừ
số tr t ợ ỳ
õ t ửt s số
ó r ừ t ổ ờ t
k ợ tr ổ tự ởt tỹ ỷ tt
t k ợ tr ổ tự
õ t ửt ữ s
K
f,K =
k=1
k22
(g , Xk )Xk ,
2
2
1 E,1 (k T )
tr õ K số ữỡ õ trỏ õ
r ú t t t ồ t số t
t số õ ữ r
s số ừ ữỡ t ú tổ tr t ồ t
số t ỵ s
✷✷
✷✳✷✳✶ ✣à♥❤ ❧þ✳ ✭❬✹❪✮ ●✐↔ sû fδ,K (x) ❧➔ ♥❣❤✐➺♠ ❝❤➾♥❤ ❤â❛ ❝õ❛ ❜➔✐ t♦→♥
✭✷✳✶✮✲✭✷✳✷✮
✈î✐ ❞ú ❦✐➺♥ ❜à ♥❤✐➵✉ gδ (x) ✈➔ f (x) ❧➔ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ✈î✐ ❞ú
❦✐➺♥ ❝❤➼♥❤ ①→❝ g(x)✳ ◆❣♦➔✐ r❛ t❛ ❣✐↔ t❤✐➳t t❤➯♠ r➡♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭✷✳✸✮
✈➔ ✭✷✳✶✼✮ ✤÷ñ❝ t❤ä❛ ♠➣♥✳ ◆➳✉ t❤❛♠ sè ❝❤➾♥❤ ❤â❛ ✤÷ñ❝ ❝❤å♥ ❧➔ K = [γ]
✭[γ] ❧➔ ❦þ ❤✐➺✉ ♣❤➛♥ ♥❣✉②➯♥ ❝õ❛ γ ✮ ✈î✐
E
δ
γ=
1
p+2
✭✷✳✷✶✮
t❤➻ ✤→♥❤ ❣✐→ s❛✉ ✤➙② ✤ó♥❣
f (·) − fδ,K (·)
1+
π2
1 − Eα,1 (−π 2 T α )
2
p
E p+2 δ p+2 .
✭✷✳✷✷✮
❈❤ù♥❣ ♠✐♥❤✳ ❚ø ✭✷✳✶✻✮ ✈➔ ✭✷✳✷✵✮ t❛ ❝â
f (·) − fδ,K (·)
∞
k2π2
(g, Xk )Xk −
1 − Eα,1 (−k 2 π 2 T α )
=
k=1
∞
k=1
K
+
k2π2
(g, Xk )Xk −
1 − Eα,1 (−k 2 π 2 T α )
k2π2
(g, Xk )Xk −
1 − Eα,1 (−k 2 π 2 T α )
k=1
∞
=
k=K+1
K
+
k=1
∞
K
k2π2
(g δ , Xk )Xk
2
2
α
1 − Eα,1 (−k π T )
k=1
K
k=1
K
k=1
k2π2
(g, Xk )Xk
1 − Eα,1 (−k 2 π 2 T α )
k2π2
(g δ , Xk )Xk
2
2
α
1 − Eα,1 (−k π T )
k2π2
(g, Xk )Xk
1 − Eα,1 (−k 2 π 2 T α )
k2π2
(g − g δ , Xk )Xk
2
2
α
1 − Eα,1 (−k π T )
−p
p
(1 + k 2 ) 2 (1 + k 2 ) 2 (f, Xk )Xk
k=K+1
k2π2
+ sup
2 2 α
1 k K 1 − Eα,1 (−k π T )
(K + 1)−p E +
K
(g − g δ , Xk )Xk
k=1
k2π2
1 − Eα,1 (−π 2 T α )
δ.
K
K + 1 t t ữủ
22
1 E,1 ( 2 T )
2
2
2
p+2 p+2 .
1+
E
1 E,1 ( 2 T )
p E +
f (ã) f,K (ã)
=
t ú t ồ p = 0 tt t
ữủ t t f
E t ú t t ữủ t
ổ t ữủ sỹ ở tử ổ ợ tố ở õ
p
C p+2 C số ữ ỵ tr
t r tỹ ữủ
f
H p (0,1)
tữớ
1
ổ ữủ t r trữớ ủ ú t ồ = ( 1 ) p+2
t s ú
f (ã) f,K (ã)
tr õ số D ử tở f
p
D p+2 ,
H p (0,1)
p ồ ữ
t ỳ tr t t ử t
t ú tổ tr t ồ t số
m
(g , Xk )Xk
Pm g =
k=1
K = K(, g ) ừ t
(I PK )g
(I PK1 )g , > 1.
ờ sỷ ữủ tọ
K ừ t t õ s
K
E
( 1)
1
p+2
.
✷✹
❈❤ù♥❣ ♠✐♥❤✳ ❚ø ✭✷✳✸✮ ✈➔ ✭✷✳✷✺✮ t❛ ❝â
PK−1 g − g = (PK−1 − I)g δ − (I − PK−1 )(g − g δ )
(PK−1 − I)g δ − (I − PK−1 )(g − g δ )
✭✷✳✷✼✮
(τ − 1)δ.
▼➦t ❦❤→❝✱ t❛ ❝â
∞
PK−1 g − g =
(g, Xk )Xk
k=K
∞
=
k=K
∞
=
k=K
1 − Eα,1 (−k 2 π 2 T α )
(f, Xk )Xk
k2π2
1 − Eα,1 (−k 2 π 2 T α )
2 −p
2 p
2 (1 + k ) 2 (f, X )X
(1
+
k
)
k
k
k2π2
1 − Eα,1 (−k 2 π 2 T α )
2 −p
sup
(1
+
k
)2E
2π2
k
k K
E
.
K p+2
❚ê ❤ñ♣ ❝→❝ ✤→♥❤ ❣✐→ ✭✷✳✷✼✮ ✈➔ ✭✷✳✷✽✮ t❛ ✤↕t ✤÷ñ❝
(τ − 1)δ
E
.
K p+2
✭✷✳✷✽✮
✭✷✳✷✾✮
✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ r➡♥❣
K
E
(τ − 1)δ
1
p+2
.
❇ê ✤➲ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤✳
✷✳✷✳✺ ✣à♥❤ ❧þ✳ ✭❬✹❪✮ ●✐↔ sû fδ,K (x) ❧➔ ♥❣❤✐➺♠ ❝❤➾♥❤ ❤â❛ ❝õ❛ ❜➔✐ t♦→♥
✭✷✳✶✮✲✭✷✳✷✮
✈î✐ ❞ú ❦✐➺♥ ❜à ♥❤✐➵✉ gδ (x) ✈➔ f (x) ❧➔ ♥❣❤✐➺♠ ❝❤➼♥❤ ①→❝ ✈î✐ ❞ú
❦✐➺♥ ❝❤➼♥❤ ①→❝ g(x)✳ ◆❣♦➔✐ r❛ t❛ ❣✐↔ t❤✐➳t t❤➯♠ r➡♥❣ ❝→❝ ✤✐➲✉ ❦✐➺♥ ✭✷✳✸✮
✈➔ ✭✷✳✶✼✮ ✤÷ñ❝ t❤ä❛ ♠➣♥✳ ◆➳✉ t❤❛♠ sè ❝❤➾♥❤ ❤â❛ K ✤÷ñ❝ ❝❤å♥ ❧➔ ♥❣❤✐➺♠
❝õ❛ ✭✷✳✷✺✮ t❤➻ t❛ ❝â ✤→♥❤ ❣✐→ s❛✉
f (·) − fδ,K (·)
2
p
CE p+2 δ p+2 ,
✭✷✳✸✵✮
✷✺
tr♦♥❣ ✤â
p
−2
π2
π 2 (τ + 1)
p+2
p+2 .
+
C=(
)
(τ
−
1)
1 − Eα,1 (−π 2 T α )
1 − Eα,1 (−π 2 T α )
❈❤ù♥❣ ♠✐♥❤✳ ❚ø ❜➜t ✤➥♥❣ t❤ù❝ t❛♠ ❣✐→❝ ❝õ❛ ❝❤✉➞♥✱ t❛ ❝â
f (·) − fδ,K (·)
✭✷✳✸✶✮
f (·) − fK (·) + fK (·) − fδ,K (·) ,
k2 π2
2 2 α gk Xk ✳
α,1 (−k π T )
tr♦♥❣ ✤â fk = ΣK
k=1 1−E
❚r÷î❝ ❤➳t✱ t❛ ✤→♥❤ ❣✐→ sè ❤↕♥❣
t❤ù ♥❤➜t ð ✈➳ ♣❤↔✐ ❝õ❛ ✭✷✳✸✶✮✳ ❉➵ t❤➜② r➡♥❣
∞
f (·) − fK (·)
H p (0,1)
=
f k Xk
k=K+1
H p (0,1)
∞
(1 + k 2 )p fk2
=
1
2
✭✷✳✸✷✮
E.
k=K+1
❙û ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ t❛♠ ❣✐→❝ ✈➔ ✭✷✳✷✺✮ t❛ ❝â
Af − AfK = (I − PK )g
(I − PK )g δ + (I − PK )(g − g δ )
(I − PK )g δ + (I − PK )(g − g δ )
✭✷✳✸✸✮
(τ + 1)δ.
❑➳t ❤ñ♣ ✈î✐ ✤→♥❤ ❣✐→ ✭✷✳✶✾✮ t❛ ❝â
f (·) − fK (·)
π2
1 − Eα,1 (−π 2 T α )
p
p+2
2
p
E p+2 ((τ + 1)δ) p+2 .
✭✷✳✸✹✮
❚✐➳♣ t❤❡♦ t❛ ✤→♥❤ ❣✐→ sè ❤↕♥❣ t❤ù ❤❛✐ ð ✈➳ ♣❤↔✐ ❝õ❛ ✭✷✳✸✶✮✳ ❚❤❡♦ ▼➺♥❤
✤➲ ✷✳✶✳✺ ✈➔ ❇ê ✤➲ ✷✳✷✳✹ t❛ ❝â
