Problems for equations with special parabolic operator of fractional differentiation
We establish the well-posedness of the Cauchy problem and the two-point boundary-value problem for an equation with an operator of fractional differentiation that corresponds to the singular parabolic Beltrami – Laplace operator on a surface of the Dini class.
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| Date: | 2009 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
2009
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3082 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509109669855232 |
|---|---|
| author | Matychuk, M. I. Матійчук, М. І. |
| author_facet | Matychuk, M. I. Матійчук, М. І. |
| author_sort | Matychuk, M. I. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:44:57Z |
| description | We establish the well-posedness of the Cauchy problem and the two-point boundary-value problem for an equation with an operator of fractional differentiation that corresponds to the singular parabolic Beltrami – Laplace operator on a surface of the Dini class. |
| first_indexed | 2026-03-24T02:35:53Z |
| format | Article |
| fulltext |
UDK 517.954
M. I. Matijçuk (Çerniv. nac. un-t)
ZADAÇI DLQ RIVNQN| ZI SPECIAL|NYM PARABOLIÇNYM
OPERATOROM DROBOVOHO DYFERENCIGVANNQ
We establish the correctness of the Cauchy problem and a two-point boundary-value problem for an
equation with operator of fractional differentiation which corresponds to the singular parabolic Beltrami
– Laplace operator on the surface from the Dini class.
Ustanovlena korrektnost\ zadaçy Koßy y dvuxtoçeçnoj kraevoj zadaçy dlq uravnenyq s opera-
torom drobnoho dyfferencyrovanyq, kotor¥j sootvetstvuet synhulqrnomu parabolyçeskomu
operatoru Bel\tramy – Laplasa na poverxnosty yz klassa Dyny.
Zadaçi Koßi i krajovi zadaçi dlq rivnqn\ z operatoramy drobovoho dyferen-
cigvannq vyvçalys\ u bahat\ox robotax (dyv., napryklad, [1 – 5]). Instrumentom
redukci] zadaç do intehral\nyx rivnqn\ [ special\ni operatory.
1. Teoremy pro dig deqkyx intehral\nyx operatoriv. 1.1. Pro funda-
mental\nyj rozv’qzok (f . r.) paraboliçnoho rivnqnnq na poverxni iz klasu
Dini. Rozhlqnemo B-paraboliçne rivnqnnq na poverxni S+ = S × +∞( , )0 u
prostori En
+
:
Λ ∆( ) ( )D u
u
t
B ub
x x
b
n
≡
∂
∂
+ − +( ) =′1 0 , n > 2, (1)
de Bxn
=
∂
∂
2
2xn
+
2ν +1
x xn n
∂
∂
, ν ≥ – 1 / 2, b ≥ 1.
Prypustymo, wo poverxnq S pokryva[t\sq vidkrytymy mnoΩynamy Sl{ } ,
S = Sl∪ i S v Sl vyznaça[t\sq rivnqnnqm xi = ϕi
l x( ) ′′( ) , i = 1 1, n − , u kryvo-
linijnyx koordynatax ′′x = ( , , )x xn1 2… − , do toho Ω ϕi
l( ) ∈ C Tb
i
( , )( )2 ω
. Opera-
tor Laplasa ∆ x na Sl ma[ vyhlqd [5]
∆l
l
ii j
n
lg x
x
g x g= ′′( )
∂
∂
′′( )−
=
−
∑( ) /
,
( )1 2
1
2
ll
ij
j
x
x
′′( ) ∂
∂
,
de gl
ij
— elementy matryci, qka obernena dlq matryci z elementiv
g x
x
x
x
ij
l k
l
ik
n
k
l
( )
( ) ( )
′′( ) =
∂ ′′( )
∂
∂ ′′(
=
−
∑ ϕ ϕ
1
1 ))
∂x j
, g x g xl
ij
l( ) ( )det′′( ) = ′′( )( ) .
Nexaj ψ l{ } — sukupnist\ funkcij, qki utvorggt\ na S rozbyttq odynyci,
ˆ ( )ψ l x′ — hladki funkci] z nosiqmy v Sl i ˆ ( )ψ ′x = 1, qkwo ′x ∈ supp ψ l ⊂ Sl .
Todi na S+
operator Λ( )D nabere vyhlqdu
Λ Λ( ) ( ) ˆD u x D ul
l
l l= ′( ) [ ]∑ ψ ψ ,
de
Λ ∆l
b
l x
b
kj
l
b k j
D
t
B
t
a
n
( ) ( ) ( )=
∂
∂
+ − +( ) =
∂
∂
−
≤ + ≤
1
2 2bb
x
k
x
jx D B
n∑ ′′( ) ′′ .
© M. I. MATIJÇUK, 2009
1088 ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 8
ZADAÇI DLQ RIVNQN| ZI SPECIAL|NYM PARABOLIÇNYM OPERATOROM … 1089
Zaznaçymo, wo tut akj
l( ) ∈ C Tb
l
( , )( )2 1− ω
, k = 2b , i pry k < 2b
akj
l( ) ∈ C Tl
( , )( )0 ω
, oskil\ky S C b∈ ( , )2 ω
.
Poznaçymo çerez T G t x x y
x
l
n
n
nξ
0
( ) , , ;′′ ′′( ) funkcig Hrina zadaçi Koßi dlq riv-
nqnnq z parametrom
∂
∂
= ′′( ) ′′
+ =
∑u
t
a y D B ukj
l
x
k
k j b
x
j
n
( )
2 2
.
Holovna çastyna f. r. rivnqnnq (1) vyznaça[t\sq formulog
G t x x T G t x xl x
l
nn
n
0 0( , , ) , , ;( )ξ β ξ ξξ= ′( ) ′′ − ′′ ′′( )) ′( )
′′( )∑ β ξ
ξ
l
ll g( )
,
de
βl
l
x2 1∑ ′( ) = , supp βl lx S′( ) ⊂ , βl lC S∈ ∞ ( ) .
Teorema 1. Qkwo poverxnq S v En−1 naleΩyt\ klasu C b( , )2 ω
, to f . r.
rivnqnnq (1) vyznaça[t\sq formulog
G t x G t x d G t x y y yn( , , ) ( , , ) ( , , ) ( , , )ξ ξ τ τ τ ξ ν= + −0 0 Φ 00
0
0
S
t
ydS G W
+
∫∫ ≡ + Λ
(2)
i dlq joho poxidnyx spravdΩugt\sq ocinky
D D B G t x Ct Tm
x
k
x
j n k j m b
xn n
n
ξ
ξξ ν( , , ) /≤ − − + + +( )1 2 2 ee c t x− ′( ){ }ρ ξ, ,
,
ν ν0 2 1= + , k j b+ ≤2 2 , m ≤ 1 ,
D D B W t x t Tm
x
k
x
j n k j m b
xn n
n
ξ
ξξ ν
Λ ( , , ) /≤ − − + + +( )2 2 2 ee c t x− ′( ){ }ρ ξ, ,
,
∆ ∆x x
k
x
j n b b
xD B G t x Ct F x t T
n n
( , , ) ,/ξ ν≤ ( )− − +( )1 2 2 ξξ ρ ξn e c t x x x− + ′( ){ }, , ,∆
,
∆ ∆ ∆t x
k
x
j n b b bD B G t x Ct F t t
n
( , , ) ,/ξ ν≤ ( )− − +( )1 2 2 2 tt T eb k j b
x
c t x
n
n2 2 2− −( ) − ′( ){ }/ , ,ξ ρ ξ
,
F t A t d
t
( ) ( )
( )
= ≡ ∫ω
ω τ
τ
τ
0
, F x t F x x t b( , ) /= ( ) + −1 2
,
n nν ν= + +2 1 , 0 2 2< + ≤k j b , 0 2< <∆t tb
.
Rozv’qzok zadaçi Koßi Λ( )D u f= , u xt= =0 ϕ( ) ,
∂
∂
=
=
u
xn xn 0
0 dlq dovil\-
nyx f C∈ ( , )0 ω
, ϕ ∈ +C S( ) vyznaça[t\sq odnoznaçno intehralamy
u t x G t x dS d G t x
S
n
S
( , ) ( , , ) ( ) ( , , )= + −
+
∫ ξ ϕ ξ ξ τ τ ξν
ξ
0
++
∫∫ f dSn
t
( , )τ ξ ξν
ξ
0
0
(3)
i naleΩyt\ klasu C b F( , ) ( )2 Γ .
Z teoremy 1 vyplyvagt\ dvi vaΩlyvi vlastyvosti f. r. rivnqnnq (1).
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 8
1090 M. I. MATIJÇUK
Vlastyvist\ 1 (formula zhortky):
G t x y G y y dS G t x
S
n y( , , ) ( , , ) ( , , )
+
∫ = +τ ξ τ ξν0
.
Vlastyvist\ 2. Qkwo P xk0
( ) — mnohoçlen stepenq k b0 2< , qkyj [ par-
nym po xn , to
G t x y P y y dS P x
S
k n y k( , , ) ( ) ( )
+
∫ =
0
0
0
ν
,
vnaslidok çoho
D B G t x y y x y dS
m r
x
m
x
j
S
r
n
p
y
n
′
+
+
∫ ′ − ′ =
>
( , , ) ( )
,
ν0 2
0 ,, ,
, , ,
( ) , ,( )
j p
C m r j p
P x x m r j
mj
r m n
p j
>
= =
′ < <
−
−
−
1
2 pp,
de
C m j jmj
j− = + +
+
1 4 1
1
1
! ! ( )
( )
Γ
Γ
ν
ν
.
Slid zaznaçyty, wo dlq pobudovy f. r. za formulog (2), tobto za konstruk-
ci[g Levi, dlq wil\nosti Φ( , , )t x ξ potencialiv WΛ pry zastosuvanni ope-
ratora Λ( )D do G t x( , , )ξ otrymu[mo intehral\ne rivnqnnq z kvazirehulqrnym
qdrom, a vyvçennq dyferencial\nyx vlastyvostej poverxnevoho intehrala
zvodyt\sq zhidno z umovamy na poverxng S do vidpovidnyx ob’[mnyx potencialiv,
qk u § 6 [5].
1.2. Teoremy pro dig intehral\nyx operatoriv typu operatoriv drobo-
voho intehruvannq ta dyferencigvannq. Rozhlqnemo funkci] G t xl( ) ( , , )± ξ =
= G t x t b l b( , , ) ( )/ξ − ±2 2
pry t > 0 i G l( )± ≡ 0 pry t < 0; ( , )τ ξ = M0 ∈ 0, T[ ) ×
× S+ = Γ.
Vvedemo operatory i klasy funkcij
u t x Ml+ ( , ; )0 =
= d G t x y f M f t x Ml
S
t
β β β τ
τ
( ) ( , , ) ( , ; ) ( , ; )+ − −[ ]
+
∫∫ 0 0 yy dS G fn
lν0 ≡ +( ) ( ) ,
u t x M d G t x y f y Ml l
S
t
− −= −
+
∫∫( , ; ) ( , , ) ( , ; )( )
0 0β β β
τ
yy dS G fn
lν0 ≡ −( ) ( ) .
Oznaçennq 1. Funkciq f t x( , ; , )τ ξ naleΩyt\ C m
µ ω
ω
,
( , ) ( , )
1
2 Γ Γ , qkwo vona
pry t > τ ma[ neperervni poxidni D fx
k ≡ D B fx
k
x
j
n
do porqdku m[ ] m[ ]( — ci-
la, a m{ } — drobova çastyna çysla m) , dlq qkyx spravdΩugt\sq ocinky
1) pry k = k + 2 j ≤ m[ ]
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 8
ZADAÇI DLQ RIVNQN| ZI SPECIAL|NYM PARABOLIÇNYM OPERATOROM … 1091
D f t x M C t t T ex
k k b b
xn
n( , ; ) ( ) /
0
2
1
2≤ − −( )− +( )τ ω τµ ξ −− ′{ }c t xρ τ ξ( , , , )
;
2) pry k = m[ ] i ∆x tb< − τ2
∆ ∆ ∆x x
k m b m
x
D f t x M C t x x T( , ; ) ( ) /
0
2
2≤ − ( )− +( ) { }τ ωµ
nn
n e c t xξ ρ τ ξ− ′{ }( , , , )
;
3) pry k ≤ m[ ] i 0 < ∆t < t – τ
∆ ∆ ∆t x
k m k b m bD f t x M C t t( , ; ) ( )/ /
0
2 2
2≤ −−( ) − +( )τ ωµ tt T eb
x
c t x
n
n2( ) { }− ′ξ ρ τ ξ( , , , )
.
C m
µ ω
ω
,
( , ) ( )
1
2 Γ — klas funkcij f t x( , ) , poxidni qkyx zadovol\nqgt\ umovy 1 – 3
pry znaçennqx τ = c = 0.
Umova K l( )±± . Budemo vvaΩaty, wo dlq modulq neperervnosti ω( )t i çysel
m ≥ l > 0 vykonu[t\sq umova K l( )±
, qkwo isnugt\ stali ε, C > 0 taki, wo dlq
τ < t
ω ω τ τε ε( ) ( )t t Cm l m l− + ±{ }+ − + ±{ }+≤1 1
.
Teorema 2. Operator G l( )+
[ vyznaçenym na mnoΩyni funkcij
C m
µ ω
ω
,
( , ) ( , )
1
2 Γ Γ i vidobraΩa[ ]] v C l
m l
µ ω
ω
+
−
,
( , ) ( , )Γ Γ , qkwo dlq ( , , )ω i m l spravd-
Ωu[t\sq umova K l( )+
i µ ≤ nν – 1 + 2 b , m ≤ 2 b – 1, de ω( )t = ω µν
1
( )( )t +
+ ω1
( )( )m l t− + ω2
( )( )m l t−
, a ω λ
1
( )( )t = ω i t( ) pry λ > 0 i ω λ
i t( ) ( ) = F ti ( ) pry
λ = 0, i = 1, 2 ; µν = nν – 1 + 2 b – µ, m{ } = 0 .
Teorema 3. Qkwo f ∈ C m
µ ω
ω
,
( , ) ( , )
1
2 Γ Γ i dlq ( , , )ω i m l vykonu[t\sq umova
K l( )−
, µ ≤ nν – 1 + 2 b , m + l[ ] < 2b , to u t x Ml( ) ( , ; )−
0 naleΩyt\ klasu
C
l
m l
µ ω
ω
−
+
∗
∗
,
( , ) ( , )Γ Γ , de t = ω µν
1
( )( )t + ω1
l
t{ }( ) + ω2
l
t{ }( ) .
Naslidok 1. Qkwo f ∈ C m( , ) ( )ω Γ , to
G f Cl m l m l
( ) ,
( ) ( )
( )
± ( )∈
−∓ ω
Γ .
Teper vvedemo operatory drobovoho intehruvannq i dyferencigvannq:
I f G f t x M
S
ba
+ = −α
α
( )
( )
( ) ( , , )( )1 2
0Γ
=
=
1 1
0Γ( )
( ) ( , , ) ( , ; )
α
β β β βα
τ
νt d G t x y f y M y
t
S
n− −−∫ ∫
+
00 dSy , 0 < α < 1,
D f D I f
S S+ += −α α( ) ( ) ( )Λ 1 =
= Λ( ) ( ) ( , , ) ( , ; )D t d G t x y f y M y d
t
S
n− −−∫ ∫
+
β β β βα
τ
ν
0
0 SSy .
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 8
1092 M. I. MATIJÇUK
Teorema 4. Qkwo funkciq f t x( , ) sumovna na Γ = (0, T) × S+
i pry t > 0
naleΩyt\ klasu Dini C( , ) ( )0 ω Γ , to D I f
S S+ +
α α ( ) = f.
Dovedennq. Rozhlqnemo vyraz I f1−α
α( ) , fα ≡ I fα
, i skorysta[mos\ for-
mulog zhortky (vlastyvistg 1). Todi
I f
d
t
G t x
t
S
1
0
1
1
− =
− −
−∫ ∫
+
α
α αα
τ
τ
τ ξ( )
( ) ( )
( , , )
Γ
×
×
1
1
0
0
Γ( ) ( )
( , , )
α
β
τ β
τ β ξ
α
τ
νd
G y f y dSn y
S
−
−
−∫ ∫
+
=
=
1
1
0
1
0
Γ Γ( ) ( ) ( ) ( )α α
τ
τ
β
τ βα α
τ
− − −∫ ∫ −
d
t
d
t
×
×
S
n
S
G t x G y dS
+ +
∫ ∫ − −
( , , ) ( , , )τ ξ τ β ξ ξν
ξ
0 ff y y dSn y( , )β ν0 =
=
1
1
0
1
0
Γ Γ( ) ( ) ( ) ( )
( , ,
α α
τ
τ
β
τ β
β
α α
τ
− − −
−∫ ∫ −
d
t
d
G t x
t
yy f y y dSn y
S
) ( , )β ν0
+
∫ .
U perßyx dvox intehralax pominq[mo porqdok intehruvannq i v intehrali
( ) ( )t d
t
− −− − +∫ τ τ β τα α
β
1
poklademo τ = β + η β( )t − . V rezul\tati otryma[mo
I f d G t x y f y y dSn y
S
t
1
0
0− = −
+
∫∫α
α
ντ β β( ) ( , , ) ( , ) .
Za teoremogO1 i prypuwennqm na f C∈ ( , )0 ω
ostannij intehral ma[ poxidni,
wo vxodqt\ v operator Λ( )D , i, krim c\oho, Λ( ) ( )D I f1−α
α = f.
Teoremu dovedeno.
Intehruvannqm çastynamy za zminnog β vyrazu D f
S+
α ( ) z uraxuvannqm vlas-
tyvostej f. r. G t x( , , )ξ moΩna nadaty vyhlqdu
D f t x M f t x M t
S+ =
−
− +
−
−α α
α
τ
α
( , ; )
( )
( , ; ) ( )
(
0 0
1
1 1Γ Γ αα
α
)
( )( )G fb+2
. (4)
Tomu di] operatoriv D
S+
α i I
S+
α
opysani teoremamy 2 i 3. Z nyx vyplyvagt\
taki tverdΩennq.
Naslidok 2. 1. Nexaj dlq velyçyn ω αi t m b( ), , 2( ) vykonu[t\sq umova
K b( )+2 α
i µ ≤ ην – 1 + 2b, m < 2b. Todi operator drobovoho dyferencigvannq
D
S+
α
vidobraΩa[ prostir C m
µ ω
ω
,
( , )( )
1
2 �Γ v C l
m l
µ ω
ω
+
−
,
( , )( )�Γ , de ω( )t = ω µν
1
( )( )t +
+ ω1
( )( )m l t− + ω2
( )( )m l t−
, a
�Γ = Γ × Γ abo
�Γ = Γ.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 8
ZADAÇI DLQ RIVNQN| ZI SPECIAL|NYM PARABOLIÇNYM OPERATOROM … 1093
2. Qkwo f C m∈
�
( , )( )ω Γ , to D f C
S
m b m b
+
−
∈ −( )α α ω α2 1
2, ( )
( )Γ .
2. Zadaça Koßi ta dvotoçkova krajova zadaça z modyfikovanym opera-
torom drobovoho dyferencigvannq. 2.1. Zadaça Koßi. Rozhlqnemo rivnqn-
nq z operatorom drobovoho dyferencigvannq
D u D u t x
t u x
A
S S
k b
+ +
∗
−
≤
≡ −
−
= ∑α α
α
αα
( , )
( , )
( ) ( )
0
1 2Γ kk
kt x D u f t x( , ) ( , )+ (5)
i budemo ßukaty rozv’qzok z poçatkovog umovog
u xt= =0 ϕ( ) , x S∈ +
. (6)
Oznaçennq 2. Rozv’qzkom zadaçi Koßi (5), (6) nazyva[t\sq funkciq u t x( , ) ,
qka ma[ taki vlastyvosti:
1) u ∈ Ct x
b
,
( , )( )2 α ω Γ , Γ = ( , )0 T S× +
;
2) fraktal\nyj intehral
W t x u d
G t x
t
t
S
( , )
( )
( , , )
(
= =
−
−
−
− ∫ ∫
+
Iλ
α
α
τ
τ ξ
τ
1
0
1
1Γ ))
( , )α
ν
ξτ ξ ξu dSn
0
naleΩyt\ klasu Ct x
b
,
( , )( )1 2 Γ ;
3) u t x( , ) zadovol\nq[ rivnqnnq (5) i umovu (6).
Rozv’qzok zadaçi (5), (6) ßuka[mo u vyhlqdi
u t x t x
S
( , ) ( , )= +I
α ν . (7)
Todi pry pidstanovci intehrala (7) u rivnqnnq (5) na pidstavi teoremy 4 dlq
ν( , )t x otrymu[mo intehral\ne rivnqnnq
ν
ϕ
α α
τ
τ
α
( , )
( )
( )
( , )
( ) ( )
t x
t x
f t x
d
t
t
=
−
+ +
−
−
∫Γ Γ1
1
0
11−α ×
×
S
nA t x D G t x dS
+
∫ −( , , ) ( , , ) ( , )τ ξ ν τ ξ ξν
ξ
0 . (8)
Joho qdro
K t x
t
A t x D G t xA( , , , )
( ) ( )
( , , ) , , , )τ ξ
α τ
τ ξα≡
−
(−
1
1Γ
dlq vypadku operatora A ≡
k b k x
kA D
≤∑ 2 ( )α
porqdku 2 2b b( )α α= [ ] zhidno z
teoremog 1 zadovol\nq[ nerivnist\
K t x C t T eA
n
b
x
cb b
n
n( , , , ) ( )τ ξ τ
ν α
ξ≤ −
− −− + −{ }
0
2
1 2 2 11 1 2
( )
( ) /
x
t b
q
− ′
−
ξ
τ
. (9)
Tomu pry 2 0bα[ ] > budu[t\sq rezol\venta [5] i rozv’qzok nabyra[ vyhlqdu
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 8
1094 M. I. MATIJÇUK
ν τ τ ξ τ ξ ξν( , ) ( , ) ( , , , ) ( , )t x F t x d R t x F
t
S
A n= + ∫ ∫
+0
0 ddsξ , (10)
de
R t x d K t x y K yA
p
t
S
p( , , , ) ( , , , ) ( , ,τ ξ β β β τ
τ
=
=
∞
∑ ∫ ∫
+1
,, )ξ νy dsn y
0 + K t xA( , , , )τ ξ , (11)
F t x
t
x f t x( , )
( )
( ) ( , )≡
−
+
−α
α
ϕ
Γ 1
.
Qdro KA [ kvazirehulqrnym, tomu dlq rezol\venty R t xA( , , , )τ ξ takoΩ vy-
konu[t\sq nerivnist\ (9) z inßymy dodatnymy stalymy C0 , c1.
Qkwo u rivnqnni (5) A ≡ E, tobto [ odynyçnym, to rezol\venta vyraΩa[t\sq
za dopomohog funkci] typu Mittah-Lefflera. Spravdi,
K t x
t
d
t
2
0
1 1
1 1
( , , )
( ) ( ) ( ) ( )
ξ
α τ
τ
α τα α=
−
∫ − −Γ Γ
×
×
S
nG t x y G y y dy
+
∫ −( , , ) ( , , )τ τ ξ ν0 . (122 )
Vykorystovugçy formulu zhortky dlq G (vlastyvist\ 1), znaxodymo
K t x
d
t
G t x
t
2 2
0
1 1
1
( , ; )
( ) ( )
( , , )ξ
α
τ
τ τ
ξα α=
−∫ − −Γ
=
=
1
22
2 1
2 1
Γ Γ( )
( , ) ( , , )
( )α
α α ξ
α
α
α
B t G t x
t
G−
−
= .
Za indukci[g dovodyt\sq, wo
K t x
t
m
G t xm
m
( , , )
( )
( , , )ξ
α
ξ
α
=
−1
Γ
. ( )12m
Zvidsy otrymu[mo
R t x
t
k
G t x E t G t
k
k
( , , )
( )
( , , ) ( ) ( ,ξ
α
ξ
α
α
α= ≡
=
∞ −
∑
1
1
Γ
xx t, )ξ α−1
, (12)
de E tα
α( ) =
k
kt
k=
∞∑ +0
α
α αΓ( )
.
ZobraΩennq funkci] ν( , )t x za formulog (10) dozvolq[ pereviryty dlq
u t x( , ) vsi umovy v oznaçenni 2 i ocinyty rozv’qzok zadaçi (5), (6). Perekona[mos\
spoçatku u vykonanni poçatkovo] umovy (6). Digçy na ν( , )t x operatorom I
S+
α
,
znaxodymo
IS
t
S
d
t
G t xα
α αν
α α
τ
τ τ
τ ξ=
− −
−∫ ∫−
+
1
1 0
1Γ Γ( ) ( ) ( )
( , , )) ( )ϕ ξ ξd +
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 8
ZADAÇI DLQ RIVNQN| ZI SPECIAL|NYM PARABOLIÇNYM OPERATOROM … 1095
+
1
0
1Γ( ) ( )
( , , ) ( , )
α
τ
τ
τ ξ τ ξα
t
S
d
t
G t x f R F∫ ∫−
− + ∗[ ]−
+
ξξν ξ
0dS .
Tut çerez R F∗ poznaçeno druhyj dodanok u formuli (10), qkyj pry t → 0
ma[ porqdok t
b
b
2
2
α[ ]
.
Vnaslidok zaminy t – τ = t h u perßomu dodanku IS
α ν budemo maty
I
S
t
S
h h dh G th x+
+
=
−
−∫ ∫− −α α αν
α α
1
1
1
0
1
Γ Γ( ) ( )
( ) ( , , ξξ ϕ ξ ξν
ξ
α
) ( ) n
b
bdS o t0
2
2+
{ }
.
OtΩe, pry t → 0 otrymu[mo spivvidnoßennq (6):
lim ( , ) lim ( , )
t t
Su t x t x
→ →
=
0 0
I
α ν =
=
B
G th x
t
S
n
( , )
( ) ( )
lim ( , , ) ( )
α α
α α
ξ ϕ ξ ξν1
1 0
−
− → +
∫Γ Γ
00 dS xξ ϕ= ( ) .
Qkwo ϕ ∈ C S( )( )ω +
, f ∈ Cx
( )( )ω Γ , Ak ∈ C ( )( )ω Γ , to z formul (11), (10), (8)
vyplyva[, wo ν( , )t x zadovol\nq[ nerivnosti
ν ϕα( , ) ( )t x C t fC S≤ +( )−
′+ ,
(13)
∆ ∆x xt x C t fν ω ϕ ωα
ω( , ) ( )≤ ( ) +
−
.
Ci nerivnosti oznaçagt\, wo ν ∈ C b2 α
ω( ) ( )Γ . Tomu za naslidkom 1 pro dig opera-
tora drobovoho intehruvannq otrymu[mo u = I
S+
α ν ∈ C b F
0
2( , )( )α Γ , pryçomu dlq
poxidnyx spravdΩugt\sq nerivnosti
D u t x C t fx
k
k
b( , ) ≤ +
−
2 ϕ ω ω ,
(14)
∆ ∆x x
b
x
k
bD u t x CF t f2 2α
ω ωϕ[ ] −
≤ ( ) +
( , ) .
Zalyßylos\ rozhlqnuty fraktal\nyj intehral w = I
S
u+
−( )( )1 α
. Oskil\ky
u = I
α ν , to za teoremogO4
w d G t x y d
S S S= ( ) = = −+ +
−
I I I
( ) ( ) ( , , ) ( , )1 α α ν ν τ τ ν τ ξ SS
S
t
ξ
+
∫∫
0
.
Ale ν ∈ C b2 α
ω( )
, tomu, qk i v teoremi 1, w ∈ Cx t
b
,
( , )( )2 1 Γ .
Na zaverßennq pidstavymo ν( , )t x iz formuly (10) u formulu (7) i pominq[-
mo porqdok intehruvannq. V rezul\tati otryma[mo ßukanyj rozv’qzok za dopo-
mohog funkci] Hrina
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 8
1096 M. I. MATIJÇUK
u t x Z t x d d Z t xn
S
( , ) ( , , ) ( ) ( , , , )= +
+
∫ 1 2
0ξ ϕ ξ ξ ξ τ τ ξν ff dn
S
t
( , )τ ξ ξ ξν0
0 +
∫∫ , (15)
de komponenty funkci] Hrina vyznaçagt\sq formulamy
Z t x y
G t x y
t
t
1
0
1
1
1
( , , )
( ) ( )
( , , )
( )
=
−
−
−∫ −Γ Γα α
τ
τ τα αα τd +
+
1
1 0
1Γ( ) ( )
( , , ) ( ,
− −
−∫ ∫ −α
β
β
τ
τ
τ ξ τ βα
β
α
t t
A
d d
t
G t x R ,, , )ξ ξν
ξy dSn
S
0∫ ,
Z t x y
t
G t x y2 1
1
( , , , )
( ) ( )
( , , )β
α β
βα=
−
−−Γ
+
+
1
1
0
Γ( ) ( )
( , , ) ( , , , )
α
τ
τ
τ ξ τ β ξ ξ
β
α
ν
t
A n
d
t
G t x R y∫ −
−− ddS
S
ξ∫ .
MiΩ komponentamy Z1 i Z2 isnu[ takyj zv’qzok:
D Z t x y Z t x yt
1
1 2
− − =α τ τ( , , ) ( , , , ) ,
Dt
α
— operator drobovoho dyferencigvannq dlq D
d
dt
t = .
OtΩe, dovedeno taku teoremu.
Teorema 5 (pro korektnist\). Nexaj poverxnq S v En−1 naleΩyt\ klasu
C b( , )2 ω
, u zadaçi (5), (6) A Ck ∈ ( )( )ω Γ , f C∈ ( )( )ω Γ , ϕ ω∈ +C S( )( ) i porqdok
rivnqnnq u (5) 2 2b b( )α α= [ ] . Todi isnu[ funkciq Hrina zadaçi ( , )Z Z1 2 , za
dopomohog qko] rozv’qzok zadaçi vyznaça[t\sq formulog (15) i neperervno za-
leΩyt\ vid danyx zadaçi (14).
2.2. Dvotoçkova krajova zadaça. V oblasti Γ = (0, T ) × S+
rozhlqnemo
zadaçu pro znaxodΩennq funkcij u t x P t x( , ), ( , )( ) dlq rivnqnnq
D u
t u x
B t P t x A
S
k b
k+ −
−
+ =
−
≤[ ]
∑α
α
αα
( , )
( )
( ) ( , )
0
1 2Γ
(( , ) ( , )t x D u f t xx
k + (16)
z umovamy
u xt= =0 ϕ( ) , u xt T= = ψ( ) . (17)
Dlq rozv’qzku zadaçi Koßi (16), (17) budemo maty spivvidnoßennq
u t x Z Z f B t P( , ) ( )= ∗ + ∗ ∗ −[ ]1 2ϕ . (18)
Teper zadovol\nymo v (17) umovu u xt T= = ψ( ) . Todi otryma[mo
Z Z f B t P xt T t T1 2∗ + ∗ ∗ −[ ] == =ϕ ψ( ) ( ) . (19)
Budemo ßukaty P t x( , ) u vyhlqdi
P t x B t Z t x C( , ) ( ) ( , , , )= 2 0 0 , C = const. (20)
Qkwo P t x( , ) pidstavyty u formulu (19), to dlq znaxodΩennq nevidomo] vely-
çyny C distanemo rivnqnnq
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 8
ZADAÇI DLQ RIVNQN| ZI SPECIAL|NYM PARABOLIÇNYM OPERATOROM … 1097
B d Z T x Z dS C
T
n
S
2
0
2 2 0 0 0( ) ( , , , ) ( , , , )τ τ τ ξ τ ξ ξν
ξ∫ ∫
+
== Φx T( , , )ϕ ψ , (21)
de
Φx
S
nT Z T x d( , , ) ( , , , ) ( )ϕ ψ ξ ϕ ξ ξ ξν≡
+
∫ 1 0 0 +
+
0
2
0
T
S
nd Z T x f dS x∫ ∫
+
−τ τ ξ τ ξ ξ ψν
ξ( , , , ) ( , ) ( ) . (22)
Z linijnoho rivnqnnq (21) znaxodymo C i vnaslidok c\oho otrymu[mo
P t x T B d Z T x Zx
T
S
( , ) ( , , ) ( ) ( , , , ) (= ∫ ∫
+
Φ ϕ ψ τ τ τ ξ
0
2
2 2 ττ ξ ξν
ξ, , , )0 0 0
1
n dS
−
×
× B t Z t x( ) ( , , , )2 0 0 . (23)
Qkwo dlq funkci] Z t x2( , , , )τ ξ pravyl\nog [ formula zhortky, to vyraz
dlq P t x( , ) sprowu[t\sq:
P t x T B t B dx
T
( , ) ( , , ) ( ) ( )=
∫
−
Φ ϕ ψ τ τ
0
2
1
. (24)
Ocinymo P t x( , ) za umov, wo ϕ ∈ +C S( ) , f C∈ ( )( )ω Γ , ψ ∈ +C S( ) , B ∈
∈ KC T0,[ ]. Todi P KC∈ ( )Γ i vykonu[t\sq nerivnist\
P t x C f B BC C C KC( , ) ≤ + +( ) −
0 0
1ϕ ψ , (25)
de B0 =
0
2T
B d∫ ( )τ τ .
Pidstavymo funkcig P t x( , ) iz (23) u formulu (18) i znajdemo komponentu
u t x( , ) , qka takoΩ zadovol\nq[ nerivnist\ (25).
1. Koçubej A. M. Zadaça Koßy dlq πvolgcyonn¥x uravnenyj drobnoho porqdka // Dyffe-
renc. uravnenyq. – 1989. – 25, # 8. – S. 1359 – 1368.
2. Eidelman S. D., Kochubei A. N. Cauchy problem for fractional diffusion equations // J. Different.
Equat. – 2004. – 199. – P. 211. – 255.
3. Samko S., Kylbas A. A., Maryçev S. Y. Yntehral¥ y proyzvodn¥e drobnoho porqdka y yx
pryloΩenyq. – Mynsk: Nauka y texnyka, 1987. – 688 s.
4. Analytic methods the theory of differential and preudo-differential equations of parabolic type / Eds
S.D. Eidelman, S. D. Ivasyshen, A. N. Kochubei. – Basel etc.: Birkhäuser, 2004. – 390 p.
5. Matijçuk M. I. Paraboliçni synhulqrni krajovi zadaçi. – Ky]v: In-t matematyky NAN Ukra]-
ny, 1999. – 176 s.
OderΩano 03.02.09
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 8
|
| id | umjimathkievua-article-3082 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:35:53Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/36/f2ce43d2aa10a82e305043d7d6ae6136.pdf |
| spelling | umjimathkievua-article-30822020-03-18T19:44:57Z Problems for equations with special parabolic operator of fractional differentiation Задачі для рівнянь зі спеціальним параболічним оператором дробового диференціювання Matychuk, M. I. Матійчук, М. І. We establish the well-posedness of the Cauchy problem and the two-point boundary-value problem for an equation with an operator of fractional differentiation that corresponds to the singular parabolic Beltrami – Laplace operator on a surface of the Dini class. Установлена корректность задачи Коши и двухточечной краевой задачи для уравнения с оператором дробного дифференцирования, который соответствует сингулярному параболическому оператору Вельтрами - Лапласа на поверхности из класса Дини. Institute of Mathematics, NAS of Ukraine 2009-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3082 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 8 (2009); 1088-1097 Український математичний журнал; Том 61 № 8 (2009); 1088-1097 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3082/2913 https://umj.imath.kiev.ua/index.php/umj/article/view/3082/2914 Copyright (c) 2009 Matychuk M. I. |
| spellingShingle | Matychuk, M. I. Матійчук, М. І. Problems for equations with special parabolic operator of fractional differentiation |
| title | Problems for equations with special parabolic operator of fractional differentiation |
| title_alt | Задачі для рівнянь зі спеціальним параболічним оператором дробового диференціювання |
| title_full | Problems for equations with special parabolic operator of fractional differentiation |
| title_fullStr | Problems for equations with special parabolic operator of fractional differentiation |
| title_full_unstemmed | Problems for equations with special parabolic operator of fractional differentiation |
| title_short | Problems for equations with special parabolic operator of fractional differentiation |
| title_sort | problems for equations with special parabolic operator of fractional differentiation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3082 |
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