(Luận văn thạc sĩ) tìm hiểu về phương trình tích phân fredholm với nhân dạng chập trên khoảng hữu hạn

ĐẠI HỌC QUỐC GIA HÀ NỘI
TRƯỜNG ĐẠI HỌC KHOA HỌC TỰ NHIÊN
------------

NGUYỄN THỊ MINH THÚY

TÌM HIỂU VỀ PHƯƠNG TRÌNH FREDHOLM
VỚI NHÂN DẠNG CHẬP TRÊN KHOẢNG HỮU HẠN

LUẬN VĂN THẠC SĨ TOÁN HỌC

Hà Nội - 2018


ĐẠI HỌC QUỐC GIA HÀ NỘI
TRƯỜNG ĐẠI HỌC KHOA HỌC TỰ NHIÊN
------------

NGUYỄN THỊ MINH THÚY

TÌM HIỂU VỀ PHƯƠNG TRÌNH FREDHOLM
VỚI NHÂN DẠNG CHẬP TRÊN KHOẢNG HỮU HẠN
Chuyên ngành: Toán giải tích
Mã số: 60 46 01 02

LUẬN VĂN THẠC SĨ TỐN HỌC

Cán bộ hướng dẫn: TS. Lê Huy Chuẩn

Hà Nội - 2018

▲❮■ ❈❷▼ ❒◆
✣➸ ❤♦➔♥ t❤➔♥❤ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ ✈➔ ❦➳t t❤ó❝ ❦❤â❛ ❤å❝✱ ✈ỵ✐ t➻♥❤ ❝↔♠ ❝❤➙♥
t❤➔♥❤ ❡♠ ①✐♥ tọ ỏ t ỡ s s tợ trữớ ❤å❝ ❑❤♦❛ ❤å❝ ❚ü ♥❤✐➯♥
✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ ❡♠ ❝â ♠ỉ✐ tr÷í♥❣ ❤å❝ t➟♣ tèt tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ồ
t t trữớ
ỷ ớ ỡ tợ t❤➛② ▲➯ ❍✉② ❈❤✉➞♥ ✤➣ t➟♥ t➻♥❤ ❣✐ó♣ ✤ï ❡♠
tr♦♥❣ sốt q tr ự trỹ t ữợ t t
tốt ỗ t❤í✐✱ ❡♠ ①✐♥ ❜➔② tä ❧á♥❣ ❝↔♠ ì♥ tỵ✐ t❤➛② ❝ỉ ❦❤♦❛
❚♦→♥ ✲ ❈ì ✲ ❚✐♥ ❤å❝✱ ❜↕♥ ❜➧ ✤➣ ❣✐ó♣ ✤ï ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ ❡♠ tr♦♥❣ s✉èt q✉→
tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳
❊♠ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦
❍➔ ◆ë✐✱ ♥❣➔② ✷✽ t❤→♥❣ ✶✶ ♥➠♠ ✷✵✶✽
❍å❝ ✈✐➯♥

◆❣✉②➵♥ ❚❤à ▼✐♥❤ ❚❤ó②

✶


▼ö❝ ❧ö❝
▲❮■ ❈❷▼ ❒◆
▲❮■ ▼Ð ✣❺❯
✶ ❚♦→♥ tû t➼❝❤ ♣❤➙♥ ✈ỵ✐ ♥❤➙♥ ❞↕♥❣ ❝❤➟♣ tr♦♥❣ L2(0, w)

✶
✸
✹

✶✳✶ ❳➙② ❞ü♥❣ t♦→♥ tû ♥❣❤à❝❤ ✤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

ỹ tỗ t tró❝ ❝õ❛ t♦→♥ tû ♥❣❤à❝❤ ✤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹
✶✳✸ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ✈ỵ✐ ✈➳ ♣❤↔✐ ✤➦❝ ❜✐➺t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

Pữỡ
tr t r ợ tr♦♥❣
p
L (0, w)

✷✳✶ ❚➼♥❤ ❝❤➜t t♦→♥ tû t➼❝❤ ♣❤➙♥ ✈ỵ✐ tr
Pữỡ tr t ợ ❞↕♥❣ ❝❤➟♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
Pữỡ tr t ợ t
Pữỡ tr t ợ ♣❤↔✐ tr♦♥❣ ❦❤æ♥❣
Wp2 (0, w) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✸ ❱➼ ❞ö →♣ ❞ö♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✳✳
✳✳✳✳
✳✳✳✳
❣✐❛♥
✳✳✳✳
✳✳✳✳

Lp (0, w)

❑➌❚ ▲❯❾◆

✷✶

✷✶
✷✺

✷✺

✷✽
✸✷

✸✽

✷


▲❮■ ▼Ð ✣❺❯
◆❤✐➲✉ ✈➜♥ ✤➲ tr♦♥❣ t♦→♥ ❤å❝✱ ❝ì ❤å❝✱ ✈➟t ❧➼ ✈➔ ❝→❝ ♥❣➔♥❤ ❦➽ t❤✉➟t ❦❤→❝ ❞➝♥
✤➳♥ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥✳ ▼ö❝ t✐➯✉ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ t➻♠ ữỡ tr
t r ợ ❝â ❞↕♥❣
w

k(x − t)f (t)dt = φ(x),

µf (x) +
0

tr♦♥❣ ✤â µ ❧➔ sè ♣❤ù❝ ✈➔ k(x) ∈ L(0, w)✳
✣➸ ♥❣❤✐➯♥ ❝ù✉ ♣❤÷ì♥❣ tr➻♥❤ ♥➔②✱ t❛ s➩ ①➨t t♦→♥ tû t➼❝❤ ♣❤➙♥ ❝â ❞↕♥❣
w

d
Sf =
dx

s(x − t)f (t)dt

0

✈➔ ①➨t ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ❧♦↕✐ ✶ t÷ì♥❣ ù♥❣
Sf = φ(x).

❇➡♥❣ ❝→❝❤ ❝❤å♥

x

s(x) =

k(u)du + µ+

(x > 0),

k(u)du + µ−

(x < 0),

0
x

s(x) =
0

µ = à+ + à ,

ữỡ tr tr tr t ữỡ tr r ợ

ở ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ tr➻♥❤ ❜➔② ❧↕✐ ♠ët sè ❦➳t q ự ữỡ

tr r ợ ố ử ừ ỗ ữỡ
ã ữỡ ❝õ❛ ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② t➼♥❤ ❦❤↔ ♥❣❤à❝❤ ❝õ❛ t♦→♥ tû S tr♦♥❣
❦❤ỉ♥❣ ❣✐❛♥ L2(0, w)❀ ❝➜✉ tró❝ ❝õ❛ t♦→♥ tû ♥❣❤à❝❤ ✤↔♦ ✈➔ ①➨t ♣❤÷ì♥❣ tr➻♥❤
t➼❝❤ ♣❤➙♥ tr♦♥❣ ❦❤ỉ♥❣ L2(0, w) ợ t
ã ữỡ ❝õ❛ ❧✉➟♥ ✈➠♥ tr➻♥❤ ❜➔② ✈➲ t➼♥❤ ❝❤➜t ❝õ❛ t♦→♥ tû S tr♦♥❣ ❦❤ỉ♥❣
❣✐❛♥ Lp(0, w)✱ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ Wp2(0, w) ✈➔ ❝✉è✐
❝ị♥❣ ❧➔ ♠ët ✈➼ ❞ư ♠✐♥❤ ❤å❛✳
◆ë✐ ❞✉♥❣ ❝õ❛ ❧✉➟♥ ✈➠♥ ✤÷đ❝ tr➻♥❤ ❜➔② ❞ü❛ t❤❡♦ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪✳
✸


ữỡ
tỷ t 2 ợ
tr L (0, w)
✶✳✶ ❳➙② ❞ü♥❣ t♦→♥ tû ♥❣❤à❝❤ ✤↔♦
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ t❛ ♥❣❤✐➯♥ ❝ù✉ ✈➲ t➼♥❤ ❦❤↔ ♥❣❤à❝❤ ❝õ❛ t♦→♥ tû S tr♦♥❣
✈ỵ✐ t♦→♥ tû S ❝â ❞↕♥❣

L2 (0, w)

w

d
Sf =
dx

s(x − t)f (t)dt, f (x) ∈ L2 (0, w),

✭✶✳✶✮

0
w

tr♦♥❣ ✤â s(x) t❤✉ë❝ L2(−w, w) ✈➔ ❤➔♠ sè g(x) = s(x − t)f (t)dt ❧➔ ♠ët ❤➔♠ sè
0
❧✐➯♥ tö❝ t✉②➺t ✤è✐✳
❚♦→♥ tû S ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ♥❤÷ tr➯♥ ❧➔ t♦→♥ tû t ợ
t ữủ t tỷ ♥❣❤à❝❤ ✤↔♦ ❝õ❛ t♦→♥ tû S t❛ ♣❤↔✐ t➻♠ ❤➔♠ sè N1(x), N2(x)
t❤ä❛ ♠➣♥
SN1 (x) = M (x), SN2 (x) = 1,

✈ỵ✐ 1 ❧➔ ❤➔♠ ❤➡♥❣ ❜➡♥❣ 1 ✈➔ M (x) = s(x), 0 ≤ x ≤ w. ❑❤✐ ✤â✱ t♦→♥ tû ♥❣❤à❝❤ ✤↔♦
T = S −1 ✤÷đ❝ ❜✐➸✉ ❞✐➵♥ q số N1 (x) N2 (x)
ỵ ❈❤♦ S ❧➔ t♦→♥ tû ❜à ❝❤➦♥ tr♦♥❣ L2(0, w)✳ õ t tỷ S ữủ
ữợ
w
Sf =

d
dx

s(x, t)f (t)dt,
0

tr♦♥❣ ✤â s(x, t) t❤✉ë❝ L2(0, w) ✈ỵ✐ ♠é✐ x ❝è ✤à♥❤✳
❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ✤à♥❤ ♥❣❤➽❛ ❤➔♠ sè
ex (t) =

1,

0 ≤ t ≤ x,

0,

x < t ≤ w.

✹


ữỡ tỷ t ợ tr♦♥❣

L2 (0, w)

◆➳✉ f ∈ L2(0, w) t❤➻ Sf ∈ L2(0, w) t ổ ữợ tr L2(0, w)
t❛ ❝â
x
Sf, ex =

(Sf )dt.
0

▲↕✐ ❝â

Sf, ex = f, S ∗ ex

✭✶✳✷✮

S ∗ ex = s(x, t),

✭✶✳✸✮

✈ỵ✐ S ∗ ❧➔ t♦→♥ tû ❧✐➯♥ ❤ñ♣ ❝õ❛ t♦→♥ tû S ✳ ✣➦t
t❛ ✤÷đ❝

w
∗

s(x, t)f (t)dt.

f, S ex =

✭✶✳✹✮

0

❚ø ✭✶✳✷✮ ✲ ✭✶✳✹✮ t❛ ❝â
x

w

(Sf )dt = Sf, ex = f, S ∗ ex =
0

❱➟②

d
Sf =
dx

s(x, t)f (t)dt.

0

w

s(x, t)f (t)dt.
0

❚ø ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ❤➔♠ ex ✈➔ ✤➥♥❣ t❤ù❝ ✭✶✳✸✮ t❛ s✉② r❛ ❤➺ q✉↔ s❛✉ ✤➙②✳
❍➺ q✉↔ ✶✳✶✳2❍➔♠ sè s(x, t) tr♦♥❣ ❝æ♥❣ t❤ù❝ ✭✶✳✶✮ ❝â t❤➸ ✤÷đ❝ ❝❤å♥ s❛♦ ❝❤♦
s(x, t) t❤✉ë❝ L (0, w) ✈ỵ✐ ♠é✐ x ✈➔
w

|s(x + ∆x, t) − s(x, t)|2 dt ≤ ||S||2 |∆x|.

s(0, t) = 0;
0

❚❛ ❦➼ ❤✐➺✉ A ❧➔ t♦→♥ tû t➼❝❤ ♣❤➙♥ tr➯♥ L2(0, w) ①→❝ ✤à♥❤ ❜ð✐
x

Af = i

f (t)dt.

✭✶✳✺✮

0

❑❤✐ ✤â✱ t♦→♥ tû ❧✐➯♥ ❤ñ♣ A∗ ❝â ❞↕♥❣
w

∗

A f = −i

f (t)dt.
x

✺

✭✶✳✻✮


ữỡ tỷ t ợ tr

L2 (0, w)

ỵ S t tỷ ❜à ❝❤➦♥ ✈ỵ✐ ♥❤➙♥ ✈✐ ♣❤➙♥ ❞↕♥❣ ✭✶✳✶✮✳ ❑❤✐ ✤â✱ t❛
❝â ❜✐➸✉ ❞✐➵♥

w

(AS − SA∗ )f = i

✭✶✳✼✮

(M (x) + N (t))f (t)dt,
0

tr♦♥❣ ✤â M (x) = s(x), N (x) = −s(−x), 0 ≤ x ≤ w.
❈❤ù♥❣ ♠✐♥❤✳ ❚ø ✭✶✳✶✮✱ ✭✶✳✺✮ ✈➔ ✭✶✳✻✮ t❛ ❝â


ASf = A 



w

d
dx

x



0

w

SA∗ f = S −i





s (x − u)

0

u



w

f (t)dts(x) −

= i −
0



w

s(x − t)d

f (t)dt 
u

0



w

u

u

w

−

f (t)dtdu

f (t)dtd(s(x − u))
0

f (t)dts(x − u)

= −i 

s(x − u)

w w

f (t)dtdu = i
w

w

0



u

 0w

w

d
f (t)dt = −i
dx
x

w

w

= −i 



w

f (t)dt = −iS 

s(−t)f (t)dt.
0

0

x



w

s(x − t)f (t)dt − i

=i

0

0

w

x

s(τ − t)f (t)dt

s(τ − t)f (t) dτ

dτ

0

0



w

d

s(x − t)f (t)dt = i

w

=i



w

s(x − t)f (t)dt = i
0

s(x) − s(x − t) f (t)dt.
0

❉♦ ✤â
(AS − SA∗ )f = ASf − SA∗ f
w

w

s(x − t)f (t)dt − i

=i
0

0

0
w

s(x − t)f (t)dt − i

=i
0

0

0

0
w

s(x − t)f (t)dt = i
0

s(x − t)f (t)dt
0

[s(x) − s(−t)]f (t)dt
0

✻

w

s(x)f (t)dt − i

s(−t)f (t)dt + i

w

s(x)f (t)dt − i

s(x − t)f (t)dt

w

0
w

w

s(x)f (t)dt − i

s(−t)f (t)dt + i

w

=i

w


ữỡ tỷ t ợ tr♦♥❣

L2 (0, w)

w

[M (x) + N (t)]f (t)dt (0 ≤ x w).

=i

0

ỵ ữủ ự
sỷ t♦→♥ tû S ❞↕♥❣ ✭✶✳✶✮ ❝â ♥❣❤à❝❤ ✤↔♦ ❜à ❝❤➦♥✳ ❑❤✐ ✤â✱ ✤➥♥❣ t❤ù❝ ✭✶✳✼✮
❧➔ ❝ì sð ✤➸ t❛ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ①➙② ❞ü♥❣ t♦→♥ tû ♥❣❤à❝❤ ✤↔♦ ❝õ❛ S ✳ ❱ỵ✐ T = S −1
t❛ ❝â
(T A − A∗ T )f = T (AS − SA∗ )T f = S −1 (AS − SA∗ )S −1 f





w

[M (x) + N (t)](S −1 f )(t)dt

= S −1 i
0



w

= S −1 i

M (x)(S −1 f )(t)dt + i
0

N (t)(S −1 f )(t)dt
0

w

w

(S −1 f )(t)dtS −1 (M (x)) + i

=i



w

0

N (t)S −1 f (t)dtS −1 (1)
0

=i S

−1

f, 1 N1 (x) + i S

−1

f, N (t) N2 (x)

= i f, (S −1 )∗ 1 N1 (x) + i f, (S −1 )∗ N (t) N2 (x)
= i f, M1 (t) N1 (x) + i f, M2 (t) N2 (x)
w

=i

[N1 (x)M1 (t) + N2 (x)M2 (t)]f (t)dt,
0

tr♦♥❣ ✤â S ∗M1 = 1,
❚❛ ❦➼ ❤✐➺✉
✈➔ t♦→♥ tû

S ∗ M2 = N (x), SN1 = M (x), SN2 = 1.
Q(x, t) = N1 (x)M1 (t) + N2 (x)M2 (t),
w

Qf (x) =

Q(x, t)f (t)dt.
0

ỵ ◆➳✉ t♦→♥ tû T ❜à ❝❤➦♥ tr♦♥❣ L2(0, w) ✈➔ t❤ä❛ ♠➣♥
T A − A∗ T = iQ

t❤➻ ❤➔♠ sè

✭✶✳✽✮

2w−|x−t|

1
φ(x, t) =

2

Q
x+t

✼

s+x−t s−x+t
,
ds
2
2

✭✶✳✾✮


ữỡ tỷ t ợ tr♦♥❣

L2 (0, w)

❧✐➯♥ tö❝ t✉②➺t ✤è✐ t❤❡♦ t ✈➔
w

∂
φ(x, t) f (t)dt.
∂t

d
T f (x) =
dx

✭✶✳✶✵✮

0

❈❤ù♥❣ ♠✐♥❤✳ ❉♦ T ❜à ❝❤➦♥ t ỵ tỗ t F (x, t) t❤✉ë❝ L2(0, w)
s❛♦ ❝❤♦ t♦→♥ tû T ✤÷đ❝ ❜✐➸♥ ❞✐➵♥ ữợ
w

d
Tf =
dx

F (x, t)f (t)dt.



0

q sè F (x, t) ❝â t❤➸ ✤÷đ❝ ❝❤å♥ s❛♦ ❝❤♦
w

|F (x + ∆x, t) − F (x, t)|2 dt ≤ ||T ||2 |∆x|.

F (w, t) = 0,

✭✶✳✶✷✮

0

❱➻ ✈➟②✱ t➼❝❤ ♣❤➙♥

w

F (x, s)ds
t

❧✐➯♥ tö❝ t❤❡♦ x ♥➯♥ t❛ ❝â t❤➸ ✤à♥❤ ♥❣❤➽❛ F1(x, t) ❜ð✐
w w

F1 (x, t) = −

F (u, s)dsdu.
x

✭✶✳✶✸✮

t

❚ø ✭✶✳✶✶✮✱ ✭✶✳✶✷✮ t❛ s✉② r❛ t♦→♥ tû T1 ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐
w

T1 f =

F1 (x, t)f (t)dt
0

t❤ä❛ ♠➣♥
❚❤❡♦ ✭✶✳✽✮✱ t❛ ❝â ✤➥♥❣ t❤ù❝ s❛✉

T1 = iA∗2 T A.

T1 A − A∗ T1 = iQ1 ,

tr♦♥❣ ✤â

w

✭✶✳✶✹✮

Q1 (x, t)f (t)dt

✭✶✳✶✺✮

Q(u, s)ds(u − x)du.

✭✶✳✶✻✮

Q1 f = iA∗2 QAf =
0
w w

Q1 (x, t) =
x

t

✽

ữỡ tỷ t ợ tr♦♥❣

❚ø ✭✶✳✶✹✮ t❛ ❝â

w

L2 (0, w)

w

F1 (x, t)ds +
t

✭✶✳✶✼✮

F1 (s, t)ds = Q1 (x, t).
x

❚ø ✭✶✳✶✸✮✱ ✭✶✳✶✻✮ t❛ ❝â ✤➥♥❣ t❤ù❝
w

∂F1 (x, s)
∂Q1
ds − F1 (x, t) =
.
∂x
∂x

✭✶✳✶✽✮

∂F1 (x, t) ∂F1 (x, t)
∂ 2 Q1 (x, t)
−
=
.
∂x
∂t
∂t∂x

✭✶✳✶✾✮

t

❚ø ✭✶✳✶✻✮ ✈➔ ✭✶✳✶✽✮ t❛ ❝â
−

❚❤❡♦ ✭✶✳✶✸✮ t❛ t❤➜②
w w

w w

F (u, s)dsdu = 0, F1 (x, w) = −

F1 (w, t) = −
w

t

F (u, s)dsdu = 0.
x w

❙✉② r❛

✭✶✳✷✵✮
❙û ❞ö♥❣ ♣❤➨♣ ✤ê✐ ❜✐➳♥ ξ = x + t, η = x − t tr♦♥❣ ✭✶✳✶✾✮✱ ✭✶✳✷✵✮ t❛ t❤✉ ✤÷đ❝
F1 (w, t) = F1 (x, w).

−∂F1

ξ+η ξ−η
,
2
2
∂x

∂F1
−

ξ+η ξ−η
,
2
2
∂t

ξ+η ξ−η
,
2
2
∂t∂x

∂ 2 Q1
=

.

❚❛ ❦➼ ❤✐➺✉
F2 (ξ, η) = F1

ξ+η ξ−η
,
2
2

∂ 2 Q1 (x, t)
, Q2 (x, t) =
=
∂x∂t

w

Q(u, t)du.
x

❑❤✐ ✤â

−∂F2 (ξ, η) ∂F2 (ξ, η)
−
= Q2
∂x
∂t

ξ+η ξ−η
,
2
2

❙û ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ ✤↕♦ ❤➔♠ ❝õ❛ ❤➔♠ ❤đ♣ t❛ ✤÷đ❝
−2

❍ì♥ ♥ú❛ tø ✭✶✳✷✵✮ s✉② r❛✱

∂F2 (ξ, η)
= Q2
∂ξ

ξ+η ξ−η
,
2
2

F2 (2w − |η|, η) = 0.

✾

.

.

✭✶✳✷✶✮

ữỡ tỷ t ợ tr♦♥❣

❉♦ ✤â

L2 (0, w)

ξ

−1
F2 (ξ, η) =
2

Q2

s+η s−η
,
ds.
2
2

2w−|η|

✣÷❛ ✈➲ ❜✐➳♥ x ✈➔ t✱ t❛ ✤÷đ❝
x+t

−1
F1 (x, t) =
2

s+x−t s−x+t
,
ds.
2
2

Q2

✭✶✳✷✷✮

2w−|x−t|

✣➦t z = s − 2x + t ✈➔ ✈✐➳t ❧↕✐ ✭✶✳✷✷✮ ữợ
F1 (x, t) =






t

Q2 (z + x t, z)dz,

x≥t

w−x+t

w



 Q2 (z + x − t, z)dz,

x ≤ t.

t

❑❤✐ ✤â✱ t❤❡♦ ✭✶✳✷✶✮✿
◆➳✉ x ≥ t t❤➻




t

∂F1 
= −
∂x

Q2 (z + x − t, z)dz 

w−x+t

x





t

= − −(w − x + t)x .Q2 (z + x − t, w − x + t) +

∂Q2 (z + x − t, z) 
dz
∂z

w−x+t
t

∂Q2 (z + x − t, z)
dz
∂x

= −Q2 (z + x − t, w − x + t) +
w−x+t
t

∂Q2 (z + x − t, z)
dz.
∂x

=
w−x+t

◆➳✉ x < t t❤➻




w

∂F1 
=
∂x

w

Q2 (z + x − t, z)dz  =
t

❱➻ ✈➟②

x

Q2 (z + x − t, z)dz.
t

2w−|x+t|

∂F1
−1
=
∂x
2

Q
x+t

✶✵

s+x−t s−x+t
,
ds.
2
2

✭✶✳✷✸✮


ữỡ tỷ t ợ tr♦♥❣

❚ø ✭✶✳✶✸✮ s✉② r❛
F (x, t) = −

L2 (0, w)

∂ 2 F1 (x, t)
.
∂t∂x

✭✶✳✷✹✮

❑➳t ❤ñ♣ ✭✶✳✶✶✮✱ ✭✶✳✷✸✮✱ ✭✶✳✷✹✮ t❛ ❝❤ù♥❣ ữủ ỵ
t tỷ
U f = f (w − x).

✭✶✳✷✺✮

U SU = S ∗ .

✭✶✳✷✻✮

❇ê ✤➲ ✶✳✶✳ ❚❛ ❝â ❜✐➸✉ ❞✐➵♥

❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ♠é✐ ❤➔♠ g(x) ❦❤↔ ✈✐ t❤ä❛ ♠➣♥ g(0) = g(w) = 0, ✈➔ f ∈ L2(0, w),
t❛ ❝â
w

w

d
dx

Sf, g =
0

s(x − t).f (t)dtg(x)dx

0
w w

=−

s(x − t)f (t)dt.g (x)dx
0

0
w

=−

w

0

❱➻ ✈➟②

s(x − t)g (x)dxdt.

f (t)
0

w

w−x

d
s(t − x)g (x)dt = −
dx

S ∗g = −

s(v).g(v + t)dv.
−x

0

✣➦t v + x = t, t❛ ✈✐➳t ❧↕✐ ❜✐➸✉ t❤ù❝ ữợ
w

d
S g =
dx

s(t x)g(t)dt.
0

t tû S, U t❛ ❝â
w

w

d
SU g =
dx

s(x − t)g(w − t)dt =
0

g(w − t)d(s(x − t))
0

w

= g(w − t)s(x − t)

w
0

s(x − t).g (w − t)dt

+
0

w

s(x − t)g (w − t)dt

=
0

✶✶


ữỡ tỷ t ợ tr♦♥❣

L2 (0, w)

w

s(w − x − t).g (w − t)dt

U SU g =
0

w

w

s(w − x − t).g (w − t)dt = −

=

s(w − x − t).d(g(w − t))
0

0





w

= − s(w − x − t)g(w − t)

w
0

d
s(w − x − t)g(w − t)dt
dt

+
0

0

d

s(u − x)g(u)du.
du

=−
w

❱➻ ✈➟②✱ ✭✶✳✷✻✮ ✤ó♥❣ ✈ỵ✐ ♠å✐ ❤➔♠ g ❦❤↔ ✈✐ ✈➔ t❤ä❛ ♠➣♥ g(0) = g(w) = 0✳ ❉♦ S ❜à
❝❤➦♥ ♥➯♥ U SU = S ∗ ✤ó♥❣ ✈ỵ✐ ♠å✐ ❤➔♠ g t❤✉ë❝ L2(0, w)✳
❇ê ✤➲ ✶✳✷✳ ❈❤♦ ❝→❝ ❤➔♠ sè N1 ✈➔ N2 tr♦♥❣ L2(0, w) t❤ä❛ ♠➣♥
SN1 = M, SN2 = 1.
✭✶✳✷✼✮
❑❤✐ ✤â
S ∗ M1 = 1, S ∗ M2 = N (x),
✭✶✳✷✽✮
tr♦♥❣ ✤â
M1 (x) = N2 (w − x), M2 (x) = 1 − N1 (w − x).
✭✶✳✷✾✮
❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â
w

w

d
s(x − t).1dt =
dx

d
S1 =
dx
0

s(x − t)dt
0

w

s (x − t)dt = −s(x − t)

=

w
0

= −s(x − w) + s(x)

0

= M (x) + N (w − x),

tù❝ ❧➔

S(1 − N1 (x)) = N (w − x).

✭✶✳✸✵✮

❚❤❡♦ ✭✶✳✷✻✮✱ ✭✶✳✷✼✮ ✈➔ ✭✶✳✸✵✮ t❛ ❝â ✈ỵ✐ M1(x) = N2(w−x) ✈➔ M2(x) = 1−N1(w − x)
t❤➻
S ∗ M1 (x) = U SU M1 (x) = U SU N2 (w − x) = U SN2 (x) = U 1 = 1;

S ∗ M2 (x) = U SU (1 − N1 (w − x)) = U S(1 − N1 (x)) = U (N (w x)) = N (x).

ỵ ✶✳✸✱ t♦→♥ tû T ✤÷đ❝ ①→❝ ✤à♥❤ ♥➳✉ T ❝â ❞↕♥❣ ✭✶✳✽✮ ✈➔ Q(x, t) =
N1 (x)M1 (t) + N2 (x)M2 (t). ❑➳t ❤đ♣ ✈ỵ✐ ❇ê ✤➲ ✶✳✷✱ t❛ ❝â ỵ s



ữỡ tỷ t ợ tr

L2 (0, w)

ỵ S ởt
t tỷ tr♦♥❣ L2(0, w) ❝â ❞↕♥❣ ✭✶✳✶✮✳ ●✐↔ sû S ❝â
−1

♥❣❤à❝❤ ✤↔♦ ❜à ❝❤➦♥ T = S ✳ ❑❤✐ ✤â T ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ✭✶✳✾✮ ✲ ✭✶✳✶✵✮✱ tr♦♥❣ ✤â
Q(x, t) = N2 (w − t)N1 (x) + (1 − N1 (w − t))N2 (x).
✭✶✳✸✶✮
❚✐➳♣ t❤❡♦✱ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ✤à♥❤ ỵ ữủ ừ ỵ
ỵ t tû S ❜à ❝❤➦♥ tr♦♥❣ L2(0, w) t❤ä❛ ♠➣♥
w
∗

(AS − SA )f = i

(M (x) + N (t))f (t)dt,

✭✶✳✸✷✮

0

✈ỵ✐ M (x) ✈➔ N (x) tr♦♥❣ L2(0, w)✳ ❑❤✐ ✤â✱ S ❧➔ ♠ët t♦→♥ tû ✈ỵ✐ ♥❤➙♥ ❞↕♥❣ ❤✐➺✉
✈➔ M (x) = s(x), N (x) = −s(−x).
❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ✈✐➳t ❧↕✐ ữợ s
w

(S1 A A S1 )f = i

M (w − x) + N (w − t) f (t)dt,
0

✈ỵ✐ S1 = U SU ;

A∗ = U AU.

❚❤➟t ✈➟②✱
w

U (SA∗ − AS)U f = U (SA∗ − AS)f (w − x) = U − i

(M (x) + N (t))f (w − t)dt
0

w

(M (w − x) + N (t))f (w − t)dt

= −i
0
w

(M (w − x) + N (t))f (w − t)dt

=i
0

0

= −i

[M (w − x) + N (w − s)]f (s)ds

w
w

[M (w − x) + N (w − s)]f (s)ds.

=i
0

ử ỵ t ữủ
w

d
S1 f =
dx


1 (x, t) .f (t)dt,
∂t
0

✶✸

✭✶✳✸✸✮


ữỡ tỷ t ợ tr♦♥❣

L2 (0, w)

tr♦♥❣ ✤â✱
2w−|x−t|

φ1 (x, t) =

1
2

M w−

s+x−t
s−x+t
+N w−
2
2

ds

x+t

tù❝ ❧➔✱
φ1 (x, t) =

w


 M (w − s)ds +

w−x+t

x
w+x−t





N (w − s)ds

♥➳✉ x > t

N (w − s)ds

♥➳✉ x < t.

t
w

M (w − s)ds +
x

t

✭✶✳✸✹✮

❚❤❛② ✭✶✳✸✹✮ ✈➔♦ ✈➳ ♣❤↔✐ ❝õ❛ ✭✶✳✸✸✮ t❛ ✤÷đ❝
w

d
S1 f =
dx

s1 (x − t)f (t)dt,
0

tr♦♥❣ ✤â✱

s1 (x) = N (x), −s1 (−x) = M (x); 0 ≤ x ≤ w.

▼➔ U S1U = S t❛ s✉② r❛

w

d
Sf =
dx

s(x − t)f (t)dt; s(x) = −s1 (−x).

✭✶✳✸✺✮

✭✶✳✸✻✮

0

❚ø ✭✶✳✸✺✮✱ ✭✶✳✸✻✮ ỵ ữủ ự

ỹ tỗ t tró❝ ❝õ❛ t♦→♥ tû ♥❣❤à❝❤ ✤↔♦
❈❤♦

❧➔ t♦→♥ tû ❞↕♥❣ ✭✶✳✶✮ ✈➔

S
RS = {Sf : f ∈ L2 (0, w)}.

RS

❧➔ t➟♣ ↔♥❤ ❝õ❛ t♦→♥ tû S ✱ tù❝ ❧➔✱

❇ê ✤➲ ✶✳✸✳ ●✐↔ sû 1 ✈➔ M (x) t❤✉ë❝ Rs✳ ❑❤✐ ✤â✱ Rs trò ♠➟t tr♦♥❣ L2(0, w)✳

❈❤ù♥❣ ♠✐♥❤✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ {1, x, x2, . . . } ❧➔ t➟♣ ❝♦♥ ❝õ❛ RS ✱ tù❝ ❧➔✱ ✈ỵ✐ ♠å✐
m ≥ 1, tỗ t Lm s SLm = xm1 tt 1 RS tỗ t
L1 s L1 = 1. sỷ tỗ t Lm s ❝❤♦
SLm = xm−1 .
✭✶✳✸✼✮
❚❤❛② f = Lm ✈➔♦ ✭✶✳✼✮ ✈➔ tø ✭✶✳✺✮ ✈➔ ✭✶✳✸✼✮ t❛ ❝â
w

(AS − SA∗ )Lm = i

[M (x) + N (t)]Lm (t)dt.
0

✶✹


ữỡ tỷ t ợ tr♦♥❣

▼➦t ❦❤→❝ ASLm = Axm−1 = i

x

xm−1 dx = i
0

xm
.
m

L2 (0, w)

❚❤❛② ✈➔♦ ✤➥♥❣ t❤ù❝ tr➯♥ t❛ ✤÷đ❝

w

xm
= SA∗ Lm + i
i
m

[M (x) + N (t)]Lm (t)dt.
0

❉♦ 1 ✈➔ M tở Rs tỗ t số N1 N2 s❛♦ ❝❤♦ SN1 = M ✈➔ SN2 = 1✳
❑❤✐ ✤â✱ t õ t t tự tr ữợ
i

xm
m





w

= S A∗ Lm + i

[N1 (x) + N (t)N2 (x)]Lm (t)dt
0



w

= S −i



w

Lm (t)dt + i
x

[N1 (x) + N (t)N2 (x)]Lm (t)dt .
0

❈❤ù♥❣ tä xm ∈ RS ✈➔ S(Lm+1) = xm ợ Lm+1 ữủ ổ tự
w

Lm+1
=
m

w

Lm (t)dt +
x

[N1 (x) + N (t)N2 (x)]Lm (t)dt.

✭✶✳✸✽✮

0

❇ê ✤➲ ✤÷đ❝ ❝❤ù♥❣
ỵ tỗ t ừ t tỷ ♥❣❤à❝❤ ✤↔♦✮✳ ●✐↔ sû 1 ✈➔ M (x) t❤✉ë❝
Rs ✈➔ t♦→♥ tû T ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ✭✶✳✾✮✱ ✭✶✳✶✵✮ ✈➔ ✭✶✳✸✶✮ ❜à ❝❤➦♥✳ ❑❤✐ ✤â✱ S ❦❤↔
♥❣❤à❝❤ ✈➔ T = S 1
ự t ữợ

w

Lm+1
=
m

w

Lm (t)dtN1 (x) +

Lm (t)N (t)N2 (x)dt −

0

❱➻ t❛ ❝â

0

w

✈➔

x

tm−1 N2 (w − t)dt = SLm , N2 (w − t) ,
0

w

w

tm−1 (1 − N1 (w − t))dt = SLm , M2

Lm (t)N (t)dt =
0

Lm (t)dt.

w

Lm (t)dt =
0

w

0

♥➯♥ ✤➥♥❣ t❤ù❝ tr➯♥ ✤÷đ❝ t ữợ
w

Lm+1
=
m

w

tm1 N2 (wt)dtN1 (x)+
0

w

tm1 (1N1 (wt))N2 (x)

Lm (t)dt.
x

0






ữỡ tỷ t ợ tr♦♥❣

❚❛ ❦➼ ❤✐➺✉

L2 (0, w)

✭✶✳✹✵✮

Lm (x) = T xm−1 , m = 1, 2, . . .

ỵ t❛ ❝â

w
∗

(T A − A T )f = i

✭✶✳✹✶✮

Q(x, t)f (t)dt.
0

❚❤❛② f = xm−1 tr♦♥❣ ✭✶✳✹✶✮ ✈➔ tø ✭✶✳✸✶✮✱ ✭✶✳✹✵✮ t❛ ❝â
w
∗

(T A − A T )x

m−1

Q(x, t)xm−1 dt.

=i
0

❙✉② r❛✱
w

1
Lm+1 (x) = i
m

N2 (w − t)N1 (x) +

1 − N1 (w − t) N2 (x) xm−1 dt,

0

❤❛②✱
w

1
Lm+1 (x) =
m

w

tm−1 N2 (w − t)dtN1 (x) +
0

w

tm−1 (1 − N1 (w − t))dtN2 (x)
x

0

ử ỵ t t ữủ
w



2wx


1 d
( φ(x, t))dt =

∂t
2 dx

d
T1 =
dx

Lm (t)dt.

Q

s+x s−x
,
ds.
2
2

x

0

❚❤❛② ✈➔♦ ✭✶✳✸✶✮✱ t❛ ✤÷đ❝
2w−x

−1 d
L1 = T 1 =
2 dx

N2 w −

s−x
s+x
s−x
s+x
N1
+ 1 − N1 (w −
) N2
2
2
2
2

x
2w−x

−1 d
=
2 dx

N2

2w−x

s+x
−d
ds =
2
dx

x

N2 (v)dv = N2 (x).
x

❚ø ✭✶✳✸✼✮✱ ✭✶✳✹✷✮ ✈➔ ✭✶✳✹✸✮ t❛ s✉② r❛ Lm = Lm. ❑❤✐ ✤â
T xm−1 = S −1 xm−1 .

✶✻

✭✶✳✹✸✮

ds


ữỡ tỷ t ợ tr♦♥❣

❙✉② r❛✱

L2 (0, w)

ST xm−1 = xm−1 .

❚ù❝ ❧➔

ST = I.

❚❛ ❝â

✭✶✳✹✹✮

✭✶✳✹✺✮
❚❤❡♦ ✭✶✳✹✹✮✱ ✭✶✳✹✺✮ t❛ s✉② r❛ t♦→♥ tû S ❦❤↔ ♥❣❤à❝❤✱ T = S −1 ✈➔ T = U T U

ỵ trú ừ t tỷ ✤↔♦✮✳ ●✐↔ sû t♦→♥ tû T ❜à ❝❤➦♥ tr♦♥❣
2
L (0, w) ✈➔ t♦→♥ tû ♥❣❤à❝❤ ✤↔♦ S ❝ô♥❣ ❜à ❝❤➦♥✳ ❑❤✐ ✤â✱ S ❧➔ t♦→♥ tû ✈ỵ✐ ♥❤➙♥
❞↕♥❣ ❤✐➺✉ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ T ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ✭✶✳✾✮✱ ✭✶✳✶✵✮ ✈➔✱ ✭✶✳✸✶✮ ✈ỵ✐ N1, N2
t❤✉ë❝ L2(0, w)✳
❈❤ù♥❣ ♠✐♥❤✳
✣✐➲✉ ❦✐➺♥ ❝➛♥✳ ỵ ữủ ự ử trữợ
ừ t ổ ữợ t õ
U ST U = U IU ⇔ S ∗ U T U = I ⇔ U T ∗ U S = I.

w

f, T ∗ 1 = T f, 1 =

f (t)

d
(φ(w, t) − φ(0, t))dt,
dt

0

tù❝ ❧➔
T ∗1 =

d
φ(w, t) − φ(0, t)

dt
2w−t

−1 d
=
2 dt

Q

s−t s+t
,
ds
2
2

t
2w−t

=

−1 d
2 dt

N2 (w −

s+t
s−t
s+t
s−t
)N1 (

) + [1 − N1 (w −
)]N2 (
) ds
2
2
2
2

t

✭✶✳✹✻✮

= N2 (w − x).

❚ø ✭✶✳✸✶✮ ✈➔ ✭✶✳✹✶✮✱ ❝❤♦ S = T −1 t❛ ❝â
(AS − SA∗ )f = i Sf, U (N2 )SN1 + Sf, U (1 − N1 ) SN2 .

❉♦ ✈➟②✱ t❤❡♦ ✭✶✳✹✸✮ ✈➔ ✭✶✳✹✻✮ t❛ s✉② r❛
w

(AS − SA∗ )f = i

(M (x) + N (t))f (t)dt,

✭✶✳✹✻✮

0

tr♦♥❣ ✤â M (x) = SN1; N (x) = S ∗(1 − N1(w − x)). ❚❤❡♦ ỵ S õ
ợ s(x) ữủ ✤à♥❤ ❜ð✐ M (x) = s(x); N (x) = −s(−x). ỵ ữủ

ự



ữỡ tỷ t ợ tr

L2 (0, w)

Pữỡ tr t ợ ✤➦❝ ❜✐➺t
❚r♦♥❣ ▼ư❝ ✶✳✶✱ ❝❤ó♥❣ t❛ ✤➣ ①➙② ❞ü♥❣ t♦→♥ tû T = S −1 tø ❤➔♠ sè N1(x) ✈➔
N2 (x)✳ ❚r♦♥❣ ♠ö❝ ♥➔②✱ t❛ sû ❞ö♥❣ ❦➳t q✉↔ ✤â ữỡ tr
Sf = eix .

ỵ S2 ❧➔ t♦→♥ tû ❜à ❝❤➦♥ ✈ỵ✐ ♥❤➙♥ ❞↕♥❣ ✭✶✳✶✮ ✈➔ ❝→❝ ❤➔♠ sè
N1 (x), N2 (x) t❤✉ë❝ L (0, w) s❛♦ ❝❤♦
SN2 = 1; SN1 = M (x).
✭✶✳✹✽✮
❑❤✐ ✤â
SB(x, λ) = eixλ ,
✭✶✳✹✾✮
tr♦♥❣ ✤â
w
B(x, λ) = u(x, λ) − iλ ei(x−t)λ u(t, λ)dt,
✭✶✳✺✵✮
x

✭✶✳✺✶✮

u(x, λ) = a(λ)N1 (x) + b(λ)N2 (x),
w

w

eiλt N2 (w − t)dt; b(λ) = eiλw − iλ

a(λ) = iλ
0

✭✶✳✺✷✮

eiλt N1 (w − t)dt.
0

❈❤ù♥❣ ♠✐♥❤✳ ❚ø ✭✶✳✸✽✮✱ t❛ t tỗ t số C s
||Lm+1 || Cm||Lm || ∀m ≥ 0.

❚ø ✤â s✉② r❛

✭✶✳✺✸✮

||Lm+1 || ≤ C m+1 m!, ∀m ≥ 0.

❚❛ ✤➦t

∞

B(x, λ) =
m=0

(iλ)m

Lm+1 .
m!

✭✶✳✺✹✮

❚ø ✭✶✳✺✸✮ t❛ t❤➜② ✈➳ ♣❤↔✐ ❝õ❛ ✭✶✳✺✹✮ ❤ë✐ tö ❦❤✐ |λ| < C −1✳ ❉♦ S ❜à ❝❤➦♥ ♥➯♥
∞

SB(x, λ) =
m=0

(iλ)m m
(iλ)m m
x = 1 + (iλx) + · · · +
x + · · · = eixλ , |λ| < C −1 .
m!
m!

✭✶✳✺✺✮

❚ø ✭✶✳✸✽✮ t❛ ❝â
w

1
Lm+1 =
m

w

tm−1 N2 (w − t)dtN1 (x) +

0

w

tm−1 (1 − N1 (w − t))dtN2 (x) −
0

✶✽

Lm (t)dt.
x


ữỡ tỷ t ợ tr♦♥❣

L2 (0, w)

❚ø ✤â s✉② r❛
∞

m=1

(iλ)m 1
Lm+1 =
(m − 1)! m

∞

w

(iλ)m
(m − 1)!

m=1

tm−1 N2 (w − t)dtN1 (x)
0
w

w

tm−1 (1 − N1 (w − t))dtN2 (x) −

+

Lm (t)dt
x

0
w ∞

= iλ
0 m=1
w ∞

+ iλ
m=1

0

w ∞

= iλ
0


+ iλ 

m=1



(iλ)m−1 m−1
t
N2 (w − t)dtN1 (x)
(m − 1)!



w

w

eiλt N1 (w − t)dt N2 (x) − iλ

− iλ

= iλ
0

m=1

B(t, λ)dt
x

0
w ∞

B(t, λ)dt
x

0


w
+ eiλt 0

w

eiλt N1 (w − t)dt N2 (x) − iλ

eiλt dt −

m=1

x m=1

(iλ)m
Lm (t)dt
(m − 1)!

(iλ)m−1 m−1
t
N2 (w − t)dtN1 (x)
(m − 1)!
w

= iλ

w ∞

(iλ)m−1 m−1
t
(1 − N1 (w − t))dtN2 (x)
(m − 1)!

w

0
w ∞

0

(iλ)m−1 m−1
t
N2 (w − t)dtN1 (x) −
(m − 1)!

(iλ)m−1 m−1
t

N2 (w − t)dtN1 (x)
(m − 1)!





w

+ eiλw − 1 − iλ

eiλt N1 (w − t)dt N2 (x).
0

❑❤✐ ✤â ❦➳t ❤ñ♣ ợ L1(x) = N2(x) t t ữủ
w

B(x, ) L1 (x) = iλ
0

m=1

(iλ)m−1 m−1
t
N2 (w − t)dtN1 (x)
(m − 1)!



w

+ eiλw − 1 − iλ

eiλt N1 (w − t)dt N2 (x),
0

✶✾




ữỡ tỷ t ợ tr♦♥❣

L2 (0, w)

tù❝ ❧➔
w ∞

B(x, λ) = iλ
0

m=1

(iλ)m−1 m−1
t
N2 (w − t)dtN1 (x)
(m − 1)!





w

+ eiλw − iλ

eiλt N1 (w − t)dt N2 (x).
0

✣➦t

w

w

0

0

❑❤✐ ✤â t❛ ✤÷đ❝✱

eiλt N1 (w − t)dt

eiλt N2 (w − t)dt, b(λ) = eiλw − iλ

a(λ) = iλ

u(x, λ) = a(λ)N1 (x) + b(λ)N2 (x).

w

B(x, λ) = u(x, λ) − iλ

B(t, λ)dt.
x

❉♦ u(x, λ) ❣✐↔✐ t➼❝❤ t❤❡♦ λ ♥➯♥ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t ❧➔
w

ei(x−t)λ u(t, λ)dt.

B(x, λ) = u(x, λ) − iλ
x

✷✵


ữỡ
Pữỡ tr t r
ợ tr Lp(0, w)
✷✳✶ ❚➼♥❤ ❝❤➜t t♦→♥ tû t➼❝❤ ♣❤➙♥ ✈ỵ✐ ♥❤➙♥ ❞↕♥❣ ❝❤➟♣
tr♦♥❣ Lp(0, w)
❳➨t t♦→♥ tû t➼❝❤ ♣❤➙♥ ✈ỵ✐ ♥❤➙♥ tr Lp(0, w)
ỵ S ♠ët t♦→♥ tû ❜à ❝❤➦♥ tr♦♥❣ Lp(0, w) ✈ỵ✐ (p 1) õ
t tỷ S ữủ ữợ ❞↕♥❣
w

d
Sf =
dx

s(x, t)f (t)dt,
0

✈ỵ✐ s(x, t) t❤✉ë❝ Lq (0, w) ✈ỵ✐ p1 + 1q = 1, ✈ỵ✐ ♠é✐ x ❝è ✤à♥❤✳

❚❛ ①➨t t♦→♥ tû tr♦♥❣ Lp(0, w) ❝â ❞↕♥❣
w

d
Sf =
dx

s(x − t)f (t)dt

✭✷✳✶✮

0

❚÷ì♥❣ tü tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ L2(0, w), t❛ ✤à♥❤ ♥❣❤➽❛ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ Lp(0, w) ❝→❝
t♦→♥ tû s❛✉
w
w
f (t)dt, A∗ f = −i

Af = i

f (t)dt.
x

0

❑❤✐ 1 ≤ p < 2 t❤➻ A ✈➔ ❦❤æ♥❣ ❧➔ t♦→♥ tû ❧✐➯♥ ❤ñ♣ ❝õ❛ ♥❤❛✉✳
❈❤♦ f ∈ Lp(0, w) ✈➔ g Lq (0, w) õ t ổ ữợ ừ f ✈➔ g ✤÷đ❝ ✤à♥❤
♥❣❤➽❛ ♥❤÷ s❛✉
w
A∗

f, g =

f (x)g(x)dx.
0

✷✶


ữỡ Pữỡ tr t r ợ tr

Lp (0, w)

ỵ S t tû ❜à ❝❤➦♥ tr♦♥❣ Lp(0, w),r 1 ≤ p ≤ 2 ❝â ❞↕♥❣ ✭✷✳✶✮✳
❑❤✐ ✤â✱ t♦→♥ tû

S ❜à ❝❤➦♥
||S||p = ||S||q ; ||S||r ≤ ||S||p .

tr♦♥❣ ♠å✐ ❦❤æ♥❣ ❣✐❛♥

L (0, w)

✈ỵ✐

p ≤ r ≤ q

✈➔

❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ f ∈ Lp(0, w), g ∈ Lq (0, w)✳ ❚♦→♥ tû ❧✐➯♥ ❤ñ♣ ❝õ❛ t♦→♥ tû S
✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ S ∗ ✈➔ ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ✤➥♥❣ t❤ù❝ s❛✉
Sf, g = f, S ∗ g .

❱➟② ♥➯♥✱
▼➔ S ∗ = U SU ♥➯♥

||S||p = ||S ∗ ||q .

✭✷✳✷✮

||S||q = ||S ∗ ||q .

✭✷✳✸✮

❚ø ✭✷✳✷✮ ✈➔ ✭✷✳✸✮ t❛ t❤✉ ✤÷đ❝ ||S||p = ||S||q ✳
✣➸ t➻♠ t♦→♥ tû ♥❣❤à❝❤ ✤↔♦ ❝õ❛ t♦→♥ tû S t❛ s➩ t➻♠ ❝→❝ ❤➔♠ sè N1(x), N2(x)
t❤ä❛ ♠➣♥
SN1 (x) = M (x), SN2 (x) = 1,

✈ỵ✐ ❧➔ ❤➔♠ ❤➡♥❣ ❜➡♥❣ 1 ✈➔ M (x) = s(x), 0 ≤ x ≤ w. ❑❤✐ ✤â✱ t♦→♥ tû ♥❣❤à❝❤ ✤↔♦
s➩ ✤÷đ❝ ❜✐➸✉ t N1(x), N2(x).
ỵ sỷ sè N1(x) ✈➔ N2(x) ❧➔ ❝→❝ ❤➔♠ ❧✐➯♥ tö❝✳ ❑❤✐ ✤â
t♦→♥ tû T ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ ✭✶✳✽✮✲ ✭✶✳✶✵✮ ✈➔ ữủ ữợ

1
T = S 1

w

T f = vf +



(x, t)f (t)dt.
0

ỡ ỳ tỗ t số h(x) t❤✉ë❝ L(−w, w) s❛♦ ❝❤♦
|γ(x, t)| ≤ h(x − t); 0 ≤ x, t ≤ w.

❈❤ù♥❣ ♠✐♥❤✳ ✣➦t
A(x) = N2 (x) 1 − N1 (x) , B(x) =

N1 (x)
.
N2 (x)

❑❤✐ ✤â
2w−|x−t|

φ(x, t) =

1
2

Q

s+x−t s−x+t
,
ds
2
2

x+t
2w−|x−t|

1
=
2

A w−
x+t

✷✷

s−x+t
s+x−t
B
ds
2
2

✭✷✳✺✮

ữỡ Pữỡ tr t r ợ ❝❤➟♣ tr♦♥❣

Lp (0, w)

2w−|x−t|

1
=
2

s
x−t s
x−t
+w−
B
+
ds.
2
2
2
2

A
x+t

✣ê✐ ❜✐➳♥ z = s + 2x − t ; s = 2z − x + t ⇒ dz = 12 ds, t❛ t❤✉ ✤÷đ❝
w− |x−t|
+ x−t
2
2

A(w − (z − x + t))B(z)dz.

φ(x, t) =
x

◆➳✉ x > t t❤➻
w

A(w − z + x − t)B(z)dz;

φ(x, t) =
x

w

∂φ
=−
∂t

A (w − z + x − t)B(z)dz;
x
w

∂φ
∂t

x=t+0

=−

A (w − z)B(z)dz.
x

◆➳✉ x < t t❤➻
w+(x−t)

A(w − z + x − t)B(z)dz;

φ(x, t) =

x
w+(x−t)

∂φ
=−
∂t

A (w − z + x − t)B(z)dz + (w + x − t) A(0).B(w + x − t);
x
w

∂φ
∂t

x=t−0

=−

A (w − z)B(z)dz − A(0)B(w).

x

❉♦ ✤â✱

∂φ
∂t

◆➳✉ x > t t❤➻
∂ 2φ
∂x∂t



w

= −

−
x=t+0

∂φ
∂t

= A(0)B(w).
x=t−0


A (w − z + x − t)B(z)dz 

x

x

✷✸

✭✷✳✻✮