Dạng chuẩn tắc của phương trình đạo hàm riêng tuyến tính cấp hai trên mặt phẳng

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼

✖✖✖✖✖✖✖✖✖✖✖✖

✣➄◆● ❚❍➚ ▲❖❆◆

❉❸◆● ❈❍❯❽◆ ❚➁❈ ❈Õ❆ P❍×❒◆● ❚❘➐◆❍
✣❸❖ ❍⑨▼ ❘■➊◆● ❚❯❨➌◆ ❚➑◆❍ ❈❻P ❍❆■
❚❘➊◆ ▼➄❚ P❍➃◆●

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✷✵


✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆

❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼

✣➦♥❣ ❚❤à ▲♦❛♥

❉❸◆● ❈❍❯❽◆ ❚➁❈ ❈Õ❆ P❍×❒◆● ❚❘➐◆❍
✣❸❖ ❍⑨▼ ❘■➊◆● ❚❯❨➌◆ ❚➑◆❍ ❈❻P ❍❆■
❚❘➊◆ ▼➄❚ P❍➃◆●
❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ●✐↔✐ ❚➼❝❤
▼➣ sè✿ ✽ ✹✻ ✵✶ ✵✷

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❈→♥ ❜ë ❤÷î♥❣ ❞➝♥ ❦❤♦❛ ❤å❝✿

❚❙✳ ❚❘➚◆❍ ❚❍➚ ❉■➏P ▲■◆❍
✐


❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✷✵

✐✐


ớ

t ừ ữỡ tr
r t t tr t tr
ự ồ ừ r tổ ữợ sỹ ữợ trỹ t ừ
r
ổ

r tr tổ ỏ sỷ ử ởt số t q t ừ

ởt số t õ ú t tr ỗ ốr q
tr ự tổ tứ t q ồ ừ
ồ ợ sỹ tr trồ t ỡ
t t ý sỹ tổ t tr
ở ừ

t

ớ ỡ
ữủ t t trữớ ồ ữ
tọ ỏ trồ t ỡ s s



r ữớ t trỹ t ữợ t t
ở t tr sốt tớ ự ứ q
tr trồ ỷ ớ ỡ t ổ
Pỏ t ồ ồ ợ ồ
t trữớ ồ ữ ổ ú ù t
t ủ t tr q tr ồ t ự t trữớ
ụ tọ t ỡ s s tợ ữớ t
ổ ở t tr sốt q tr ồ t

ữủ ỳ ỵ õ õ qỵ ừ t
ổ ồ ữủ t ỡ

t
ữớ tỹ

▼ö❝ ❧ö❝
❚r❛♥❣ ❜➻❛ ♣❤ö
▲í✐ ❝❛♠ ✤♦❛♥
▲í✐ ❝↔♠ ì♥
▲í✐ ♥â✐ ✤➛✉
✶ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✐
✐✐
✐✐✐
✶
✹

✶✳✶

▼ët sè ❦❤→✐ ♥✐➺♠ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ✳ ✳ ✳ ✳ ✳

✹

✶✳✷

P❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝➜♣ ❤❛✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼

✶✳✷✳✶

P❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✼

✶✳✷✳✷

❉↕♥❣ ❝❤✉➞♥ t➢❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤②♣❡r❜♦❧✐❝ ✳ ✳ ✳ ✳

✷✵

✶✳✷✳✸

❉↕♥❣ ❝❤✉➞♥ t➢❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ♣❛r❛❜♦❧✐❝ ✳ ✳ ✳ ✳

✷✸

✶✳✷✳✹

❉↕♥❣ ❝❤✉➞♥ t➢❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❡❧✐♣t✐❝ ✳ ✳ ✳ ✳ ✳ ✳

✷✺

✷ ❉↕♥❣ ❝❤✉➞♥ t➢❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ t✉②➳♥
t➼♥❤ ❝➜♣ ❤❛✐ tr➯♥ ♠➦t ♣❤➥♥❣
✷✽
✷✳✶

P❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ t✉②➳♥ t➼♥❤ ❝➜♣ ❤❛✐ ✈î✐ ❤❛✐ ❜✐➳♥
✤ë❝ ❧➟♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✽

✷✳✷

❉↕♥❣ ❝❤✉➞♥ t➢❝ ❦❤æ♥❣ ✤à❛ ♣❤÷ì♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✶

✷✳✸

❉↕♥❣ ❝❤✉➞♥ t➢❝ trì♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✺

✷✳✸✳✶

✣à♥❤ ❧➼ rót ❣å♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✾

✷✳✸✳✷

❉↕♥❣ ❝❤✉➞♥ t➢❝ trì♥ ❝❤♦ ❝→❝ ✤✐➸♠ ❦➻ ❞à ❣➜♣ ✳ ✳ ✳ ✳

✹✼

✐✈


❑➳t ❧✉➟♥
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✺✶
✺✷

✈


ớ õ
ỹ ừ ỵ tt t ừ ữỡ tr
r t t tr t ữủ ự
ỳ t tớ õ rt r t
ữỡ tr sõ ữỡ tr ổ t sỹ ở ừ
sỹ t t tố ừ t ọ ổ ữủ tữỡ ự
t ỳ t ữỡ tr
t ữỡ tr r ữủ sỷ ử tr
t ử tr qt t
ữủ ữớ q t tữớ ữủ ự tr
ỹ ữỡ tr r t ữỡ tr tờ qt

a(x, y)uxx + b(x, y)uxy + c(x, y)uyy = 0,



ợ a, b, c số trỡ õ t ữủ ữ ữỡ
t ý ừ ữỡ tr r tữỡ ự tự t t
tự D ợ D = b2 4ac ừ ữỡ tr t tự tỹ ữỡ
t ờ tồ ở trỡ tỹ tr
ởt trỡ t t ủ ố ợ ởt ở tờ qt
trỡ trỡ ừ tr tổổ t t tự rộ ởt
ữớ trỡ ữủ ú tr t ữ ởt ữỡ
tr tờ qt ừ ữỡ tr sõ uxx uyy = 0 ữỡ

tr uxx + uyy = 0 t ừ ữỡ
tr ởt ữớ t
ữớ t ữủ ồ ữớ t ờ t ý
õ ữỡ tr ỗ õ ừ r Pữỡ




tr t ờ tr ữủ ồ ữỡ tr ộ
ủ
r ự ừ r t ởt ữỡ tr
P ừ ữớ t ờ õ ữỡ tr ổ s ừ
t tự tự D(P ) = 0 dD(P ) = 0 t õ ữỡ trữ

dy : dx ữủ ữỡ tr
a(x, y)dy 2 b(x, y)dxdy + c(x, y)dx2 = 0.



Pữỡ tr ổ t t ợ ữớ t ởt ữ
r ữ r t ữủ

uyy + yuxx = 0.



t ờ tồ ở trỡ tỹ tr ởt
trỡ t é ữỡ tr t ờ tr trử õ tở
ữỡ tr t tr y > 0 r tr y < 0
ỡ ỳ ữớ t ự r ữỡ tr t

õ t t ộ x0 ừ trử t tr
y 0 õ 9(x x0 )2 = 4y 3
ố ợ ữỡ tr r tr ợ
t tr tr trữ
tứ ừ ữớ t ờ ởt trỡ tr
y > 0 ố ố ợ ừ rt
ữủ tr tr ởt tr trữ ổ
ự ỵ sỹ tỗ t t t ừ
ữủ t t r
r ự r ụ t
t ữ ự ừ ổ ữ ừ õ ự
ú ữủ tỹ rr ữ ữ ỵ tr
ỳ t q ừ r ữủ sỷ ử t ỹ tr ự ỵ
tt t ừ ữỡ tr r tr t





❇÷î❝ t✐➳♣ t❤❡♦ tr♦♥❣ ❧þ t❤✉②➳t ✈➲ ❝→❝ ❞↕♥❣ ❝❤✉➞♥ t➢❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
✤↕♦ ❤➔♠ r✐➯♥❣ ❤é♥ ❤ñ♣ tê♥❣ q✉→t tr➯♥ ♠➦t ♣❤➥♥❣ ❝❤õ ②➳✉ ✤÷ñ❝ t❤ü❝ ❤✐➺♥
s❛✉ ✤â✳ ▲✉➟♥ ✈➠♥ ♥➔② ✤÷ñ❝ tr➻♥❤ ❜➔② t❤❡♦ t➔✐ ❧✐➺✉ ❬✸❪✱ ❬✹❪✱ ❬✼❪ ♥❤ú♥❣ ❦➳t
q✉↔ ♥❤➟♥ ✤÷ñ❝ ❣➛♥ ✤➙② ✈➔ ✤à♥❤ ❧þ rót ❣å♥✱ ❝→❝ ❜✐➳♥ t❤ù❝ ❦❤→❝ ♥❤❛✉ ✤÷ñ❝
sû ❞ö♥❣ ✤➸ t❤✉ ✤÷ñ❝ ❝→❝ ❦➳t q✉↔ ♥➔②✳

✸


❈❤÷ì♥❣ ✶
❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à

✶✳✶ ▼ët sè ❦❤→✐ ♥✐➺♠ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ P❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ❧➔ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ❝â
❝❤ù❛ ❝→❝ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝õ❛ ❤➔♠ ➞♥✳
P❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ✤÷ñ❝ ✈✐➳t ❞÷î✐ ❞↕♥❣

F (x1 , x2 , ..., xn , u, ux1 , ..., uxn , ux1 x1 , ...) = 0,

x ∈ Ω ⊂ Rn ,

✭✶✳✶✮

tr♦♥❣ ✤â x = (x1 , ..., xn ) ❧➔ ❝→❝ ❜✐➳♥ ✤ë❝ ❧➟♣✱ u ❧➔ ❤➔♠ ➞♥ ❝õ❛ ❝→❝ ❜✐➳♥ ✤â✳
◆❣❤✐➺♠ ❝õ❛ ✭✶✳✶✮ tr➯♥ Ω ❧➔ ♠ët ❤➔♠ u ①→❝ ✤à♥❤✱ ❦❤↔ ✈✐ ✤➳♥ ❝➜♣ ❝➛♥
t❤✐➳t tr➯♥ Ω ✈➔ t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ✤â t↕✐ ♠å✐ ✤✐➸♠ t❤✉ë❝ Ω✳
◆â✐ ❝❤✉♥❣ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ t❤÷í♥❣ ❝â ✈æ ❤↕♥ ♥❣❤✐➺♠✳
❱➼ ❞ö ❝→❝ ❤➔♠

u(x, t) = ex−ct ,
u(x, t) = cos(x − ct),
❧➔ ❝→❝ ♥❣❤✐➺♠ ❝õ❛ ut + cux = 0✳ ❍ì♥ ♥ú❛✱ ♠å✐ ❤➔♠ ❦❤↔ ✈✐ ❝õ❛ c − ct ✤➲✉
❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤â✳
P❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ t❤÷í♥❣ ✤÷ñ❝ ♣❤➙♥ ❧♦↕✐ t❤❡♦ ❝→❝ t✐➯✉ ❝❤➼
s❛✉✿
❛✮ ❚❤❡♦ ❝➜♣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭♥â✐ ❝❤✉♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❝â ❝➜♣ ❝➔♥❣ ❝❛♦
❝➔♥❣ ♣❤ù❝ t↕♣✮✳
❜✮ ❚❤❡♦ ♠ù❝ ✤ë ♣❤✐ t✉②➳♥✱ t✉②➳♥ t➼♥❤ ✭♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ♥â✐ ❝❤✉♥❣
✤ì♥ ❣✐↔♥ ❤ì♥ ♣❤÷ì♥❣ tr➻♥❤ ♣❤✐ t✉②➳♥✱ ♠ù❝ ✤ë ♣❤✐ t✉②➳♥ ❝➔♥❣ ❝❛♦ t❤➻ ❝➔♥❣

✹

♣❤ù❝ t↕♣✮✳
❝✮ ❚❤❡♦ sü ♣❤ö t❤✉ë❝ ✈➔♦ t❤í✐ ❣✐❛♥ ✭♣❤÷ì♥❣ tr➻♥❤ ❜✐➳♥ ✤ê✐ t❤❡♦ t❤í✐ ❣✐❛♥
t❤➻ ✤÷ñ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ t✐➳♥ ❤â❛✱ ♥❣÷ñ❝ ❧↕✐ ✤÷ñ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤
❞ø♥❣✮✳ ❚r♦♥❣ t➻♥❤ ❤✉è♥❣ ♥➔② ♥❣÷í✐ t❛ t❤÷í♥❣ ❦➼ ❤✐➺✉ ❜✐➳♥ t❤í✐ ❣✐❛♥ ❧➔ t✱
❝→❝ ❜✐➳♥ ❝á♥ ❧↕✐ ❧➔ ❜✐➳♥ ❦❤æ♥❣ ❣✐❛♥✳
❈ö t❤➸ ❤ì♥ t❛ ❝â ❝→❝ ❦❤→✐ ♥✐➺♠ s❛✉ ✤➙②✿

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✳ ❈➜♣ ❝õ❛ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ❧➔ ❝➜♣ ❝❛♦
♥❤➜t ❝õ❛ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝â ♠➦t tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤✳
❈❤➥♥❣ ❤↕♥✱
✭✶✳✷✮

ut + cux = 0
❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝➜♣ ✶✱ ❝á♥ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤

uxx + uyy = f (x, y).

✭✶✳✸✮

α(x, y)uxx + 2uxy + 3x2 uyy = 4ex .

✭✶✳✹✮

ux uxx + (uy )3 = 0.

✭✶✳✺✮

(uxx )2 + uyy + a(x, y)ux + b(x, y)u = 0,

✭✶✳✻✮

❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝➜♣ ❤❛✐✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✸✳ ▼ët ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ❧➔ t✉②➳♥ t➼♥❤ ♥➳✉
♥â ❝â ❞↕♥❣

L[u] = f (x),

✭✶✳✼✮

tr♦♥❣ ✤â L[u] ❧➔ ♠ët tê ❤ñ♣ t✉②➳♥ t➼♥❤ ❝õ❛ u ✈➔ ❝→❝ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝õ❛ u
✈î✐ ❝→❝ ❤➺ sè ❧➔ ❝→❝ ❤➔♠ ❝õ❛ ❜✐➳♥ ✤ë❝ ❧➟♣ x✱ tù❝ ❧➔

aα (x) Dα u.

L[u] =
α

◆➳✉ f ≡ 0 t❤➻ t❛ ♥â✐ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ✭✶✳✼✮ ❧➔
❧↕✐ t❤➻ t❛ ♥â✐ ♣❤÷ì♥❣ tr➻♥❤ ✤â ❧➔

❦❤æ♥❣ t❤✉➛♥ ♥❤➜t✳

t❤✉➛♥ ♥❤➜t✱ tr→✐

❈❤➥♥❣ ❤↕♥✱ ✭✶✳✷✮✲✭✶✳✹✮ ❧➔ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤✱ tr♦♥❣ ✤â ✭✶✳✷✮
❧➔ t❤✉➛♥ ♥❤➜t✱ ✭✶✳✹✮ ❧➔ ❦❤æ♥❣ t❤✉➛♥ ♥❤➜t✳

✺

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✹✳ ▼ët ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ❦❤æ♥❣ t✉②➳♥ t➼♥❤
t❤➻ ✤÷ñ❝ ❣å✐ ❧➔

♣❤✐ t✉②➳♥✳

❈❤➥♥❣ ❤↕♥ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✺✮ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ♣❤✐ t✉②➳♥✳
◆â✐ ❝❤✉♥❣ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ♣❤ù❝ t↕♣ ❤ì♥ ❝→❝ ♣❤÷ì♥❣
tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ✈➻ ✈î✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣✱ ✤➸ t➻♠ ♠ët
♥❣❤✐➺♠ r✐➯♥❣ tø ♥❣❤✐➺♠ tê♥❣ q✉→t t❛ ❝❤➾ ♣❤↔✐ t➻♠ ❝→❝ ❣✐→ trà ❝õ❛ ❝→❝ ❤➡♥❣
sè tò② þ✱ tr♦♥❣ ❦❤✐ ✤â ✈î✐ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣✱ ✈✐➺❝ ❝❤å♥ ♥❣❤✐➺♠
r✐➯♥❣ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ ❜ê s✉♥❣ ❝â ❦❤✐ ❝á♥ ❦❤â ❤ì♥ ❝↔ ✈✐➺❝ t➻♠
♥❣❤✐➺♠ tê♥❣ q✉→t ❞♦ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠
r✐➯♥❣ ♣❤ö t❤✉ë❝ ✈➔♦ ❝→❝ ❤➔♠ tò② þ ✭①❡♠ ✈➼ ❞ö s❛✉ ✤➙②✮ ✈➔ ♥â ❝â t❤➸ ❝â
✈æ ❤↕♥ ❝→❝ ♥❣❤✐➺♠ ✤ë❝ ❧➟♣ t✉②➳♥ t➼♥❤✳

❱➼ ❞ö ✶✳✶✳✶✳ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ t✉②➳♥ t➼♥❤ ❝➜♣ ❤❛✐
uξη (ξ, η) = 0.

✭✶✳✽✮

❚➼❝❤ ♣❤➙♥ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② t❤❡♦ η ✭❣✐ú ξ ❝è ✤à♥❤✮ t❛ ❝â

uξ = f (ξ) ,
✭❞♦ ξ ❝è ✤à♥❤ ♥➯♥ ❤➡♥❣ sè t➼❝❤ ♣❤➙♥ ❝â t❤➸ ♣❤ö t❤✉ë❝ ξ ✮✳
❚➼❝❤ ♣❤➙♥ t❤❡♦ ξ ✭❣✐ú η ❝è ✤à♥❤✮ t❛ ♥❤➟♥ ✤÷ñ❝

u(ξ, η) =

f (ξ)dξ + G(η).

❉♦ t➼❝❤ ♣❤➙♥ ð tr➯♥ ❧➔ ♠ët ❤➔♠ ❝õ❛ ξ ♥➯♥ ♥❣❤✐➺♠ ❝õ❛ ✭✶✳✽✮ ❧➔

u(ξ, η) = F (ξ) + G(η),
tr♦♥❣ ✤â F, G ❧➔ ❤❛✐ ❤➔♠ ❦❤↔ ✈✐ ❜➜t ❦ý✳
◆❤÷ ✈➟②✱ ✤➸ ♥❤➟♥ ✤÷ñ❝ ♠ët ♥❣❤✐➺♠ r✐➯♥❣ t❤ä❛ ♠➣♥ ♠ët sè ✤✐➲✉ ❦✐➺♥
♥➔♦ ✤â t❛ s➩ ♣❤↔✐ ①→❝ ✤à♥❤ ❤❛✐ ❤➔♠ F, G✳

✻


✶✳✷ P❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣ ❝➜♣ ❤❛✐
✶✳✷✳✶ P❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤
❉↕♥❣ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ❝➜♣ ❤❛✐ ✈î✐ ❤❛✐ ❜✐➳♥ sè
✤ë❝ ❧➟♣ x ✈➔ y ❝â ❞↕♥❣

a(x, y)uxx +2b(x, y)uxy +c(x, y)uyy +d(x, y)ux +e(x, y)uy +f (x, y)u = g(x, y),
✭✶✳✾✮
tr♦♥❣ ✤â a, b, c, d, e, f, g ∈ C 2 (Ω) , Ω ⊆ R2 ✈➔ a2 + b2 + c2 = 0 tr♦♥❣ Ω✳
❳➨t t♦→♥ tû ✤↕♦ ❤➔♠ r✐➯♥❣

∂ ∂
,
L
∂x ∂y

∂2
∂2
∂2

∂
∂
:= a 2 + 2b
+c 2 +d
+ e + f.
∂x
∂x∂y
∂y
∂x
∂y

❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✾✮ ✤÷ñ❝ ✈✐➳t ❞÷î✐ ❞↕♥❣

Lu = g,
✈➔ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t t÷ì♥❣ ù♥❣ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✾✮ ❧➔

Lu = 0.

✭✶✳✶✵✮

❚♦→♥ tû L ❧➔ t✉②➳♥ t➼♥❤ tø ✤✐➲✉ ❦✐➺♥ L(c1 u1 + c2 u2 ) = c1 Lu1 + c2 Lu2 t❤ä❛
♠➣♥ ❝❤♦ ♠å✐ ❝➦♣ ❝õ❛ ❤➔♠ sè u1 , u2 ∈ C 2 (Ω) ✈➔ ❤➡♥❣ sè ❜➜t ❦ý u1 , u2 ∈ R✳
❉↕♥❣ t✉②➳♥ t➼♥❤ ❝õ❛ t♦→♥ tû ♥❤÷ s❛✉ ♥➳✉ u1 , ..., un ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣
tr➻♥❤ t❤✉➛♥ ♥❤➜t ✭✶✳✶✵✮✱ ❦❤✐ ✤â ✈î✐ ❜➜t ❦ý ❝→❝❤ ❝❤å♥ ❤➡♥❣ sè c1 , ..., cn ❝❤♦
❤➔♠ sè c1 u1 + ... + cn un ❝ô♥❣ ❧➔ ♠ët ♥❣❤✐➺♠ ❝õ❛ ✭✶✳✶✵✮✳ ❍ì♥ ♥ú❛✱ ♥➳✉ up
❧➔ ♥❣❤✐➺♠ r✐➯♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✾✮✱ ❦❤✐ ✤â

L(c1 u1 + ... + cn un + up ) = L(c1 u1 + ... + cn un ) + Lup = Lup = g.
❱➻ ✈➟②

u = c1 u1 + ... + cn un + up ,
❝ô♥❣ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✾✮ ❝❤♦ ✈î✐ ❤➡♥❣ sè c1 , ..., cn ❜➜t ❦ý✳
❳➨t tr÷í♥❣ ❤ñ♣ ✤ì♥ ❣✐↔♥ ♥❤➜t ❦❤✐ ❝→❝ ❤➺ sè ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✾✮ ❧➔

✼


❤➡♥❣ sè t❤ü❝✳ ●✐↔ t❤✐➳t g ❧➔ ❤➔♠ ❝❤♦ tr÷î❝ ❧➔ ❤➔♠ ❣✐↔✐ t➼❝❤ ❝â ❣✐→ trà t❤ü❝
tr♦♥❣ Ω✳ ❑❤✐ ✤â tr♦♥❣ ♠ët sè tr÷í♥❣ ❤ñ♣✱ ❝â t❤➸ ♥❤➟♥ ✤÷ñ❝ ♥❣❤✐➺♠ tê♥❣
q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✵✮ ❧➔ ♠è✐ q✉❛♥ ❤➺ ❤❛✐ ❝❤✉ ❦ý t✉➛♥ ❤♦➔♥ C 2 (Ω)✱
♥➯♥

u = uh + up ,
✤÷ñ❝ ❣å✐ ❧➔ sè ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❦❤æ♥❣ t❤✉➛♥ ♥❤➜t✳
P❤➙♥ ❧♦↕✐ ✤↕♦ ❤➔♠ t♦→♥ tû t✉②➳♥ t➼♥❤ L
✭✐✮ L

∂ ∂
∂x , ∂y

✱

❧➔ ❦❤↔ q✉② ❤♦➦❝ ♣❤➙♥ t➼❝❤ ✤÷ñ❝ ♥➳✉ ♥â ❝â ❞↕♥❣ t➼❝❤ ❝õ❛

∂ ∂
∂x , ∂y

∂
∂
t✉②➳♥ t➼♥❤ t❤ù ♥❤➜t✲ ❝➜♣ t❤ø❛ sè ❝õ❛ a ∂x

+ b ∂y
+ c✳

✭✐✐✮ L

❧➔ ❜➜t ❦❤↔ q✉② ❤♦➦❝ ❦❤æ♥❣ ♣❤➙♥ t➼❝❤ ✤÷ñ❝ ♥➳✉ ♥â ❦❤æ♥❣

∂ ∂
∂x , ∂y

❝â ❞↕♥❣ tr➯♥✳
❛✮ P❤÷ì♥❣ tr➻♥❤ ❦❤↔ q✉②
❚r♦♥❣ tr÷í♥❣ ❤ñ♣ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝â t❤➸ ✤÷ñ❝ t➻♠ t❤➜②✱ ❣✐↔ sû

L = L1 L2
∂
∂
= a1
+ b1
+ c1
∂x
∂y
❚ø ✤â ❤➺ sè

∂2
∂x∂y

=

∂2

∂y∂x

a2

∂
∂
+ b2
+ c2 .
∂x
∂y

❧➔ ❤➡♥❣ sè ✈➔ t♦→♥ tû L1 L2 ❣✐❛♦ ❤♦→♥✱ tù❝

❧➔ L1 L2 = L2 L1 ✳ ◆➳✉ u1 ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ❝➜♣ ♠ët

L1 u = 0✱ ❞♦ ✤â
Lu1 = (L1 L2 )u1 = (L2 L1 )u1 = L2 (L1 u1 ) = L2 (0) = 0,
tù❝ u1 ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✶✳✶✵✮✳ ❚÷ì♥❣ tü✱ ♥➳✉ u2 ❧➔ ♥❣❤✐➺♠ ❝õ❛ L2 (u) = 0✱ ❦❤✐
✤â u2 ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✶✳✶✵✮✳ ❚ø ✤â L ❧➔ t♦→♥ tû t✉②➳♥ t➼♥❤ ❦❤✐ u = u1 + u2
❝ô♥❣ ❧➔ ♥❣❤✐➺♠✳ ❉♦ ✤â✱ ♥➳✉ a = a1 a2 = 0 ✈➔ ❝→❝ t❤ø❛ sè L1 , L2 ❧➔ r✐➯♥❣
❜✐➺t✱ t❤➻ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ✭✶✳✶✵✮ ✤÷ñ❝ t➼♥❤ ❜➡♥❣ ❝→❝❤
c1

c2

uh = e− a1 x ϕ(b1 x − a1 y) + e− a2 x ψ(b2 x − a2 y),

✭✶✳✶✶✮

tr♦♥❣ ✤â ϕ ✈➔ ψ ❧➔ ❤➔♠ ❧➜② ✤↕♦ ❤➔♠ ♠ët ❝→❝❤ ❧✐➯♥ tö❝ ❣➜♣ ✤æ✐ t✉ý þ✳ ◆➳✉

L1 = L2 tù❝ ❧➔
∂
∂
L = L1 L1 = a1
+ b1
+ c1
∂x
∂y

✽

2

,


❦❤✐ ✤â ♥❣❤✐➺♠ tê♥❣ q✉→t ❧➔
c1

uh = e− a1 x (xϕ(b1 x − a1 y) + ψ(b1 x − a1 y)).
❚♦→♥ tû L ❧✉æ♥ ❦❤↔ q✉② ❦❤✐ ♥â ❧➔ t♦→♥ tû t❤✉➛♥ ♥❤➜t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
❝â ❞↕♥❣

∂2
∂2
∂2
L = a 2 + 2b
+ c 2.
∂x

∂x∂y
∂y

◆➳✉ a = 0 ✈➔ λ1 , λ2 ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❜➟❝ ❤❛✐ aλ2 + 2bλ + c = 0✳
❑❤✐ ✤â

L=a

∂
∂
− λ1
∂x
∂y

◆➳✉ a = 0 t❤➻

L=

∂
∂y

2b

∂
∂
− λ2
∂x
∂y

∂

∂
+c
∂x
∂y

.

.

❈❤ó þ r➡♥❣ ❝➠♥ ❝õ❛ λ1 , λ2 ❧➔ t❤ü❝ ♥➳✉ b2 − ac ≥ 0✳

❱➼ ❞ö ✶✳✷✳✶✳ ❚➻♠ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
uxx + ux = uyy + uy .
P❤÷ì♥❣ tr➻♥❤ ♥➔② ❝â ❞↕♥❣ Lu = 0 tr♦♥❣ ✤â L ❧➔ t♦→♥ tû

∂2
∂2
∂
∂
L= 2− 2+
− .
∂x
∂y
∂x ∂y
❚♦→♥ tû q✉② ✈➲

L = L1 L2 =

∂
∂

−
∂x ∂y

∂
∂
+
+1 ,
∂x ∂y

✈➔ tø ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✶✮ ♥❣❤✐➺♠ tê♥❣ q✉→t ❧➔

u = ϕ(x + y) + e−x ψ(x − y),
♠➦t ❦❤→❝ ❝ô♥❣ ❝â t❤➸ ❧➔ ✈✐➳t ❞÷î✐ ❞↕♥❣

u = ϕ(x + y) + e−x ex−y h(x − y)
= ϕ(x + y) + e−y h(x − y),
tr♦♥❣ ✤â ϕ✱ ψ ✈➔ h ❧➔ ❝→❝ ❤➔♠ t✉ý þ✳

✾


▼ët ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤ ❝➜♣ ❤❛✐ tr♦♥❣ n ❜✐➳♥ ✤ë❝ ❧➟♣ x1 , ...xn ❝â
❞↕♥❣

n

∂ 2u
Aij
+
∂x

∂x
i
j
i,j=1

n

Bi
i=1

∂u
+ Cu = G.
∂xi

✭✶✳✶✷✮

◆➳✉ ①➨t t♦→♥ tû
n

∂2
L=
Aij
+
∂x
∂x
i
j
i,j=1

n

Bi
i=1

∂
+ C,
∂xi

❦❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✷✮ ✤÷ñ❝ ✈✐➳t ♥❤÷ s❛✉

Lu = G,
✈➔ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t t÷ì♥❣ ù♥❣ ❧➔
✭✶✳✶✸✮

Lu = 0.

●✐↔ sû r➡♥❣ ❤➺ sè Aij , Bi , C tr♦♥❣ L ❧➔ sè t❤ü❝✱ ✈➔ Aij = Aji ; i, j = 1, ..., n✳
❑❤✐ L ❧➔ ❦❤↔ q✉②

L = L1 L2
= a1

∂
∂
+ ... + an
+c
∂x1
∂xn

b1

∂
∂
+ ... + bn
+d .
∂x1
∂xn

❑❤✐ ✤â ❜✐➳♥ ✤ê✐ t÷ì♥❣ tü tr♦♥❣ tr÷í♥❣ ❤ñ♣ ❝õ❛ ❤❛✐ ❜✐➳♥ ✤ë❝ ❧➟♣✳ ❱➻
✈➟② ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✸✮ ❧➔
c

d

uh = e− a1 x1 ϕ(a2 x1 −a1 x2 , ..., an x1 −a1 xn )+e− b1 x1 ψ(b2 x1 −b1 x2 , ..., bn x1 −b1 xn ),
tr♦♥❣ ✤â ϕ✱ ψ ❧➔ ❝→❝ ❤➔♠ t✉ý þ✳ ◆➳✉ a1 ❤♦➦❝ b1 ❜➡♥❣ 0 t❤➻ ♥❣❤✐➺♠ ❝õ❛
♣❤÷ì♥❣ tr➻♥❤ tê♥❣ q✉→t ✤÷ñ❝ ❜✐➳♥ ✤ê✐ t❤➼❝❤ ❤ñ♣✳ ◆❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛
♣❤÷ì♥❣ tr➻♥❤ ❦❤æ♥❣ t❤✉➛♥ ♥❤➜t ✭✶✳✶✷✮ ❧➔ u = uh + up ✱ tr♦♥❣ ✤â up ❧➔
♥❣❤✐➺♠ r✐➯♥❣✳
❜✮ P❤÷ì♥❣ tr➻♥❤ ❜➜t ❦❤↔ q✉②
❈❤♦ t♦→♥ tû L

∂ ∂
∂x , ∂y

❜➜t ❦❤↔ q✉②✱ ❦❤æ♥❣ ♣❤↔✐ ❧ó❝ ♥➔♦ ❝ô♥❣ ❝â t❤➸ t➻♠

✤÷ñ❝ ♥❣❤✐➺♠ tê♥❣ q✉→t✱ ♥❤÷♥❣ ❝â t❤➸ ①➙② ❞ü♥❣ ♥❣❤✐➺♠ ❜❛♦ ❤➔♠ ♥❤✐➲✉

✶✵

❤➡♥❣ sè t✉ý þ ♥❤÷ ♠♦♥❣ ♠✉è♥✳ ✣✐➲✉ ♥➔② ✤↕t ✤÷ñ❝ ❜➡♥❣ ❝→❝❤ t❤û ♥❣❤✐➺♠
❝â ❞↕♥❣ ♠ô

u = eαx+βy ,
tr♦♥❣ ✤â α✱ β ❧➔ ❤➡♥❣ sè ①→❝ ✤à♥❤✳ ❚ø ✤â

∂u
= αu,
∂x
❚❤➜② r➡♥❣

L=

∂ ∂
,
∂x ∂y

∂u
= βu.
∂y

eαx+βy = L (α, β) eαx+βy ,

✈➔ tr÷î❝ ✤â u = eαx+βy ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ t❤✉➛♥ ♥❤➜t ✭✶✳✶✵✮ ❦❤✐

L (α, β) = 0.
●✐↔ sû q✉❛♥ ❤➺ ❝✉è✐ ❝ò♥❣ ❧➔ ❦➳t q✉↔ ❝õ❛ β t❤✉ ✤÷ñ❝ ❤➔♠ sè β = h(α)✳
❑❤✐ ✤â ❤➔♠ sè

u = eαx+h(α)y ,
❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✵✮✳ ◆❣♦➔✐ r❛

u = ϕ (α) eαx+h(α)y ,
❝❤å♥ ❤➔♠ ϕ ❧➔ ♥❣❤✐➺♠ ❜➜t ❦ý✳ ❑❤→✐ q✉→t ❤ì♥

ϕ (α) eαx+h(α)y ,

u=

u=

ϕ (α) eαx+h(α)y dα,

α

❧➔ ❝→❝ ♥❣❤✐➺♠ ♠é✐ ❦❤✐ ✤à♥❤ ♥❣❤➽❛ ❤➔♠ C 2 (Ω)✱ ✈➔ ♣❤➨♣ ❧➜② ✤↕♦ ❤➔♠ tr♦♥❣
❦➼ ❤✐➺✉ tê♥❣ ❤♦➦❝ tr♦♥❣ ❦➼ ❤✐➺✉ t➼❝❤ ♣❤➙♥ ❧➔ ❤ñ♣ ❧➼✳ ❑❤→✐ ♥✐➺♠ tr÷î❝ ♠ð
rë♥❣ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✸✮ ❦❤✐ ❤➺ sè ❦❤æ♥❣ ✤ê✐✳
❱➼ ❞ö✱ ①➨t ♣❤÷ì♥❣ tr➻♥❤ ♥❤✐➺t ❧÷ñ♥❣

1
xxx − ut = 0, ❤➡♥❣ sè k > 0.
k
❚♦→♥ tû

∂2
1∂
L= 2−

∂x
k ∂t

✶✶

✭✶✳✶✹✮


❧➔ ❜➜t ❦❤↔ q✉②✳
❚➻♠ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ u = eαx+βt ✱ t❤✉ ✤÷ñ❝
1
α2 − β = 0.
k
❉♦ ✤â β = kα2 ✈➔ ✈î✐ ♠å✐ ❣✐→ trà ❝õ❛ ❤➔♠ sè α ✈î✐

u = eαx+kα

2

t
2

❧➔ ♥❣❤✐➺♠✳ ◆➳✉ ❧➜② α = in ❦❤✐ ✤â ❤➔♠ u = einx−kn t ❧➔ ♥❣❤✐➺♠ ✈➔ ❝â ❞↕♥❣
∞

Cn einx−kn

u=

2

t

n=1
∞
2

(An cos nx + Bn sin nx)e−kn t ,

=
n=1

❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✹✮✳

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳✶✳ P❤÷ì♥❣ tr➻♥❤
auxx + 2buxy + cuyy F (x, y, u, ux , uy ) = 0,

✭✶✳✶✺✮

t↕✐ ✤✐➸♠ P (x, y) ∈ Ω ❧➔
✭✐✮ ❤②♣❡r❜♦❧✐❝✱ ♥➳✉ ∆ (x, y) > 0✳
✭✐✐✮ ♣❛r❛❜♦❧✐❝✱ ♥➳✉ ∆ (x, y) = 0✳
✭✐✐✐✮ ❡❧✐♣t✐❝✱ ♥➳✉ ∆ (x, y) < 0✳
P❤÷ì♥❣ tr➻♥❤ ❧➔ ❤②♣❡❜♦❧✐❝ ✭♣❛r❛❜♦❧✐❝✱ ❡❧✐♣t✐❝✮ tr♦♥❣ t➟♣ ❝♦♥ G ⊂ Ω ♥➳✉
♥â ❧➔ ❤②♣❡❜♦❧✐❝ ✭♣❛r❛❜♦❧✐❝✱ ❡❧✐♣t✐❝✮ t↕✐ ♠å✐ ✤✐➸♠ ❝õ❛ G✳
❚✐➳♣ t❤❡♦✱ t➻♠ t♦↕ ✤ë ♠î✐ ξ ✈➔ η s❛♦ ❝❤♦ ✈î✐ ✤✐➲✉ ❦✐➺♥ ❝õ❛ t♦↕ ✤ë ♠î✐
♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✺✮ ❝â ♣❤➛♥ ❝❤➼♥❤ ✤➦❝ ❜✐➺t ✤ì♥ ❣✐↔♥✳ ❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤
❧➔ ❞↕♥❣ ❝❤✉➞♥ t➢❝✳

✣à♥❤ ❧þ ✶✳✷✳✷✳ ●✐↔ sû r➡♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✺✮ ❧➔ ❤②♣❡r❜♦❧✐❝✱ ♣❛r❛❜♦❧✐❝

❤♦➦❝ ❡❧❧✐♣t✐❝ tr♦♥❣ ❧➙♥ ❝➟♥ ✤✐➸♠ P0(x0, y0)✳ ❑❤✐ ✤â tç♥ t↕✐ t❤❛② ✤ê✐ ❦❤↔
♥❣❤à❝❤ ❝õ❛ ❜✐➳♥ sè
Φ:

ξ = ξ(x, y)
η = η(x, y)

✶✷


①→❝ ✤à♥❤ tr♦♥❣ ❧➙♥ ❝➟♥ ✤✐➸♠ P0(x0, y0) s❛♦ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✺✮ ❝â t❤➸
rót ❣å♥ t❤➔♥❤ ♠ët tr♦♥❣ ❜❛ ❞↕♥❣ ❞÷î✐ ✤➙②
✭✐✮ ◆➳✉ P0(x0, y0) ❧➔ ♠ët ✤✐➸♠ ❤②♣❡r❜♦❧✐❝
uξη + Ψ(ξ, η, u, uξ , uη ) = 0.

✭✶✳✶✻✮

✭✐✐✮ ◆➳✉ P0(x0, y0) ❧➔ ♠ët ✤✐➸♠ ♣❛r❛❜♦❧✐❝
uηη + Ψ(ξ, η, u, uξ , uη ) = 0.

✭✶✳✶✼✮

✭✐✐✐✮ ◆➳✉ P0(x0, y0) ❧➔ ♠ët ✤✐➸♠ ❡❧❧✐♣t✐❝
uξξ + uηη + Ψ(ξ, η, u, uξ , uη ) = 0.

✭✶✳✶✽✮

❚r♦♥❣ tr÷í♥❣ ❤ñ♣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤②♣❡❜♦❧✐❝ ♣❤➨♣ ❜✐➳♥ ✤ê✐
α = ξ + η,
β = ξ − η,

rót ❣å♥ ✭✶✳✶✻✮ ✤➸ uαα −uββ +θ(α, β, u, uα, uβ ) = 0, ✤÷ñ❝ ❣å✐ ❧➔ ❞↕♥❣ ❝❤✉➞♥
t➢❝ t❤ù ❤❛✐ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ❤②♣❡r❜♦❧✐❝✳
❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ●✐↔ sû P0(x0, y0) ❧➔ ♠ët ✤✐➸♠ t❤✉ë❝ ❤②♣❡r❜♦❧✐❝✳ ❈❤å♥
ξ ✈➔ η ❝â t❤ù tü
A = aξx2 + 2bξx ξy + cξy2 = 0,
C = aηx2 + 2bηx ηy + cηy2 = 0,
❝â ξ ✈➔ η ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❦❤æ♥❣ t✉②➳♥ t➼♥❤ ❝➜♣ ♠ët ❞↕♥❣

aϕ2x + 2bϕx ϕy + cϕ2y = 0.
❇➡♥❣ ❧þ t❤✉②➳t ✤➣ tr➻♥❤ ❜➔② t❛ ❝â

dϕ
= pFp + qFq = 2(ap2 + 2bpq + cq 2 ) = 0,
dt
✈➻ ✈➟② ❝ò♥❣ ✈î✐ ♥❤ú♥❣ ✤➦❝ tr÷♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✾✮ t❤➻

ϕ (x, y) = const.

✶✸

✭✶✳✶✾✮


◆➳✉ ❣✐↔ sû r➡♥❣ ϕy (x0 , y0 ) = 0 ①→❝ ✤à♥❤ y = y(x) ♥❤÷ ❤➔♠ ➞♥ tr♦♥❣
❧➙♥ ❝➟♥ ❝õ❛ ✤✐➸♠ x0 ✈➔

y =

ϕx (x, y)

dy
=−
.
dx
ϕy (x, y)

❉♦ ✭✶✳✶✾✮ ❤➔♠ y(x) t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣

ay 2 − 2by + c = 0.

✭✶✳✷✵✮

◆➳✉ ❣✐↔ sû r➡♥❣ ϕx (x0 , y0 ) = 0 ①→❝ ✤à♥❤ x = x(y) ♥❤÷ ❤➔♠ ➞♥ tr♦♥❣
❧➙♥ ❝➟♥ ✤✐➸♠ y0 ✈➔

x =

dx
ϕy (x, y)
=−
.
dy
ϕx (x, y)

❑❤✐ ✤â ❤➔♠ x(y) t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣
2

cx −2bx +a= 0.

✭✶✳✷✶✮

❈↔ ❤❛✐ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✵✮ ✈➔ ✭✶✳✷✶✮ ❝â t❤➸ ✤÷ñ❝ tr➻♥❤ ❜➔② ❞÷î✐ ❞↕♥❣
✤↕♦ ❤➔♠

a(dy)2 − 2bdxdy + c(dx)2 = 0.

✭✶✳✷✷✮

❑❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t✱ ❣✐↔ sû a(x0 , y0 ) = 0 ❤♦➦❝ c(x0 , y0 ) = 0
❜ð✐ ✈➻ ♥➳✉ a(x0 , y0 ) = c(x0 , y0 ) = 0 t❤➻ b(x0 , y0 ) = 0 ✈➔ ❝❤✐❛ ✭✶✳✶✺✮
❝❤♦ b(x0 , y0 ) ❝â ✤÷ñ❝ ❞↕♥❣ ✭✶✳✶✻✮✳
●✐↔ sû a(x0 , y0 ) = 0 ✈➔ a(x, y) = 0 tr♦♥❣ ❧➙♥ ❝➟♥ ◆ ❝õ❛ ✤✐➸♠ (x0 , y0 )✳
P❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✵✮ rót ❣å♥ ✈➲ ❤❛✐ ♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ r✐➯♥❣

√
√
b− ∆
b+ ∆
, y2 =
, ∆ = b2 − ac.
y1 =
a
a
●✐↔ sû ξ (x, y) = C1 ✈➔ η (x, y) = C2 ❧➛♥ ❧÷ñt ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛
❝❤ó♥❣ ✤÷ñ❝ ①→❝ ✤à♥❤ tr♦♥❣ ♠ët ♠✐➲♥ N1 ⊂ N ✳ ❑❤✐ ✤â

ξy = 0 ✈➔ ηy = 0 ❝❤♦ (x, y) ∈ N1 .

✶✹

❚❤❛② ✤ê✐ ❜✐➳♥ sè

ξ = ξ (x, y)
η = η (x, y) ,

Φ:

rót ❣å♥ ✭✶✳✶✺✮ trð ✈➲ ❞↕♥❣ ✭✶✳✶✻✮✳ ◆â ❧➔ ❦❤↔ ♥❣❤à❝❤✱ ✈➻

y1 =
y2 =

−ξx
ξy
−ηx
ηy

=
=

√
b+ ∆
a√ ,
b− ∆
a .

❙✉② r❛

√

2 ∆
ξx ηy − ξy ηx = −
ξy ηy = 0.
a
❚r♦♥❣ tr÷í♥❣ ❤ñ♣ c(x0 , y0 ) = 0 ✤÷ñ❝ ❜✐➳♥ ✤ê✐ t÷ì♥❣ tü✳

❚✐➳♣ t❤❡♦✱ ♠æ t↔ tr÷í♥❣ ❤ñ♣ ❡❧✐♣t✐❝ ✈➔ ♣❛r❛❜♦❧✐❝✳
✭✐✐✮ ●✐↔ sû P0 (x0 , y0 ) ❧➔ ✤✐➸♠ t❤✉ë❝ ♣❛r❛❜♦❧✐❝✳ ❈❤å♥ ξ ✈➔ η s❛♦ ❝❤♦ A =

B = 0✳ ❑❤✐ ✤â b2 − ac = 0 ♥➯♥ ♠ët tr♦♥❣ ❤❛✐ ❤➺ sè a ❤♦➦❝ c ❦❤→❝
0✳ ❍❛② ♥â✐ ❝→❝❤ ❦❤→❝ b ❜➡♥❣ 0 ❧➔ ♠➙✉ t❤✉➝♥ (a, b, c) = (0, 0, 0)✳ ◆➳✉
a = 0 t❤➻ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✷✮ trð t❤➔♥❤
b
y = .
a
●✐↔ sû ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♥â ❧➔

ξ (x, y) = K.
▲➜② η = η(x, y) ♠ët ❤➔♠ ✤ì♥ ❣✐↔♥ ♥❤÷

JΦ(P ) =

∂(ξ, η)
= 0,
∂ (x, y)

tr♦♥❣ ✤â A = B = 0 ✈➔ C = 0✳ ❚❤❛② ✤ê✐ ❝õ❛ ❜✐➳♥ sè

Φ:

ξ = ξ (x, y)
η = η (x, y) ,

rót ❣å♥ ✭✶✳✶✺✮ ✈➲ ❞↕♥❣ ❝❤✉➞♥ t➢❝ ✭✶✳✶✼✮✳ ❚r♦♥❣ tr÷í♥❣ ❤ñ♣ a = 0✱ t❤➻

c = 0 ✈➔ sû ❞ö♥❣ ♣❤÷ì♥❣ ♣❤→♣ t÷ì♥❣ tü✳

✶✺


✭✐✐✐✮ ●✐↔ sû P0 (x0 , y0 ) ❧➔ ✤✐➸♠ t❤✉ë❝ ❡❧✐♣t✐❝✳ ❈❤å♥ ξ ✈➔ η s❛♦ ❝❤♦ A = C
✈➔ B = 0✳ ❑❤✐ ✤â b2 − ac < 0 ❝❤♦ ♥➯♥ a = 0 ✈➔ ✭✶✳✷✷✮ trð t❤➔♥❤
♣❤÷ì♥❣ tr➻♥❤ ✤↕♦ ❤➔♠ ❝õ❛ ❞↕♥❣ ♣❤ù❝

y1 =

√
b+i −∆
,
a

y2 =

√
b−i −∆
.
a

●✐↔ sû ϕ (x, y) = ξ (x, y) + iη (x, y) = K ❧➔ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛
♣❤÷ì♥❣ tr➻♥❤ t❤ù ♥❤➜t✳ ❚ø ✭✶✳✶✾✮ ❝â A = C ✈➔ B = 0 ✳ ❑❤✐ t❤❛② ✤ê✐

❜✐➳♥ sè

Φ:

ξ = ξ (x, y)
η = η (x, y) ,

rót ❣å♥ ✭✶✳✶✺✮ ✈➲ ❞↕♥❣ ✭✶✳✶✽✮✳

P❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✷✮ ✤÷ñ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣ ❝õ❛ ✭✶✳✶✺✮ tr♦♥❣
✤â ♥❣❤✐➺♠ ❝õ❛ ♥â ❧➔ ✤➦❝ tr÷♥❣✳ ❚r♦♥❣ ♠✐➲♥ ❤②♣❡r❜♦❧✐❝ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✺✮
t❤ø❛ ♥❤➟♥ ❤❛✐ ❤å ❝õ❛ ✤➦❝ t➼♥❤ t❤ü❝ ❝➢t ♥❤❛✉ t❤❡♦ ❝❤✐➲✉ ♥❣❛♥❣✳ ❚r♦♥❣ ♠✐➲♥
♣❛r❛❜♦❧✐❝ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✺✮ t❤ø❛ ♥❤➟♥ ♠ët ❤å ❝õ❛ ✤➦❝ tr÷♥❣ t❤ü❝ tr♦♥❣
✤â ♠✐➲♥ ❝õ❛ ❡❧✐♣t✐❝ ❦❤æ♥❣ ❝â ✤➦❝ tr÷♥❣ t❤ü❝✳

❱➼ ❞ö ✶✳✷✳✸✳ ❳→❝ ✤à♥❤ ❞↕♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
x2 uxx − y 2 uyy − 2yuy = 0,
✤÷❛ ♣❤÷ì♥❣ tr➻♥❤ trð t❤➔♥❤ ❞↕♥❣ ❝❤✉➞♥ t➢❝ tr♦♥❣ ♠✐➲♥ ❤②♣❡❜♦❧✐❝ ✈➔ t➻♠
♥❣❤✐➺♠ tê♥❣ q✉→t✳

●✐↔✐✿ ❇✐➺t t❤ù❝ ❧➔ b2 − ac = x2y2 ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ❧➔ ❤②♣❡r❜♦❧✐❝ tr♦♥❣
R2 \ {(x, y) : x = 0, y = 0}✳ ❚r➯♥ ✤÷í♥❣ t❤➥♥❣ x = 0 ✈➔ y = 0 ♣❤÷ì♥❣
tr➻♥❤ ❧➔ ♣❛r❛❜♦❧✐❝ ✳
❳➨t ♠✐➲♥ ❤②♣❡r❜♦❧✐❝✳
◆❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣✳ ❚❤❛② t❤➳ y ❜ð✐ λ tr♦♥❣ ✭✶✳✷✵✮ ♥❤➟♥
✤÷ñ❝ ♣❤÷ì♥❣ tr➻♥❤

aλ2 − 2bλ + c = 0,

✶✻

✈î✐

a = x2 ,
❚ø ✤â ❝â ♥❣❤✐➺♠

λ1 =

c = −y 2 .

b = 0,

y
λ2 = − .
x

y
,
x

⑩♣ ❞ö♥❣ t➻♠ r❛ ❧í✐ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥ t❤÷í♥❣ ✈➔ ♥❣❤✐➺♠ ❝õ❛
♣❤÷ì♥❣ tr➻♥❤

dy
y
= ,
dx x

❝â ♥❣❤✐➺♠ ✤ó♥❣ ❧➔

y
x

= C1 ✈➔
dy
y
=− ,
dx
x

♥❣❤✐➺♠ ✤ó♥❣ ❧➔ xy = C2 ✳
❇✐➳♥ sè ♠î✐ ❧➔

ξ = xy
η = xy.

❈❤♦ ξ ✈➔ η →♣ ❞ö♥❣ ♣❤➨♣ t➼♥❤ ✈❡❝tì ✈➔ ❏❛❝♦❜✐❛♥

y
, xy , ❏❛❝♦❜✐❛♥ ❧➔
x

− xy2 x1
y x.

.

❈❤♦ ξ ✈➔ η →♣ ❞ö♥❣ ♣❤➨♣ t➼♥❤ ✈❡❝tì ✈➔ ❍❡ss✐❛♥

y
,
x

❍❡ss✐❛♥ ❧➔

xy,

2 xy3 − x12
− x12 0.

❍❡ss✐❛♥ ❧➔

.

0 1
1 0.

❈â ♥❣❤➽❛ ❧➔

α = − x12 y,
α1 = x23 y,
β1 = 0,

β = x1 ,
α2 = − x12 ,
β2 = 1,

✶✼

γ = y,
α3 = 0
β3 = 0.

δ=x


❘ót ❣å♥ t➼♥❤ t♦→♥ ✿

−uξ + uη x2
,
ux = uξ α + uη γ = y
x2
uξ + uη x2
uy = uξ β + uη δ =
,
x2
uxx = uξξ α2 + 2uξη αγ + uηη γ 2 + uξ α1 + uη β1
uξ 2 y − 2uξη yx2 + uη2 yx4 + 2uξ x
=y
,
x4
uxy = uξξ αβ + uξη (αδ + βγ) + uηη γδ + uξ α2 + uη β2
−uξ 2 y + uη2 yx4 − uξ x + uη x3
,
=
x3
uyy = uξξ β 2 + 2uξη βδ + uηη δ 2 + uξ α3 + uη β3
uξ 2 + 2uξη x2 + uη2 x4
=

x2
❑❤✐ ✤â

.

x2 uxx − y 2 uyy − 2yuy = −2y(2uξη y + uη x)
= −4uξη y 2 − 2uη xy
= −4ξηuξη − 2ηuη
1
= −4ξη uξη + uη .
2ξ

❱➻ ❞♦ ξ =

y
x

✱ η = xy ♥➯♥ ✤➝♥ ✤➳♥ y 2 = ξη ✳

❉↕♥❣ ❝❤✉➞♥ t➢❝ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ tr♦♥❣ ♠✐➲♥ ❤②♣❡r❜♦❧✐❝ ❧➔

uξη +

1
uη = 0.
2ξ

❚❤❛② t❤➳ uη = v t❛ ✤÷ñ❝ ♣❤÷ì♥❣ tr➻♥❤ ❝✉è✐ ❝ò♥❣ trð t❤➔♥❤ ♣❤÷ì♥❣
tr➻♥❤ ❝➜♣ ♠ët✳

vξ +

1
v = 0,
2ξ

✈î✐ ♥❣❤✐➺♠ tê♥❣ q✉→t
1

v(ξ, η) = ξ − 2 f (η).
▲➜② t➼❝❤ ♣❤➙♥ ✤è✐ ✈î✐ η ✱ t❛ ❝â
1

u(ξ, η) = ξ − 2 ϕ(η) + ψ(ξ).

✶✽