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Վ. Ժ. ԴՈՒՄԱՆՅԱՆ

ՄԱԹԵՄԱՏԻԿԱԿԱՆ ՖԻԶԻԿԱՅԻ

ՀԱՎԱՍԱՐՈՒՄՆԵՐ

ԵՐԵՎԱՆ

ԵՊՀ ՀՐԱՏԱՐԱԿՉՈՒԹՅՈՒՆ

ՀՏԴ 517.9:530.1 ԳՄԴ 22.311 Դ 940

Երաշխավորված է ՀՀ ԿԳ նախարարության կողմից որպես դասագիրք՝ բուհերի ֆիզիկամաթեմատիկական մասնագիտությունների ուսանողների համար Հրատարակության է երաշխավորել ԵՊՀ ինֆորմատիկայի և կիրառական մաթեմատիկայի ֆակուլտետի գիտական խորհուրդը Մասնագետ խմբագիր՝ ֆիզմաթ. գիտ. դոկտոր, պրոֆեսոր Ա. Հ. Հովհաննիսյան

Վահրամ Ժորայի Դումանյան Դ 940 Մաթեմատիկական ֆիզիկայի հավասարումներ /Վ. Ժ. Դումանյան: - Եր., ԵՊՀ հրատ., 2017, 132 էջ: Ուսումնասիրվում են երկրորդ կարգի մասնական ածանցյալներով գծային դիֆերենցիալ հավասարումները և դրանց լուծումների կառուցման եղանակները:

ՀՏԴ 517.9:530.1 ԳՄԴ 22.311

ISBN 978-5-8084-2196-7  ԵՊՀ հրատ., 2017  Վ. Ժ. Դումանյան, 2017

Bovandakowyown Nera owyown Glowx 1. Erkrord kargi havasarowmneri dasakargowm: Bnowagri maker owyner

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

§ 1.

Havasarowmneri dasakargowm

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

§ 2. Dasakargman invariantowyown oork oxmiareq artapatkerowmneri nkatmamb

§ 3.

Bnowagri maker

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

owyner

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Glowx 2. Hiperbolakan tipi havasarowmner § 1. Koii xndir mar

§ 2.

Koii lokalacva xndir aliqayin havasarman ha-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Fowryei

a oxowyan

kira owm

aliqayin

Koii xndri low owm stanalow hamar

§ 3.

. . . . . . . . . . . . . . . . . . . . . . . . . 27

havasarman

hamar

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Koii xndir lari tatanman havasarman hamar: Dalamberi ba-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

§ 4.

Koii xndri low man miakowyown aliqayin havasarman hamar

§ 5.

Aliqayin havasarman hamar Koii xndri low man goyowyown ereq

tara akan o oxakanneri depqowm

§ 6.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Aliqayin havasarman hamar Koii xndri low man goyowyown erkow

mek tara akan o oxakanneri depqowm

. . . . . . . . . . . . . . . . . . . . . . . . 47

§ 7.

Aliqneri difowziayi masin

§ 8.

Xa xndir hiperbolakan havasarman hamar

§ 9.

o oxakanneri anjatman meod

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Fowryei

. . . . . . . . . . . . . . . . . . . 55

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Glowx 3. Parabolakan tipi havasarowmner § 1.

. . . 38

. . . . . . . . . . . . . . . . . . . . . . . . . 70

a oxowyan kira owm jermahaordakanowyan

havasarman hamar Koii xndri low owm stanalow hamar

. . . . . . . . . 71

§ 2. Fowndamental low owm: Jermahaordakanowyan havasarman hamar Koii xndri low man goyowyown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

§ 3. Low man miakowyown: Maqsimowmi skzbownq: Low man anndhat kaxva owyown skzbnakan fownkciayic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

§ 4. Xa xndir parabolakan havasarman hamar . . . . . . . . . . . . . . . . . . . 86 § 5. o oxakanneri anjatman meod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Glowx 4. lipsakan tipi havasarowmner

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

§ 1. Harmonik fownkcianer: Laplasi havasarman fowndamental low owm: Grini bana er . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

§ 2. Potencialner: Oork fownkciayi nerkayacowm potencialneri gowmari tesqov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

§ 3. Mijini masin eorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 § 4. Maqsimowmi skzbownq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 § 5. Dirixlei xndir: Low man miakowyown

anndhat kaxva owyown

ezrayin fownkciayic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

§ 6. Oork fownkciayi nerkayacowm gndowm: Grini fownkcian gndi hamar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

§ 7. Laplasi havasarman hamar Dirixlei xndri low man goyowyown gndowm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

§ 8. Mijini masin hakadar eorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 § 9. Veracneli ezakiowyan masin eorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 § 10. Liowvili eorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 § 11. Neymani xndir Laplasi havasarman hamar` gndowm . . . . . . . . . . . . 124

Grakanowyown

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Nera owyown Diferencial havasarowmner kovowm en ayn havasarowmner, oroncowm anhaytner mek kam mi qani o oxakanneric kaxva fownkcianer en, nd orowm havasarowmneri mej masnakcowm en inpes anhayt fownkcianer, aynpes l nranc a ancyalner: Ee anhayt fownkcianer kaxva en mek

o oxakanic, apa havasarowmner kovowm en sovorakan diferencial havasarowmner, isk ee anhayt fownkcianer kaxva en mi qani (erkow kam aveli) o oxakanneric, apa havasarowmner kovowm en masnakan a ancyalnerov diferencial havasarowmner: Havasarman mej masnakco oroneli fownkciayi a ancyali maqsimal karg kovowm havasarman karg: Dasagrqowm owsowmnasirvowm en miayn erkrord kargi g ayin diferencial havasarowmner: Nermow enq oro nanakowmner: Rn -ov nanakenq n-a ani vklidesyan tara owyown, x = (x1 , ..., xn )-ov nanakenq Rn tara owyan ket, |x| = (x21 +...+x2n )1/2 : Rn tara owyan tirowy kam n-a ani tirowy aselov khaskananq Rn tara owyan keteri bac kapakcva bazmowyown (o datark): Dicowq Q-n n-a ani tirowy : E ⊂ Q bazmowyown kovowm Q-i hamar xist nerqin, ee E ⊂ Q, orte E -ov nanakva E bazmowyan akowm Rn tara owyan metrikayov: C k (Q)-ov nanakenq Q tirowyowm min k -rd karg nera yal anndhat masnakan a ancyalner owneco bolor fownkcianeri bazmowyown, orte k -n obacasakan amboj iv : C k (Q)-ov nanakenq C k (Q) bazmowyan enabazmowyown, or bakaca bolor ayn fownkcianeric, oronc min k-rd karg nera yal

masnakan a ancyalner anndhat en Q-owm: C 0 (Q) Q-owm

C 0 (Q) bazmowyownneri hamar, oronq hamapatasxanabar

Q-owm anndhat fownkcianeri bazmowyownnern en, kgtagor enq

na C(Q)

C(Q) nanakowmner:

C ∞ (Q)-ov nanakenq ayn fownkcianeri bazmowyown, oronq patkanowm ∞ C k (Q): en bolor C k (Q), k = 0, 1, ..., bazmowyownnerin` C ∞ (Q) = k=0

C ∞ (Q)-ov nanakenq ayn fownkcianeri bazmowyown, oronq patkanowm ∞ C k (Q): en bolor C k (Q), k = 0, 1, ..., bazmowyownnerin` C ∞ (Q) = f (x) fownkciayi a ajin

k=0

erkrord kargi masnakan a ancyalneri

hamar kgtagor enq na fxi , fxi xj nanakowmner. f xi ≡

∂f (x) , ∂xi

f xi xj ≡

∂ 2 f (x) : ∂xi ∂xj

f ∈ C 1 (Q) fownkciayi (fx1 , ..., fxn ) gradient knanakenq ∇f : (n − 1)-a ani S ak maker owy aselov` khaskananq C k , k ≥ 1, dasi

a anc ezri (n − 1)-a ani sahmana ak ak maker owy, aysinqn` Rn -owm nka kapakcva sahmana ak ak maker owy ( S = S ), orn tva het yal hatkowyamb. cankaca x0 ∈ S keti hamar goyowyown owni ayd keti Ux0 rjakayq (n-a ani)

C k (Ux0 )-in patkano aynpisi Fx0 (x) fownkcia, nd orowm

∇Fx0 (x0 ) = 0, or S ∩ Ux0 bazmowyown nkaragrvowm Fx0 (x) = 0 havasa-

rowmov (S ∩ Ux0 bazmowyan bolor keter bavararowm en Fx0 (x) = 0 havasarman,

Ux0 -in patkano cankaca ket, or bavararowm Fx0 (x) = 0

havasarman, patkanowm S -in): Q tirowyi ezr knanakenq ∂Q-ov: Aysowhet kenadrenq, ee haka a-

k hatowk nvi, or ditarkvo tirowyneri ezrer bakaca en verjavor vov irar het hatvo (n − 1)-a ani C 1 dasi ak maker owyneric: Nkatenq, or ee S ak maker owy patkanowm C k dasin, apa ayd maker owyi cankaca x0 ∈ S keti hamar goyowyown owni ayd keti ayn6

qan oqr Ux0 rjakayq, or S ∩ Ux0 hatowm miareqoren proyektvowm koordinatakan harowyownneric mekowm nka C k dasin patkano ezr owneco or (n − 1)-a ani Dx0 tirowyi vra. goyowyown owni aynpisi i, i = 1, ..., n, or ayd hatowm nkaragrvowm xi = ϕx0 (x1 , ..., xi−1 , xi+1 , ..., xn ),

havasarowmov

(x1 , ..., xi−1 , xi+1 , ..., xn ) ∈ Dx0 ,

ϕx0 ∈ C k (Dx0 ): S ∩ Ux0 hatowm kanvanenq S maker owyi

parz ktor (kam ktor):

Qani or S maker owy sahmana ak

ak, apa {Ux , x ∈ S}

a kowyic kareli ntrel verjavor ena a kowy: Aydpisi verjavor ena a kowyin hamapatasxano S1 , ..., SN parz ktorneri hamaxowmb kanvanenq S maker owyi parz ktornerov a kowy: (n − 1)-a ani C k , k ≥ 1, dasi S maker owy aselov` khaska-

nanq kapakcva maker owy, or kareli aynpes a kel verjavor vov Ui , i = 1, ..., N , tirowynerov (n-a ani), or Si

= S ∩ Ui ,

i = 1, ..., N , bazmowyownneric yowraqanyowr miareqoren proyektvowm

koordinatakan harowyownneric mekowm nka C k dasin patkano ezrov or (n − 1)-a ani Di tirowyi vra. or p-i hamar, p = p(i), 1 ≤ p ≤ n, ayd hatowm nkaragrvowm xp = ϕi (x1 , ..., xp−1 , xp+1 , ..., xn ),

havasarowmov

(x1 , ..., xp−1 , xp+1 , ..., xn ) ∈ Di ,

ϕi ∈ C k (Di ): S maker owyi U1 , ..., UN a kowyin hama-

patasxano Si , i = 1, ..., N , parz ktorneri hamaxowmb kanvanenq S maker owyi parz ktornerov a kowy: Aysowhet (n − 1)-a ani ma-

ker owy aselov` khaskananq C k , k ≥ 1, dasi (n − 1)-a ani maker owy: Dicowq S - Q-owm nka C k , k ≥ 1, dasi or maker owyi parz ktor dicowq

xn = ϕ(x1 , ..., xn−1 ) = ϕ(x ), x ∈ D,

ϕ(x ) ∈ C k (D),

ayd ktori havasarowmn : Kasenq, or S -i vra trva f (x) = f (x1, ..., xn), x ∈ S , fownkcian patkanowm C k (S) bazmowyan, f ∈ C k (S), ee f (x , ϕ(x )) fownkcian patkanowm C k (D) bazmowyan: Enadrenq S - Q-owm nka C k , k ≥ 1, dasi ak maker owy (masnavorapes S = ∂Q) S1, ..., SN nra parz ktornerov a kowy : Kasenq, or S -i vra trva f (x), x ∈ S , fownkcian patkanowm C k (S) bazmowyan, f ∈ C k (S), ee f ∈ C k (Si), i = 1, ..., N : Dvar nkatel, or f (x) fownkciayi patkanelowyown C k (S) bazmowyan kaxva S maker owyi parz ktornerov a kowyic: Verada nanq diferencial havasarowmnerin: Kira owyownnerowm anhraetowyown a ajanowm owsowmnasirel diferencial havasarowmner oroaki tirowyowm gtnel grit, kam motavor, low owmner kam owsowmnasirel low man orakakan hatkowyownner: nd orowm, ditarkvowm en diferencial havasarman o bolor low owmner, ayl ayn low owmner, oronq, orpes kanon, tirowyi ezri vra bavararowm en owsowmnasirvo xndri bnowyic bxo lracowci paymanneri: Berenq mi qani tipayin rinakner:

Xndir kayanowm het yalowm` gtnel oweri azdecowyan tak gtnvo membrani havasarak owyan dirq: Enadrvowm , or membrani cankaca owylatreli dirq irenic nerkayacnowm (x, u) = (x1, x2, u) tara owyan maker owy, or miareqoren proyektvowm x1Ox2 harowyan or Q tirowyi vra trvowm u = u(x), x ∈ Q, havasarowmov, orte u ∈ C 1 (Q): Enadrvowm , or ee u = ϕ(x), x ∈ Q, membrani or owylatreli dirq , apa cankaca ayl u = u(x) owylatreli dirq stacvowm u = ϕ(x) dirqic 1.

Membrani

havasarak owyan

arman

xndir:

membrani yowraqanyowr keti Ou a ancqin zowgahe tea oxowyamb: Enadrvowm na , or membrani vra azdo artaqin ow owva Ou a ancqin zowgahe

owni f (x) anndhat xtowyown: Membran ϕ dirqic

u dirq tea oxelow hamar ayd owi katara axatanq havasar u(x) f (x)dudx = f (x) u(x) − ϕ(x) dx : Q ϕ(x)

Baci ayd, membrani vra azdowm na nerqin ow: Khamarenq, or membran ϕ dirqic u dirq tea oxelow nacqowm ayd owi katara axatanq

havasar

−

k(x)

1 + |∇u|2 − 1 + |∇ϕ|2 dx

(x1 , x1 + Δx1 ) × (x2 , x2 + Δx2 ) tarrin ayd owi hatkacra axatanq

hamematakan membrani maker owyi ayn masi makeresi o oxowyan, or proyektvowm ayd tarri vra, k(x) > 0 gor akic kovowm membrani kva owyown, ∇u = (ux1 , ux2 ) : Ee membrani ezri keterowm kira va g1 (x, u) = g1 (x) − σ1 (x)u (σ1 (x) ≥ 0 ezri a agakan amracman gor akicn ) g ayin xtowyamb ow, apa membran ϕ(x) dirqic u(x) dirq tea oxelow hamar ayd owi katara axatanq havasar u(x) σ1 (x) 2 g1 (x, u) du dS = g1 (x) (u(x) − ϕ(x)) − u (x) − ϕ2 (x)

∂Q ϕ(x)

dS :

∂Q

u(x) dirqowm membrani potencial nergian havasar 1 + |∇u|2 − 1 + |∇ϕ|2 dx − f (x)(u − ϕ) dx+ U (u) = U (ϕ) + k(x) Q

σ1 2 u − ϕ2 − g1 (u − ϕ) dS, +

∂Q

orte U (ϕ)-n ϕ dirqowm membrani potencial nergian :

Parzowyan hamar enadrenq, or membrani owylatreli u(x) dirqeri hamar ∇u(x) bavakanaa

oqr |∇u|4 kargi andamner karo enq havi a nel: Ayd depqowm u(x) dirqowm membrani potencial nergian kndowni

U (u) = U (ϕ) + Q

k |∇u|2 − |∇ϕ|2 dx −

f (x)(u − ϕ) dx+ Q

σ1 2 u − ϕ2 − g1 (u − ϕ) dS + ∂Q

tesq: Ee u(x)- membrani havasarak owyan dirqn , apa cankaca or ayl owylatreli v(x) dirqi depqowm t = 0 ket ⎡

P (t) = U (u + tv) = U (u) + t ⎣ ⎡

t2 ⎣

(k∇u∇v − f v) dx +

⎤

(σ1 uv − g1 v) dS ⎦ +

∂Q

k|∇v|2 dx +

⎤ σ1 v 2 dS ⎦

∂Q

bazmandami (st t-i) minimowmi ket (ayste ∇u∇v-ov nanakva ∇u ∇v vektorneri skalyar artadryal` ∇u∇v = ux vx +ux vx ): Het abar,

dP (0) = 0, dt

orteic stacvowm , or cankaca v ∈ C 1(Q) fownkciayi hamar membrani havasarak owyan dirq nkaragro u(x) fownkcian bavararowm

k∇u∇v dx + Q

σ1 uv dS = Q

∂Q

f v dx +

g1 v dS

(0.1)

∂Q

integral nowynowyan: Ee membrani ezr anar , aysinqn` kot amracva , apa membrani bolor owylatreli u(x) dirqer bavararowm en u

∂Q

= ϕ

∂Q

(0.2)

paymanin, ayd depqowm kamayakan nergian havasar

U (u) = U (ϕ) + Q

u(x)

dirqowm membrani potencial

k |∇u|2 − |∇ϕ|2 − f (u − ϕ) dx :

Dicowq u-n kot amracva membrani havasarak owyan dirqn : Ayd depqowm cankaca v ∈ C 1(Q) fownkciayi hamar, or bavararowm v|∂Q = 0

(0.3)

paymanin, u + tv fownkcian kbavarari (0.2) paymanin: Het abar, bolor aydpisi v fownkcianeri hamar (k∇u∇v − f v) dx +

P (t) = U (u + tv) = U (u) + t Q

k|∇v|2 dx

bazmandam t = 0 ketowm ndownowm oqragowyn areq: Owsti, (0.3) paymanin bavararo cankaca v ∈ C 1(Q) fownkciayi hamar kot amracva membrani havasarak owyan dirq nkaragro u(x) fownkcian bavararowm k∇u∇v dx = Q

(0.4)

f v dx Q

integral nowynowyan: Ee hamarenq, or membrani havasarak owyan u oroneli dirq trvowm o e mek, ayl erkow angam diferenceli fownkciayi mijocov, u ∈ C 2(Q), apa (0.1) (0.4) integral paymanner kareli oxarinel lokal paymannerov` enadrelov k(x) ∈ C 1(Q), k(x) > 0, x ∈ Q, σ1 , g1 , ϕ ∈ C(∂Q): Hamaayn strogradskow bana i

k∇u∇v dx = − Q

v div (k∇u) dx +

k ∂Q

∂u v dS, ∂ν

∂u = (∇u, ν)|∂Q = ∂ν ∂Q (ux1 ν1 + ux2 ν2 )|∂Q , ν = (ν1 , ν2 )- ∂Q-in tarva Q-i nkatmamb artaqin

orte A = (A1 , A2 ) vektori hamar divA = A1x1 +A2x2 ,

miavor normal vektorn : Het abar, (0.1) grel

(0.4) nowynowyownner kareli

∂u div (k∇u) + f v dx − + σ1 u − g1 v dS = 0 k ∂ν

(0.1 )

∂Q

div (k∇u) + f v dx = 0

(0.4 )

tesqov:

Qani or div(k∇u) + f fownkcian anndhat , apa (0.4 ) nowynowyownic stanowm enq div (k∇u) + f = 0,

x ∈ Q,

(0.5)

in kot amracva membrani depqowm (0.2) ezrayin paymani het miasin handisanowm ayn lokal payman, orin petq bavarari oroneli u(x) fownkcian: (0.2) ezrayin paymanin bavararo (0.5) havasarman low owm gtnelow xndir kovowm a ajin ezrayin xndir ( kam Dirixlei xndir) (0.5) havasarman hamar:

Qani or (0.1 )-owm v(x) kamayakan fownkcia C 1 (Q) dasic, apa ditarkelov masnavorapes (0.3) paymanin bavararo v(x) fownkcianer` kstananq, or ays depqowm

s u(x) fownkcian bavararowm (0.5)

havasarman: Het abar, (0.1 ) nowynowyown kareli grel ∂u + σ1 u − g1 v dS = 0 k ∂ν ∂Q

tesqov: Dicowq σ1 , g1 ∈ C 1 (∂Q): Qani or C 1 (∂Q), ∂Q ⊂ C 1 , bazmowyan patkano cankaca fownkcia owni C 1 (Q)-in patkano arownakowyown, apa verjin nowynowyownic stanowm enq ∂u + σu = g ∂ν ∂Q

(0.6)

ezrayin payman , (0.6)

orte

g1 σ1 ≥ 0, g = : k k bavararo (0.5) havasarman

σ=

ezrayin paymanin

xndir kovowm

errord ezrayin xndir (0.5)

depqowm errord ezrayin xndir kovowm

Neymani xndir):

low owm gtnelow

σ≡0

havasarman hamar:

erkrord ezrayin xndir ( kam

Ays depqowm ezrayin paymann owni

∂u =g ∂ν ∂Q tesq:

Ayspisov

nkaragrvowm

stacanq,

(0.5)

(0.7)

membrani

havasarman

low man

havasarak owyan

mijocov,

dirq

bavararowm

oroaki ezrayin paymani:

Aym ditarkenq membrani arman xndir:

Dicowq

u(x, t)

Ayd depqowm

fownkcian oroowm membrani dirq amanaki

ut (x, t)

aragowyown

(0.8)

(0.9)

aragacowm (enadrvowm , or ayd a ancyalner goyowyown

t = t0

u|t=t0 = ψ0 (x),

x ∈ Q,

(0.8)

ut |t=t0 = ψ1 (x),

x∈Q:

(0.9)

paymanner kovowm en

skzbownqi

oxarinva

xtowyownn

skzbnakan paymanner :

hamaayn`

(0.5)

membrani havasarak owyan

ketowm,

membrani

arman

havasarowmn , orowm

−ρ(x)utt + f (x, t)

fownkciayov

f (x, t)-n

ndhanrapes asa , kaxva

artaqin

(

havasarowm

f (x)

−ρ(x)utt -n

owi

fownkcian

inerciayi

xtowyownn

or,

t-ic).

divx (k∇x u) + f (x, t) − ρ(x)utt = 0, ∂Q

(x, t)

pahin trva en membrani

aragowyown.

Dalamberi

owi

pahin:

utt (x, t) fownkcianer oroowm en membrani x ∈ Q keti

ownen): Dicowq amanaki oroaki

keti dirq

x ∈ Q, t > t0 :

ezri vra trva paymanneric kaxva , inpes

(0.10)

stacionar depqowm,

ezrayin paymanner ndownowm en (0.2), (0.6) kam (0.7) tesq

tei ownen

ditarkvo amanaki bolor t ≥ t0 areqneri hamar: (0.2), (0.8), (0.9) kam (0.7), (0.8), (0.9) (kam (0.6), (0.8), (0.9))

paymannerin bavararo (0.10) havasarman low owm gtnelow xndir

a ajin

kovowm , hamapatasxanabar,

xa xndir (0.10) havasarman hamar:

erkrord (kam errord)

kam

Ayspisov, membrani arowm nkaragrvowm (0.10) havasarman low man mijocov, or bavararowm skzbnakan

oroaki ezrayin paymanneri:

Anverj tara va membrani depqowm ( Q = R2 ) arowm nkaragro u(x, t), x ∈ R2 , t > 0, fownkcian (0.10) havasarman low owm

bavararowm

(0.8), (0.9) skzbnakan paymannerin: Ayd depqowm asowm en, or u(x, t) fownkcian (0.10) havasarman hamar

skzbnakan xndri (Koii xndri)

low owm : Ee (0.5)

(0.10) havasarowmnerowm gor akicner hastatownner en,

k(x) ≡ k > 0, ρ(x) ≡ ρ > 0, apa ayd havasarowmner hamapatasxana-

bar kovowm en`

Powasoni havasarowm. Δu = −

f (x) , k

(0.5 )

x ∈ Q,

aliqayin havasarowm. f (x, t) utt − Δu = − , a20 k

orte Δ ≡

∂2 ∂2 + 2 ∂x1 ∂x2

kovowm

x ∈ Q, t > t0 ,

a0 =

k , ρ

(0.10 )

Laplasi perator:

Mek tara akan o oxakani depqowm (0.10 ) havasarowmn owni f (x, t) utt − uxx = − , a20 k

x ∈ (α, β), t > t0 ,

(0.10 )

tesq: Ays havasarowm nkaragrowm (α, β) mijakayqi vra teakayva

lari arowm: E aa

x = (x1 , x2 , x3 )

depqowm

f (x, t) , u − Δu = − 2 tt a0 k havasarowm, orte

tirowyowm (

x ∈ Q, t > t0 ,

(0.10 )

Δu = ux1 x1 + ux2 x2 + ux3 x3 , nkaragrowm gazi arowm

u(x, t)

fownkcian bnowagrowm amanaki

pahin

x ∈ Q

a0 -n

gazowm

ketowm gazi nman eowm hastatown nowmic): Ays depqowm ayni tara man aragowyownn :

2. Jermowyan tara man xndir:

ρ>0

tirowyowm ownenq nyow, orn owni

k(x) > 0 x∈Q

Dicowq

e aa

xtowyown,

c>0

u(x, t)-ov

jermahaordakanowyan gor akic:

tara owyan

jermownakowyown

nanakenq

pahin

t = t0

skzbnakan pahin jer-

x ∈ Q,

(0.11)

ketowm jermastian: Enadrenq, or

mastian haytni ,

u(x, t)|t=t0 = ψ0 (x), pahanjvowm gtnel jermastian Dicowq

tirowy

(t1 , t2 ), t0 ≤ t1 < t2 ,

Q-i

t > t0

hamar:

enatirowy : Fowryei renqi hamaayn`

∂Q

amanakahatva owm

ezrov

tirowy mtno jer-

mowyan qanak havasar

t2 dt t1 orte

ν -n ∂Q -in

(t1 , t2 )

tarva

∂Q

Q -i

tirowyowm a ka

amanakahatva owm

f (x, t)

xtowyamb jermowyan abyowr, apa

- owm jermowyan a havasar

t2 f (x, t) dx +

∂u dS, ∂ν

nkatmamb artaqin miavor normaln :

k(x)

∂Q

Ayd jermowyown axsvowm yowraqan yowr

k(x)

∂u dS : ∂ν

x ∈ Q

ketowm jermastiani

areq u(x, t1)-ic min

t2 t1

dt t1

o oxelow vra tei owni

t2 f (x, t) dx +

u(x, t2 )

k(x)

∂Q

∂u dS = ∂ν

c(x)ρ(x) u(x, t2 ) − u(x, t1 ) dx

jermayin havasarak owyan havasarowm: Havi a nelov, or t2 u(x, t2 ) − u(x, t1 ) = t1

∂u dt ∂t

gtvelov strogradskow bana ic` kstananq t2

c(x)ρ(x) Q

∂u − div (k(x)∇u) − f (x, t) dx = 0, ∂t

orte ∇u = (ux , ux , ux ): Ee enaintegralayin fownkcian anndhat Q tirowyowm, apa (t1 , t2 ) mijakayqi kamayakan linel` havi a nelov Q tirowyi kstananq, or verjin havasarowyown hamareq

c(x)ρ(x)

∂u − div (k(x)∇u) = f (x, t), ∂t

x ∈ Q, t > t0 ,

(0.12)

diferencial havasarman: Ayn depqowm, erb c(x), ρ(x) k(x) fownkcianer hastatownner en` c(x) = c, ρ(x) = ρ, k(x) = k, (0.12) havasarowm kovowm jermahaordakanowyan havasarowm.

f (x, t) ut − Δu = , a2 cρ

(0.12 )

orte a2 = cρk , Δu = ux x + ux x + ux x : Nkatenq, or (0.12) havasarowm tei owni miayn t > t0 miayn Q tirowyi nerqin keteri hamar: u(x, t) fownkciayi varq t = t0 pahin trvowm (0.11) skzbnakan paymanov, isk x ∈ ∂Q keterowm petq trvi lracowci: Ayn eladrvowm fizikakan konkret xndrov, or jermayin kap hastatowm Q tirowyi artaqin mijavayri mij : 1 1

2 2

3 3

Parzagowyn depqowm ∂Q ezri vra t-i bolor ditarkvo areqneri hamar trvowm u(x, t) jermastian` u|∂Q = f0 (x, t) :

(0.13)

Ayd depqowm jermastian knkaragrvi (0.12) havasarman ayn low man mijocov, or bavararowm (0.11) (0.13) paymannerin: Ee haytni ∂Q ezrov jermayin hosqi q0(x, t) xtowyown, apa Fowryei renqi hamaayn` ezrayin paymann owni k(x)

∂u = q0 (x, t) ∂ν ∂Q

(0.14)

tesq: Ee haytni Q tirowyic dowrs nka mijavayri u0(x, t) jermastian, ∂Q ezrov jermayin hosqi q0 (x, t) xtowyown hamematakan u|∂Q u0 |∂Q jermastianneri tarberowyan, apa ezrayin paymann ndownowm k(x)

∂u + k1 u = k1 u 0 ∂ν ∂Q ∂Q

(0.15)

tesq, orte k1(x) > 0 rjaka mijavayri het marmni jerma oxanakowyan gor akicn : (0.11), (0.13) kam (0.11), (0.14) (kam (0.11), (0.15)) paymannerin bavararo (0.12) havasarman low owm gtnelow xndir kovowm , hamapatasxanabar, a ajin kam erkrord (kam errord) xa xndir (0.12) havasarman hamar: Ayn depqowm, erb nyow lcnowm amboj R3 tara owyown (Q = R3), u(x, t) jermastian bavararowm (0.12) havasarman, erb t > t0 (0.11) skzbnakan paymanin, erb t = t0: Ayd depqowm asowm en, or u(x, t) fownkcian (0.12) havasarman hamar skzbnakan xndri (Koii xndri) low owm :

Glowx 1 Erkrord kargi havasarowmneri dasakargowm Bnowagri maker owyner n-a ani Rn , n > 1, x = (x1 , ..., xn ), tara owyan Q bac bazmowyan vra ditarkenq n

aij (x)uxi xj +

i,j=1

erkrord

kargi

ai (x)uxi + a(x)u = f (x),

x ∈ Q,

(1.1)

i=1

masnakan

a ancyalnerov

g ayin

diferencial

havasarowm, orte havasarman aij , ai , a, (i, j = 1, ..., n)

gor akicner

trva irakan areqani fownkcianer en C(Q)-ic, havasarman f (x)

azat andam (aj mas) trva fownkcia C(Q)-ic: u(x) fownkcian kovowm (1.1) havasarman

low owm, ee u ∈ C 2 (Q)

bavararowm (1.1)

havasarman: Barr kargi a ancyalneri aij (i, j = 1, ..., n) gor akicner avorowm en

avag gor akicneri A(x) = aij (x) qa akowsayin matric:

Hetagayowm, a anc ndhanrowyown xaxtelow, kenadrenq, or A(x) matric simetrik : Iroq, havi a nelov ayn ast, or C 2 (Q) dasin patkano fownkcianeri hamar

uxi xj = uxj xi , n i,j=1

aij uxi xj =

i, j = 1, ..., n,

aij + aji u xi xj , i,j=1

apa A(x) matrici oxaren mit kareli vercnel

aij (x) + aji (x) = (A(x) + A∗ (x)) simetrik matric ( oxelov havasarowm):

§ 1. Havasarowmneri dasakargowm

Vercnenq kamayakan x ket Q-ic: A(x) matrici se akan areqner, aysinqn` det A(x) − λE = 0 havasarman armatner, nanakenq λ1 (x), ..., λn (x) (yowraqanyowr λi krknvowm aynqan angam, orqan nra patikowyownn ): Qani or A(x) matric simetrik , apa bolor λi, i = 1, ..., n, irakan en: Dicowq drancic n− = n− (x) bacasakan en, n0 = n0 (x) zroyakan en n+ = n+ (x) drakan en. n− + n0 + n+ = n: 1. n0 = 0 depq. a) ee n+ = n, n− = 0, kam n+ = 0, n− = n, apa x ketowm (1.1) havasarowm kovowm lipsakan tipi, b) ee n+ = n − 1, n− = 1, kam n+ = 1, n− = n − 1, apa x ketowm (1.1) havasarowm kovowm hiperbolakan tipi, g) ee n+ > 1 n− > 1 (da hnaravor miayn n ≥ 4 depqowm), apa x ketowm (1.1) havasarowm kovowm owltrahiperbolakan tipi: 2. n0 > 0 depq. Ays depqowm x ketowm (1.1) havasarowm kovowm parabolakan tipi: Ditoowyown: Havasarman dasakargman sahmanowmic bxowm , or havasarman tip oroelow hamar partadir gtnel λi(x) armatneri areqner, ayl bavarar imanal armatneri nanner, aveli it` n−(x), n0 (x) n+ (x) amboj ver: Ayd npatakov ditarkenq (A(x)ξ, ξ) =

ai j (x)ξi ξj ,

ξ = (ξ1 , ..., ξn ) ∈ Rn ,

i, j=1

qa akowsayin (x ∈ Q fiqsa ket ): Hanrahavi dasnacic haytni , or veraservo irakan a oxowyan mijocov ayn kareli berel kanonakan (ankyownag ayin) tesqi qa akowsayin eri inerciayi renqi hamaayn` ayd kanonakan tesqi drakan nanov andamneri qanak n+(x) , bacasakan nanov andamneri qanak n−(x) , isk n0 (x) = n − n+ (x) − n− (x):

Ee (1.1) havasarowm or E ⊂ Q bazmowyan bolor keterowm lipsakan (hiperbolakan ayln), apa ayn kovowm lipsakan (hiperbolakan ayln) E bazmowyan vra :

rinakner.

Powasoni havasarowm.

Δu = f (x), ∂2

x ∈ Q,

∂2

orte Δ = 2 + ... + 2 Laplasi peratorn , Δu = ux x + ... + ux x , ∂x1 ∂xn (erb f = 0 ays havasarowm kovowm Laplasi havasarowm ) Q tirowyowm lipsakan tipi , qani or ays depqowm A(x) = E bolor x ∈ Q keteri hamar λ1 (x) = ... = λn (x) = 1: 1 1

n n

Aliqayin havasarowm.

ux1 x1 + ... + uxn−1 xn−1 − uxn xn = f (x),

x ∈ Q,

Q tirowyowm hiperbolakan tipi , qani or ays depqowm bolor x ∈ Q kete-

ri hamar λ1 (x) = ... = λn−1 (x) = 1,

λn (x) = −1:

Jermahaordakanowyan havasarowm.

ux1 x1 + ... + uxn−1 xn−1 − uxn = f (x),

x ∈ Q,

Q tirowyowm parabolakan tipi , qani or ays depqowm bolor x ∈ Q kete-

ri hamar λ1 (x) = ... = λn−1 (x) = 1,

λn (x) = 0:

Trikomii havasarowm.

x2 ux1 x1 + ux2 x2 = f (x),

x ∈ Q ⊂ R2 ,

gndowm xa tipi , qani or ays depqowm havasarowm {|x| < 1, x2 > 0} kisarjanowm lipsakan tipi , {|x| < 1, x2 < 0} kisarjanowm hiperbolakan tipi , isk {|x| < 1, x2 = 0} tramag i vra parabolakan tipi : Q = {|x| < 1}

§ 2. Dasakargman invariantowyown oork oxmiareq artapatkerowmneri nkatmamb

Dicowq U -n or tirowy Q bazmowyownic, U ⊂ Q, dicowq y = ϕ(x),

x ∈ U,

(1.2)

kam st koordinatneri. y1 = ϕ1 (x1 , ..., xn ), ..., yn = ϕn (x1 , ..., xn ), x = (x1 , ..., xn ) ∈ U, ϕ(x) = (ϕ1 (x), ..., ϕn (x)), U -owm

trva a oxowyown :

Enadrenq a) (1.2) a oxowyown U tirowy oxmiareqoren artapatkerowm V tirowyi vra (y ∈ V ), b) ϕ(x) ∈ C 2 (U ), aysinqn` ϕi (x) ∈ C 2 (U ), i = 1, ..., n, i Yakobii matric U tirowyowm i veraservowm, aysinqn g) J(x) = ∂ϕ ∂xj (1.2) a oxowyan yakobyan` det J(x) = 0, x ∈ U : Haytni , or a) - g) paymanneri depqowm (1.2) a oxowyan hakadar x = ψ(y),

y ∈ V,

(1.2 )

a oxowyown nowynpes tva nman hatkowyownnerov: V tirowyowm sahmanenq v(y) fownkcian. v(y) = u(ψ(y)),

y∈V :

(1.3)

Cowyc tanq, or ee a) - g) paymanner tei ownen, apa havasarowm, orin V tirowyowm bavararowm v(y) fownkcian, y ∈ V ketowm klini nowyn tipi, inpisin (1.1) havasarowm y ∈ V ketin hamapatasxano x ∈ U ketowm (st (1.2) kam (1.2 )): Henc ays pndowm khaskananq orpes dasakargman invariantowyown nva a oxowyownneri nkatmamb:

st (1.3)-i u(x) = v(y) = v(ϕ(x)),

x ∈ U (y ∈ V ) :

Bolor i, j = 1, ..., n hamar u xi =

vyk ϕkxi ,

k=1

u xi xj =

n n

vyk ys ϕkxi ϕsxj +

k=1 s=1

vyk ϕkxi xj :

k=1

Het abar, v(y) fownkcian V tirowyowm bavararowm n

aij

i,j=1

vyk ys ϕkxi ϕsxj +

k,s=1

n n vyk ϕkxi xj + ai vyk ϕkxi + av = f, (1. 1) i=1

k=1

havasarman, ori avag andamneri gor akicnern ownen aks (y) =

ϕkxi (x)aij (x)ϕsxj (x)

i,j=1

x=ψ(y)

k, s = 1, ..., n

tesq: Verjin bana ic het owm , or (1.1) havasarman avag gor a kicneri A(y) matric J(x), A(x) J ∗ (x) ereq matricneri artadryal . A(y) = J(x)A(x)J ∗ (x) = JAJ ∗ A

x=ψ(y)

= A(y) ,J ∗ −1 , orte A kam A = J −1 AJ

= J(ψ(y)), A = A(ψ(y)),

y ∈V:

Dicowq P -n aynpisi veraservo matric , or ξ = P η a oxowyown A matrici (Aξ, ξ) qa akowsayin berowm kanonakan tesqi` (Aξ, ξ) = (Λη, η), orte Λ-n ankyownag ayin matric , ori ankyownagi 1, ..., 1, 0, ..., 0, −1, ..., −1 vektorn : Ayd depqowm n+

n−

Λ = P ∗ AP,

∗−1 P, Λ = P ∗ J −1 AJ

matrici (Aξ, ξ) qa akowsayin ξ = (J ∗ −1 P )η orteic bxowm , or A

a oxowyan mijocov bervowm nowyn (Λη, η) tesqi, in tesqi bervel r A matrici qa akowsayin : Het abar, A(x) drakan, zroyakan

A(y), y = ϕ(x), matricneri

bacasakan se akan areqneri qanakner nowynn en:

Pndowmn apacowcva :

§ 3. Bnowagri maker owyner Dicowq Q tirowyowm nka (n − 1)-a ani S oork maker owy ( S ⊂ Q) trva F (x) = 0

(1.4)

havasarowmov, orte F ∈ C 1 (Q) irakan areqani fownkcia , ∇F S = 0: x0 ∈ S

ket kovowm

(1.1)

havasarman hamar

∇F (x)A(x), ∇F (x) ≡

aij (x)

i,j=1 erb

bnowagri

ket, ee

∂F (x) ∂F (x) = 0, ∂xi ∂xj

(1.5)

x = x0 :

Ee vowm

x ∈ S

bolor keter bnowagri keter en, apa

(1.1)

havasarman

hamar

bnowagri: Nanakenq owyown vektori

(1.5)

maker

bnowagri maker owy

∇F (x) : ν vektorn |∇F (x)| |ν| = ν1 + ... + νn2 = 1 (νi -n

ν(x) = (ν1 (x), ..., νn (x)) =

miavor erkarowyown,

owowyan kazma ankyan

owy ko-

kam

owaki

owni

∇F

ow-

havasar

cos-in` νi = cos(ν, xi )): Ayd depqowm

havasarowyown hamareq

aij (x)νi (x)νj (x) = 0

i,j=1 havasarowyan:

(1.5 )

Berenq bnowagri maker owyneri rinakner hastatown gor akicnerov oro tipayin havasarowmneri hamar ( Q = Rn ): Powasoni havasarowm: Powasoni havasarowm bnowagriner owni, qani or ays depqowm A(x) = E (1.5) havasarowyown kndowni |∇F |2 = 0 tesq: Jermahaordakanowyan havasarowm: Jermahaordakanowyan havasarman hamar (1.5 ) havasarowyown ndownowm het yal tesq`

=0: ν12 + ... + νn−1

Het abar, νn2 = 1, νn = ±1, bnowagriner en handisanowm ayn maker owyner, oronc gradienti xn owowyan kazma ankyown 0◦ kam 180◦ , aysinqn` xn = C harowyownner, orte C -n kamayakan hastatown (F = xn − C ): Aliqayin havasarowm: Aliqayin havasarman hamar (1.5 ) havasarowyown ndownowm het yal tesq`

− νn2 = 0 : ν12 + ... + νn−1 √ Het abar, νn = , νn = ± , bnowagriner en handisanowm ayn maker owyner, oronc gradienti xn owowyan kazma ankyown 45◦

kam 135◦ : Nkatenq, or n = 2 masnavor depqowm aliqayin havasarman bnowagriner en handisanowm miayn x1 + x2 = C x1 − x2 = C owiner, orte C -n kamayakan hastatown : Nenq bnowagrineri mi kar or hatkowyown: Haytni , or oork gor akicnerov aj masov u + a1 (x)u + a2 (x)u = f (x), a < x < b, erkrord kargi sovorakan diferencial havasarowmneri low owmner s oork en. rinak` hastatown gor akicnerov hamase havasarowm ( f (x) ≡ 0) owni miayn anverj diferenceli low owmner: Masnakan a ancyalnerov diferencial havasarowmneri hamar iraviak ayl :

Dicowq mek o oxakanic kaxva f (t) g(t) fownkcianer patkanowm en C 2 (R1 )-in en patkanowm C 3 (R1 )-in, rinak. f (t) = 0, erb t ≤ t1 , f (t) = (t − t1 )3 , erb t ≥ t1 , isk g(t) = (t − t2 )5 , erb t ≤ t2 , g(t) = 2(t − t2 )3 erb t ≥ t2 , orte t1 t2 irakan ver en: u(x1 , x2 ) = f (x1 ) + g(x2 ) fownkcian amboj R2 -owm u x1 x2 = 0

hastatown gor akicnerov hamase havasarman low owm , nd orowm u-n i patkanowm C 3 (R1 )-in: Ayd fownkciayi errord kargi a ancyalnern ownen xzowmner x1 = t1 x2 = t2 owineri vra, oronq ayd havasarman bnowagriner en: Ays er owy krowm ndhanowr bnowy: Dicowq n-a ani Q tirowy (n − 1)-a ani L oork maker owyov baanva erkow Q1 Q2 enatirowyneri. L = {x ∈ Q : Φ(x) = 0},

orte Φ ∈ C 1 (Q),

(1.6)

∇ΦL = 0,

Q1 = Q ∩ {x ∈ Q : Φ(x) > 0}, Q2 = Q ∩ {x ∈ Q : Φ(x) < 0}, Q = Q1 ∪ Q2 ∪ L :

Ditarkenq Q \ L bazmowyan vra orova h(x) ∈ C(Q1 ) ∩ C(Q2 ) fownkcian: L maker owyi vra h(x) fownkciayi iq kanvanenq het yal fownkcian` [h](x) = lim h(y) − lim h(y), y∈Q1 y→x

y∈Q2 y→x

x∈L:

ketowm h(x) fownkciayi anndhatowyan hamar anhraet bavarar, or [h](x0 ) = 0, nd orowm`

x0 ∈ L

h(x0 ) = lim h(y) : y∈Q1 y→x0

Tei owni het yal pndowm, or knerkayacnenq a anc apacowyci:

eorem 1.1.1 Dicowq Q tirowyowm u fownkcian (1.1) havasarman

low owm , nd orowm` or amboj k > 0 hamar

u ∈ C k+1 (Q) ∩ C k+2 (Q1 ∪ L) ∩ C k+2 (Q2 ∪ L),

isk havasarman gor akicner azat andam patkanowm en C k (Q)-in: Ee L maker owyi or ketowm low man (k + 2)-rd kargi a ancyalneric gone mek goyowyown owni (hiatakva a ancyali iq zroyic tarber ), apa ayd ket bnowagri ket : Masnavorapes, ee L maker owyi yowraqanyowr ketowm goyowyown owni low man (k + 2)-rd kargi a ancyalneric or mek, apa L maker owy bnowagri : Low man havasarman kargic aveli barr kargi a ancyalneri xzowmner kovowm en low man

owyl xzowmner:

eorem 1.1.1- kareli vera akerpel het yal kerp` eoremowm nva

low owmneri owyl xzowmner teakayva en bnowagrineri vra:

Glowx 2 Hiperbolakan tipi havasarowmner Erkrord kargi hiperbolakan havasarowmner a avel haax handipowm en tatanoakan procesneri het kapva fizikakan xndirnerowm: § 1. Koii xndir

Koii lokalacva xndir aliqayin

havasarman hamar u(x, t), x ∈ Rn , t > 0,

fownkcian kovowm

utt − x u = f (x, t), ut=0 = ϕ(x),

ut t=0 = ψ(x),

x ∈ Rn , t > 0,

(2.1)

x ∈ Rn ,

(2.2)

x ∈ Rn ,

(2.3)

Koii xndri low owm, ee u-n patkanowm C 2(x ∈ Rn, t ≥ 0) bazmowyan bavararowm (2.1), (2.2), (2.3) havasarowyownnerin: Low man sahmanowmic aknhaytoren het owm , or (2.1), (2.2), (2.3) xndri low man goyowyan hamar anhraet en f ∈ C(x ∈ Rn , t ≥ 0), ϕ ∈ C 2 (Rn ), ψ ∈ C 1 (Rn )

paymanner: xndri het mekte kareli ditarkel na het yal aveli ndhanowr xndir: (2.1) − (2.3)

Dicowq Q-n {x ∈ Rn , t = 0} harowyan or n-a ani tirowy : Q-ov knanakenq na (n+1)-a ani {x ∈ Rn , t ∈ R1 } tara owyan bazmowyown, or kazmva Q-i keteric: Vercnenq Q bazmowyan kamayakan (x0 , 0) ket kamayakan t0 > 0 iv aynpes, or Qx0 ,t0 = {|x − x0 | ≤ t0 , t0 > 0}

a avov n-a ani gownd nka lini Q-i mej: gagaov (n + 1)-a ani kon nanakenq Ωx ,t . t0

Qx0 ,t0

himqov

(x0 , t0 )

0 0

Ωx0 ,t0 = {|x − x0 | < t0 − t, 0 < t < t0 } :

koni Sx ,t = {|x − x0 | = t0 − t, 0 ≤ t ≤ t0 } komnayin maker owy (2.1) havasarman konayin bnowagrii ktor : (2.1) havasarman Q himqov bnowagri konoid kam owaki Q himqov konoid kanvanenq bolor Ωx ,t koneri miavorowm handisaco ΩQ tirowy.

Ωx0 ,t0

0 0

ΩQ =

Ωx0 ,t0 :

(2.4)

(x0 ,0)∈Q, t0 >0: Q 0 0 ⊂Q x ,t

harowyan mej vercnenq kamayakan ΩQ hamapatasxan konoidn : {x ∈ Rn , t = 0}

utt − u = f (x, t),

tirowy

dicowq

(x, t) ∈ ΩQ ,

(2.5)

ut=0 = ϕ(x),

x ∈ Q,

(2.6)

= ψ(x),

x ∈ Q,

(2.7)

t=0

xndir, orte f, ϕ, ψ hamapatasxan bazmowyownneri vra trva

fownkcianer en, kovowm Koii lokalacva xndir : (2.5)−(2.7) xndri low owm kovowm (2.5), (2.6), (2.7) havasarowyownnerin bavararo u ∈ C 2 (ΩQ ∪ Q) fownkcian:

Koii lokalacva xndir Koii (2.1) − (2.3) xndri ndhanracowmn . (2.1) − (2.3) xndir hamnknowm (2.5) − (2.7) xndri het Q = Rn depqowm: Sahmanowmic het owm , or (2.5), (2.6), (2.7) xndri low man goyowyan hamar anhraet f ∈ C(ΩQ ∪ Q),

ϕ ∈ C 2 (Q),

ψ ∈ C 1 (Q) :

§ 2. Fowryei a oxowyan kira owm aliqayin havasarman hamar Koii xndri low owm stanalow hamar

Dicowq trva (2.1) − (2.3) xndir: Mer owsowmnasirowyan plan het yaln . nax, Fowryei a oxowyan formal kira mamb, a anc xist himnavorowmneri, menq <kk ahenq> ayn bana , orov (2.1) − (2.3) xndri low owm artahaytvowm ϕ(x), ψ(x) f (x, t) fownkcianeri mijocov: Aynowhet (stacva bana i gnowyamb) kandrada nanq xndri xist himnavorva owsowmnasirowyan, inpes na stacva bana i xist himnavorman: Parzowyan hamar ditarkenq utt − x u = 0,

(2.10 )

x ∈ Rn , t > 0,

hamase havasarowm ereq tara akan o oxakani depqowm, n = 3, x = (x1 , x2 , x3 ): Dicowq u-n (2.10 ) havasarman low owm : Bazmapatkenq (2.10 )-n st x-i integrenq R3 -ov: e−i(x,ξ) -ov, orte ξ = (ξ1 , ξ2 , ξ3 ) ∈ R3 , Kstananq orte

u (ξ, t) =

(ξ, t) = 0, u tt + |ξ|2 u e−i(x,ξ) u(x, t) dx

handisanowm

a oxowyown st tara akan o oxakanneri:

( 2.10 )

t > 0, u(x, t)

low man Fowryei

(2.2), (2.3)

skzbnakan

paymanneric stanowm enq u (ξ, t)t=0 = ϕ(ξ),

(2. 2)

: u t (ξ, t)t=0 = ψ(ξ)

(2. 3)

( 2.10 ), (2. 2), (2. 3)

xndir yowraqanyowr fiqsa ξ ∈ R3-i depqowm handisanowm Koii xndir hastatown gor akicnerov sovorakan diferencial havasarman hamar, ori low owmn owni u (ξ, t) = ϕ(ξ) cos |ξ|t +

ψ(ξ) sin |ξ|t |ξ|

tesq: Het abar, Fowryei hakadar a oxowyan mijocov xndri low owm nerkayacvowm u(x, t) =

(2π)3

tesqov, orte

= (2π)3 = R3

α(y) ⎝ (2π)3

Ayste K(z, t) =

ϕ(ξ) cos |ξ|tei(x,ξ) dξ+

sin |ξ|t ei(x,ξ) dξ = − ∂ uϕ (x, t) − ∂ uψ (x, t) ψ(ξ) |ξ| ∂t2 ∂t

uα (x, t) =

⎛

(2π)3

u (ξ, t)ei(x,ξ) dξ =

(2.1) − (2.3)

(2π)3

α (ξ) R3

⎛

cos |ξ|t i(x,ξ) e dξ = |ξ|2 ⎞

⎝

(2.8)

α(y)e

−i(y,ξ)

dy ⎠ ei(x,ξ)

cos |ξ|t dξ = |ξ|2

⎞ cos |ξ|t ⎠ dy = K(x − y, t)α(y) dy : ei(x−y,ξ) dξ |ξ|2

(2.9)

ei(z,ξ)

cos |ξ|t dξ, |ξ|2

z = (z1 , z2 , z3 ) ∈ R3 :

(2.10)

Qani or (2.10) integral kaxva miayn t-ic ayn havel miayn z = (0, 0, |z|), |z| = 0, ketowm K(z, t) = K(0, 0, |z|, t) = (2π)3

(2π)2 |z|

∞

2π

r dr

dϕ

∞

|z|-ic,

eir cos θ|z|

apa bavarar

cos rt sin θ dθ = r2

cos rt sin r|z| dr = r

⎛∞ ⎞ ∞ sin r(|z| + t) sin r(|z| − t) ⎝ dr + dr⎠ = = (2π)2 |z| r r

⎧ ⎪ ⎪ ⎪ ⎨1,

π (sgn(|z| + t) + sgn(|z| − t)) = (2π)2 |z| 4π|z| ⎪ ⎪ ⎪ ⎩0,

|z| > t, |z| < t :

Teadrelov stacva artahaytowyown (2.9)-i mej` kstananq uα (x, t) =

K(x − y, t)α(y) dy = R3

Het abar

4π

∞ t

4π

|x−y|>t

α(y) dy = |x − y|

dr r

α(y) dSy : |x−y|=r

∂uα =− ∂t 4πt

α(y) dSy , |x−y|=t

orteic, hamaayn (2.8)-i, stanowm enq ⎛

u(x, t) =

∂ ⎜ 1 ⎝ ∂t 4πt

|x−y|=t

⎞

⎟ ϕ(y) dSy ⎠ + 4πt

ψ(y) dSy : |x−y|=t

Ayspisov, menq stacanq (2.10), (2.2), (2.3) xndri low man tesq: artahaytowyown kovowm Kirxhofi bana :

(2.11) (2.11)

§ 3. Koii xndir lari tatanman havasarman hamar: Dalamberi bana

Ditarkenq hiperbolakan tipi hamase havasarman parzagowyn rinak` utt − a2 uxx = 0,

−∞ < x < +∞, t > 0,

orte a > 0 hastatown : Ays havasarowm nkaragrowm anverj lari tatanowm, orte u(t, x) fownkcian x koordinat owneco keti dirqn amanaki t pahin: Qani or grva havasarowm y = at o oxakani oxarinmamb ndownowm uyy − uxx = 0 kanonakan tesq, apa, a anc ndhanrowyown xaxtelow, khamarenq a = 1. utt − uxx = 0,

−∞ < x < +∞, t > 0 :

(2.12)

Koii xndir (2.12) havasarman hamar kayanowm het yalowm` gtnel (2.12) havasarman ayn u ∈ C 2 (x ∈ R1 , t ≥ 0) low owm, or bavararowm ut=0 = ϕ(x),

ut t=0 = ψ(x),

−∞ < x < +∞,

(2.13)

−∞ < x < +∞,

(2.14)

skzbnakan paymannerin, orte ϕ(x) ψ(x) haytni fownkcianer en (skzbnakan tvyalner): (2.13) payman nkaragrowm skzbnakan t = 0 pahin lari x koordinat owneco keti dirq, isk (2.14) payman` aragowyown: Low enq (2.12), (2.13), (2.14) Koii xndir: Nax gtnenq (2.12) havasarman ndhanowr low owm: (2.12) havasarman bnowagriner en handisanowm x ± t = C , C = const, owiner: (x, t) o oxakanneric ancnenq (ξ, η), ξ = x + t,

η = x − t,

o oxakannerin: Nanakenq u(x, t) = v(ξ, η) = v(x+t, x−t): Nor o oxakanneri hamakargowm (2.12) havasarowm kndowni vξη = 0

tesq: Gtnenq ays havasarman ndhanowr low owm: Aknhayt , or havasarman cankaca low man hamar

(2.15) (2.15)

vη (ξ, η) = f ∗ (η),

orte f ∗(η) fownkcian kaxva miayn η o oxakanic: Yowraqanyowr ξ -i hamar integrelov ays havasarowm st η o oxakani` kstananq v(ξ, η) =

f ∗ (η) dη + f1 (ξ) = f1 (ξ) + f2 (η),

(2.16)

orte f1 f2 miayn mek o oxakanic, hamapatasxanabar ξ -ic η -ic, kaxva fownkcianer en: Tei owni na haka ak` kamayakan f2 erkow angam anndhat diferenceli fownkcianeri hamar (2.16) f1 bana ov orova v(ξ, η) fownkcian (2.15) havasarman low owm : Qani or (2.15) havasarman cankaca low owm kareli nerkayacnel (2.16) tesqov` hamapatasxan f1 f2 ntrowyamb, apa (2.16) bana ov trvowm (2.15) havasarman ndhanowr low owm: Het abar, u(x, t) = f1 (x + t) + f2 (x − t)

(2.17)

fownkcian (2.12) havasarman ndhanowr low owmn : Enadrenq (2.12) − (2.14) xndri low owm goyowyown owni: Ayd depqowm ayn nerkayacvowm (2.17) tesqov: Gtnenq aynpisi f1 f2 fownkcianer, or bavararven (2.13), (2.14) skzbnakan paymanner. ut=0 = u(x, 0) = f1 (x) + f2 (x) = ϕ(x),

(2.18)

ut t=0 = ut (x, 0) = f1 (x) − f2 (x) = ψ(x) :

(2.19)

Integrelov erkrord havasarowyown

x0 -ic x

kstananq

x ψ(α) dα + C,

f1 (x) − f2 (x) = x0

orte C = f1 (x0 ) − f2 (x0 ): Ayspisov ownenq, or f1 (x) + f2 (x) = ϕ(x), x f1 (x) − f2 (x) =

ψ(α) dα + C, x0

orteic gtnowm enq f1 (x) = ϕ(x) + f2 (x) = ϕ(x) −

x ψ(α) dα +

C ,

ψ(α) dα −

C :

x x0

Ayspisov, ϕ ψ fownkcianeri mijocov menq oroecinq fownkcianer, oronq teadrelov (2.17)-i mej` kstananq ϕ(x − t) + ϕ(x + t) 1 + u(x, t) =

x+t ψ(α) dα :

(2.20)

x−t

artahaytowyown kovowm Dalamberi bana : Enadrelov, or (2.12) − (2.14) xndri low owm goyowyown owni, menq stacanq Dalamberi bana , inn apacowcowm low man miakowyown: Iroq, ee goyowyown ownenar (2.12) − (2.14) xndri mek ayl low owm, apa ayn knerkayacver (2.20) Dalamberi bana ov khamnkner naxord low man het: Dicowq ϕ fownkcian erkow angam, isk ψ fownkcian mek angam anndhat diferenceli en: Aknhayt , or ayd depqowm (2.20) bana ov nerkayacva fownkcian bavararowm (2.12) havasarman (2.13), (2.14) skzbnakan paymannerin (2.20)

(kareli stowgel anmijakan teadrmamb): Ayspisov, apacowcva

(2.12), (2.13), (2.14)

eorem 2.3.1

Koii xndri low man goyowyown

miakowyown:

Dicowq ϕ ∈ C (R1), ψ ∈ C (R1): Ayd depqowm (2.12), (2.13), (2.14) Koii xndir owni low owm, or trvowm Dalamberi (2.20) bana ov ayd low owm miakn :

Dalamberi bana ic bxowm en Koii xndri low man oro owagrav hatkowyownner: Dicowq

M (x0 , t0 )-

x − t = x0 − t0 N (x0 − t0 , 0) kovowm

u(x0 , t0 )

or fiqsa ket , tanenq ayd ketov ancno

x + t = x0 + t0

bnowagriner, oronq

P (x0 + t0 , 0) keterowm: M N P

bnowagri e ankyown:

areq orovowm

keterowm

u(M ) =

a ancq khaten

e ankyown (N P himqov konoid)

(2.20)

bana i`

ketowm low man

skzbnakan fownkciayi areqnerov

ϕ(N ) + ϕ(P ) 1 +

hatva i vra.

ψ(α) dα :

(2.20 )

Ditarkenq erkow rinak:

rinak 1: aragowyown.

Dicowq skzbnakan pahin bolor keter ownen zroyakan

ψ(x) ≡ 0: Ayd depqowm (2.20) bana u(x, t) =

Lracowci enadrelov, or

x ≤ x1

kam

x ≥ x2 ),

u(x0 , t0 ) =

kndowni het yal tesq`

ϕ(x − t) + ϕ(x + t) :

ϕ(x) = 0

miayn

(x1 , x2 )

mijakayqowm ( ϕ(x)

≡ 0,

kownenanq

ϕ(x0 − t0 ) + ϕ(x0 + t0 ) ,

ϕ(x0 − t0 ) , ϕ(x0 + t0 ) u(x0 , t0 ) = , u(x0 , t0 ) =

u(x0 , t0 ) = 0,

erb

(x0 − t0 , x0 + t0 ) ⊂ (x1 , x2 ),

erb

x0 − t0 ∈ (x1 , x2 ), x0 + t0 ∈ (x2 , +∞),

erb

x0 − t0 ∈ (−∞, x1 ), x0 + t0 ∈ (x1 , x2 ),

erb

(x0 − t0 , x0 + t0 ) ⊂ (−∞, x1 )

erb

kam

(x0 −t0 , x0 +t0 ) ⊂ (x2 , +∞)

kam

x0 −t0 ∈ (−∞, x1 ), x0 +t0 ∈ (x2 , +∞) :

rinak 2: Aym enadrenq, or skzbnakan eowm zroyakan . ϕ(x) ≡ 0, ψ skzbnakan aragowyown zroyic tarber miayn (x1 , x2 ) mijakayqowm: (2.20) bana ic kstananq

u(x0 , t0 ) = u(x0 , t0 ) = u(x0 , t0 ) = u(x0 , t0 ) =

x 0 +t0

ψ(α) dα,

erb

(x0 − t0 , x0 + t0 ) ⊂ (x1 , x2 ),

x0 −t0

x2 ψ(α) dα,

erb

x0 − t0 ∈ (x1 , x2 ), x0 + t0 ∈ (x2 , +∞),

erb

x0 − t0 ∈ (−∞, x1 ), x0 + t0 ∈ (x1 , x2 ),

x0 −t0 x 0 +t0

ψ(α) dα, x1

erb

ψ(α) dα,

x0 − t0 ∈ (−∞, x1 ), x0 + t0 ∈ (x2 , +∞),

u(x0 , t0 ) = 0, erb (x0 −t0 , x0 +t0 ) ⊂ (−∞, x1 ) kam (x0 −t0 , x0 +t0 ) ⊂ (x2 , +∞) :

akerpenq erkow pndowm, oronq anmijapes stacvowm en Dalamberi bana ic orpes het anq:

eorem 2.3.2 Ee ϕ

skzbnakan fownkcianer kent en or x0 keti nkatmamb, apa low owm ayd ketowm zro cankaca t-i hamar` u(x0 , t) ≡ 0: x0 keti nkatmamb ϕ

ψ fownkcianeri kent linel nanakowm

ϕ(x0 − x) = −ϕ(x0 + x),

ψ(x0 − x) = −ψ(x0 + x) :

Hamaayn Dalamberi bana i` ownenq u(x0 , t) =

ϕ(x0 − t) + ϕ(x0 + t) 1 +

eorem 2.3.3 Ee ϕ

x 0 +t

ψ(α) dα = 0 : x0 −t

skzbnakan fownkcianer zowyg en or x0 keti nkatmamb, apa low man masnakan a ancyal ayd ketowm zro cankaca t-i hamar` ux (x0 , t) = 0: ψ

x0 keti nkatmamb ϕ

ψ fownkcianeri zowyg linel nanakowm

ϕ(x0 − x) = ϕ(x0 + x),

ψ(x0 − x) = ψ(x0 + x) :

A ancelov (2.20) bana st x-i

havi a nelov ayn ast, or zowyg

fownkciayi a ancyal kent fownkcia , kstananq ϕ (x0 − t) + ϕ (x0 + t) 1 + ψ(x0 + t) − ψ(x0 − t) = 0 :

ux (x0 , t) =

Low man kayownowyown: Inpes arden cowyc enq tvel, (2.12) havasarman low owm, or bavararowm (2.13)

(2.14) skzbnakan

paymannerin, orovowm miareqoren: Aym apacowcenq, or skzbnakan paymanneri anndhat o oxvelow depqowm low owm

s anndhat

o oxvowm: Tei owni het yal pndowm:

Kamayakan [0, t0] amanakahatva i hamar kamayakan ε towyan hamar goyowyown owni aynpisi δ(ε, t0) > 0, or (2.12) havasarman cankaca erkow u1(x, t) u2(x, t) low owmner t0 amanakahatva i nacqowm iraric ktarberven ε-ic oqr a ov` eorem 2.3.4

|u1 (x, t) − u2 (x, t)| < ε,

u1 t=0 = ϕ1 (x),

0 ≤ t ≤ t0 ,

u2 t=0 = ϕ2 (x),

u1t t=0 = ψ1 (x),

u2t t=0 = ψ2 (x),

skzbnakan fownkcianer iraric tarberven δ-ic oqr a ov` |ϕ1 (x) − ϕ2 (x)| < δ,

|ψ1 (x) − ψ2 (x)| < δ :

Apacowyc: Ays pndman apacowyc anmijapes bxowm Dalamberi bana ic: Ownenq |u1 (x, t) − u2 (x, t)| ≤

ϕ1 (x + t) − ϕ2 (x + t) ϕ1 (x − t) − ϕ2 (x − t) + +

x+t |ψ1 (α) − ψ2 (α)| dα, x−t

orteic stanowm enq |u1 (x, t) − u2 (x, t)| ≤

δ δ 1 + + δ2t ≤ δ(1 + t0 ) : 2 2 2

ε , kownenanq ayn, in pahanjvowm r apacowcel: 1 + t0 Ee maematikakan xndri low owm anndhat kaxva lracowci

Vercnelov δ =

paymanneric (skzbnakan, ezrayin

ayln), apa asowm en, or xndir kayown

: Kasenq, or maematikakan xndir drva ko ekt, ee xndri low owm goyowyown owni, xndir owni miak low owm

xndri low owm anndhat

kaxva naxnakan tvyalneric (kayown ): Ayspisov, menq cowyc tvecinq, or Koii xndir lari tatanman havasarman hamar drva ko ekt: O ko ekt drva xndri rinak kberenq parabolakan

lipsakan

havasarowmnerin nvirva glowxnerowm:

§ 4. Koii xndri low man miakowyown aliqayin havasarman hamar Ays paragrafowm kowsowmnasirenq Koii xndri low man miakowyown: Lemma 2.4.1 Dicowq u ∈ C 2 (Ωx0 ,t0 ), x0 ∈ Rn , t0 > 0, utt − x u = 0, ut=0 = 0,

(x, t) ∈ Ωx0 ,t0 ,

(2.21)

x ∈ Qx0 ,t0 ,

(2.22)

x ∈ Qx0 ,t0 :

(2.23)

ut t=0 = 0,

Ayd depqowm Ωx0 ,t0 -owm u(x, t) ≡ 0:

Apacowyc: Dicowq u(x, t)-n (2.21) − (2.23) xndri low owmn : Vercnenq kamayakan (ξ, τ ) ∈ Ωx ,t ket: Cowyc tanq, or 0 0

u(ξ, τ ) = 0 :

(2.24)

Ditarkenq Ωx ,t konin <zowgahe > (ξ, τ ) gagaov Ωξ,τ enakon. 0 0

Ωξ,τ = {|x − ξ| < τ − t, 0 < t < τ } ⊂ Ωx0 ,t0 :

Aknhayt , or u ∈ C 2 (Ωξ,τ ): Bazmapatkenq (2.21) nowynowyown ut -ov. ut utt − ut

uxi xi = 0,

(x, t) ∈ Ωξ,τ :

i=1

Qani or ut utt =

apa

1 2 u , 2 t t

ut uxi xi = (uxi ut )xi −

1 2 u , 2 xi t

1 2 2 ut + u xi − (uxi ut )xi = 0, t i=1 i=1 n

i = 1, ..., n,

(x, t) ∈ Ωξ,τ :

(2.25)

Hetaga aradranqi parzowyan hamar t o oxakan nanakenq xn+1 -ov, t = xn+1 , (2.25) havasarowyown grenq divA = A1x1 + ... + Anxn + An+1xn+1 = 0

(2.25 )

tesqov, orte A = (A1 , ..., An , An+1 ) vektor owni het yal komponentner` Ai = −uxi uxn+1 , i = 1, ..., n, " # n uxi : uxn+1 + An+1 = i=1

Integrelov kstananq

(2.25 )- Ωξ,τ -ov

kira elov strogradskow bana `

(A, ν) dS = 0, ∂Ωξ,τ

(2.26)

orte ν = (ν1 , ..., νn , νn+1 ) vektor ∂Ωξ,τ -in tarva Ωξ,τ -i nkatmamb artaqin miavor normaln : ∂Ωξ,τ maker owy kazmva erkow ktorneric` ∂Ωξ,τ = S ξ,τ ∪ Qξ,τ : Gtnenq ν vektor ayd maker owyneric yowraqanyowri

vra: Aknhayt , or Qξ,τ -i vra ν = (0, ..., 0, −1): Qani or Sξ,τ maker owy trvowm F (x1 , ..., xn+1 ) ≡

(xi − ξi )2 − (xn+1 − τ )2 = 0

i=1

havasarowmov, apa nra artaqin miavor normal owni het yal tesq` Fx1 , ..., Fxn+1 (x1 − ξ1 , ..., xn − ξn , τ − xn+1 ) = = ν= 1/2 (x1 − ξ1 )2 + ... + (xn − ξn )2 + (τ − xn+1 )2 Fx1 + ... + Fxn+1 x1 − ξ1 xn − ξ n , ..., √ ,√ : (2.27) = √ 2(τ − xn+1 ) 2(τ − xn+1 ) 2 st (2.22)

(2.23) paymanneri` A|xn+1=0 = 0, owsti Qξ,τ -i vra A = 0

(2.26)

havasarowyown kareli grel het yal tesqov` " #% $ n n xi − ξ i uxn+1 + √ uxn+1 + − dS = u xi √ u xi 0= 2(τ − xn+1 ) 2 2 i=1 i=1 Sξ,τ

= √ 2 2

$ n Sξ,τ

i=1

= √ 2 2

u xi −

Sξ,τ i=1

xi − ξi ux u xi − τ − xn+1 n+1

u2xn+1

n (xi − ξi )2 1− (τ − xn+1 )2 i=1

xi − ξi ux τ − xn+1 n+1

Ayste gtagor ecinq ayn ast, or Sξ,τ -i vra (2.28)-ic anmijapes het owm , or Sξ,τ -i vra

n & i=1

dS :

(ux1 , ..., uxn , uxn+1 ) = uxn+1

x1 − ξ 1 xn − ξn , ..., ,1 , τ − xn+1 τ − xn+1

(2.28)

(xi − ξi )2 = (τ − xn+1 )2 :

x1 − ξ 1 , τ − xn+1 xn − ξ n = uxn+1 , τ − xn+1

aysinqn` Sξ,τ -i vra

dS =

ux1 = uxn+1 u xn

orteic, hamaayn (2.27)-i, stanowm enq, or Sξ,τ -i vra √ (ux1 , ..., uxn , uxn+1 ) = uxn+1 2ν :

Vercnenq Sξ,τ konayin maker owyi cankaca

ni, l-ov nanakenq ayd nii owowyan (n + 1)-a ani miavor vektor: Hamaayn verjin havasarowyan` ayd nii erkaynqov √ ∂u = uxn+1 2(ν, l) = 0, ∂l

in nanakowm , or u = u(x1 , ..., xn , xn+1 ) fownkcian Sξ,τ konayin maker owyi nii owowyamb i o oxvowm: Owsti, (2.24) havasarowyown bxowm (2.22) paymanic: Lemman apacowcva :

Apacowcva lemmayic bxowm (2.5) − (2.7) Koii lokalacva xndri low man miakowyown, het abar na miakowyown: Iroq, dicowq u1 (x, t)-

(2.1) − (2.3) Koii xndri low man u2 (x, t)- (2.5)−(2.7) xndri low owmner

en: Ayd depqowm u = u1 (x, t) − u2 (x, t) tarberowyown khandisana utt − x u = 0, ut=0 = 0,

ut t=0 = 0,

(x, t) ∈ ΩQ ,

(2.50 )

x ∈ Q,

(2.60 )

x∈Q:

(2.70 )

xndri low owm: Da nanakowm , or u(x, t) fownkcian ΩQ -owm bavararowm (2.21), (2.22), (2.23)-in:

Hamaayn apacowcva lemmayi` cankaca

(x0 , t0 ) ∈ ΩQ keti depqowm u(x, t) ≡ 0 Ωx0 ,t0 -owm, het abar u1 (x, t) ≡ u2 (x, t) ΩQ -owm: Ayspisov, apacowcva het yal pndowm:

eorem 2.4.1 (2.5)−(2.7) xndir i karo ownenal mekic aveli low owm:

Inpes arden nvel , ays eoremic, masnavorapes, het owm (2.1) − (2.3) Koii xndri low man miakowyown:

eorem 2.4.2 (2.1)−(2.3) xndir i karo ownenal mekic aveli low owm:

§ 5. Aliqayin havasarman hamar Koii xndri low man goyowyown ereq tara akan o oxakanneri depqowm

Ays paragraf nvirva n = 3 depqowm (2.1) aliqayin havasarman hamar Koii xndri low man goyowyan: Nanakenq Aϕ (x, t) =

4πt

ϕ(y) dSy ,

x ∈ R3 , t > 0 :

(2.29)

|y−x|=t

havasarowyown ereq o oxakanneric kaxva yowraqanyowr ϕ(x), x ∈ R3 , anndhat fownkciayi hamapatasxanowyan mej dnowm ors a ani tara owyan {(x, t) : x ∈ R3 , t > 0} kisatara owyownowm oroa

Aϕ fownkcia, nd orowm` (x0 , t0 ) ketowm Aϕ fownkciayi areq kaxva miayn x0 ∈ R3 kentronov t0 > 0 a avov sferayi vra ϕ fownkciayi ndowna areqneric: Da nanakowm , or Ωx ,t koni (x0 , t0 ) gagaowm, owsti Ωx ,t koni cankaca ayl ketowm, Aϕ fownkciayi areq orovowm miayn ayd koni Qx ,t himqi vra ϕ fownkciayi ndowna areqnerov: Het abar, ee ϕ(x) fownkcian trva o e amboj {x ∈ R3 , t = 0} harowyan, ayl or Q ⊂ {x ∈ R3 , t = 0} tirowyi vra, apa Aϕ fownkcian orova Q himqov ΩQ konoidi vra: eorem 2.5.1' Dicowq ϕ ∈ C 3 (Q), ψ ∈ C 2 (Q): Ayd depqowm (2.29)

0 0

u(x, t) =

∂Aϕ (x, t) + Aψ (x, t), ∂t

(x, t) ∈ ΩQ ,

(2.30)

fownkcian utt − (ux1 x1 + ux2 x2 + ux3 x3 ) = 0,

(x, t) ∈ ΩQ ,

(2.31)

ut=0 = ϕ(x),

x ∈ Q,

(2.32)

= ψ(x),

x ∈ Q,

(2.33)

t=0

Koii lokalacva xndri low owm : (2.30) artahaytowyown kovowm Kirxhofi bana :

Min eoremi apacowycin ancnel` nax apacowcenq het yal andak pndowm: Dicowq (x0, t0) kamayakan ket {x ∈ R3, t > 0} kisatara owyownic: Ayd depqowm 1. ee ϕ ∈ C k Qx ,t , k ≥ 0, apa Aϕ(x, t) ∈ C k Ωx ,t , k = 0, 1, 2, ..., 2. ee ϕ ∈ C k Qx ,t , k ≥ 0, apa Aϕ(x, t)t=0 = 0, x ∈ Qx ,t , 3. ee ϕ ∈ C k Qx ,t , k ≥ 1, apa ∂Aϕ∂t(x, t) t=0 = ϕ(x), x ∈ Qx ,t , Lemma 2.5.1:

0 0

0 0 0 0

0 0

4. ee ϕ ∈ C

5. ee ϕ ∈ C k

0 0

Qx0 ,t0 , k ≥ 2,

apa

Qx0 ,t0 , k ≥ 2,

apa

Apacowyc: (2.29)-owm

∂ Aϕ (x, t) − Aϕ (x, t) = 0, x ∈ Qx0 ,t0 , ∂t2 ∂ Aϕ (x, t) = 0, x ∈ Qx0 ,t0 : ∂t2 t=0

katarenq y = x + ηt

o oxakani oxarinowm, orte x = (x1, x2, x3) fiqsa ket , t-n fiqsa

drakan iv : st koordinatneri` ays o oxakani oxarinowm owni het yal tesq` y1 = x1 + η1 t, y2 = x2 + η2 t, y 3 = x 3 + η3 t :

Ayd depqowm Aϕ (x, t) = =

t 4π

Lemma 2.5.1-i 1.

4πt

ϕ(y1 , y2 , y3 ) dSy = |y−x|=t

ϕ(x1 + η1 t, x2 + η2 t, x3 + η3 t) dSη : |η|=1

(2.34)

2. pndowmner het owm en (2.34) nerkayacowmic {(x, t) ∈ Ωx ,t , |η| = 1} ak bazmowyan vra enaintegralayin fownkciayi anndhatowyownic: 0 0

3. pndowm apacowcelow hamar havenq Aϕ (x, t) fownkciayi a ajin kargi a ancyal st t o oxakani.

∂Aϕ (x, t) = ∂t 4π =

4π

ϕ(x + ηt) dSη + |η|=1

t 4π

ϕ(x + ηt) dSη + |η|=1

= 4π

ϕ(x + ηt) dSη +

|η|=1

∂ϕ(y) ηi dSη = ∂yi y=x+ηt i=1

t 4π

|η|=1

(∇ϕ(y), η)y=x+ηt dSη =

|η|=1

t 4π

∂ϕ(y) dSη , ∂ν y=x+ηt

(2.35)

|η|=1

orte ν = η sferayin tarva artaqin miavor normaln : 3. pndowm anmijapes het owm (2.35) nerkayacowmic, erb t = 0: Aym a oxenq (2.35) havasarowyown (ancnelov y o oxakani), kstananq

∂Aϕ (x, t) = ∂t 4πt2 =

4πt2

4πt

ϕ(y) dSy + |y−x|=t

4πt

∂ϕ(y) dSy = ∂ν

|y−x|=t

ϕ(y) dy : |y−x|≤t

Ayste menq gtvecinq strogradskow bana ic.

∂ϕ(y) dSy = ∂ν

|y−x|=t

(∇ϕ(y), ν) dSy =

div∇ϕ(y) dy = |y−x|≤t

Ayspisov`

∂Aϕ = Aϕ + ∂t t 4πt

ϕ(y) dy : |y−x|≤t

ϕ(y) dSy :

dρ

|y−x|=ρ

A ancenq stacva havasarowyown st t-i.

∂ 2 Aϕ 1 ∂Aϕ = − 2 Aϕ + − ∂t2 t t ∂t 4πt2

dρ

|y−x|=ρ

ϕ(y) dSy +

−

4πt

⎛

|y−x|=t

4πt2

1 ⎜1 ϕ(y) dSy = − 2 Aϕ + ⎝ Aϕ + t t t 4πt

ϕ(y) dSy +

dρ

|y−x|=ρ

4πt

Myows komic ownenq

ϕ(y) dSy = |y−x|=t

4πt

⎟ ϕ(y) dSy ⎠ −

dρ

x Aϕ (x, t) = Aϕ (x, t) =

⎞

|y−x|=ρ

4πt

ϕ(y) dSy : |y−x|=t

ϕ(y) dSy , |y−x|=t

orteic het owm 4. pndowm: 5. pndowm het owm 2. 4. pndowmneric: Ee ϕ ∈ C 2 Qx ,t , apa Aϕ (x, t)t=0 = 0: Lemman apacowcva : ϕ ∈ C Qx ,t Aym apacowcenq eorem: Hamaayn (2.4)-i` bavarar apacowcel, or eorem tei owni, ee Q tirowyi oxaren ditarkenq Qx ,t ⊂ Q ak gownd, isk ΩQ konoidi oxaren` hamapatasxan Ωx ,t ak kon: Anhraet cowyc tal, or ee ϕ ∈ C 3 Qx ,t , ψ ∈ C 2 Qx ,t , apa 0 0

0 0

u∈C

0 0

Ωx0 ,t0 utt − (ux1 x1 + ux2 x2 + ux3 x3 ) = 0, ut=0 = ϕ(x),

ut t=0 = ψ(x),

(x, t) ∈ Ωx0 ,t0 ,

(2.31 )

x ∈ Qx0 ,t0 ,

(2.32 )

x ∈ Qx0 ,t0 :

(2.33 )

fownkciayi patkanelowyown C 2 Ωx ,t dasin het owm Lemma 2.5.1-i pndowm 1.-ic, (2.31 ) havasarowyown` pndowm 4.-ic, (2.32 ) (2.33 ) paymanner` pndowmner 2., 3., 5.-ic: eoremn apacowcva : u(x, t)

0 0

Het anq 2.5.1' Ee ϕ(x), ψ(x)

fownkcianer sahmana ak en Q-owm, apa (2.30) Kirxhofi bana ov trva (2.31) − (2.33) xndri low owm bavararowm het yal anhavasarowyan` |∇ϕ(x)|

|u(x, t)| ≤ Φ0 + t(Φ1 + Ψ0 ),

(x, t) ∈ (ΩQ ∪ Q),

orte Φ0 = sup |ϕ|,

Φ1 = sup |∇ϕ|,

Ψ0 = sup |Ψ| :

Iroq, cankaca (x0 , t0 ) ∈ Q keti hamar (st (2.29), (2.30) (2.35), stanowm enq |u(x0 , t0 )| ≤

4π

sup |ϕ| dSη + Q

|η|=1

t0 4π

(2.4)-i),

t0 4π

sup |∇ϕ| dSη + |η|=1

havi a nelov

sup |ψ| dSη = |η|=1

= Φ0 + t0 (Φ1 + Ψ0 ) :

akerpenq hamapatasxan pndowmner utt − (ux1 x1 + ux2 x2 + ux3 x3 ) = 0,

Ω = R3

masnavor depqowm

x = (x1 , x2 , x3 ) ∈ R3 , t > 0,

(2.36)

ut=0 = ϕ(x),

x ∈ R3 ,

(2.37)

= ψ(x),

x ∈ R3 ,

(2.38)

t=0

Koii xndri hamar:

eorem 2.5.1 Dicowq ϕ ∈ C ⎛

u(x, t) =

∂ ⎜ 1 ⎝ ∂t 4πt

|y−x|=t

(R3 ), ψ ∈ C 2 (R3 ): Ayd depqowm ⎞ ⎟ ϕ(y) dSy ⎠ + ψ(y) dSy 4πt

(2.39)

|y−x|=t

Kirxhofi bana ov trva fownkcian (2.36), (2.37), (2.38) Koii xndri low owm :

Het anq 2.5.1 Ee ϕ(x), ψ(x)

|∇ϕ(x)| fownkcianer sahmana ak

en R3 -owm, apa (2.39) Kirxhofi bana ov trva (2.31) − (2.33) xndri low owm bavararowm het yal anhavasarowyan` |u(x, t)| ≤ Φ0 + t(Φ1 + Ψ0 ),

x ∈ R3 , t ≥ 0,

orte Φ0 = sup |ϕ|, R3

Φ1 = sup |∇ϕ|, R3

Ψ0 = sup |Ψ| : R3

Ditoowyown: eorem 2.5.1-owm enadrvowm

, , minde low man goyowyan hamar anhraet en ϕ ∈ C (R3), ψ ∈ C (R3) paymanner: Nenq miayn, or ays anhraet paymanner bavarar en low man goyowyan hamar, aysinqn` eoremowm nva paymanner akan en. ee skzbnakan pahi paymanner C 2-ic en, apa oro pahi low owm karo patkanel C 2 dasin: ϕ ∈ C 3 (R3 ) ψ ∈ C 2 (R3 )

§ 6. Aliqayin havasarman hamar Koii xndri low man goyowyown erkow

mek tara akan o oxakanneri

depqowm

Naxord paragrafowm n = 3 depqi hamar (2.1) − (2.3) xndri owsowmnasirowyown himnva r Aϕ (x, t) =

4πt

ϕ(y) dSy ,

x = (x1 , x2 , x3 ) ∈ R3 , t > 0,

(2.29)

|y−x|=t

fownkciayi hatkowyownneri vra, oronq akerpva en Lemma 2.5.1-owm: arownakelov ays fownkciayi owsowmnasirowyown, n = 2 n = 1 depqeri hamar Koii xndri low man goyowyown kareli stanal n = 3 depqic <vayrjqi> eanakov: Tei owni het yal pndowm: Lemma 2.6.1 Dicowq ϕ ∈ C(R3): 1. Ee ϕ(x1, x2, x3) ≡ ϕ(x1, x2), aysinqn` ϕ fownkcian kaxva

o oxakanic, apa Aϕ(x1, x2, x3, t) fownkcian nowynpes kaxva

o oxakanic, ayd depqowm Aϕ =

2π

(y1 −x1

)2 +(y

ϕ(y1 , y2 ) dy1 dy2 2 −x2

)2 ≤t2

(t2 − (y1 − x1 )2 − (y2 − x2 )2 )1/2

x3 x3

(2.40)

2. Ee ϕ(x1, x2, x3) ≡ ϕ(x1), aysinqn` ϕ fownkcian kaxva x2 x3

o oxakanneric, apa Aϕ(x1, x2, x3, t) fownkcian nowynpes kaxva

x3 o oxakanneric,

ayd depqowm

Aϕ =

x 1 +t

ϕ(y1 ) dy1 : x1 −t

Apacowyc: (2.29) bana

kareli grel het yal kerp` ⎞ ⎛ 1 ⎝ ϕ(y1 , y2 ) dSy + ϕ(y1 , y2 ) dSy ⎠ , Aϕ (x1 , x2 , x3 , t) = 4πt S+

S−

orte S + = {|y − x| = t} ∩ {y3 ≥ x3 }- {|y − x| = t} sferayi verin kisasferan , isk S − = {|y − x| = t} ∩ {y3 ≤ x3 }-` storin kisasferan: Ays erkow kisasferaneri proyekcianer {y3 = 0} harowyan vra nerkayacnowm en Kt (x1 , x2 ) rjan: Het abar,

⎛

1 ⎜ Aϕ (x1 , x2 , x3 , t) = ⎝ 4πt

Kt (x1 ,x2 )

dy1 dy2 + ϕ(y1 , y2 ) |ν3 |

Kt (x1 ,x2 )

⎞ dy1 dy2 ⎟ ϕ(y1 , y2 ) ⎠, |ν3 |

orte ν3 - hamapatasxan kisasferayi ν = (ν1 , ν2 , ν3 ) artaqin miavor normali errord koordinatn : Qani or {|y − x| = t} sferayi artaqin miavor normaln owni

ν=

y−x t

tesq, apa S +

S − kisasferaneri hamar t2 − (y1 − x1 )2 − (y2 − x2 )2 |y3 − x3 | |ν3 | = = t t

Aϕ (x1 , x2 , x3 , t) =

2π

Kt (x1 ,x2 )

ϕ(y1 , y2 ) dy1 dy2 : t − (y1 − x1 )2 − (y2 − x2 )2

Lemmayi a ajin pndowmn apacowcva : Enadrenq ϕ fownkcian kaxva na

x2 o oxakanic: Ayd depqowm

(2.40)-ic kstananq Aϕ =

2π

Kt (x1 ,x2 )

ϕ(y1 , y2 ) dy1 dy2 = t − (y1 − x1 )2 − (y2 − x2 )2

2π

⎛

x 1 +t

⎜ ϕ(y1 ) ⎜ ⎝

x1 −t

2π

x2 +

x2 −

⎞

√2 t −(y1 −x1 )2

dy2

− (y1 − x1

√2

t −(y1 −x1 )2

x 1 +t

y2 − x2

ϕ(y1 ) arcsin

t2 − (y1 − x1 )

x1 −t

2π

− (y2 − x2

⎟ ⎟ dy1 =

)2 ⎠

y2 =x2 +√t2 −(y1 −x1 )2 dy1 = √2 y2 =x2 −

t −(y1 −x1 )

x 1 +t

ϕ(y1 ) dy1 : x1 −t

Lemman ambojowyamb apacowcva :

n = 3 depqowm (2.39) bana talis (2.36)−(2.38) Koii xndri low owm, erb ϕ ∈ C 3 (R3 ), ψ ∈ C 2 (R3 ): Masnavor depqowm, erb ϕ kaxva en x3 kam x3

ψ fownkcianer

x2 o oxakanneric, ayd bana noric talis

(2.36) − (2.38) xndri low owm: Bayc ayd depqowm, hamaayn Lemma 2.6.1-i, ayd low owmner nowynpes kaxva en hamapatasxanabar x3 kam x3

o oxakanneric, aysinqn` (2.1) − (2.3) xndri low owm en (f (x, t) ≡ 0), erb

n=2

n = 1:

Ayspisov, n = 2 depqowm

utt − (ux1 x1 + ux2 x2 ) = 0,

x ∈ R2 ,

(2.42)

= ψ(x),

x ∈ R2 ,

(2.43)

t=0

Koii xndri low owm trvowm ⎛ ∂ ⎜1 u(x, t) = ⎝ ∂t 2π

|y−x|≤t

2π

(2.41)

ut=0 = ϕ(x),

x = (x1 , x2 ) ∈ R2 , t > 0,

|y−x|≤t

⎞ ϕ(y1 , y2 ) dy1 dy2 ⎟ ⎠+ t2 − (y1 − x1 )2 − (y2 − x2 )2

ψ(y1 , y2 ) dy1 dy2 t2 − (y1 − x1 )2 − (y2 − x2 )2

bana ov: (2.44) artahaytowyown kovowm Powasoni bana :

(2.44)

n = 1 depqowm, inpes arden gitenq, Koii xndri low owm nerkayacvowm

(2.20) Dalamberi bana ov, owaki ays paragrafowm menq ayn stacanq mek ayl` <vayrjqi> eanakov: Owadrowyown darnenq mek hangamanqi vra: n = 2

n = 1 depqerowm

low owmner stacanq` gtvelov n = 3 depqowm low man goyowyownic, erb enadrvowm r ϕ ∈ C 3 (R3 ), ψ ∈ C 2 (R3 ): Het abar, stacva bana er tei ownen, erb ϕ ∈ C 3 (R2 ), ψ ∈ C 2 (R2 )

ϕ ∈ C 3 (R1 ), ψ ∈ C 2 (R1 ) hamapatas-

xanabar: Sakayn, inpes arden cowyc enq tvel, n = 1 depqowm oorkowyan payman kareli owlacnel, bavarar enadrel ϕ ∈ C 2 (R1 ), ψ ∈ C 1 (R1 ): Nenq, or utt − a2 x u = 0,

x ∈ R3 , t > t 0 ,

ut=t0 = ϕ(x),

x ∈ R3 ,

ut t=t0 = ψ(x),

x ∈ R3 ,

Koii xndir, orte a > 0 hastatown , t0 ∈ R1 , τ = a(t − t0 ) o oxakani

oxarinmamb bervowm ver owsowmnasirva xndrin ⎞ ⎛ ∂ ⎜ ⎟ ϕ(y) dSy ⎠ + u(x, t) = ⎝ ∂t 4πa2 (t − t0 ) |y−x|=a(t−t0 )

4πa2 (t − t0 )

ψ(y) dSy

(2.45)

|y−x|=a(t−t0 )

fownkcian ays xndri low owm :

§ 7. Aliqneri difowziayi masin Ays paragrafowm khamozvenq, or Koii xndri low owmnern ownen tarber hatkowyownner, oronq kaxva en tara owyan n a oakanowyownic: Ditarkenq utt − a2 x u = 0,

x ∈ Rn , t > 0,

(2.46)

ut=0 = 0,

ut t=0 = ψ(x),

x ∈ Rn ,

(2.47)

x ∈ Rn ,

(2.48)

Koii xndir, orte ψ(x) ∈ C 2 (Rn ), a > 0 hastatown : Enadrenq goyowyown owni aynpisi R > 0, or ψ(x) > 0, erb ψ(x) ≡ 0, erb |x| ≥ R: Menq kditarkenq n = 3 n = 2 depqer: n = 3 depqowm (2.46) − (2.48) xndri low owm trvowm u(x, t) =

4πa2 t

ψ(y) dSy , x ∈ R3 , t > 0, |y−x|=at

Kirxhofi bana ov, oric het owm , or cankaca fiqsa x ∈ R3 , keti hamar u(x, t) > 0, erb at ∈ (|x| − R, |x| + R), u(x, t) = 0,

erb

|x| < R,

at ≤ |x| − R

kam

|x| > R,

at ≥ |x| + R :

Ayl kerp asa , ee |x| > R, apa at − R < |x| < at + R

gndayin ertowm u(x, t) > 0 u(x, t) = 0 ayd ertic dowrs: Da nanakowm , or skzbnakan pahin ψ(x) fownkciayi a ajacra azdecowyown amanaki nacqowm tara vowm het yal kerp. a ajanowm gndayin aliq |x| = at+R a ajnayin akatov |x| = at − R hetin akatov, oronq arvowm en d(at ± R) = a aragowyamb: Ayd gndayin aliqi laynowyown 2R : Cankaca

dt x, |x| > R, ketowm u(x, t) = 0 (dadari viak ) qani de at ≤ |x| − R (min a ajnayin akat khasni ayd ketin), aynowhet |x| − R < at < |x| + R amanakahatva owm u(x, t) > 0 (tatanvo viak ) heto krkin u(x, t) = 0 (dadari viak ), erb at ≥ |x| + R (erb aliqi hetin akat arden ancel x ketov): A ajnayin akati a kayowyan hatkowyown bnoro hiperbolakan havasarowmneri hamar Koii xndrin, in paymanavorva nman xndrov

nkaragrvo er owynerowm azdecowyan tara man verjavor aragowyamb: Sakayn hetin akati a kayowyown, ori ancnelowc heto ketowm verakangnvowm dadari viak, e aa xndri a annahatkowyown : n = 2 depqowm nman er owy ka: Iroq, n = 2 depqowm (2.46)−(2.48) xndri

low owm trvowm u(x, t) =

2πa

|y−x|≤at

ψ(y1 , y2 ) dy1 dy2 , x ∈ R2 , t > 0, t − (y1 − x1 )2 − (y2 − x2 )2

Powasoni bana ov: Cankaca x ∈ R2 , |x| > R, keti hamar u(x, t) = 0,

erb

at ≤ |x| − R,

u(x, t) > 0,

erb

at > |x| − R :

Ayspisov, u(x, t) > 0, erb |x| < at + R

u(x, t) = 0, erb |x| ≥ at + R:

Da nanakowm , or skzbnakan pahin ψ(x) fownkciayi a ajacra

azdecowyown amanaki nacqowm tara vowm het yal kerp. x, |x| > R, |x| − R pah, ketowm u(x, t) = 0 dadari viak arownakvowm min t = a erb ayd ketov ancnowm a aragowyamb arvo |x| = at + R a ajnayin |x| − R paherin u(x, t) > 0 akat, amanaki bolor hajord t > a x ketowm ayl s dadar i lini: Sakayn dvar nkatel, or ayd ketowm u(x, t) → 0, erb t → ∞, in nanakowm , or keti eowm anverj oqr ,

erb t → ∞: Ayd kapakcowyamb asowm en, or n = 2 depqowm tei ownenowm aliqi hetin akati difowzia (hetin akat bacakayowm ):

Dyowameli skzbownq:

Aym owsowmnasirenq Koii xndir anhamase

aliqayin havasarman hamar: Aknhayt , or bavarar ditarkel hamase

skzbnakan paymannerov depq` utt − a2 x u = f (x, t), ut=0 = 0,

x ∈ Rn , t > 0,

x ∈ Rn ,

(2.49) (2.50)

ut t=0 = 0,

x ∈ Rn :

(2.51)

Ayd npatakov ditarkenq vtt − a2 x v = 0,

x ∈ Rn , t > τ ≥ 0,

v t=τ = 0,

x ∈ Rn ,

= f (x, τ ),

t=τ

(2.52) (2.53)

x ∈ Rn ,

(2.54)

xndir, ori low owm kaxva x, t o oxakanneric τ parametric` v = v(x, t, τ ): Tei owni het yal pndowm, or kovowm Dyowameli skzbownq: eorem 2.7.1 (Dyowameli skzbownq) Dicowq v(x, t, τ ) fownkcian (2.52), (2.53), (2.54) xndri low owm : Ayd depqowm t u(x, t) =

v(x, t, τ ) dτ

(2.55)

fownkcian (2.49), (2.50), (2.51) xndri low owm : Apacowyc: A ancelov (2.55) bana erkow angam st t o oxakani, havi a nelov (2.53), (2.54) paymanner` stanowm enq t ut = v(t, x, t) +

t vt (x, t, τ ) dτ =

vt (x, t, τ ) dτ,

t utt = vt (t, x, t) +

(2.56)

t vtt (x, t, τ ) dτ = f (x, t) +

vtt (x, t, τ ) dτ :

Qani or st xi o oxakanneri a ancman gor oowyown kareli tanel integrali nani tak, apa t x u = x

t v(x, t, τ ) dτ =

x v(x, t, τ ) dτ :

Het abar, utt − a2 x u = f (x, t) +

vtt (x, t, τ ) dτ − a2

t x v(x, t, τ ) dτ =

t = f (x, t) +

vtt (x, t, τ ) − a2 x v(x, t, τ ) dτ = f (x, t) :

Ayspisov, u fownkcian bavararowm (2.49) havasarman: (2.55) (2.56) nerkayacowmneric anmijapes het owm , or u fownkcian bavararowm (2.50) (2.51) paymannerin: eoremn apacowcva : Ev ayspes, menq karo enq grel utt − a2 x u = f (x, t), u

x ∈ Rn , t > 0,

= ϕ(x),

x ∈ Rn ,

ut t=0 = ψ(x),

x ∈ Rn ,

t=0

Koii xndri low man tesq, ee ownenq hamase havasarman hamar Koii xndri low owm: Grenq ayd low owmneri tesq mez haytni depqeri hamar: Enadrvowm , or f (x, τ ) fownkcian yowraqanyowr τ ≥ 0 hamar, orpes x o oxakanic kaxva fownkcia, patkanowm hamapatasxan dasin ( n = 1 depqowm C 1 dasin, n = 2, 3 depqerowm C 2 dasin): n=1

(Dalamberi bana ).

ϕ(x − at) + ϕ(x + at) + u(x, t) = 2a

x+at

x−at

ψ(ξ) dξ + 2a

⎛ ⎝kam, n=2

tes (2.20 ), u(M ) = ϕ(N ) +2 ϕ(P ) + 21

x+a(t−τ )

f (ξ, τ ) dξ

dτ

x−a(t−τ )

ψ(α) dα +

⎞ f (x, t) dxdt⎠ ,

MNP

(Powasoni bana ).

⎛ u(x, t) =

∂ ⎜ 1 ⎝ ∂t 2πa

⎞

ϕ(y1 , y2 ) dy1 dy2

|y−x|≤at

a2 t2

− (y1 − x1

− (y2 − x2

⎟ ⎠+

+ n=3

2πa

a2 t2 − (y1 − x1 )2 − (y2 − x2 )2

|y−x|≤at

t dτ

ψ(y1 , y2 ) dy1 dy2

|y−x|≤a(t−τ )

f (y1 , y2 , τ ) dy1 dy2 , a (t − τ )2 − (y1 − x1 )2 − (y2 − x2 )2

(Kirxhofi bana ).

⎛ u(x, t) =

∂ ⎜ 1 ⎝ ∂t 4πa2 t t +

dτ

⎞

⎟ ϕ(y) dSy ⎠ +

|y−x|=at

4πa2 (t − τ )

4πa2 t

ψ(y) dSy + |y−x|=at

f (y, τ ) dSy : |y−x|=a(t−τ )

§ 8. Xa xndir hiperbolakan havasarman hamar

Kisaanverj lari tatanowmner: Ditarkenq het yal xndir. gtnel utt − a2 uxx = 0,

0 < x < ∞, 0 < t < ∞,

havasarman ayn low owm, or bavararowm u(0, t) = μ(t)

kam ux (0, t) = ν(t)

(a > 0)

t ≥ 0,

ezrayin u(x, 0) = ϕ(x),

ut (x, 0) = ψ(x),

0 ≤ x < ∞,

skzbnakan paymannerin: Ays xndir kovowm xa xndir lari tatanman havasarman hamar: Parzowyan hamar kditarkenq u(0, t) = 0 hamase ezrayin payman, in hamapatasxanowm ayn depqin, erb lari ayr amracva , inpes na ux (0, t) = 0 hamase ezrayin payman, in hamapatasxanowm ayn depqin, erb lari ayr azat : Nax ditarkenq amracva

ayrov kisaanverj lari tatanowmner` u(0, t) = 0: Tara enq xndir amboj a ancqi vra` arownakelov skzbnakan fownkcianer bacasakan kisaa ancqi vra kent ov: Nanakenq

ϕ(x) =

ψ(x) =

⎧ ⎪ ⎪ ⎪ ⎨ϕ(x), x ≥ 0, ⎪ ⎪ ⎪ ⎩−ϕ(−x), x < 0, ⎧ ⎪ ⎪ ⎪ ⎨ψ(x), x ≥ 0, ⎪ ⎪ ⎪ ⎩−ψ(−x), x < 0 :

Aknhayt , or anverj lari tatanowm nkaragro ϕ(x − at) + ϕ(x + at) u (x, t) = + 2a

x+at

dξ ψ(ξ)

x−at

fownkcian {x > 0, t > 0} tirowyowm kisaanverj lari hamar akerpva

xndri low owm . ayd fownkcian bavararowm havasarman, skzbnakan paymannerin u(0, t) = 0 ezrayin paymanin (eorem 2.3.2): Verada nalov naxnakan xndri ϕ(x) ψ(x) skzbnakan fownkcianerin` low owm knerkayacvi het yal ov. x ezri a kayowyown de s i drs orvowm), x − at > 0 tirowyowm (t < a kownenanq ϕ(x − at) + ϕ(x + at) + u(x, t) = 2a

x+at

ψ(ξ) dξ, x−at

tirowyowm (t > xa arden a ka ezri azdecowyown), havi a nelov simetrik sahmannerowm kent fownkciayi integrali zro linel, kownenanq

x − at < 0

ϕ(x + at) − ϕ(at − x) + u(x, t) = 2a

x+at

ψ(ξ) dξ : at−x

Aym ditarkenq azat ayrov kisaanverj lari tatanowmner` ux (0, t) = 0: Nowyn kerp, xndir tara enq amboj a ancqi vra, bayc aym skzbnakan

fownkcianer zowyg ov arownakelov bacasakan kisaa ancqi vra: Nanakenq ⎧ ϕ(x) =

ψ(x) =

⎪ ⎪ ⎪ ⎨ϕ(x), x ≥ 0

⎪ ⎪ ⎪ ⎩ϕ(−x), x < 0, ⎧ ⎪ ⎪ ⎪ ⎨ψ(x), x ≥ 0 ⎪ ⎪ ⎪ ⎩ψ(−x), x < 0 :

Ays depqowm s aknhayt , or anverj lari tatanowm nkaragro u (x, t) =

ϕ(x − at) + ϕ(x + at) + 2a

x+at

dξ ψ(ξ)

x−at

fownkcian {x > 0, t > 0} tirowyowm kisaanverj lari hamar akerpva

xndri low owm , sakayn i tarberowyown kent arownakowyan depqi` ays depqowm bavararowm ux (0, t) = 0 ezrayin paymanin (eorem 2.3.3): Verada nalov naxnakan xndri ϕ(x) ψ(x) skzbnakan fownkcianerin` low owm knerkayacvi het yal ov. x ezri a kayowyown de s i drs orvowm, inpes naxord depqowm, erb t < a kownenanq u(x, t) =

ϕ(x − at) + ϕ(x + at) + 2a

x+at

ψ(ξ) dξ, x−at

x t > a

arden a ka ezri azdecowyown, havi a nelov simetrik erb sahmannerowm zowyg fownkciayi integrali hatkowyown, kownenanq ⎛ at−x ⎞ x+at ϕ(x + at) + ϕ(at − x) 1 ⎝ u(x, t) = + ψ(ξ) dξ + ψ(ξ) dξ ⎠ : 2a

Verjavor lari tatanowmner: Ditarkenq het yal xndir. gtnel utt − a2 uxx = 0,

0 < x < l, 0 < t < ∞,

(a > 0)

(2.57)

havasarman low owm, or bavararowm u(x, 0) = ϕ(x),

ut (x, 0) = ψ(x),

0 ≤ x ≤ l,

(2.58)

skzbnakan paymannerin u(0, t) = u(l, t) = 0,

t ≥ 0,

(2.59)

ezrayin paymannerin, in hamapatasxanowm ayn depqin, erb lari

ayrer amracva en: Inpes

naxord depqerowm, xndir tara enq amboj a ancqi

vra: Skzbnakan fownkcianer arownakenq` arownakva fownkcianer nanakelov ϕ, ψ, het yal ov. nax kent ov arownakenq [0, l]-ic [−l, l], ϕ(x) = ϕ(x),

0 ≤ x ≤ l,

ψ(x) = ψ(x),

0 ≤ x ≤ l,

ϕ(x) = −ϕ(−x),

−l ≤ x ≤ 0,

= −ψ(−x), ψ(x)

−l ≤ x ≤ 0,

aynowhet arownakenq 2l parberowyamb amboj a ancqi vra`

ϕ(x ± 2lk) = ϕ(x),

−l ≤ x ≤ l,

k = ± 1, 2, ...

± 2lk) = ψ(x), ψ(x

−l ≤ x ≤ l,

k = ± 1, 2, ... :

Aknhayt , or anverj lari tatanowm nkaragro u (x, t) =

ϕ(x − at) + ϕ(x + at) + 2a

x+at

dξ ψ(ξ)

x−at

fownkcian (2.57), (2.58), (2.59) xndri low owm ((2.59) ezrayin payman bavararva ` hamaayn eorem 2.3.2-i): Stacva bana owm naxnakan

xndri ϕ(x) ψ(x) skzbnakan fownkcianerin verada nalow hamar, inpes naxord depqerowm, anhraet {0 < x < l, t > 0} tirowyowm ϕ ψ fownkcianeri areqner artahaytel ϕ(x) ψ(x) fownkcianeri areqneri mijocov, in onowm enq nercoin: § 9. o oxakanneri anjatman meod

o oxakanneri anjatman kam Fowryei meod masnakan a ancyalnerov diferencial havasarowmneri low man himnakan meodneric , or karadrenq nax (2.57), (2.58), (2.59) amracva

ayrerov lari tatanman xndri hamar: Qani or (2.57) havasarowm g ayin hamase , owsti erkow masnavor low owmneri gowmar s ayd havasarman low owm : orenq gtnel (2.57) havasarman aynpisi masnavor low owmner, oronc gowmar klini (2.57), (2.58), (2.59) xndri low owm: Nax low enq het yal andak xndir. gtnel (2.57) havasarman ayn o trivial (o nowynabar zro) low owmner, oronq bavararowm en (2.59) ezrayin paymannerin ownen Hamase havasarowmner:

u(x, t) = X(x)T (t)

(2.60)

tesq, orte X(x) fownkcian kaxva miayn x o oxakanic, T (t) fownkcian kaxva miayn t o oxakanic: Teadrelov (2.60) tesqi u(x, t) fownkcian (2.57) havasarman mej` stanowm enq

X(x)T (t) − a2 X (x)T (t) = 0,

orteic, havi a nelov X(x) ≡ 0, T (t) ≡ 0, stanowm enq

1 T (t) X (x) = 2 : X(x) a T (t)

(2.61)

Orpeszi (2.60) fownkcian lini (2.57) havasarman low owm, anhraet , or (2.61) havasarowyown tei ownena {(x, t) : 0 < x < l, t > 0} tirowyi

bolor keteri hamar: (2.61) havasarowyan ax mas kaxva miayn x-ic, isk aj mas` miayn t-ic: Fiqselov x o oxakani or areq

o oxelov

t o oxakan (kam haka ak)` stanowm enq, or (2.61) havasarowyan

ax maser nowynabar havasar en mi nowyn hastatownin: Ayd

hastatown nanakenq −λ-ov.

1 T (t) X (x) = 2 = −λ : X(x) a T (t)

Aysteic X(x)

T (t) fownkcianeri hamar stanowm enq

X (x) + λX(x) = 0,

X(x) ≡ 0,

T (t) + λa2 T (t) = 0,

T (t) ≡ 0,

0 < x < l,

(2.62)

t > 0,

(2.63)

sovorakan diferencial havasarowmner: (2.59) ezrayin paymanneric ownenq u(0, t) = X(0)T (t) = 0, u(l, t) = X(l)T (t) = 0,

orteic het owm , or X(0) = X(l) = 0 :

(2.64)

Ayspisov X(x) fownkciayi hamar stacanq het yal xndir` gtnel λ hastatowni ayn areqner, oronc hamar

X (x) + λX(x) = 0,

0 < x < l,

X(0) = X(l) = 0,

(2.65)

xndirn owni o zroyakan low owm: Aydpisi λ hastatownner kovowm en (2.65) xndri se akan areqner, isk hamapatasxan low owmner` se akan fownkcianer: Ays xndirn anvanowm en na xndir: Ditarkenq λ < 0, λ = 0

λ > 0 hnaravor depqer:

towrm-Liowvili

1. λ < 0 depqowm (2.65) xndir owni o trivial low owm: Iroq, (2.62) havasarman ndhanowr low owmn owni

X(x) = C1 e

√

−λx

+ C2 e −

√

−λx

tesq: (2.64) ezrayin paymanneric ownenq

⎧ ⎪ ⎪ ⎪ ⎨X(0) = C1 + C2 = 0, ⎪ √ √ ⎪ ⎪ ⎩X(l) = C1 e −λl + C2 e− −λl = 0, orteic C1 = C2 = 0

X(x) ≡ 0:

2. λ = 0 depqowm (2.65) xndir krkin owni o trivial low owm: Iroq, ays depqowm (2.62) havasarman ndhanowr low owmn

X(x) = C1 x + C2 : (2.64) ezrayin paymanneric ownenq ⎧ ⎪ ⎪ ⎪ ⎨X(0) = C2 = 0, ⎪ ⎪ ⎪ ⎩X(l) = C1 l + C2 = 0, orteic C1 = C2 = 0

X(x) ≡ 0:

3. λ > 0 depqowm (2.62) havasarman ndhanowr low owmn

X(x) = C1 cos

√

λx + C2 sin

√

λx :

(2.64) ezrayin paymanneric ownenq ⎧ ⎪ ⎪ ⎪ ⎨X(0) = C1 = 0, ⎪ √ √ ⎪ ⎪ ⎩X(l) = C1 cos λl + C2 sin λl = 0,

orteic C1 = 0

C2 sin

√

λl = 0: Ee X(x) ≡ 0, apa C2 = 0

aysinqn`

√

λ=

sin

√

λl = 0,

πn , l

orte n = 1, 2, ...: Het abar, (2.65) xndir karo ownenal o trivial low owmner miayn λ = λn =

πn 2 l

n = 1, 2, ...

areqneri (se akan areqneri) depqowm: Yowraqanyowr λn -in hamapatasxanowm Xn (x) = Dn sin

πn x l

low owm (se akan fownkcian), orte Dn - kamayakan hastatown : λn -in hamapatasxano (2.63) havasarman ndhanowr low owmn owni Tn (t) = An cos

tesq, orte An -

πan πan t + Bn sin t l l

Bn - kamayakan hastatownner en:

Ayspisov, πan πan πn t + Bn sin t sin x, un (x, t) = Xn (x)Tn (t) = An cos l l l

fownkcianer

(2.57)

havasarman

masnavor

bavararowm en (2.59) ezrayin paymannerin

low owmner

n = 1, 2, ...

en,

oronq

nerkayacvowm en erkow

fownkcianeri artadryali tesqov, oroncic mek kaxva miayn x-ic, myows` miayn t-ic: Ays low owmner karo en bavararel naxnakan xndri (2.58) skzbnakan paymannerin miayn masnavor ϕ

ψ fownkcianeri

hamar: Aym verada nanq (2.57), (2.58), (2.59) ndhanowr xndrin: Enadrenq An , Bn gor akicner aynpisin en, or u(x, t) ≡

∞ n=1

un (x, t) =

∞ n=1

An cos

πan πan πn t + Bn sin t sin x l l l

(2.66)

arq, inpes na

ayn arqer, oronq stacvowm en ays arq erkow angam

andam a andam a ancelis st zowgamet en Parz

{0 ≤ x ≤ l, t ≥ 0}

paymannerin,

u(x, t) aynpes

paymanneric gtnenq

An , Bn

o oxakanneri, havasaraa

bazmowyan vra:

fownkcian l

(2.57)

kbavarari

∞

un (x, 0) =

∞

n=1

An sin

n=1

∞ ∂un

∂t

n=1

(2.58)

havasarman:

ezrayin

skzbnakan

gor akicner: Ownenq

u(x, 0) = ϕ(x) =

ut (x, 0) = ψ(x) =

(2.59)

inpes

(x, 0) =

∞ πn n=1

πn x, l

aBn sin

(2.67)

πn x: l

(2.68)

fownkcianer kareli nerkayacnel Fowryei arqov, apa ∞

ϕn = l

πn ϕn sin x, ϕ(x) = l n=1 ∞

ψn = l

πn x, ψn sin ψ(x) = l n=1 (2.69), (2.70)

(2.67), (2.68)

arqeri

l ϕ(ξ) sin

πn ξ dξ, l

(2.69)

ψ(ξ) sin

πn ξ dξ : l

(2.70)

bana eri hamematowyownic er owm ,

or skzbnakan paymanneri bavararman hamar anhraet vercnel

An = ϕ n , ayd depqowm

(2.66)

(2.57), (2.58), (2.59)

Bn =

ψn , πna

arqov nerkayacva

u(x, t)

(2.71) fownkcian khandisana

xndri low owm:

Ayspisov, menq low owm nerkayacrinq

(2.66)

arqi tesqov: Ee

(2.66)

arq taramitowm kam ayd arqov nerkayacva fownkcian diferenceli , apa, iharke, ayn i karo linel Aym

tesnenq,

fownkcianer, orpeszi arq

(2.57) diferencial havasarman low owm:

paymanneri

(2.71)

petq

bavararen

bana ov orova gor akicnerov

(2.66)

ayn arqer, oronq stacvowm en ayd arq erkow angam andam

a andam a ancelis st x t o oxakanneri, linen havasaraa

zowgamet {0 ≤ x ≤ l, t ≥ 0}-owm: Qani or |un (x, t)| ≤ |An | + |Bn |,

apa

∞

(|An | + |Bn |)

(2.72)

n=1

arq (2.66) arqi maorant (2.72) arqi zowgamitowyownic het owm (2.66) arqi havasaraa zowgamitowyown: ut (x, t) fownkciayi anndhatowyown stanalow npatakov owsowmnasirenq ut (x, t) ∼

∞ ∂un

∂t

n=1

∞ πan n=1

−An sin

πn πan πan t + Bn cos t sin x l l l

(2.73)

arqi havasaraa zowgamitowyown: (2.73) arqi hamar maorant πa n (|An | + |Bn |) l n=1 ∞

(2.74)

arq: Ev verjapes, utt(x, t) uxx(x, t) fownkcianeri anndhatowyown stanalow npatakov owsowmnasirenq utt (x, t) ∼

∞ ∂ 2 un n=1

=−

∂t2

πan πan πn n2 An cos t + Bn sin t sin x, l l l n=1

∞ πa 2

uxx (x, t) ∼

∞ ∂ 2 un n=1

=−

∞ π 2

∂x2

(2.75)

πan πan πn n2 An cos t + Bn sin t sin x l l l n=1

(2.76)

arqeri havasaraa zowgamitowyown: Ays arqeri hamar hastatown bazmapatkii towyamb maorant arq handisanowm

∞

n2 (|An | + |Bn |)

(2.77)

n=1

arq: Aynowhet , havi a nelov

(2.72), (2.73), (2.77)

An , Bn

gor akicneri

(2.71) nerkayacowm,

arqeri zowgamitowyan xndir bervowm ∞

nk |ϕn |,

k = 0, 1, 2,

(2.78)

k = −1, 0, 1,

(2.79)

n=1 ∞

nk |ψn |,

n=1

arqeri zowgamitowyan xndrin: Hamaayn Fowryei arqeri haytni hatkowyownneri ∞

nk |ϕn |,

k = 0, 1, 2,

n=1

arqeri zowgamitowyan hamar bavarar enadrel, or

ϕ ∈ C 2 [0, l]

cian owni errord kargi ktor a ktor anndhat a ancyal

ϕ(0) = ϕ(l) = 0, payman, isk

∞

nk |ψn |,

ϕ (0) = ϕ (l) = 0,

fownk-

tei owni

(2.80)

k = −1, 0, 1,

n=1

arqeri zowgamitowyan hamar bavarar enadrel, or

ψ ∈ C 1 [0, l] fownk-

cian owni erkrord kargi ktor a ktor anndhat a ancyal

ψ(0) = ψ(l) = 0

tei owni

(2.81)

payman: Ayspisov, apacowcvec het yal eorem:

eorem 2.9.1

Dicowq ϕ ∈ C 2[0, l] fownkcian owni errord kargi ktor a ktor ahndhat a ancyal, ψ ∈ C 1[0, l] fownkcian owni erkrord kargi ktor a ktor ahndhat a ancyal tei ownen (2.80), (2.81) paymanner: Ayd depqowm (2.66) arqov nerkayacva u(x, t)

fownkcian, orte An , Bn gor akicner orovowm en (2.71) bana ov, (2.57), (2.58), (2.59)

xndri low owm :

Stacionar anhamase owyamb havasarowmner: Ditarkenq het yal xndir. gtnel utt − a2 uxx = f (x),

0 < x < l, 0 < t < ∞,

(a > 0)

(2.82)

anhamase havasarman low owm, or bavararowm u(x, 0) = ϕ(x),

ut (x, 0) = ψ(x),

0 ≤ x ≤ l,

(2.83)

skzbnakan paymannerin u(0, t) = u(l, t) = 0,

t ≥ 0,

(2.84)

hamase ezrayin paymannerin: Low owm ntrenq u(x, t) = v(x, t) + w(x)

(2.85)

erkow fownkcianeri gowmari tesqov, oronq bavararowm en (2.84) ezrayin paymanin: w(x) fownkcian kovowm low man stacionar mas: Pahanjenq, or v(x, t) fownkcian bavarari vtt = a2 vxx

hamase havasarman: Hamaayn (2.83) paymani v(x, 0) = u(x, 0) − w(x) = ϕ(x) − w(x),

(2.86)

vt (x, 0) = ut (x, 0) = ψ(x) :

Ayspisov, v(x) fownkciayi hamar stacanq xndir, or kareli low el

o oxakanneri anjatman meodov: Teadrelov (2.85) nerkayacowm (2.82) havasarman mej` w(x) fownkciayi hamar kstananq a2 wxx + f (x) = 0,

havasarowm, or low vowm erkow angam integrelov st x o oxakani` havi a nelov w(0) = w(l) = 0 ezrayin paymanner: Teadrelov w(x) fownkcian (2.86) skzbnakan paymani mej karo enq gtnel v(x) fownkcian:

ndhanowr anhamase owyamb havasarowmner: utt − a2 uxx = f (x, t),

Dicowq trva

0 < x < l, 0 < t < ∞,

(a > 0)

(2.87)

anhamase havasarowm, in hamapatasxanowm ayn depqin, erb azdo artaqin ow o oxvowm amanaki nacqowm: Ditarkenq (2.87), (2.83), (2.84) xndir: Low owm ntrenq u(x, t) =

∞

un (t) sin

n=1

πn x l

(2.88)

tesqov: Nerkayacnenq havasarman f (x, t) aj mas

(2.83) skzbnakan

fownkcianer Fowryei hamapatasxan arqerov. ∞

fn (t) = l

πn f (x, t) = fn (t) sin x, l n=1 ϕ(x) =

∞

ϕn sin

n=1

ψ(x) =

∞

ψn sin

n=1

Teadrelov (2.88)

πn x, l πn x, l

ϕn =

ψn =

l l

f (ξ, t) sin

πn ξ dξ, l

(2.89)

ϕ(ξ) sin

πn ξ dξ, l

(2.90)

ψ(ξ) sin

πn ξ dξ : l

(2.91)

(2.89) artahaytowyownner (2.87) havasarman mej`

stanowm enq ∞

un (t) +

πan 2

n=1

orteic

un (t) +

πan 2 l

πn πn fn (t) sin x= x, l l n=1 ∞

un (t) sin

un (t) = fn (t),

n = 1, 2, ... :

(2.92)

Ayspisov, un (t) fownkcian oroelow hamar stacanq hastatown gor akicnerov sovorakan diferencial havasarowm: (2.83) skzbnakan paymanneric stanowm enq u(x, 0) = ϕ(x) =

∞ n=1

ut (x, 0) = ψ(x) =

orteic

πn πn ϕn sin x= x, l l n=1 ∞

un (0) sin

∞

πn πn ψn sin x= x, l l n=1 ∞

un (0) sin

n=1

un (0) = ϕn ,

un (0) = ψn :

(2.93)

Low elov (2.92), (2.93) xndir kstananq un (t) fownkcian: Anhamase ezrayin paymanner: Ditarkenq a ajin xa xndir lari tatanman havasarman hamar ndhanowr depqowm. utt − a2 uxx = f (x, t), u(x, 0) = ϕ(x), u(0, t) = μ1 (t),

0 < x < l, 0 < t < ∞, ut (x, 0) = ψ(x),

(a > 0)

0 ≤ x ≤ l,

u(l, t) = μ2 (t),

t≥0:

Ays xndir low elow hamar nermow enq nor v(x, t) anhayt fownkcia. u(x, t) = v(x, t) + U (x, t),

orte enadrvowm , or U (x, t) fownkcian haytni : v(x, t) fownkcian petq lini vtt − a2 vxx = f(x, t)

havasarman low owm, orte f(x, t) = f (x, t) − het yal skzbnakan ezrayin paymannerin`

Utt − a2 Uxx

v(x, 0) = ϕ(x) = ϕ(x) − U (x, 0),

bavarari

vt (x, 0) = ψ(x) = ψ(x) − Ut (x, 0), v(0, t) = μ 1 (t) = μ1 (t) − U (0, t), v(l, t) = μ 2 (t) = μ2 (t) − U (l, t) :

ntrenq U (x, t) fownkcian aynpes, or μ 1 (t) = μ 2 (t) = 0 :

Ayd npatakov karo enq vercnel U (x, t) = μ1 (t) +

x μ2 (t) − μ1 (t) : l

Ayspisov, u(x, t) fownkciayi hamar ndhanowr ezrayin paymannerov xndir bervec v(x, t) fownkciayi hamar hamase ezrayin paymannerov xndrin:

Glowx 3 Parabolakan tipi havasarowmner u(x, t), x ∈ Rn , t > 0,

fownkcian kovowm

Lu ≡ ut − x u = f (x, t), ut=0 = ϕ(x), Koii xndri low owm, ee

u-n

x ∈ Rn , t > 0,

(3.1)

x ∈ Rn ,

(3.2)

patkanowm

C 2 (x ∈ Rn , t > 0) ∩ C(x ∈ Rn , t ≥ 0) bazmowyan,

{x ∈ Rn , t > 0}

tirowyowm bavararowm

(3.1)

havasarman

t = 0 depqowm (3.2) paymanin: (3.2) payman kovowm skzbnakan payman, isk

ϕ(x)

fownkcian kovowm skzbnakan fownkcia:

Low man

sahmanowmic

aknhaytoren

low man goyowyan hamar anhraet en

het

owm

(3.1), (3.2)

f ∈ C(x ∈ Rn , t > 0)

xndri

ϕ ∈ C(Rn )

paymanner:

Inpes

het

aliqayin havasarman depqowm, mer owsowmnasirowyan plan

yaln . skzbowm Fowryei

(3.1), (3.2)

a oxowyan mijocov (a anc himnavorman)

xndri low man bana

, aynowhet

xndri maematikakan xist hetazotman:

kandrada nanq

§ 1. Fowryei a oxowyan kira owm jermahaordakanowyan havasarman hamar Koii xndri low owm stanalow hamar

Parzowyan hamar ditarkenq hamase havasarman depq. ut − x u = 0,

x ∈ Rn , t > 0,

(3.10 )

x ∈ Rn :

(3.2)

ut=0 = ϕ(x),

Ev ayspes, dicowq u(x, t) fownkcian (3.10), (3.2) xndri low owm : Bazmapatkenq (3.10) nowynowyan aj ax maser e−i(x,ξ)-ov, orte ξ -n Rn tara owyan kamayakan ket , stacva havasarowyown integrenq Rn -ov: Kstananq u t (ξ, t) + |ξ|2 u (ξ, t) = 0,

orte u(ξ, t)-n t > 0 parametric kaxva

a oxowyownn st x ∈ Rn o oxakani.

u (ξ, t) =

(3. 10 )

x ∈ Rn , t > 0, u(x, t)

fownkciayi Fowryei

u(x, t)e−i(x,ξ) dx :

Nowyn ov, (3.2) skzbnakan paymanic kstananq u t (ξ, t)t=0 = ϕ(ξ) :

(3. 2)

Yowraqanyowr fiqsa ξ ∈ Rn depqowm (3.10), (3.2) xndir u(ξ, t) fownkciayi hamar handisanowm Koii xndir hastatown gor akicnerov sovorakan diferencial havasarman hamar, ori low owmn owni

u (ξ, t) = e−|ξ| t ϕ(ξ),

t > 0, ξ ∈ Rn ,

tesq: Het abar, Fowryei hakadar a oxowyan mijocov, xndri low owm nerkayacvowm u(x, t) =

(2π)n

u (ξ, t)ei(x,ξ) dξ =

(2π)n

(3.10 ), (3.2)

i(x,ξ) e−|ξ| t ϕ(ξ)e dξ =

= = Rn

(2π)n

⎛ ϕ(y) ⎝ (2π)n

tesqov, orte K(z, t) = K(z, t)

(2π)n

e−|ξ| t ei(x,ξ)

ϕ(y)e−i(y,ξ) dy dξ =

Rn −|ξ|2 t i(x−y,ξ)

⎞

dξ ⎠ dy =

K(x − y, t)ϕ(y) dy Rn

ei(z,ξ) e−|ξ| t dξ,

z = (z1 , ..., zn ) ∈ Rn , t > 0 :

fownkciayi hamar tei owni K(z, t) =

n n ' ' eizj ξj e−ξj t dξj = k(zj , t), 2π j=1 j=1 −∞

havasarowyown, orte k(α, t) =

2π

eiαξ e−ξ t dξ = e−ξ t cos αξ dξ, 2π

−∞

α ∈ R1 , t > 0 :

−∞

Havi a nelov, or ∂k =− ∂α 2π

α α ξ sin αξe−ξ t dξ = − cos αξe−ξ t dξ = − k(α, t), 4πt 2t

−∞

kstananq

α2

k(α, t) = e− 4t k(0, t) :

Myows komic ownenq k(0, t) =

het abar

2π

e−ξ t dξ = √ , 2 πt

−∞

α2

e− 4t k(α, t) = √ , 2 πt z2

α ∈ R1 , t > 0,

|z| j n ' e− 4t e− 4t √ = √ , K(z, t) = 2 πt (2 πt)n j=1

z ∈ Rn , t > 0,

owsti

(3.10 ), (3.2) xndri low man hamar karo enq grel K(x − y, t)ϕ(y) dy =

u(x, t) = Rn

√ (2 πt)n

e−

|x−y|2 4t

(3.3) artahaytowyown kovowm

ϕ(y) dy,

x ∈ Rn , t > 0 :

Powasoni bana

(3.3)

§ 2. Fowndamental low owm: Jermahaordakanowyan havasarman hamar Koii xndri low man goyowyown Ditarkenq

⎧ ⎪ ⎪ |x|2 ⎪ ⎪ ⎨ √ n e− 4t , x ∈ Rn , t > 0, U (x, t) = (2 πt) ⎪ ⎪ ⎪ ⎪ ⎩0, x ∈ Rn , t ≤ 0,

fownkcian: {0} = {x = 0, t = 0} ketowm U (x, t) fownkcian xzvowm , isk mnaca bolor keterowm bavararowm jermahaordakanowyan havasarman: Dvar stowgel, or U ∈ C ∞ (Rn+1 \{0}):

fownkcian kovowm jermahaordakanowyan havasarman fowndamental low owm : Sahmanowm:

U (x, t)

Tei owni het yal pndowm:

eorem 3.2.1 Dicowq ϕ ∈ C(Rn ) fownkcian sahmana ak ` sup |ϕ(x)| = M < ∞ :

x∈Rn

Ayd depqowm (3.3) bana ov trva fownkcian (3.10), (3.2) xndri low owm : Ayd low owm sahmana ak ` sup

x∈Rn ,t≥0

|u(x, t)| ≤ M,

(3.4)

avelin, tei ownen het yal anhavasarowyownner` inf ϕ ≤ u(x, t) ≤ sup ϕ, Rn

Apacowyc: Vercnenq kamayakan nanakenq

x ∈ Rn , t ≥ 0 :

R > 0, 0 < δ < T < ∞

(3.4 ) QR,δ,T -ov

QR,δ,T = {|x| < R, δ < t < T }

glan: Nax apacowcenq, or (3.3) bana ov trva u(x, t) fownkcian QR,δ,T -owm bavararowm (3.10 ) patkanowm C 2(QR,δ,T ) dasin havasarman: (3.3) integral nerkayacnenq u(x, t) = I1 (x, t) + I2 (x, t)

(3.5)

erkow integralneri gowmari tesqov, orte I1 (x, t) =

K(x − y, t)ϕ(y) dy, |y|≤2R

I2 (x, t) =

K(x − y, t)ϕ(y) dy : |y|>2R

nra Qani or I1(x, t) integrali enaintegralayin fownkcian cankaca kargi a ancyalner st x t o oxakanneri anndhat en {(x, t) ∈ QR,δ,T , |y| ≤ 2R} ak sahmana ak bazmowyan vra, apa a ancman gor oowyown kareli tea oxel integrali nani tak: Het abar, I1 ∈ C ∞(QR,δ,T ) I1(x, t) fownkcian QR,δ,T -owm bavararowm (3.10 ) havasarman: nra cankaca

I2 (x, t) integrali enaintegralayin fownkcian kargi a ancyalner st x t o oxakanneri anndhat en {(x, t) ∈ QR,δ,T , |y| > 2R} bazmowyan vra: Sakayn I2 -owm integrowm katarvowm {|y| > 2R} ansahmana ak bazmowyownov: I2(x, t) fownkciayi hamar I1(x, t) fownkciayi hamapatasxan hatkowyownner apacowcelow

hamar bavarar cowyc tal, or I2 (x, t) fownkcian a ancyalner st x

nra cankaca kargi

t o oxakanneri {|y| > 2R} bazmowyan vra ownen

integreli maorantner, oronq kaxva en (x, t)-ic, (x, t) ∈ QR,δ,T : Qani or {(x, t) ∈ QR,δ,T , |y| > 2R} keteri hamar |x − y| ≥ |y| − |x| ≥ |y| − R,

apa

|x−y|2

|K(x − y, t)ϕ(y)| ≤

(|y|−R)2

e− 4t e− 4T √ |ϕ(y)| ≤ √ M: n (2 πt) (2 πδ)n

Aj masowm grva fownkcian oroneli maorantn . |I2 (x, t)| ≤ |y|>2R

M |K(x − y, t)ϕ(y)| dy ≤ √ 2 πδ

e−

(|y|−R)2 4T

dy < ∞ :

|y|>2R

Nman ov kareli gtnel maorantner a ancyalneri hamar: rinak, gtnenq maorant a ajin kargi a ancyalneri (st xi , i = 1, ..., n,

o oxakanneri) hamar . −2(xi − yi ) − |x−y|2 ∂ = √ ≤ 4t (K(x − y, t)ϕ(y)) ϕ(y) e (2 πt)n 4t ∂xi ≤

|x| + |y| − |x−y|2 R + |y| − (|y|−R)2 4t 4T √ √ e |ϕ(y)| ≤ M, e (2 πt)n 2t (2 πδ)n 2δ

i = 1, ..., n,

∂ + |y|)2 − (|y|−R)2 (K(x − y, t)ϕ(y)) ≤ (R 4T M: (2√πδ)n 4δ 2 e ∂t

Ayspisov, I1 (x, t)

I2 (x, t) fownkcianer anverj diferenceli en

∞

QR,δ,T -owm bavararowm en (3.10 ) havasar-

QR,δ,T -owm, I1 , I2 ∈ C (QR,δ,T ),

man: Het abar, u(x, t) fownkcian (st (3.5))

s tva nowyn

hatkowyownnerov: Qani or R > 0, T > 0, δ > 0 kamayakan en, apa u(x, t) fownkcian bavararowm (3.10 ) havasarman {x ∈ Rn , t > 0} tirowyowm u ∈ C ∞ (x ∈ Rn , t > 0):

Dicowq x ∈ Rn kamayakan ket , σ > 0 kamayakan iv : Ayd depqowm

e−

|x−y|2 σ

e−

(xj −yj )2 σ

dyj =

j=1−∞

Rn n '

n '

+∞

dy =

⎛ +∞ ⎞ n ' √ √ √ ⎝ e−ηj2 dηj ⎠ σ = ( πσ) = ( πσ)n :

j=1

−∞

Het abar, cankaca x ∈ Rn , t > 0, σ > 0 depqowm √ (2 πt)n

e−

|x−y|2 4tσ

dy = σ n/2 :

(3.6)

dy ≡ 1,

(3.7)

Masnavorapes, erb σ = 1, Rn

K(x − y, t) dy = √ n (2 πt)

e−

|x−y|2 4t

x ∈ Rn , t > 0 :

Aym apacowcenq, or u ∈ C(x ∈ Rn , t ≥ 0) tei owni (3.2) payman: Dra hamar bavarar cowyc tal, or kamayakan x0 ∈ Rn hamar lim

(x,t)→(x0 ,0) (t>0)

u(x, t) = ϕ(x0 ) :

(3.8)

Vercnenq kamayakan ε > 0: Qani or ϕ(x) fownkcian anndhat ketowm, apa goyowyown owni aynpisi δ > 0, or |ϕ(y) − ϕ(x0 )| ≤ ε

Ownenq u(x, t) − ϕ(x0 ) = =

erb

|y − x0 | ≤ δ :

K(x − y, t) ϕ(y) − ϕ(x0 ) dy =

K(x − y, t) ϕ(y) − ϕ(x0 ) dy+

|y−x0 |≤δ

K(x − y, t) ϕ(y) − ϕ(x0 ) dy = I1,δ + I2,δ :

|y−x0 |>δ

(3.9)

st (3.7) havasarowyan |I1,δ | ≤

K(x − y, t)|ϕ(y) − ϕ(x0 )| dy ≤

|y−x0 |≤δ

K(x − y, t) dy ≤ ε

≤ε

K(x − y, t) dy = ε

(3.10)

|y−x0 |≤δ

Gnahatenq (3.9) nerkayacman erkrord gowmarelin. |I2,δ | ≤

K(x − y, t)|ϕ(y) − ϕ(x0 )| dy ≤

|y−x0 |>δ

|x−y|2

K(x − y, t) |ϕ(y)| + |ϕ(x )| dy ≤ 2M

≤ |y−x0 |>δ

|y−x0 |>δ

Vercnenq aynpisi x, or |x − x0 | <

|x−y|2 e− 8t √ e− 8t dy : (2 πt)n

δ :

Ayd depqowm, erb |y − x0 | > δ, |x − y| = |(y − x0 ) − (x − x0 )| ≥ |y − x0 | − |x − x0 | > δ −

δ δ = :

st (3.6)-i ownenq

|x−y|2

|I2,δ | ≤ 2M |y−x0 |>δ

≤ 2M e (3.9),

δ − 32t

(3.10),

δ2 e− 8t √ e− 32t dy ≤ n (2 πt)

|x−y|2

e− 8t √ dy = 2M e−δ /32t 2n/2 → 0, (2 πt)n

(3.11)

erb

t → +0 :

a nowyownneric anmijapes het owm

(3.11) (3.8)

havasarowyown: tirowyowm u(x, t) low man sahmana akowyown anmijapes stacvowm (3.3) nerkayacowmic` {x ∈ Rn , t > 0}

|u(x, t)| ≤

K(x − y, t)|ϕ(y)| dy ≤ M

K(x − y, t) dy = M, Rn

isk t = 0 depqowm tei owni (3.2)-, orteic het owm (3.4) gnahatakan: Nman ov apacowcvowm en (3.4 ) anhavasarowyownner: Erb x ∈ Rn , t > 0 tei owni inf ϕ = K(x − y, t) inf ϕ dy ≤ u(x, t) ≤ K(x − y, t) sup ϕ dy = sup ϕ, Rn

isk t = 0 depqowm tei owni (3.2)-, orteic het owm (3.4 ) gnahatakan: eoremn apacowcva :

§ 3. Low man miakowyown: Maqsimowmi skzbownq: Low man anndhat kaxva owyown skzbnakan fownkciayic Nanakenq B = B(x ∈ Rn , t ≥ 0) bolor g(x, t) fownkcianeri bazmowyown, oronq orova en {x ∈ Rn , t ≥ 0}-owm

sahmana ak en kamayakan

{x ∈ Rn , 0 ≤ t ≤ T } ertowm`

cankaca T > 0 hamar goyowyown owni aynpisi C(T ) > 0 iv, or |g(x, t)| ≤ C(T ),

erb

x ∈ Rn , 0 ≤ t ≤ T :

Tei owni het yal pndowm:

eorem 3.3.1 B bazmowyan patkano Lu ≡ ut − x u = f (x, t), ut=0 = ϕ(x),

x ∈ Rn , t > 0, x ∈ Rn ,

(3.1) (3.2)

xndri low owm miakn : Min eoremi apacowycin ancnel` nenq miayn, or (3.1), (3.2) xndri low man miakowyown kareli apacowcel na

B dasic aveli layn dasowm:

Nanakenq Bα -ov, orte α ≥ 0 obacasakan iv , bolor g(x, t) fownkcianeri bazmowyown, oronq orova en {x ∈ Rn , t ≥ 0}-owm bavararowm en het yal paymanin`

kamayakan T > 0 hamar goyowyown owni aynpisi C(T ) > 0, or

|g(x, t)| ≤ C(T )eα|x| ,

erb

x ∈ Rn , 0 ≤ t ≤ T :

Tei owni het yal pndowm, or nerkayacnowm enq a anc apacowyci:

eorem 3.3.1| (3.1)

(3.2)

xndri low owm miakn

B α , α ≥ 0,

dasowm

Aknhayt , or B = B0 ⊂ Bα : Nenq na , or sahmana akowyan paymani bacakayowyan depqowm (a anc or ayl paymani) (3.1), (3.2) xndri low owm miak :

eorem 3.3.1-i apacowyc:

u2 (x, t) fownkcianer

Dicowq u1 (x, t)

(3.1), (3.2) xndri low owmner en

patkanowm en B dasin: Ayd depqowm

u(x, t) = u1 (x, t) − u2 (x, t) fownkcian Lu ≡ ut − x u = 0, ut=0 = 0,

xndri low owm

x ∈ Rn , t > 0,

(3.10 ) (3.20 )

x ∈ Rn ,

patkanowm B dasin: Da nanakowm , or cankaca

T > 0 hamar goyowyown owni aynpisi C(T ) > 0 iv, or |u(x, t)| ≤ C(T ),

x ∈ Rn , 0 ≤ t ≤ T :

(3.12)

x ∈ Rn , t > 0 :

(3.13)

Cowyc tanq, or u(x, t) ≡ 0,

(3.13)- apacowcelow hamar fiqsenq kamayakan (x0 , t0 ) ∈ {x ∈ Rn , t > 0}

ket

cowyc tanq, or u(x0 , t0 ) = 0 :

Vercnenq kamayakan ε > 0

(3.14)

ditarkenq

w± (x, t) = ε(|x|2 + (2n + 1)t) ± u(x, t),

x ∈ Rn , t ≥ 0,

fownkcianer, orte w+ - hamapatasxanowm havasarowyan aj masowm + nanin, w− - hamapatasxanowm − nanin: w+

w−

fownkcianer patkanowm en bavararowm en

C 2 (x ∈ Rn , t > 0) ∩ C(x ∈ Rn , t ≥ 0)

Lw± = ε,

x ∈ Rn , t > 0,

dasin (3.15)

havasarman: w± fownkcianer bavararowm en w± t=0 = ε|x|2 ,

x ∈ Rn ,

(3.16)

skzbnakan paymanin: Hamaayn (3.12)-i` goyowyown owni aynqan me R, or {|x| = R, 0 ≤ t ≤ t0 } glanayin maker owyi vra w±

|x|=R 0≤t≤t0

= εR2 + (2n + 1)tε ± u

≥ εR2 − |u|

|x|=R 0≤t≤t0

≥

≥ εR2 − C(t0 ) ≥ 0

(3.17)

erb R → ∞): nd orowm, karo enq (x0 , t0 ) ket nka R- aynqan me , or R > |x0 | ΩR, t = {|x| < R, 0 < t < t0 } glani verin himqi nersowm: gtvenq het yal pndowmic, or heto kapacowcenq:

(qani or enadrel

εR2 − C(t0 ) → +∞,

Lemma 3.3.1

Dicowq

fownkcian patkanowm C 2(ΩR ) ∩ C(ΩR ) = {|x| < R, 0 < t < ∞}, R > 0, tva het yal z(x, t)

dasin, orte ΩR hatkowyownnerov` 1. Lz ≥ 0 ΩR -owm, 2. z t=0 ≥ 0, ≥ 0, orte t0 -n or drakan iv : 3. z |x|=R 0≤t≤t0

Ayd depqowm z(x, t) ≥ 0,

(x, t) ∈ ΩR, t0 = {|x| ≤ R, 0 ≤ t ≤ t0 } :

Havi a nelov (3.15), (3.16), (3.17) w± (x, t) fownkcianeri hamar kira elov Lemma 3.3.1-` stanowm enq, or ΩR, t -owm w± (x, t) ≥ 0, masnavorapes w± (x0 , t0 ) ≥ 0: Het abar

−ε |x0 |2 + (2n + 1)t0 ≤ u(x0 , t0 ) ≤ ε |x0 |2 + (2n + 1)t0 ,

orteic

|u(x0 , t0 )| ≤ ε |x0 |2 + (2n + 1)t0 :

Qani or ε > 0 kamayakan , stanowm enq (3.14) havasarowyown: eoremn apacowcva : Lemma 3.3.1-i apacowyc: Katarenq hakaso enadrowyown: Enadrenq goyowyown owni aynpisi (x1 , t1 ) ∈ ΩR, t0 ket, or z(x1 , t1 ) < 0: Ditarkenq v(x, t) = e−t z(x, t) fownkcian: Parz , or v(x1 , t1 ) < 0 :

(3.18)

v(x, t) anndhat fownkcian ΩR, t0 ak glanowm ndownowm ir oqragowyn

areq. goyowyown owni aynpisi (x2 , t2 ) ∈ ΩR, t0 ket, or v(x2 , t2 ) =

min

(x,t)∈ΩR, t0

v(x, t) :

Hamaayn (3.18)-i` v(x2 , t2 ) < 0 :

Lemmayi 2.

(3.19)

3. paymanneri hamaayn v t=0 = (e−t z)t=0 ≥ 0,

|x|=R 0≤t≤t0

= (e−t z)

|x|=R 0≤t≤t0

≥ 0,

owsti (x2 , t2 ) ket i karo patkanel o glani {|x| ≤ R, t = 0} storin himqin, o l glani {|x| = R, 0 ≤ t ≤ t0 } komnayin maker owyin: Het abar (x2 , t2 ) ket kam ΩR, t0 glani nerqin ket kam patkanowm glani verin

himqin` |x2 | < R, t2 = t0 : A ajin depqowm, erb (x2 , t2 )- minimowmi ket , v(x2 , t2 ) < 0,

vt (x2 , t2 ) = 0,

vxi xi (x2 , t2 ) ≥ 0,

i = 1, ..., n,

erkrord depqowm v(x2 , t2 ) < 0,

vt (x2 , t2 ) ≤ 0,

vxi xi (x2 , t2 ) ≥ 0,

i = 1, ..., n :

Erkow depqowm l Lz(x2 , t2 ) = L(et v)(x2 , t2 ) = et (v + vt − x v) x=x2 < 0, t=t2

in hakasowm lemmayi 1. paymanin: Lemman apacowcva : Dicowq u(x, t) fownkcian patkanowm B bazmowyan

(3.10 ), (3.2)

xndri low owm : Miakowyan eoremic het owm , or ayd low owm petq hamnkni (3.3) Powasoni integralov trva low man het: Het abar ayn sahmana ak amboj {x ∈ Rn , t > 0} kisatara owyownowm

bavararowm

(3.4 ) anhavasarowyownnerin: Ayspisov, tei owni het yal pndowm:

eorem 3.3.2 (Maqsimowmi skzbownq)

B bazmowyan patkano

(3.1 ), (3.2) xndri low owm bavararowm (3.4 ) anhavasarowyownne0

rin: eorem 3.3.1-i apacowycic bxowm , or jermahaordakanowyan havasarman hamar Koii xndri sahmana ak low owm tva het yal hatkowyownnerov` 1.

amanaki skzbnakan t = 0 pahin linelov miayn anndhat, low owm

anmijapes da nowm anverj diferenceli bolor t > 0 hamar: 2. Ee amanaki skzbnakan t = 0 pahin low owm havasar zroyi amenowreq, baca owyamb or keti inqan ases oqr rjakayqi, orte ayn drakan , apa cankaca t > 0 hamar low owm da nowm drakan bolor keterowm: Da nanakowm , or (3.1), (3.2) xndrov nkaragrvo jermowyan tara man aragowyown anverj , in, iharke, cowyc talis xndri o liareq hamapatasxanowyown bnowyan er owyin: Jermowyan tara man er owyneri a avel grit nkaragrowyan hamar anhraet ditarkel Koii aveli bard xndir o g ayin diferencial havasarman hamar:

eorem 3.3.3 (Skzbnakan fownkciayic low man anndhat kaxva owyan masin) Dicowq u (x, t) u (x, t) fownkcianer patkanowm

en B bazmowyan

⎧ ⎪ ⎪ ⎪ ⎨u1t − x u1 = f (x, t), ⎪ ⎪ ⎪ ⎩ u1

t=0

= ϕ1 (x),

x ∈ Rn ,

⎧ ⎪ ⎪ ⎪ ⎨u2t − x u2 = f (x, t), ⎪ ⎪ ⎪ ⎩ u1

t=0

= ϕ2 (x),

x ∈ Rn , t > 0,

x ∈ Rn ,

xndirneri low owmner en: Ayd depqowm, ee or ε > 0 hamar |ϕ1 (x) − ϕ2 (x)| ≤ ε,

x ∈ Rn ,

apa |u1 (x, t)) − u2 (x, t)| ≤ ε,

x ∈ Rn , t ≥ 0 :

(3.20)

Apacowyc: Ditarkenq u(x, t) = u (x, t) − u (x, t) tarberowyown: u(x, t)

fownkcian patkanowm B bazmowyan ⎧ ⎪ ⎪ ⎪ ⎨ut − x u = 0, ⎪ ⎪ ⎪ ⎩ u

t=0

= ϕ(x),

x ∈ Rn , t > 0, x ∈ Rn ,

Koii xndri low owm , orte ϕ(x) = ϕ1 (x) − ϕ2 (x), |ϕ(x)| ≤ ε: Ver aradrva ic het owm , or u(x, t) fownkcian sahmana ak {x ∈ Rn , t > 0} kisatara owyownowm hamaayn maqsimowmi skzbownqi` bavararowm −ε ≤ inf (ϕ1 − ϕ2 ) = inf ϕ ≤ u(x, t) = u1 (x, t) − u2 (x, t) ≤ Rn

≤ sup ϕ = sup(ϕ1 − ϕ2 ) ≤ ε, Rn

x ∈ Rn , t ≥ 0,

anhavasarowyownnerin, orteic het owm eoremn apacowcva :

anhavasarowyown:

(3.20)

Skzbnakan fownkciayic low man anndhat kaxva owyan bacakayowyan rinak: Ditarkenq Koii het yal xndirner (<haka-

dar> jermahaordakanowyan havasarman hamar)` (K0 )

(Kn )

u0 (x, t) ≡ 0

⎧ ⎪ ⎪ ⎪ ⎨u0t + u0xx = 0, ⎪ ⎪ ⎪ ⎩ u0

t=0

= 0,

x ∈ R1 ,

⎧ ⎪ ⎪ ⎪ ⎨unt + unxx = 0, ⎪ ⎪ ⎪ ⎩ un

un (x, t) = e−n e

t=0

n2 t

x ∈ R1 , t > 0,

= e−n cos nx,

x ∈ R1 :

cos nx, n = 1, 2, ..., x ∈ R1 , t ≥ 0, fownkcianer

hamapatasxanabar (K0) (Kn) xndirneri low owmner en patkanowm en B bazmowyan: (Kn) xndrowm skzbnakan fownkcian (bolor a ancyalneri het miasin) havasaraa st x ∈ R1 gtowm zroyi, erb n → ∞, aysinqn` (K0) xndri skzbnakan fownkciayin: Sakayn, rinak, erb x = 0, un (x, t) − u0 (x, t) low owmneri tarberowyown cankaca t > 0 hamar gtowm anverji, erb n → ∞: Iroq.

|un (0, t) − u0 (0, t)| = un (0, t) = e−n en t → ∞,

erb

n→∞:

Dyowameli skzbownq: Aym owsowmnasirenq Koii xndir anhamase

jermahaordakanowyan havasarman hamar: Aknhayt , or bavarar ditarkel hamase skzbnakan paymanov depq. ut − a2 x u = f (x, t), x ∈ Rn , t > 0, u

t=0

= 0, x ∈ Rn :

(3.21) (3.22)

Ditarkenq vt − a2 x v = 0, x ∈ Rn , t > τ ≥ 0,

(3.23)

v t=τ = f (x, τ ), x ∈ Rn ,

(3.24)

xndir, ori low owm kaxva x, t o oxakanneric τ parametric. v = v(x, t, τ ): Nenq miayn, or (3.21), (3.22) xndir, inpes aliqayin havasarman depqowm, o oxakani hamapatasxan oxarinowmov bervowm a = 1 depqin: Tei owni het yal pndowm, or kovowm Dyowameli skzbownq: eorem 3.3.4 (Dyowameli skzbownq) Dicowq v(x, t, τ ) fownkcian (3.23), (3.24) xndri low owm : Ayd depqowm t v(x, t, τ ) dτ

u(x, t) =

(3.25)

fownkcian (3.21), (3.22) xndri low owm : Apacowyc: A ancelov u(x, t) fownkcian st a nelov (3.23) payman` stanowm enq t ut = v(t, x, t) +

o oxakani, havi

t vt (x, t, τ ) dτ = f (x, t) +

vt (x, t, τ ) dτ :

(3.26)

Qani or st xi o oxakanneri a ancman gor oowyown kareli tea oxel integrali nani tak, apa t x u = x

t v(x, t, τ ) dτ =

x v(x, t, τ ) dτ :

Het abar ut − a2 x u = f (x, t) +

vt (x, t, τ ) dτ − a2

t = f (x, t) +

t x v(x, t, τ ) dτ =

vt (x, t, τ ) − a2 x v(x, t, τ ) dτ = f (x, t) :

Ayspisov, u fownkcian bavararowm (3.21) havasarman: (3.25) nerkayacowmic anmijapes het owm , or u fownkcian bavararowm (3.22) paymanin: eoremn apacowcva : Ayspisov, ee ownenq hamase havasarman hamar Koii xndri low owm, menq karo enq grel na ut − a2 x u = f (x, t), x ∈ Rn , t > 0, ut=0 = ϕ(x), x ∈ Rn ,

Koii xndri low owm: Enadrenq f (x, τ ) fownkcian anndhat patkanowm B bazmowyan, isk ϕ fownkcian anndhat sahmana ak: Ayd depqowm low owmn owni |x−y|2 n e− 4a2 t ϕ(y) dy+ u(x, t) = √ 2 πa2 t R n

t +

|x−y|2 − n e 4a2 (t−τ ) f (y, τ ) dy, dτ 2 πa2 (t − τ ) R

x ∈ Rn , t > 0,

tesq, nd orowm u ∈ B : § 4. Xa xndir parabolakan havasarman hamar

Ditarkenq het yal xndir. gtnel Lu ≡ ut − uxx = f (x, t),

0 < x < l, 0 < t < T,

havasarman ayn low owm, or bavararowm u(0, t) = μ1 (t),

u(l, t) = μ2 (t),

0 ≤ t ≤ T,

ezrayin paymannerin u(x, 0) = ϕ(x),

0 ≤ x ≤ l,

skzbnakan paymanin: Ays xndir kovowm a ajin xa xndir jermahaordakanowyan havasarman hamar: Nanakenq QT = {0 < x < l, 0 < t < T }, Γ1,T = {x = 0, 0 < t < T },

Γ0 = {0 < x < l, t = 0}, Γ2,T = {x = l, 0 < t < T },

ΓT = ΓT = Γ0 ∪ Γ1,T ∪ Γ2,T : ΓT -n kovowm QT owankyan parabolakan ezr: A ajin xa xndri u(x, t)

low owm patkanowm C 2 (QT ) ∩ C(QT ) bazmowyan. u ∈ C 2 (QT ) ∩ C(QT ): Tei owni het yal pndowm:

eorem 3.4.1

Lu = f (x, t),

(x, t) ∈ QT ,

uΓ = ϕ, T

(3.26) (3.27)

a ajin xa xndri low owm miakn :

Apacowyc: Dicowq u (x, t)

u2 (x, t) fownkcianer mi nowyn xa xndri

low owmner en.

⎧ ⎪ ⎪ ⎪ ⎨Lu1 = f (x, t), ⎪ ⎪ ⎪ ⎩ u1

ΓT

= ϕ,

⎧ ⎪ ⎪ ⎪ ⎨Lu2 = f (x, t), ⎪ ⎪ ⎪ ⎩ u2

ΓT

(x, t) ∈ QT ,

=ϕ:

Ayd depqowm v(x, t) = u1 (x, t) − u2 (x, t) fownkcian ⎧ ⎪ ⎪ ⎪ ⎨Lv = 0, ⎪ ⎪ ⎪ ⎩v

ΓT

(x, t) ∈ QT ,

= 0,

xndri low owm : Lemma 3.3.1-ic het owm , or (x, t) ∈ QT :

v(x, t) ≥ 0,

(3.28)

Katarelov nowyn datoowyownner −v(x, t) = u2(x, t) − u1(x, t) fownkciayi hamar` kstananq (x, t) ∈ QT :

−v(x, t) ≥ 0, (3.28), (3.29)-ic

(3.29)

het owm , or v(x, t) = u1 (x, t) − u2 (x, t) ≡ 0,

(x, t) ∈ QT :

eoremn apacowcva : Ditarkenq het yal xndir`

ut − uxx = 0, u

(x, t) ∈ QT , =ϕ:

ΓT

(3.30) (3.31)

Tei owni het yal pndowm:

eorem 3.4.2 (Maqsimowmi skzbownq) Dicowq u(x, t) fownkcian (3.30), (3.31)

xndri low owm : Ayd depqowm

min ϕ ≤ u(x, t) ≤ max ϕ, ΓT

ΓT

(x, t) ∈ QT :

(3.32)

Apacowyc: Nanakenq m = min ϕ,

M = max ϕ :

ΓT

Ditarkenq z(x, t) = u(x, t) − m fownkcian: Aknhayt , or z(x, t) fownkcian bavararowm (3.30) havasarman z Γ ≥ 0 : T

st Lemma 3.3.1-i, z(x, t) ≥ 0,

(x, t) ∈ QT ,

u(x, t) ≥ m,

(x, t) ∈ QT :

aysinqn` Katarelov nowyn datoowyownner M −u(x, t) fownkciayi hamar` kstananq (x, t) ∈ QT :

u(x, t) ≤ M,

eoremn apacowcva : (3.32) anhavasarowyownneric bxowm het yal pndowm: eorem 3.4.3 (Modowli maqsimowmi skzbownq) Dicowq u(x, t) fownkcian (3.30), (3.31) xndri low owm : Ayd depqowm max |u(x, t)| ≤ max |ϕ| : ΓT

(x,t)∈QT

(3.33)

eorem 3.4.3-ic het owm (3.30), (3.31) xndri low man anndhat kaxva owyown ezrayin fownkciayic:

eorem 3.4.4 (Ezrayin fownkciayic low man anndhat kaxva owyan masin) Dicowq u (x, t) u (x, t) fownkcianer

⎧ ⎪ ⎪ ⎪ ⎨u1t − u1xx = f (x, t), ⎪ ⎪ ⎪ ⎩ u1

ΓT

= ϕ1 ,

⎧ ⎪ ⎪ ⎪ ⎨u2t − u2xx = f (x, t), ⎪ ⎪ ⎪ ⎩ u2

ΓT

(x, t) ∈ QT ,

= ϕ2 ,

xndirneri low owmner en: Ayd depqowm, ee or ε > 0 hamar |ϕ1 − ϕ2 |

ΓT

≤ ε,

apa |u1 (x, t) − u2 (x, t)| ≤ ε,

Apacowyc: Ditarkenq u(x, t) fownkcian ⎧

= u1 (x, t) − u2 (x, t)

⎪ ⎪ ⎪ ⎨ut − uxx = 0, ⎪ ⎪ ⎪ ⎩ u

ΓT

(x, t) ∈ QT :

tarberowyown:

u(x, t)

(x, t) ∈ QT ,

= ϕ,

xndri low owm , orte ϕ = ϕ1 − ϕ2 : Hamaayn eorem 3.4.3-i` ownenq |u1 (x, t) − u2 (x, t)| = |u(x, t)| ≤ max |u(x, t)| ≤ max |ϕ| = (x,t)∈QT

= max |ϕ1 − ϕ2 | ≤ ε, ΓT

ΓT

(x, t) ∈ QT :

eoremn apacowcva : § 5. o oxakanneri anjatman meod

Inpes arden nel enq naxord glxowm, o oxakanneri anjatman kam Fowryei meod masnakan a ancyalnerov diferencial havasarowmneri low man himnakan meodneric : Ays paragrafowm katarelov nman datoowyownner, inpes lari tatanman havasarman depqowm, menq karadrenq Fowryei meod jermahaordakanowyan havasarman a ajin xa xndri hamar QT = {0 < x < l, 0 < t < T } tirowyowm:

Hamase havasarowmner: Ditarkenq het yal xndir` ut − a2 uxx = 0,

0 < x < l, 0 < t < T,

u(0, t) = u(l, t) = 0, u(x, 0) = ϕ(x),

0 ≤ t ≤ T, 0≤x≤l:

(3.34) (3.35) (3.36)

Qani or (3.34) havasarowm g ayin hamase , apa erkow masnavor low owmneri gowmar s ayd havasarman low owm : orenq gtnel

(3.34) havasarman aynpisi masnavor low owmner, oronc gowmar klini (3.34), (3.35), (3.36) xndri low owm: Nax low enq het yal andak xndir`

gtnel (3.34) havasarman ayn o trivial (o nowynabar zro) low owmner, oronq bavararowm en (3.35) ezrayin paymannerin

ownen

u(x, t) = X(x)T (t)

(3.37)

tesq, orte X(x) fownkcian kaxva miayn x o oxakanic, T (t) fownkcian kaxva miayn t o oxakanic: Teadrelov (3.37) tesqi u(x, t) fownkcian (3.34) havasarman mej` stanowm enq

X(x)T (t) − a2 X (x)T (t) = 0,

orteic, havi a nelov X(x) ≡ 0, T (t) ≡ 0, stanowm enq

X (x) 1 T (t) = 2 : X(x) a T (t)

(3.38)

Qani or (3.38) havasarowyan ax mas kaxva miayn x-ic, isk aj mas` miayn t-ic, apa (3.38) havasarowyan aj

ax maser nowynabar

havasar en mi nowyn hastatownin: Ayd hastatown nanakenq −λ-ov.

1 T (t) X (x) = 2 = −λ : X(x) a T (t)

Aysteic X(x)

T (t) fownkcianeri hamar stanowm enq

X (x) + λX(x) = 0, X(x) ≡ 0,

T (t) + λa2 T (t) = 0, T (t) ≡ 0,

0 < x < l,

(3.39)

t > 0,

(3.40)

sovorakan diferencial havasarowmner: (3.35) ezrayin paymanneric ownenq X(0) = X(l) = 0 :

(3.41)

Ayspisov, X(x) fownkciayi hamar stacanq

X (x) + λX(x) = 0, 0 < x < l, X(0) = X(l) = 0,

(3.42)

towrm-Liowvili xndir, orn owsowmnasirel enq lari tatanman havasarowm low elis cowyc enq tvel, or miayn λn =

πn 2 l

n = 1, 2, ...

areqneri depqowm (3.42) xndirn owni πn x l

Xn (x) = Dn sin

o zroyakan low owm, orte Dn - kamayakan hastatown : λn -in hamapatasxano (3.40) havasarman ndhanowr low owmn Tn (t) = An e

−

πan 2 l

orte An - kamayakan hastatown : Ayspisov. un (x, t) = Xn (x)Tn (t) = An e

−

πan 2 l

sin

πn x, l

n = 1, 2, ...

fownkcianer (3.34) havasarman masnavor low owmner en, oronq bavararowm en (3.35) ezrayin paymannerin nerkayacvowm en erkow fownkcianeri artadryali tesqov: Ayd fownkcianeric mek kaxva miayn x o oxakanic, myows` miayn t o oxakanic: Ays low owmner karo en bavararel naxnakan xndri (3.36) skzbnakan paymanin miayn masnavor ϕ fownkcianeri hamar: Aym verada nanq (3.34), (3.35), (3.36) ndhanowr xndrin: Enadrenq An gor akicner aynpisin en, or u(x, t) ≡

∞ n=1

un (x, t) =

∞ n=1

An e

−

πan 2 l

sin

πn x l

(3.43)

arq, inpes na ayn arqer, oronq stacvowm en ays arq erkow angam st x-i mek angam st t-i andam a andam a ancelis, havasaraa

zowgamet en hamapatasxan bazmowyownneri vra:

Parz , or u(x, t) fownkcian kbavarari inpes (3.35) ezrayin paymannerin, aynpes l (3.34) havasarman: (3.36) skzbnakan paymanic gtnenq An gor akicner: Qani or u(x, 0) = ϕ(x) =

∞

un (x, 0) =

∞

n=1

An sin

n=1

πn x, l

(3.44)

apa An gor akicner ϕ fownkciayi st sinowsneri arqi verlow owyan Fowryei gor akicnern en (enadrvowm , or ϕ fownkcian aynpisin , or ayn kareli verlow el st sinowsneri Fowryei arqi). An = ϕ n = l

l ϕ(ξ) sin

πn ξ dξ : l

(3.45)

Ayspisov, menq low owm nerkayacrinq (3.43) arqi tesqov: Ee ayd arq taramitowm , kam ayd arqov nerkayacva fownkcian diferenceli , apa, iharke, ayn i karo linel (3.34) diferencial havasarman low owm: Qani or |un (x, t)| ≤ |An |,

apa

∞

|An |

(3.46)

n=1

vayin arq (3.43) fownkcional arqi hamar maorant , (3.46) arqi zowgamitowyownic het owm (3.43) arqi havasaraa zowgamitowyown {0 ≤ x ≤ l, 0 ≤ t ≤ T } bazmowyan vra: Hamaayn Fowryei arqeri haytni hatkowyownneri` (3.46) arqi zowgamitowyan hamar bavarar enadrel, or ϕ ∈ C[0, l], owni ktor a ktor anndhat a ancyal tei owni ϕ(0) = ϕ(l) = 0

payman:

(3.47)

Owsowmnasirenq Ditarkenq ut (x, t) ∼

ut (x, t)

∞ ∂un n=1

uxx (x, t) ∼

∂t

=−

∞ ∂ 2 un n=1

∂x2

uxx (x, t)

∞ πa 2

=−

fownkcianeri anndhatowyown:

n2 An e

−

πan 2 l

sin

n=1

∞ π 2

n2 An e

−

πan 2 l

πn x, l

(3.48)

πn x l

(3.49)

sin

n=1

arqer: Dicowq |ϕ| ≤ M : Ayd depqowm

l 2 πn |An | = ϕ(ξ) sin ξ dξ ≤ 2M, l l

orteic het owm , or 2 πan 2 − πan t ∂un l , e ∂t ≤ 2M l 2 2 2 − πan t ∂ un ≤ 2M πn e l : ∂x2 l

Aknhayt , or cankaca t0 -i hamar, orte 0 < t0 ≤ T ,

Het abar, {0 ≤ x ≤ l,

2 πan 2 − πan t0 ∂un l e , ∂t ≤ 2M l

t ≥ t0 ,

2 2 πn 2 − πan t0 ∂ un l e , ∂x2 ≤ 2M l

t ≥ t0 :

(3.48),

arqer havasaraa zowgamitowm en t0 ≤ t ≤ T } bazmowyan vra: Avelin. (3.49)

πan 2 k+m π 2m+k ∂ t0 − u n 2m+k 2m l n a e , ∂xk ∂tm ≤ 2M l

t ≥ t0 ,

arq kareli cankaca angam a ancel st x-i st t-i stacva arqer klinen havasaraa zowgamet {0 ≤ x ≤ l, t0 ≤ t ≤ T } bazmowyan vra: Qani or t0 -n kamayakan , apa {0 < x < l, 0 < t < T } (3.43)

tirowyowm (3.43) arq (3.34) havasarman low owm anverj diferenceli : Menq apacowcecinq het yal pndowm: eorem 3.5.1 Dicowq ϕ fownkcian anndhat ` ϕ ∈ C[0, l], owni ktor a ktor anndhat a ancyal tei owni (3.47) payman: Ayd depqowm (3.43) arqov nerkayacva u(x, t) fownkcian, orte An gor akicner orovowm en (3.45) bana ov, (3.34), (3.35), (3.36) xndri low owm : Anamase havasarowmner: Ditarkenq ut − a2 uxx = f (x, t),

0 < x < l, 0 < t < T,

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

0 ≤ t ≤ T,

(3.50) (3.51)

0≤x≤l:

(3.52)

xndir: Inpes lari tatanman havasarman depqowm, low owm ntrenq u(x, t) =

∞

un (t) sin

n=1

πn x l

(3.53)

tesqov` t o oxakan ditarkelov orpes parametr: Havasarman f (x, t) aj mas nerkayacnenq Fowryei arqov. ∞

πn fn (t) sin x, f (x, t) = l n=1

Teadrelov (3.53) stanowm enq ∞

orteic

πan 2

n=1

un (t) +

l f (ξ, t) sin

artahaytowyownner

(3.54)

un (t) +

fn (t) = l

(3.50)

(3.54)

havasarman mej`

πn πn x= x, fn (t) sin l l n=1 ∞

un (t) sin

πan 2 l

πn ξ dξ : l

un (t) = fn (t),

n = 1, 2, ... :

(3.55)

fownkcian oroelow hamar stacanq hastatown gor akicnerov sovorakan diferencial havasarowm: st (3.52) skzbnakan paymani` un (t)

u(x, 0) =

∞

un (0) sin

n=1

πn x = 0, l

het abar un (0) = 0 :

(3.56)

Low elov (3.55) sovorakan diferencial havasarowm (3.56) zroyakan skzbnakan paymanov` kstananq πan 2 t (t − τ ) − l fn (τ ) dτ : un (t) = e

(3.57)

Teadrelov (3.57) artahaytowyown (3.53)-i mej` stanowm enq (3.50), (3.51), (3.52) xndri low owm`

u(x, t) =

∞ n=1

⎛ ⎝

t e

−

πan 2 l

(t − τ )

⎞ fn (τ ) dτ ⎠ sin

Anhamase ezrayin paymanner:

πn x: l

Ditarkenq a ajin xa xndir

jermahaordakanowyan havasarman hamar ndhanowr depqowm.

ut − a2 uxx = f (x, t),

0 < x < l, 0 < t < T,

u(x, 0) = ϕ(x), u(0, t) = μ1 (t),

(a > 0)

0 ≤ x ≤ l,

u(l, t) = μ2 (t),

t≥0:

Ays xndir low elow hamar nermow enq nor v(x, t) anhayt fownkcia. u(x, t) = v(x, t) + U (x, t),

orte enadrvowm , or U (x, t) fownkcian haytni : Ayd v(x, t) fownkcian petq lini vt − a2 vxx = f(x, t) havasarman low owm, orte f(x, t) = f (x, t) − Ut − a2 Uxx ,

het yal skzbnakan

ezrayin paymannerin` v(x, 0) = ϕ(x) = ϕ(x) − U (x, 0),

bavarari

v(0, t) = μ 1 (t) = μ1 (t) − U (0, t), v(l, t) = μ 2 (t) = μ2 (t) − U (l, t) :

ntrenq U (x, t) fownkcian aynpes, or μ 1 (t) = μ 2 (t) = 0 :

Ayd npatakov karo enq vercnel (inpes lari tatanman havasarman depqowm) U (x, t) = μ1 (t) +

x μ2 (t) − μ1 (t) : l

Ev ayspes, u(x, t) fownkciayi hamar ndhanowr ezrayin paymannerov xndir bervec v(x, t) fownkciayi hamar hamase ezrayin paymannerov xndrin:

Glowx 4 lipsakan tipi havasarowmner § 1. Harmonik fownkcianer: Laplasi havasarman fowndamental low owm: Grini bana er

Dicowq

Q ⊂ Rn , n ≥ 1 ,

tirowy : Kasenq, or u(x), x harmonik Q tirowyowm, ee u ∈ C 2 (Q) bavararowm

∈ Q,

u ≡ ux1 x1 + ... + uxn xn = 0

fownkcian (4.1)

Laplasi havasarman: Nenq, or goyowyown owni fownkcia, or yowraqanyowr ketowm bavararowm Laplasi havasarman, sakayn harmonik , qani or anndhat : n = 2 depqowm aydpisi fownkciayi rinak u(x1 , x2 ) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎨Re exp − ⎪ ⎪ ⎪ ⎪ ⎩0,

(x1 + ix2 )4

x21 + x22 = 0,

fownkcian, or xzvowm (0, 0) ketowm: n = 1 depqowm (a, b) ⊂ R1 mijakayqi vra orova harmonik fownk2 cianer ddxu2 = 0 havasarman low owmner en ownen u(x) = c1 x + c2 tesq, orte c1 - c2 - kamayakan hastatownner en: Mez hetaqrqrelow n > 1 depq: Ays depqowm harmonik fownkcianeri bazmowyown apes harowst : Harmonik fownkcianeri tesowyan mej kar or der ownen hatowk tesqi harmonik fownkcianer:

Dicowq ξ -n Rn , n ≥ 2, tara owyan kamayakan ket : x ∈ Rn keti he avorowyown ξ ketic nanakenq ρ-ov. ρ = ρ(x) = |x − ξ|: Gtnenq bolor u(x) harmonik fownkcianer, oronq kaxva en miayn ρ(x)-ic: Ee u = u(ρ), apa u xi = u ρ ρxi = u ρ uxi xi = uρρ

(xi − ξi ) , ρ

(xi − ξi )2 (xi − ξi )2 + uρ − uρ , ρ ρ ρ3 u = uρρ + (n − 1)

i = 1, ..., n,

uρ : ρ

Het abar, mez hetaqrqro harmonik fownkcianer

uρρ + (n − 1)

uρ = 0, ρ

ρ > 0,

sovorakan diferencial havasarman low owmner en: havasarman ndhanowr low owmn

n = 2

depqowm ayd

u(ρ) = c1 ln ρ + c2 ,

isk n > 2 depqowm` u(ρ) =

c1 + c2 , ρn−2

orte c1 c2 kamayakan hastatownner en: Ayspisov, n = 2 depqowm c1 ln |x − ξ| fownkcianer, isk c1 fownkcianer Rn \ {ξ} tirowyowm harmonik en: |x − ξ|n−2

n> 2

depqowm

Laplasi havasarman fowndamental low owm ξ ketowm ezakiow-

yamb

kovowm

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ln |x − ξ|, x ∈ R2 \ {ξ}, 2π U (x − ξ) = ⎪ ⎪ −1 ⎪ ⎪ , x ∈ Rn \ {ξ}, n > 2, ⎩ (n − 2)σn |x − ξ|n−2

fownkcian, orte σn miavor sferayi maker owyi makeresn Masnavorapes, n = 3 depqowm fowndamental low owmn owni U (x − ξ) =

−1 , 4π|x − ξ|

Rn -owm:

x ∈ R3 \ {ξ},

tesq: gtvelov strogradskow bana ic` dowrs berenq bana er, oronq menq kgtagor enq hetagayowm: Dicowq Q-n Rn tara owyan sahmana ak tirowy , ∂Q ∈ C 1 , u ∈ C 2 (Q), v ∈ C 1 (Q): Ayd depqowm

vu dx = Q

v div(∇u) dx =

(v∇u, ν) dS −

div(v∇u) dx − Q

∇u∇v dx =

∂Q

∇u∇v dx =

v ∂Q

∂u dS − ∂ν

∇u∇v dx, Q

orte ν -n ∂Q-in tarva Q-i nkatmamb artaqin miavor normal vektorn : Ayspisov, menq stacanq Grini a ajin bana .

vu dx =

v ∂Q

∂u dS − ∂ν

∇u∇v dx : Q

Aym enadrenq u, v ∈ C 2 (Q): (4.2) havasarowyan mej oxelov fownkcianeri derer` kstananq

uv dx = Q

u ∂Q

∂v dS − ∂ν

(4.2)

∇v∇u dx : Q

havasarowyownic andam a andam hanelov stacva havasarowyown` kstananq Grini erkrord bana . (4.2)

(vu − uv) dx = Q

v ∂Q

∂v ∂u −u ∂ν ∂ν

Stacva bana eric bxowm het yal pndowm:

dS :

(4.3)

eorem 4.1.1 Dicowq Q-n sahmana ak tirowy , ∂Q ∈ C , u(x) fownk1

cian harmonik Q-owm

u ∈ C 2 (Q): ∂Q

Apacowyc: Apacowyc het

Ayd depqowm

∂u dS = 0 : ∂ν

owm , rinak, (4.2) bana ic, vercnelov v ≡ 1:

§ 2. Potencialner: Oork fownkciayi nerkayacowm potencialneri gowmari tesqov

Dicowq Q-n Rn tara owyan sahmana ak tirowy , ∂Q ∈ C 1 , u(x) fownkcian patkanowm C 2 (Q) bazmowyan, u ∈ C 2 (Q): Parzowyan hamar enadrenq n = 3: Vercnenq kamayakan ξ ∈ Q ditarkenq Qε = Q \ {|x − ξ| ≤ ε} tirowy, orte ε > 0 kamayakan iv , or oqr ξ keti ∂Q ezric owneca he avorowyownic. 0 < ε < r(ξ) = min |ξ − y| : y∈∂Q

Kira enq

(4.3)

bana

u(x)

|x − ξ|

v(x) =

fownkcianeri hamar

(ndhanowr depqowm orpes v(x) fownkcia petq vercnel |x −1ξ|n−2 fownkcian, erb n = 2, ln |x − ξ| fownkcian, erb n = 2): Qani or v(x) fownkcian Qε tirowyowm harmonik , kstananq Qε

u dx = |x − ξ|

+ ∂Q

∂Qε

= |x−ξ|=ε

∂u ∂ −u ∂ν |x − ξ| ∂ν x

|x − ξ|

dSx =

dSx +

dSx :

(4.4)

Nanakenq M = max |u(x)|: Qani or x∈Q

u dx dx ≤ M = |x − ξ| |x − ξ| |x−ξ|≤ε |x−ξ|≤ε 2π =M

apa

r dr = 2M πε2 ,

u dx → |x − ξ|

Qε

ε sin θ dθ

dϕ

u dx, |x − ξ|

erb

ε→0:

(4.5)

Nanakenq M1 = max |∇u(x)|: Qani or x∈Q

∂u dSx dSx |(∇u, ν)| ≤ ≤ ∂ν |x − ξ| |x − ξ| |x−ξ|=ε |x−ξ|=ε ≤

|∇u| dSx ≤

M1 4πε2 = 4πM1 ε, ε

|x−ξ|=ε

apa

|x−ξ|=ε

∂u dSx →0 ∂ν |x − ξ|

erb

ε→0:

(4.6)

Qani or {|x − ξ| = ε} sferayi x ketowm tarva Qε -i nkatmamb artaqin miavor normal ξ −ε x vektorn , apa ayd sferayi vra ∂ ∂νx

|x − ξ|

u(x) |x−ξ|=ε

∂ ∂νx

∇x

|x − ξ|

ξ−x , |x − ξ| ε

dSx =

ε2

=−

x−ξ ξ−x , |x − ξ|3 ε

u(x) dSx =

ε2

4πε2 u(θ) = 4πu(θ), ε2

|x−ξ|=ε

orte θ ∈ {|x − ξ| = ε}: Owsti

u(x) |x−ξ|=ε

∂ ∂νx

|x − ξ|

dSx → 4πu(ξ),

erb

ε→0:

(4.7)

Ancnelov sahmani (4.4) havasarowyan mej, erb ε → 0, havi a nelov (4.5), (4.6), (4.7), kstananq u(ξ) = −

4π

u dx + |x − ξ| 4π

∂Q

∂u ∂ −u ∂ν |x − ξ| ∂ν x

|x − ξ|

dSx :

Havi a nelov, or n = 3 depqowm 4π|x−1− ξ| = U (x − ξ), orte U -n Laplasi havasarman fowndamental low owmn , stacva havasarowyown karo enq grel het yal tesqov.

U (x − ξ)u(x) dx +

u(ξ) = Q

u(x) ∂Q

∂u ∂ U (x − ξ) dSx : U (x − ξ) − ∂ν x ∂ν

Verjapes, ξ -n oxarinelov x-ov, x- oxarinelov y-ov` stacva

havasarowyown kndowni het yal tesq. U (x − ξ)u(y) dy+

u(x) = Q

u(y)

∂Q

(4.8)

∂u ∂ U (x − y) dSy , U (x − y) − ∂ν y ∂ν

x∈Q:

(4.8)

bana tei owni cankaca n ≥ 2 a oakanowyan depqowm: u0 (x) =

U (x − y)ρ0 (y) dy,

x ∈ Q,

(4.9)

fownkcian, orte yamb:

ρ0 ∈ C(Q),

kovowm

avalayin potencial ρ0

xtow-

u1 (x) =

U (x − y)ρ1 (y) dSy ,

x ∈ Q,

(4.10)

∂Q

fownkcian, orte ρ1 ∈ C(∂Q), kovowm parz erti potencial ρ1 xtowyamb: u2 (x) = ∂Q

∂U (x − y) ρ2 (y) dSy , ∂νy

x ∈ Q,

(4.11)

fownkcian, orte ρ2 ∈ C(∂Q), kovowm krknaki erti potencial xtowyamb: Dvar nkatel, or parz erti krknaki erti potencialner tirowyowm anverj diferenceli harmonik fownkcianer en: Menq apacowcecinq (n = 3 depqowm) het yal pndowm:

eorem 4.2.1 Dicowq Q-n R

ρ2

tara owyan sahmana ak tirowy ,

∂Q ∈ C : Ayd depqowm cankaca u ∈ C 2 (Q) fownkcia nerkayacvowm

avalayin potenciali ( u xtowyamb), parz erti potenciali ∂u xtowyamb) krknaki erti potenciali ( u xtowyamb) gowmari (− ∂ν tesqov:

Het anq: Ee eoremowm hiatakva u(x) fownkcian harmonik

Q tirowyowm, apa Q-owm ayn karo nerkayacvel parz

krknaki

erteri potencialneri gowmari tesqov:

§ 3. Mijini masin eorem

eorem 4.3.1 (Maker owayin mijini masin)

Dicowq

Q-n

tara owyan kamayakan tirowy , u(x) fownkcian harmonik Q tirowyowm, x0 ∈ Q kamayakan ket : Ayd depqowm cankaca R-i hamar, 0 < R < r(x0 ), orte r(x0 )-n x0 keti he avorowyownn ∂Q ezric, tei owni

u(x ) = σn Rn−1

u(y) dSy |x0 −y|=R

havasarowyown, orte σn miavor sferayi maker owyi makeresn Rn -owm:

Ayl xosqov, x0 ∈ Q ketowm harmonik fownkciayi areq havasar x0 kentronov

R a avov sferayi vra ayd fownkciayi ndowna

areqneri mijin vabanakanin:

Apacowyc: eoremi apacowyc aradrenq

depqi hamar: Qani or BR(x0) = {|y − x0| < R} gownd apes nka Q tirowyi mej` BR (x0 ) = {|y − x0 | < R} Q, apa u(x) ∈ C 2 (B R (x0 )) karo enq kira el (4.8) bana u(x) fownkciayi hamar BR (x0 ) gndowm. u(x) = − 4π

|x0 −y|=R

4π

∂u(y) dSy − ∂ν |x − y|

|x0 −y|=R

∂ u(y) ∂νy

|x − y|

Masnavorapes, erb x = x0, kstananq u(x0 ) =

4πR

|x0 −y|=R

=−

∂u(y) dSy − ∂ν 4π 4π

u(y) |x0 −y|=R

dSy ,

u(y)

|x0 −y|=R

∂ ∂νy

qani or st eorem 4.1.1-i ownenq

n = 3

x ∈ BR (x0 ) :

∂ ∂νy

|x0 − y|

dSy =

dSy .

∂u(y) dSy = 0 {|x0 − y| = R} ∂ν

|x0 −y|=R

sferayi y ketowm tarva BR(x0) gndi nkatmamb artaqin miavor normal y − x0 vektorn : Het abar, ayd sferayi vra R ∂ ∂νy

|x0 − y|

=−

u(x ) = 4πR2

(y − x0 , y − x0 ) =− 2 R |x0 − y|3 R u(y) dSy :

|x0 −y|=R

eoremn apacowcva : eorem 4.3.1 - ic bxowm het yal pndowm:

eorem 4.3.2 ( avalayin mijini masin) Dicowq Q-n Rn tara owyan kamayakan tirowy , u(x) fownkcian harmonik Q tirowyowm, x0 ∈ Q kamayakan ket : Ayd depqowm cankaca R-i hamar, 0 < R < r(x0),

tei owni u(x0 ) =

n σn R n

u(y) dy |x0 −y|≤R

havasarowyown, orte σnn miavor gndi avaln Rn-owm: Ayl xosqov, x0 ∈ Q ketowm harmonik fownkciayi areq havasar x0 kentronov R a avov gndowm ayd fownkciayi ndowna areqneri mijin vabanakanin: Apacowyc: Dicowq 0 < ρ < r(x0 ),

n = 3:

st eorem 4.3.1-i, kamayakan ρ-i hamar,

4πρ2 u(x0 ) =

u(y) dSy : |x0 −y|=ρ

Integrelov ays havasarowyown st ρ-i 0-ic R, stanowm enq 4π 3 R u(x0 ) =

u(y) dSy =

dρ

|x0 −y|=ρ

u(y) dy : |x0 −y|≤R

eoremn apacowcva : § 4. Maqsimowmi skzbownq

Kasenq, or Q ⊂ Rn tirowyowm u(x) anndhat fownkcian tva mijini hatkowyamb, ee cankaca x0 ∈ Q keti hamar cankaca R > 0 hamar, 0 < R < r(x0 ), tei owni het yal havasarowyown` u(x0 ) =

n σn R n

u(y) dy :

(4.12)

|x0 −y|≤R

eorem 4.3.2-ic het owm , or harmonik fownkcianer tva en mijini hatkowyamb: Irakanowm ayd hatkowyamb bnowagrvowm en bolor harmonik fownkcianer. hetagayowm menq kapacowcenq, or tei owni na mijini veraberyal hakadar eorem:

Mijini hatkowyamb tva fownkcianeri hamar tei owni het yal pndowm: Dicowq Q-n Rn tara owyan sahmana ak tirowy , u(x) fownkcian patkanowm C(Q)-in tva mijini hatkowyamb: Ayd depqowm kam Lemma 4.4.1

u(x) ≡ const,

x ∈ Q,

kam min u < u(x) < max u, Q

x∈Q:

(4.13)

Nanakenq M = max u: Cowyc tanq, or ee goyowyown owni Q aynpisi x ∈ Q ket, or u(x0) = M , apa u(x) = M , x ∈ Q: Vercnenq kamayakan y ∈ Q ket cowyc tanq, or u(y) = M : Miacnenq y x0 keter L = L verjavor bekyalov, orn ambojowyamb nka Q tirowyi mej: L bekyali ∂Q ezri he avorowyown nanakenq d d = min |x − y| > 0 L bekyal a kenq Bi = {|x − xi | < }, i = 0, 1, ..., N , verjavor qanaki gnderov, orte xi ∈ L ∩ ∂Bi−1, i = 1, ..., N , nd orowm y ∈ BN : Dicowq n = 3: st (4.12)-i ownenq Apacowyc:

x∈L y∈∂Q

u(x0 ) =

4π(d/2)3

u(x) dx, B0

or kareli artagrel

u(x0 ) − u(x) dx = 0

tesqov: Qani or u(x0) − u(x) enaintegralayin fownkcian anndhat B 0 -owm obacasakan , apa B 0-owm u(x0) − u(x) ≡ 0, aysinqn` B 0-owm B1 gndi u(x) ≡ u(x0 ) = M , masnavorapes, u(x1 ) = M : x1 keti hamar krknelov nowyn datoowyownner` kstananq, or B 1-owm u(x) ≡ M ,

masnavorapes` u(x2 ) = M : Krkin katarelov nowyn datoowyownner` ardyownqowm kstananq, or B N -owm u(x) ≡ M , masnavorapes` u(y) = M : Ev ayspes, apacowcecinq, or kam Q-owm u(x) ≡ const kam Q-owm tei owni (4.13) anhavasarowyan aj mas: Kira elov apacowcva pndowm −u(x) fownkciayi nkatmamb` kstananq, or kam Q-owm u(x) ≡ const kam Q-owm tei owni (4.13) anhavasarowyan ax mas: Lemman apacowcva

: Lemma 4.4.1-ic het owm , or lemmayi paymannerin bavararo hastatownic tarber u(x) fownkcian Q tirowyi nersowm i karo ndownel aynpisi areqner, oronq havasar en Q-owm ayd fownkciayi me agowyn kam

oqragowyn areqnerin: Het abar, aydpisi fownkcian ir me agowyn

oqragowyn areqner ndownowm ∂Q ezri vra: Tei owni het yal pndowm:

Dicowq Q-n Rn tara owyan sahmana ak tirowy , u(x) fownkcian patkanowm C(Q)-in tva mijini hatkowyamb: Ayd depqowm kam Lemma 4.4.2

u(x) ≡ const,

x ∈ Q,

kam min u < u(x) < max u, ∂Q

∂Q

x∈Q:

Iharke, tei owni na het yal aveli owyl pndowm:

Dicowq Q-n Rn tara owyan sahmana ak tirowy , u(x) fownkcian patkanowm C(Q) tva mijini hatkowyamb: Ayd depqowm Lemma 4.4.3

min u ≤ u(x) ≤ max u, ∂Q

∂Q

x∈Q:

Qani or Q tirowyowm harmonik fownkcian tva mijini hatkowyamb, apa Lemma 4.4.2-ic

Lemma 4.4.3-ic anmijapes het owm en

het yal pndowmner:

eorem 4.4.1 (Me agowyn areqi skzbownq)

tara owyan sahmana ak tirowy , C(Q)-in harmonik : Ayd depqowm kam u(x) ≡ const,

u(x)

Dicowq Q-n Rn fownkcian patkanowm

x ∈ Q,

kam min u < u(x) < max u, ∂Q

∂Q

x∈Q:

eorem 4.4.2 (Me agowyn areqi owyl skzbownq)

tara owyan sahmana ak tirowy , u(x) C(Q) bazmowyan harmonik : Ayd depqowm min u ≤ u(x) ≤ max u, ∂Q

∂Q

Dicowq Q-n Rn fownkcian patkanowm

x∈Q:

§ 5. Dirixlei xndir: Low man miakowyown

anndhat

kaxva owyown ezrayin fownkciayic

Dicowq Q-n Rn tara owyan sahmana ak tirowy : bazmowyan patkano u(x) fownkcian kovowm u = f (x),

x ∈ Q,

u∂Q = ϕ(x),

C 2 (Q) ∩ C(Q)

(4.14) (4.15)

Dirixlei xndri low owm (f (x) ϕ(x) trva fownkcianer en), ee ayn Q tirowyowm bavararowm (4.14) havasarman, isk ∂Q ezri vra (4.15) ezrayin paymanin: Low man sahmanowmic aknhaytoren het owm , or (4.14), (4.15) xndri low eliowyan hamar anhraet , or havasarman aj mas ezrayin fownkcian linen anndhat. f ∈ C(Q), ϕ ∈ C(∂Q): eorem 4.4.2 - ic bxowm en het yal erkow pndowmner:

eorem 4.5.1 (Miakowyan eorem)

(4.14), (4.15)

xndir

karo

ownenal mekic aveli low owm:

Apacowyc: Enadrenq haka ak: Dicowq

fownkcianer (4.14), (4.15) xndri low owmner en: Ayd depqowm u(x) = u1 (x)−u2 (x) fownkcian u = 0,

u1 (x)

u2 (x)

x ∈ Q,

u∂Q = 0,

hamase xndri low owm : Qani or u ∈ C(Q) fownkcian harmonik Q-owm max u = min u = 0, apa st eorem 4.4.2-i` u(x) ≡ 0, x ∈ Q: eoremn ∂Q ∂Q apacowcva :

eorem 4.5.2 (Ezrayin fownkciayic low man anndhat kaxva owyan masin) u (x) u (x) Dicowq

fownkcianer

⎧ ⎪ ⎪ ⎪ ⎨u1 = f (x), ⎪ ⎪ ⎪ ⎩ u1

∂Q

x ∈ Q,

= ϕ1 (x),

⎧ ⎪ ⎪ ⎪ ⎨u2 = f (x), ⎪ ⎪ ⎪ ⎩ u2

∂Q

x ∈ Q,

= ϕ2 (x),

xndirneri low owmner en: Ayd depqowm, ee or

ε>0

hamar

|ϕ1 (x) − ϕ2 (x)| ≤ ε,

x ∈ ∂Q,

(4.16)

|u1 (x) − u2 (x)| ≤ ε,

x∈Q:

(4.17)

apa

Apacowyc: Ditarkenq u(x) = u (x) − u (x) tarberowyown: u(x) fownk1

cian

⎧ ⎪ ⎪ ⎪ ⎨u = 0, ⎪ ⎪ ⎪ ⎩ u

∂Q

x ∈ Q,

= ϕ(x),

xndri low owm , orte ϕ(x) = ϕ1(x) − ϕ2(x): st eorem 4.4.2-i havasarowyan` −ε ≤ min ϕ ≤ u(x) ≤ max ϕ ≤ ε, ∂Q

∂Q

(4.16) an-

x ∈ Q,

orteic het owm (4.17): eoremn apacowcva : Min Dirixlei xndri low man goyowyan harci owsowmnasirowyann ancnel` nerkayacnenq Adamari rinak, or cowyc talis, or Koii xndir Laplasi havasarman hamar drva o ko ekt, orovhet bacakayowm low man anndhat kaxva owyown skzbnakan fownkcianeric: Adamari rinak: Ditarkenq Koii het yal xndirner Laplasi havasarman hamar`

u0 (x, t) ≡ 0

(K0 )

⎧ ⎪ ⎪ ⎪ ⎪ u0tt + u0xx = 0, x ∈ R1 , t > 0, ⎪ ⎪ ⎪ ⎪ ⎨ u0 t=0 = 0, x ∈ R1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩u0t = 0, x ∈ R1 , t=0

(Kn )

⎧ ⎪ ⎪ ⎪ ⎪ untt + unxx = 0, x ∈ R1 , t > 0, ⎪ ⎪ ⎪ ⎪ ⎨ un t=0 = 0, x ∈ R1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩unt = 1 sin nx, x ∈ R1 : t=0 n

un (x, t) =

sh nt sin nx, n = 1, 2, ..., x ∈ R1 , t ≥ 0, fownkcianer n2 (K0 ) (Kn ) xndirneri low owmner en: (Kn ) xndrowm

hamapatasxanabar skzbnakan fownkcian havasaraa st x ∈ R1 gtowm zroyi, erb n → ∞, aysinqn` (K0 ) xndri skzbnakan fownkciayin: Sakayn, erb x = πj , j = 0, ±1, ..., un (x, t) − u0 (x, t) tarberowyown i gtowm zroyi, erb n → ∞:

§ 6. Oork fownkciayi nerkayacowm gndowm: Grini fownkcian gndi hamar

Ditarkenq n = 3 depq: Dicowq u ∈ C 2 (|x| ≤ R): Ayd depqowm, st (4.8) nerkayacman, cankaca x, |x| < R, keti hamar tei owni het yal havasarowyown. u(x) = −

4π

|y|≤R

−

u(y) dy + |x − y| 4π

4π

u(y) |y|=R

∂ ∂ν y

|y|=R

∂u(y) dSy − ∂ν |x − y|

|x − y|

dSy :

(4.18)

Vercnenq kamayakan ξ ket, or i patkanowm {|x| ≤ |ξ| > R: Ayd depqowm tei owni het yal havasarowyown. 0=−

4π

|y|≤R

−

u(y) dy + |ξ − y| 4π

4π

u(y) |y|=R

|y|=R

∂ ∂ν y

ak gndin.

∂u(y) dSy − ∂ν |ξ − y|

|ξ − y|

dSy :

(4.19)

Iroq, (4.19) havasarowyown stanalow npatakov {|y| ≤ R} gndowm kira enq Grini (4.3) erkrord bana u(y) 4π|ξ−1− y| fownkcianeri hamar. u(y)y |y|≤R

4π

|y|=R

−1 4π|ξ − y|

∂u(y) 1 dSy − ∂ν |ξ − y| 4π

u(y) dy = 4π|ξ − y|

u(y) |y|=R

∂ ∂ν y

|ξ − y|

dSy ,

havi a nenq, or 4π|ξ1− y| fownkcian harmonik {|y| < R} gndowm: Bazmapatkenq (4.19) havasarowyown kamayakan d(ξ) (|ξ| > R) anndhat fownkciayov stacva havasarowyown andam a andam hanenq (4.18) havasarowyownic: Kstananq, or cankaca x, |x| < R, keti

hamar tei owni het yal havasarowyown` d(ξ) − u(y) dy+ u(x) = 4π |ξ − y| |x − y| |y|≤R

4π

+ 4π

∂u(y) ∂ν

|y|=R

u(y) |y|=R

∂ ∂νy

d(ξ) − |x − y| |ξ − y|

dSy +

d(ξ) − |ξ − y| |x − y|

dSy :

(4.20)

Mer npatakn ` yowraqanyowr x, |x| < R, keti hamar gtnel aynpisi ξ , |ξ| > R, ket (ξ = ξ(x))

tei ownena

d(ξ) = d (ξ(x)) fownkcia, or {|y| = R} sferayi vra d(ξ) ≡ , |x − y| |ξ − y|

|y| = R,

(4.21)

nowynowyown: Ayd depqowm (4.20) havasarowyan aj masi erkrord gowmarelin havasar klini zroyi: ntrenq ξ = ξ(x) ket ξ = a(x) x

tesqov: Gtnenq a(x) fownkcian: st (4.21)-i ownenq |ax − y|2 ≡ d2 |x − y|2 ,

|y| = R,

orteic (a2 − d2 )|x|2 + R2 (1 − d2 ) ≡ 2(x, y)(a − d2 ),

|y| = R :

Vercnelov a = d2 , kownenanq d2 (d2 − 1)|x|2 + R2 (1 − d2 ) ≡ 0, (d2 − 1) d2 |x|2 − R2 ≡ 0,

orteic het owm , or d=

R |x|

|y| = R, |y| = R,

(hamaayn

(4.21)

paymani

d > 0):

Nkatenq, or

d2 ≡ 1

depq mer

pahanjnerin i bavararowm, qani or ayd depqowm stacvowm

ξ(x) = x, or

het abar

|ξ| > R:

|ξ| = |x| < R,

a ≡ 1,

in hakasowm mer ayn enadrowyan,

Ev ayspes, ee vercnenq

d(ξ(x)) = d(x) = (nkatenq, or ays depqowm

R , |x|

R2 R2 |x| = > R), |x|2 |x| (4.20)-i, tei owni

(4.21)

nowynowyown

P (x, y)u(y) dSy

(4.22)

|ξ| =

tei owni: Het abar, st

apa

G(x, y)u(y) dy +

u(x) =

R2 x |x|2

ξ=

|y|≤R

|y|=R

havasarowyown, or

u = f (x), |x| < R, u|x|=R = ϕ,

(4.23) (4.24)

Dirixlei xndri low man hamar kndowni het yal tesq (ayste en-

u ∈ C 2 (|x| ≤ R)) G(x, y)f (y) dy + u(x) =

adrvowm , or

|y|≤R

orte

G(x, y) =

P (x, y)ϕ(y) dSy ,

|y|=R

$ 2 R − + R |x| 2 x − y 4π |x − y| |x| P (x, y) =

(4.25)

∂ G(x, y), ∂νy

−1

% ,

|y| ≤ R, |x| < R,

|y| = R, |x| < R :

(4.26)

G(x, y) fownkcian kovowm (4.23), (4.24) xndri Grini fownkcia, isk P (x, y) fownkcian kovowm

(4.23), (4.24) xndri Powasoni mijowk : Powasoni mijowki

hamar kareli stanal, oro imastov, aveli parz tesq: Hamaayn

(4.21)-i, P (x, y) =

y ∂ = G(x, y) = ∇y G(x, y), ∂νy R

(4.26)-i

4π

−

4π

−∇y

d y + ∇y , |x − y| |ξ − y| R

d(ξ − y) y x−y + , |x − y|3 |ξ − y|3 R

4π

−

ξ−y y x−y + , |x − y|3 d2 |x − y|3 R

⎛ ⎞ R2 x − y y |x| ⎜ ⎟ |x|2 , ⎠ = = ⎝−x + y + 4π|x − y|3 R2 R = 4π|x − y|3 =

Kamayakan het yal tesq`

G(x, y) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σ ⎪ n ⎪ ⎪ ⎪ ⎪ ⎨

n ≥ 2

|x|2 y 1− 2 R

R2 − |x|2 , 4πR|x − y|3

depqowm

y R

|y| = R, |x| < R :

(4.23), (4.24)

⎛

R |x|

n−2

(4.27)

xndri Grini fownkcian owni

⎞

⎜ ⎟ ⎜ ⎟ + ⎜− ⎟ , |y| ≤ R, |x| < R, n−2 ⎠ ⎝ |x − y|n−2 R2 x − y |x|2

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ R2 ⎪ ⎪ y − |x| x ⎪ ⎪ |x|2 ⎪ ⎪ ⎩− ln , 2π R|x − y|

|y| ≤ R, |x| < R,

erb

erb n > 2,

n = 2,

isk Powasoni mijowk` P (x, y) =

R2 − |x|2 , σn R|x − y|n

|y| = R, |x| < R,

orte σn miavor sferayi maker owyi makeresn Rn -owm: Menq cowyc tvecinq, or ee (4.23), (4.24) xndri low owm goyowyown owni patkanowm C 2 (|x| ≤ R) bazmowyan, apa ayd low owmn owni (4.25) tesq: Hiatakva xndri low man goyowyan veraberyal nenq miayn, or

ee f (x)

ϕ(x) fownkcianer bavararowm en oroaki paymanneri, apa

(4.25) bana ov trva u(x) fownkcian ayd xndri low owm : Tei owni

het yal pndowm, or nerkayacnowm enq a anc apacowyci:

eorem 4.6.1 Ee f ∈ C(|x| ≤ R) ∩ C (|x| < R), ϕ ∈ C(|x| = R), apa

(4.23), (4.24)

Dirixlei xndri low owm goyowyown owni

trvowm

(4.25)

bana ov: § 7. Laplasi havasarman hamar Dirixlei xndri low man goyowyown gndowm

Ays paragrafowm kapacowcenq eorem 4.6.1- f (x) ≡ 0 masnavor depqowm: Tei owni het yal pndowm:

eorem 4.7.1 Ee ϕ ∈ C(|x| = R), apa u = 0,

|x| < R,

(4.28)

u|x|=R = ϕ,

(4.29)

Dirixlei xndri low owm goyowyown owni trvowm P (x, y)ϕ(y) dSy ,

u(x) =

|x| < R,

(4.30)

|y|=R

bana ov:

Apacowyc: Apacowyc katarenq n = 3 depqi hamar: Nax cowyc tanq,

or u(x) fownkcian patkanowm C 2 (|x| < R) bazmowyan

harmonik :

(4.27) Powasoni mijowk nerkayacnenq het yal tesqov` P (x, y) =

R2 − |x − y|2 − |y|2 − 2(x − y, y) R2 − |(x − y) + y|2 = = 4πR|x − y| 4πR|x − y|3

=−

(x − y, y) , − 4πR|x − y| 2πR|x − y|3

|y| = R, |x| < R :

Qani or, erb |y| = R, |x| < R, tei owni

y (x − y, y) , = R ∇y |x − y|3 |x − y| R

∂ ∂νy

|x − y|

havasarowyown (nkatenq, or νy = Ry ), apa 1 ∂ − P (x, y) = − 4πR|x − y| 2π ∂νy (4.30)

|x − y|

bana kareli grel het yal tesqov`

u(x) = −

4πR

|y|=R

ϕ(y) dSy − |x − y| 2π

ϕ(y) |y|=R

∂ ∂νy

|y| = R, |x| < R,

|x − y|

dSy ,

|x| < R :

Stacva bana cowyc talis, or u(x) fownkcian parz erti krknaki erti potencialneri gowmar : Het abar, u-n patkanowm C ∞(|x| < R) bazmowyan harmonik {|x| < R} gndowm: Aym cowyc tanq, or u(x) fownkcian patkanowm C(|x| ≤ R) bazmowyan bavararowm (4.29) ezrayin paymanin: Vercnenq kamayakan x0 ket, |x0 | = R, cowyc tanq, or u(x) → ϕ(x0 ),

erb

x → x0 , |x| < R :

(4.31)

Vercnenq kamayakan ε > 0: Qani or ϕ(x) fownkcian anndhat x0 ketowm, apa goyowyown owni aynpisi δ > 0, or |ϕ(y) − ϕ(x0 )| ≤ ε,

|y − x0 | ≤ δ, |y| = R :

bana ic anmijapes bxowm (kira elov ayn fownkciayi hamar), or (4.22)

(4.32)

u(x) ≡ 1, |x| ≤ R,

P (x, y) dSy = 1 : |y|=R

Het abar, u(x) − ϕ(x0) tarberowyown (|x| < R) karo enq nerkayacnel het yal tesqov` u(x) − ϕ(x0 ) =

P (x, y)ϕ(y) dSy − |y|=R

|y|=R

P (x, y)ϕ(x0 ) dSy =

P (x, y) ϕ(y) − ϕ(x0 ) dSy =

= |y|=R

P (x, y) ϕ(y) − ϕ(x0 ) dSy +

= S1 (δ)

P (x, y) ϕ(y) − ϕ(x0 ) dSy =

S2 (δ)

= I1 (x) + I2 (x),

orte S1 (δ) = {|y| = R} ∩ {|y − x0 | ≤ δ}, S2 (δ) = {|y| = R} ∩ {|y − x0 | > δ}: I2 (x) integralner: st (4.32)-i` ownenq

Gnahatenq I1 (x) |I1 (x)| ≤

P (x, y) ϕ(y) − ϕ(x0 ) dSy ≤ ε

S1 (δ)

P (x, y) dSy ≤

S1 (δ)

P (x, y) dSy = ε,

≤ε

|x| < R

(4.33)

|y|=R

(gtvecinq na ayn astic, or Powasoni mijowk obacasakan ): Nanakenq M = max |ϕ(x)|: |x|=R

|I2 (x)| ≤

P (x, y) ϕ(y) − ϕ(x0 ) dSy ≤

S2 (δ)

P (x, y) dSy ,

≤ 2M

|x| < R :

(4.34)

S2 (δ)

δ Vercnenq x − x0 < ,

|x| < R: Ee y ∈ S2 (δ), apa

δ δ |x − y| = (y − x0 ) − (x − x0 ) ≥ y − x0 − x − x0 ≥ δ − = : (4.34) gnahatakanic stanowm enq |I2 (x)| ≤ 2M

P (x, y) dSy = 2M

S2 (δ)

≤

M 2πR

S2 (δ)

R2 − |x|2 M (R2 − |x|2 ) dSy ≤ 4πR2 , (δ/2) 2πR(δ/2)3

R2 − |x|2 dSy ≤ 4πR|x − y|3 x − x0 < δ ,

|x| < R :

Vercnelov x0-in bavakanaa mot x, |x| < R, kstananq |I2 (x)| ≤ ε : (4.33) x-eri

(4.35)

(4.35) gnahatakanneric stanowm enq, or x0 -in bavakanaa mot

hamar (|x| < R)

|u(x) − ϕ(x0 )| ≤ |I1 (x)| + |I2 (x)| ≤ 2ε,

orteic het owm (4.31)-: eoremn apacowcva : Hajord paragrafner nvirva en apacowcva eoremi oro kar or kira owyownnerin: § 8. Mijini masin hakadar eorem

eorem 4.8.1 (Mijini masin hakadar eorem)

Dicowq Q-n Rn tara owyan kamayakan tirowy , u(x) fownkcian anndhat Q-owm tva mijini hatkowyamb: Ayd depqowm u(x) fownkcian harmonik Q-owm: Apacowyc: Vercnenq kamayakan x0 ∈ Q ket o R > 0 aynpisin , or x0 kentronov R a avov ak gownd nka Q tirowyowm. BR (x0 ) = {|x − x0 | < R} Q: Qani or x0 ∈ Q ket kamayakan , apa eoremn apacowcelow hamar bavarar apacowcel, or u(x) fownkcian harmonik BR (x0)-owm: v(x)-ov nanakenq BR (x0) gndowm x ∈ BR (x0 ),

v = 0, v ∂B

R (x

= u∂B

R (x

Dirixlei xndri low owm: Qani or u∂B (x ) ∈ C(∂BR (x0)), apa st eorem 4.7.1-i` v(x) low owm goyowyown owni: Ditarkenq u(x) − v(x), x ∈ B R (x0), fownkcian: Ays fownkcian patkanowm C(B R (x0)) bazmowyan BR (x0) R

gndowm tva mijini hatkowyamb, qani or u(x) fownkcian tva mijini hatkowyamb` st eoremi paymani, na v(x) fownkcian tva

mijini hatkowyamb, qani or harmonik : st Lemma 4.4.3-i` tei ownen het yal anhavasarowyownner. min (u − v) ≤ u(x) − v(x) ≤ max0 (u − v),

∂BR (x0 )

∂BR (x )

x ∈ B R (x0 ) :

Myows komic (u − v)∂BR (x0 ) = 0, het abar B R (x0 )-owm u(x) ≡ v(x), aysinqn` u(x) fownkcian harmonik BR (x0 ) gndowm: eoremn apacowcva :

Ditoowyown: Nkatenq, or eoremi apacowyci nacqowm pahanjvec Dirixlei xndri low man bacahayt tesq, ayl pahanjvec miayn Dirixlei xndri low man goyowyown:

Ditarkowm: Ditarkenq Laplasi peratori (dicowq n = 2) parzagowyn tarberakan motarkowm: Bavakanaa

oqr h-i depqowm u = ux1 x1 + ux2 x2 ≈ +

[u(x1 − h, x2 ) − 2u(x1 , x2 ) + u(x1 + h, x2 )] + h2

[u(x1 , x2 − h) − 2u(x1 , x2 ) + u(x1 , x2 + h)] : h2

Da nanakowm , or u = 0 Laplasi havasarowm oxarinvowm

u(x1 , x2 ) =

[u(x1 − h, x2 ) + u(x1 + h, x2 ) + u(x1 , x2 − h) + u(x1 , x2 + h)]

havasarowyamb, or o ayl in , qan mijini hatkowyan artahaytowyown, orovhet u(x) fownkciayi areq (x1 , x2 ) kentronakan ketowm havasar ayd keti (x1 −h, x2 ), (x1 +h, x2 ), (x1 , x2 −h), (x1 , x2 +h) har an ors keterowm fownkciayi areqneri mijin vabanakanin:

§ 9. Veracneli ezakiowyan masin eorem

eorem 4.9.1 (Veracneli ezakiowyan masin) Dicowq Q-n Rn tara owyan tirowy , x0 ∈ Q or ket , isk u(x) fownkcian harmonik

Q \ {x0 }-owm:

u(x) = o U (x − x0 ) , orte

ketowm kareli oroel aynpes, or stacva fownkcian

lini harmonik amboj depqowm

(4.36)

tirowyowm:

paymann owni het yal tesq,

n=2

|x − x0 |

erb

x → x0 ,

u(x) = o ln |x − x0 | ,

erb

x → x0 :

u(x) = o isk

(4.36)

U -n Laplasi havasarman fowndamental low owmn , apa u(x)

fownkcian

n=3

x → x0 ,

erb

(4.37)

depqowm

Apacowyc: Apacowyc katarenq n = 3 depqi hamar: Vercnenq aynpisi R > 0, or x0 kentronov R a avov ak gownd nka lini Q tirowyowm. BR (x0 ) = {|x − x0 | < R} Q, ditarkenq u(x) fownkcian BR (x0 ) \ {x0 }-owm: v(x)-ov nanakenq BR (x0 ) gndowm v = 0, v ∂B

R (x )

x ∈ BR (x0 ), = u∂B

R (x

(4.38)

Dirixlei xndri low owm: v(x) fownkcian goyowyown owni patkanowm C(B R (x0 ))-in: eoremn apacowcelow hamar bavarar cowyc tal, or u(x) v(x) fownkcianer hamnknowm en irenc oroman tirowyneri ndhanowr masowm` B R (x0 ) \ {x0 }-owm, aysinqn` kamayakan x1 ∈ BR (x0 ) \ {x0 } keti hamar u(x1 ) = v(x1 ) :

(4.39)

Ev ayspes, dicowq x1 ∈ BR (x0 ) \ {x0 } kamayakan ket : Vercnenq cankaca ε > 0 iv {ρ < |x−x0 | < R} tirowyowm, orte 0 < ρ < |x1 −x0 |,

ditarkenq het yal w± (x) = ± (u(x) − v(x)) + ε

|x1 − x0 | |x − x0 |

erkow fownkcianer, orte w+ - hamapatasxanowm havasarowyan aj masowm + nanin, w− - hamapatasxanowm − nanin: Ankhayt , or w+ (x), w− (x) fownkcianer harmonik en {ρ < |x − x0 | < R} tirowyowm w± ∈ C(ρ ≤ |x − x0 | ≤ R): {ρ < |x − x0 | < R} tirowyi {|x − x0 | = R} artaqin ezri vra, st (4.38)-i, |x1 − x0 | w± (x)|x−x0 |=R = ε >0: |x − x0 | {ρ < |x − x0 | < R} tirowyi {|x − x0 | = ρ} nerqin ezri vra, st (4.37)-i, |x1 − x0 | = w± (x)|x−x0 |=ρ = ± (u(x) − v(x)) |x−x0 |=ρ + ε ρ |x1 − x0 | =ε +o , erb ρ → 0 ρ ρ qani or v ∈ C(B R (x0 )), apa v(x) = o , erb x → x0 : |x − x | ntrenq ρ iv (ρ < |x1 − x0 |) aynqan oqr, or w± (x)∂{ρ<|x−x0 |<R} > 0 :

Ayd depqowm, hamaayn me agowyn areqi skzbownqi` w± (x) > 0,

ρ ≤ |x − x0 | ≤ R,

masnavorapes, w± (x1 ) > 0,

orteic het owm , or |u(x1 ) − v(x1 )| < ε :

Qani or ε > 0 kamayakan , apa tei owni (4.39) havasarowyown: eoremn apacowcva :

§ 10. Liowvili eorem

Kasenq, or amboj tara owyan mej orova u(x) fownkcian sahmana ak ver ic (nerq ic) , ee goyowyown owni aynpisi M hastatown, or u(x) ≤ M

(u(x) ≥ M ) ,

eorem 4.10.1 (Liowvili eorem)

x ∈ Rn :

Amboj Rn tara owyan mej

orova ver ic kam nerq ic sahmana ak harmonik fownkcian hastatown :

Apacowyc:

Dicowq u(x) harmonik fownkcian sahmana ak ver ic (nerq ic): Ayd depqowm M − u(x) fownkcian (u(x) − M fownkcian) harmonik obacasakan: Owsti, Liowvili eorem bavarar apacowcel miayn masnavor depqi hamar, erb u(x) ≥ 0: Cowyc tanq, or Rn -owm obacasakan harmonik fownkcian hastatown : Dicowq Rn -owm harmonik u(x) fownkcian obacasakan ` u(x) ≥ 0 : Vercnenq kamayakan x0 ∈ Rn , |x0 | = 0, ket cowyc tanq, or u(x0 ) = u(0) :

(4.40)

Dicowq R > |x0 |: st eorem 4.7.1-i ownenq P (x, y)u(y) dSy ,

u(x) =

|x| < R,

|y|=R

masnavorapes, u(x0 ) =

P (x0 , y)u(y) dSy :

|y|=R

Parzowyan hamar enadrenq n = 3: Qani or {|y| = R} sferayi y keteri hamar R − |x0 | ≤ |x0 − y| ≤ R + |x0 |,

apa

R2 − |x0 |2 R2 − |x0 |2 ≤ P (x , y) ≤ : 4πR(R + |x0 |)3 4πR(R − |x0 |)3

Stacva anhavasarowyownner bazmapatkelov integrelov {|y| = R} sferayov` kstananq R2 − |x0 |2 4πR(R + |x0 |)3

orteic

u(y) dSy ≤ u(x0 ) ≤

|y|=R

u(y)-ov (u(y) ≥ 0)

R2 − |x0 |2 4πR(R − |x0 |)3

u(y) dSy , |y|=R

R(R2 − |x0 |2 ) R(R2 − |x0 |2 ) u(0) ≤ u(x ) ≤ u(0) : (R + |x0 |)3 (R − |x0 |)3

Stacva anhavasarowyownner tei ownen cankaca R > |x0| hamar: Ancnelov sahmani, erb R → ∞, kstananq (4.40) havasarowyown: eoremn apacowcva : § 11. Neymani xndir Laplasi havasarman hamar gndowm

Ays paragrafowm kditarkenq Laplasi havasarman hamar s mek ezrayin xndir` Neymani xndir (kam erkrord ezrayin xndir): Menq ayn kowsowmnasirenq miayn ayn depqowm, erb tirowy gownd : Ayd xndir het yaln . u = 0, |x| < R, ∂u = f, ∂ν |x|=R

(4.41) (4.42)

orte ν -n {|x| = R} sferayin tarva {|x| < R} gndi nkatmamb artaqin miavor normaln : Qani or kamayakan 0 < ρ ≤ R hamar {|x| = ρ} sferayi vra ν = xρ , apa ∂u = (∇u, x) = ur , ∂ν |x|=ρ ρ |x|=ρ |x|=ρ

masnavorapes,

∂u = (∇u, x) = ur : ∂ν |x|=R R |x|=R |x|=R

(4.43)

Kasenq, or u(x) fownkcian (4.41), (4.42) Neymani xndri low owm , ee u ∈ C 2 (|x| < R) ∩ C(|x| ≤ R) ∩ {(∇u, x) ∈ C(|x| ≤ R)},

(4.44)

bavararowm (4.41) havasarman (4.42) ezrayin paymanin: Aknhayt , or low man goyowyan hamar anhraet , or f ∈ C(|x| = R) :

(4.45)

Baci ayd, st eorem 4.1.1-i, cankaca ρ < R hamar tei owni

∂u dS = ∂ν

|x|=ρ

(∇u, x) dS = 0 ρ

|x|=ρ

havasarowyown: Qani or (∇u, x)- anndhat {|x| ≤ R} gndowm tei owni (4.42) ezrayin payman, apa verjin havasarowyan mej ancnelov sahmani, erb ρ → R, kstananq low man goyowyan hamar s mek anhraet payman` f dS = 0 :

(4.46)

|x|=R

Hetagayowm cowyc ktanq, or (4.45) (4.46) paymanner Neymani xndri low man goyowyan hamar o miayn anhraet en, ayl na bavarar en: Nax andrada nanq Neymani xndri low man miakowyan harcin: Aknhayt , or Neymani xndri low owm miak , qani or ee u(x)- low owm , apa u(x) + C fownkcian s nowyn xndri low owm , orte C -n cankaca

hastatown : Tei owni het yal pndowm:

eorem 4.11.1 (Low man ndhanowr tesqi masin) Ee

u1 (x)

u2 (x)

fownkcianer mi nowyn Neymani xndri low owmner en, apa goyowyown owni aynpisi C hastatown, or u1 (x) = u2 (x) + C,

|x| < R,

ayl xosqov, Neymani xndri low owm miakn hastatown gowmarelii towyamb:

Apacowyc: Ee

fownkcianer mi nowyn Neymani xndri low owmner en, apa u(x) = u1 (x) − u2 (x) fownkcian u1 (x)

u2 (x)

u = 0,

|x| < R,

∂u = 0, ∂ν |x|=R

(4.47)

hamase paymannerov Neymani xndri low owm : Cowyc tanq, or u(x) ≡ C,

Qani or cankaca

a ajin bana i

depqowm

ρ < R

uu dx = |x|≤ρ

|x| < R :

(4.48)

u ∈ C 2 (|x| ≤ ρ),

∂u dS − ∂ν

|x|=ρ

apa st Grini

∇u∇u dx, |x|≤ρ

orteic stanowm enq ρ

|∇u|2 dx,

u(x)(∇u, x) dS = |x|=ρ

ρ<R:

|x|≤ρ

Qani or u(x) (∇u, x) fownkcianer patkanowm en C(|x| ≤ R) bazmowyan, apa st (4.47) paymani ρ→R ρ

u(x)(∇u, x) dS =

lim

|x|=ρ

u(x)(∇u, x) dS = 0, |x|=R

orteic het owm

lim

ρ→R |x|≤ρ

(4.49)

|∇u|2 dx = 0 :

(4.49)

havasarowyown karo tei ownenal miayn ayn depqowm, erb

|∇u|2 dx = 0

cankaca ρ < R hamar :

(4.50)

|x|≤ρ

(4.50)-ic

het owm , or apacowcva :

|∇u(x)| ≡ 0, |x| < R,

tei owni

(4.48)-:

eoremn

Aym andrada nanq Nanakenq v(x)-ov

(4.41), (4.42) v = 0, v

|x|=R

Neymani xndri low man goyowyan: |x| < R,

(4.51)

= f,

(4.52)

Dirixlei xndri low owm: Tei owni het yal pndowm: eorem 4.11.2 (Goyowyan masin) Dicowq tei ownen (4.45) (4.46) paymanner: Ayd depqowm (4.41), (4.42) Neymani xndri low owm goyowyown owni trvowm 1 u(x) = R

v(x · t) dt + const, t

|x| < R,

(4.53)

bana ov, orte v(x) fownkcian (4.51), (4.52) Dirixlei xndri low owmn : Apacowyc: (4.45) paymanic het owm , or (4.51), (4.52) Dirixlei xndri v(x) low owm goyowyown owni: (4.46) paymanic het owm , or v(0) = 0 :

Iroq (enadrenq n = 3), st maker owayin mijini masin eoremi, cankaca ρ < R depqowm v(0) =

v(x)

4πρ2

ρ→R 4πρ2

v(x) dS = lim |x|=ρ

v(x) dS = |x|=ρ

4πR2

f dS = 0 : |x|=R

fownkcian patkanowm C 2 (|x| < R)-in st eylori bana i n Ci xi + o |x|2 , v(x) = v(0) + (∇v(0), x) + o |x|2 = i=1

∂v(x) Ci = , i = 1, ..., n: ∂xi x=0

orte Het abar

v(x · t) Ci xi + o |x|2 t = t i=1 n

t = 0 depqowm (4.53) enaintegralayin artahaytowyown ezakiowyown

owni: Nkatenq, or ee {|x| ≤ ρ} gndi x keteri hamar ditarkenq y = xt, 0 ≤ t ≤ 1, keter, apa y keter k oxven nowyn {|y| ≤ ρ} gndowm. |y| = |x · t| = |x|t ≤ ρt ≤ ρ: Orpes (4.51), (4.52) Dirixlei xndri low owm` v(x)

fownkcian patkanowm C(|x| ≤ R)-in, het abar (4.53) bana ov trva

u(x) fownkcian s patkanowm C(|x| ≤ R)-in: Baci ayd, cankaca ρ < R

hamar v ∈ C 2 (|x| ≤ ρ)

1 ∂v(y) ∂v(y) v(xt) = ·t= , t t ∂yi y=xt ∂yi y=xt v(xt) ∂2 ∂ 2 v(y) · t, i, j = 1, ..., n, = ∂xi ∂xj t ∂yi ∂yj y=xt ∂ ∂xi

het abar, (4.53) bana ov trva u(x) fownkcian

s patkanowm

C (|x| ≤ ρ)-in

1 u(x) = R

y v(y)

y=xt

· t dt ≡ 0,

|x| ≤ ρ :

Ayspisov, u(x) fownkcian patkanowm C 2 (|x| < R)∩C(|x| ≤ R) bazmowyan

harmonik {|x| < R} gndowm: Aym cowyc tanq, or (∇u, x) fownkcian patkanowm C(|x| ≤ R)-in tei

owni (4.42) ezrayin payman: (4.53) bana grenq sferik koordinatakan hamakargowm: Ee x = (x1 , x2 , x3 ) = (r, ϕ, θ), apa y = xt keti sferik koordinat klini (rt, ϕ, θ): Owsti, sferik koordinatakan hamakargowm (4.53) bana kownena het yal tesq. 1 u(r, ϕ, θ) = R

v(rt, ϕ, θ) dt + const = R t

v(ρ, ϕ, θ) dρ + const : ρ

(4.54)

Qani or v ∈ C(|x| ≤ R), apa, hamaayn (4.54) bana i, rur (ρ, ϕ, θ) fownkcian patkanowm C(|x| ≤ R)-in, rur (ρ, ϕ, θ) ∈ C(|x| ≤ R): Het abar, (∇u, x) ∈ C(|x| ≤ R) (rur (ρ, ϕ, θ) = (∇u, x))

tei owni (4.42) ezrayin pay-

man.

∂u Rv = ur = = v =f : ∂ν |x|=R ρ ρ=R |x|=R |x|=R

eoremn apacowcva :

Grakanowyown 1. В . П . М и х а й л о в, Лекции по уравнениям математической

физики, Учеб. пособие для вузов, М.: Издательство физикоматематической литературы, 2001 2. В . П . М и х а й л о в, Дифференциальные уравнения в частных про-

изводных, М.: Наука, 1983 3. А . Н . Т и х о н о в , А . А . С а м а р с к и й, Уравнения матема-

тической физики, М.: Наука, 1977 4. И . Г . П е т р о в с к и й, Лекции об уравнениях с частными произ-

водными, М.: Наука, 1970 5. В . С . В л а д и м и р о в, Уравнения математической физики, М.:

Наука, 1988 6. В . П . М и х а й л о в , А . К . Г у щ и н, Дополнительные главы

курса "Уравнения математической физики" , Лекционные курсы НОЦ, Вып. 7, М.: МИАН, 2007 7. Л . Д . К у д р я в ц е в, Основы математического анализа, Т. 1-3,

М.: Высшая школа, 1988 8. Г . М . Ф и х т е н г о л ь ц, Курс дифференциального и интегрально-

го исчисления, Т. 1-3, М.: Наука, 1970 9. А . М . М а л ь ц е в, Основы линейной алгебры, М.: Наука, 1975 10. Л . С . П о н т р я г и н, Обыкновенные дифференциальные уравне-

ния, М.: Наука, 1982 11.

B.G. Ararqcyan, A.H. Hovhannisyan, .L. ahb a y a n, Maematikakan fizikayi havasarowmner , E P H,

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