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AdvancesinAppliedMathematics
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,2023,12(6),2861-2875
PublishedOnlineJune2023inHans.https://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2023.126288
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AQuasi-InverseMethodforInverse
ProblemsofParabolicEquations
oftheTimeFractionalOrder
YuxinWang
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Received:May25
th
,2023;accepted:Jun.19
th
,2023;published:Jun.27
th
,2023
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DOI:10.12677/aam.2023.126288
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Abstract
Inthispaper,thefractionalquasi-inversemethodisusedtosolvetheinversesource
problemofpolynomialtimefractionalparabolicequations,whichisill-posed.First-
ly,theconditionalstabilityoftheinverseproblemisgiven,andthenthefractional
quasi-inversemethodisproposed,thatis,theperturbationtermrelatedtoelliptic
differentialoperatorisintroducedintotheoriginalequation.Finally,basedonsome
properties ofMittag-Leffler function,the correspondingconvergencerateof theregu-
larsolutionunderthepriorselectionruleisgivenintheory.
Keywords
TheInverseSourceProblemofMultipleTimeFractionalParabolicEquations,Quasi
InverseRegularizationMethod,ErrorEstimation
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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J
α
=(
α
1
,...,α
m
)
Ú
§
X
ê
q
=(
q
1
,...,q
m
)
÷
v
(2)
Ú
(3),
·
‚
k
e
¡
{
E
(
p
)
α
0
,β
(
t
) :=
E
(
α
1
,α
1
−
α
2
,
···
,α
1
−
α
m
)
,β
(
−
λ
p
t
α
1
,
−
q
2
t
α
1
−
α
2
,
···
,
−
q
m
t
α
1
−
α
m
)
,t>
0
,p
= 1
,
2
,...,
DOI:10.12677/aam.2023.1262882864
A^
ê
Æ
?
Ð
…
!
Ù
¥
α
0
=(
α
1
,α
1
−
α
2
,
···
,α
1
−
α
m
)
Ú
λ
p
L
«
à
g
Dirichlet
>
.
^
‡
e
ý
Ž
f
A
1
p
‡
A
Š
.
Ú
n
2.3
[23]
‰
½
β>
0
Ú
1
>α
1
>
···
>α
m
>
0.
α
1
π/
2
<µ<α
1
π,µ
≤|
arg(
z
1
)
|≤
π
Ú
π
−
≤|
arg
z
j
|≤
π,j
=2
,
···
,m
,
ù
p
´
v
,
K
•
3
˜
‡
•
6
u
µ,α
j
(
j
=1
,
···
,m
)
Ú
β
~
ê
C
1
>
0
¦
|
E
(
α
1
,α
1
−
α
2
,
···
,α
1
−
α
m
)
,β
(
z
1
,
···
,z
m
)
|≤
C
1
1+
|
z
1
|
.
(5)
Ú
n
2.4
[18]
-
0
<α
m
<α
m
−
1
<
···
<α
1
<
1.
K
d
dt
n
t
α
1
E
(
p
)
α
0
,
1+
α
1
(
t
)
o
=
t
α
1
−
1
E
(
p
)
α
0
,α
1
(
t
)
,t>
0
.
Ú
n
2.5
[18]
-
0
<α
m
<α
m
−
1
<
···
<α
1
<
1,
¼
ê
t
α
1
−
1
E
(
p
)
α
0
,α
1
(
t
)
é
t>
0
´
.
·
K
2.6
-
λ
p
>
0
Ú
0
<α
m
<α
m
−
1
<
···
<α
1
<
1,
K
é
t>
0
k
0
<λ
p
t
α
1
E
(
p
)
α
0
,
1+
α
1
(
t
)
<
1,
¿
…
λ
p
t
α
1
E
(
p
)
α
0
,
1+
α
1
(
t
)
3
t>
0
´
î
‚
O
¼
ê
.
y
d
dt
n
λ
p
t
α
1
E
(
p
)
α
0
,
1+
α
1
(
t
)
o
=
λ
p
t
α
1
−
1
E
(
p
)
α
0
,α
1
(
t
)
>
0
.
·
‚
5
¿
λ
p
t
α
1
E
(
p
)
α
0
,
1+
α
1
(
t
)
3
t
þ
´
ë
Y
¼
ê
.
Ï
d
,
·
‚
k
lim
t
→
0
(
λ
p
t
α
1
E
(
p
)
α
0
,
1+
α
1
(
t
))=0
Ú
lim
t
→∞
(
λ
p
t
α
1
E
(
p
)
α
0
,
1+
α
1
(
t
)) =
C
1
.
y
.
.
Ú
n
2.7
[18]
é
λ
p
>
0
Ú
0
<α
s
<α
s
−
1
<
···
<α
1
<
1,
•
3
˜
‡
•
6
u
α,T
~
ê
C<
1
¦
C
λ
p
T
α
1
≤
E
(
p
)
α
0
,
1+
α
1
(
T
)
≤
1
T
α
1
λ
p
.
½
Â
2.8
-
b
≥
β
.
é
z
˜
‡
v
∈
H
,
B
α
v
½
Â
X
e
B
α
v
:=
∞
X
n
=1
1
1+
αλ
b
n
h
v,φ
n
i
φ
n
.
b
H
.
·
‚
•
Ä
˜
‡
¼
ê
h
:[0
,T
]
→
R
,
X
J
h
3
[0
,T
]
´
ë
Y
,
•
3
˜
‡
~
ê
T
0
∈
[0
,T
)
¦
é
u
˜
~
ê
η>
0,
k
|
h
(
t
)
|≥
η>
0,
t
∈
[
T
0
,T
]
¤
á
,
K
h
Ò
¡
•
÷
v
b
H
.
d
,
„
I
‡
÷
v
±
e
ü
^
‡
ƒ
˜
:
H1
h
(
t
)
3
[0
,T
]
þ
ØC
Ò
.
H2
X
J
h
(
t
)
3
[0
,T
]
þ
C
Ò
,
K
h
(
t
)
´
Œ
‡
,
…
•
3
˜
‡
~
ê
θ
¦
|
h
t
(
t
)
|≤
q
,
t
∈
[0
,T
].
¿
…
k
|
h
(
t
)
|≤
η
(
T
−
T
0
)
T
0
,
t
∈
I
,
Ù
¥
I
=
{
t
:
h
(
t
)
h
(
T
)
<
0
}
.
5
1
b
H2
#
N
h
(
t
)
3
[0
,T
]
þ
C
Ò
,
3ù
«
œ
¹
,
b
H2
‡
(
f
L
ˆ
ª
m
ý
©
1
Ø
•
"
,
=
Z
T
0
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
ds
6
= 0
.
DOI:10.12677/aam.2023.1262882865
A^
ê
Æ
?
Ð
…
!
½
n
2.9
‡
¯
K
(1)
k
X
e
•
˜
)
f
(
x
) =
∞
X
n
=1
h
g,φ
n
i
φ
n
R
T
0
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
h
(
s
)
ds
.
y
Š
â
©
z
[18],
¯
K
(1)
)
k
X
e
L
ˆ
ª
u
(
x,t
) =
∞
X
n
=1
f
n
Q
n
(
t
)
φ
n
(
x
)
,
(6)
Ù
¥
H
n
(
t
)=
R
t
0
h
(
s
)(
t
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
t
−
s
)
ds
,
f
n
=(
f,φ
n
).
t
=
T
,
d
u
(
x,T
)=
g
,
^
þ
ª
Ú
φ
n
‰
S
È
,
·
‚
k
h
g,φ
n
i
=
Z
T
0
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
h
(
s
)
ds
h
f,φ
n
i
.
?
˜
Ú
,
·
‚
k
f
(
x
) =
∞
X
n
=1
h
g,φ
n
i
φ
n
R
T
0
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
h
(
s
)
ds
.
y
.
.
3.
‡
¯
K
^
‡
-
½
5
½
n
3.1
b
h
(
t
)
÷
v
b
H
Ú
f
(
x
)
´
¯
K
(1)
)
.
é
˜
~
ê
E
Ú
p
,
X
J
k
u
(
x,T
)
k
=
k
g
k≤
ε
Ú
k
f
k
p
≤
E
,
K
•
3
˜
‡
~
ê
C
1
>
0
¦
k
f
k≤
C
1
ε
p
p
+
β
E
β
p
+
β
.
(7)
•
y
²
½
n
3.1,
·
‚
I
‡±
e
(
J
.
Ú
n
3.2
X
J
h
(
t
)
÷
v
b
H
,
K
•
3
˜
‡
~
ê
C
2
>
0
¦
λ
β
n
Z
T
0
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
h
(
s
)
ds
≥
C
2
,n
= 1
,
2
,
···
y
œ
¹
1.
h
(
t
)
÷
v
b
H1
.
du
h
(
t
)
3
[0
,T
]
þ
ØC
Ò
,
·
‚
k
λ
β
n
Z
T
0
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
h
(
s
)
ds
=
λ
β
n
Z
T
0
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
|
h
(
s
)
|
ds
≥
η
Z
T
T
0
λ
β
n
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
ds
=
ηλ
β
n
(
T
−
T
0
)
α
1
E
(
n
)
α
0
,
1+
α
1
(
T
−
T
0
)
≥
ηλ
β
1
(
T
−
T
0
)
α
1
E
(
n
)
α
0
,
1+
α
1
(
T
−
T
0
)
(8)
DOI:10.12677/aam.2023.1262882866
A^
ê
Æ
?
Ð
…
!
œ
¹
2.
h
(
t
)
÷
v
b
H2
.
-
D
=
t
:
h
(
t
)
h
(
T
)
≥
0,
T
1
= max
{
t
:
t
∈
[0
,T
]
,h
(
t
) = 0
}
Ú
C
3
=
C
(
−|
h
(0)
|
+
q
)
η
1
β
,
(9)
Ù
¥
C
1
3
Ú
n
2.3
¥
®
²
‰
½
.
·
‚
k
0
<T
1
<T
0
,[
T
1
,T
]
⊆
D
Ú
I
⊆
[0
,T
1
].
λ
n
≥
C
3
,
Ï
L
·
K
2.5,
·
‚
k
λ
β
n
Z
T
0
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
h
(
s
)
ds
=
λ
β
n
Z
D
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
h
(
s
)
ds
+
Z
I
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
h
(
s
)
ds
≥
λ
β
1
Z
D
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
|
h
(
s
)
|
ds
−
λ
β
1
Z
I
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
|
h
(
s
)
|
ds
≥
λ
β
1
Z
T
T
1
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
|
h
(
s
)
|
ds
−
η
(
T
−
T
0
)
T
0
λ
β
1
Z
T
1
0
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
ds
≥
λ
β
1
(
T
−
T
1
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
Z
T
T
1
|
h
(
s
)
|
ds
−
η
(
T
−
T
0
)
T
0
Z
T
1
0
ds
≥
λ
β
1
(
T
−
T
1
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
Z
T
T
1
|
h
(
s
)
|
ds
+
Z
T
T
0
ηds
−
ηT
1
(
T
−
T
0
)
T
0
≥
λ
β
1
(
T
−
T
1
)
α
1
−
1
E
(
C
3
)
α
0
,α
1
(
T
−
s
)
Z
T
T
1
|
h
(
s
)
|
ds
+
η
(
T
−
T
0
)(
T
0
−
T
1
)
T
0
.
(10)
λ
n
≤
C
3
,
Ï
L
©
Ü
È
©
·
‚
k
λ
β
n
Z
T
0
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
h
(
s
)
ds
=
h
(0)
λ
β
n
T
α
1
E
(
n
)
α
0
,
1+
α
1
(
T
)+
Z
T
0
h
s
(
s
)
λ
β
n
(
T
−
s
)
α
1
E
(
n
)
α
0
,
1+
α
1
(
T
−
s
)
≥|
h
(0)
|
λ
β
n
T
α
1
E
(
n
)
α
0
,
1+
α
1
(
T
)
−
Z
T
0
λ
β
n
(
T
−
s
)
α
1
E
(
n
)
α
0
,
1+
α
1
(
T
−
s
)
|
h
s
(
s
)
|
ds
≥|
h
(0)
|
λ
β
n
T
α
1
E
(
n
)
α
0
,
1+
α
1
(
T
)
−
q
Z
T
0
λ
β
n
(
T
−
s
)
α
1
E
(
n
)
α
0
,
1+
α
1
(
T
−
s
)
ds
=
|
h
(0)
|
λ
β
n
T
α
1
E
(
n
)
α
0
,
1+
α
1
(
T
)+
qλ
β
n
T
α
1
+1
E
(
n
)
α
0
,
2+
α
1
(
T
)
ds.
(11)
?
˜
Ú
,
·
‚
k
λ
β
n
Z
T
0
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
h
(
s
)
ds
≥
|
h
(0)
|
λ
β
n
T
α
1
C
λ
β
n
T
α
1
+
qT
α
1
+1
Cλ
β
n
λ
β
n
T
α
1
≥
C
|
h
(0)
|
+
Cq
≥
η.
(12)
DOI:10.12677/aam.2023.1262882867
A^
ê
Æ
?
Ð
…
!
d
(10)
Ú
(12),
•
3
˜
‡
~
ê
C
2
>
0
¦
λ
β
n
Z
T
0
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
h
(
s
)
ds
≥
C
2
,n
= 1
,
2
,
···
(13)
y
.
.
y
3
·
‚
5
y
²
½
n
3.1.
^
Holder
Ø
ª
,
·
‚
k
k
f
k
2
=
∞
X
n
=1
h
f,φ
n
i
2
=
∞
X
n
=1
λ
2
pβ
p
+
β
n
|h
f,φ
n
i|
2
β
p
+
β
λ
−
2
pβ
p
+
β
n
|h
f,φ
n
i|
2
p
p
+
β
≤
∞
X
n
=1
λ
2
p
n
|h
f,φ
n
i|
2
!
β
p
+
β
∞
X
n
=1
λ
−
2
β
n
|h
f,φ
n
i|
2
!
p
p
+
β
≤
E
2
p
p
+
β
∞
X
n
=1
h
g,φ
n
i
2
λ
β
n
R
T
0
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
h
(
s
)
ds
2
p
p
+
β
.
(14)
d
Ú
n
3.2,
•
3
˜
‡
~
ê
C
1
>
0
¦
k
f
k
2
≤
C
1
2
E
2
p
p
+
β
k
g
k
2
p
p
+
β
.
(15)
Ï
d
,
k
f
k≤
C
1
E
p
p
+
β
ε
p
p
+
β
.
(16)
y
.
.
5
2
‡
¯
K
(1)
´
¾
.(1)
)
X
J
•
3
,
Œ
U
Ø
ë
Y
•
6
u
ª
à
ê
â
.
du
h
(
t
)
3
[0
,T
]
þ
˜
‡
ë
Y
¼
ê
,
•
3
˜
‡
~
ê
C
4
>
0
¦
C
4
=sup
t
∈
[0
,T
]
|
h
(
t
)
|
<
+
∞
.
·
‚
k
Z
T
0
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
h
(
s
)
ds
≤
C
4
Z
T
0
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
ds
=
C
4
T
α
1
E
(
n
)
α
0
,
1+
α
1
(
T
)
≤
C
4
T
α
1
1+
λ
β
n
T
α
1
≤
C
4
λ
β
n
.
(17)
Ï
d
Z
T
0
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
h
(
s
)
ds
−
1
≥
λ
β
n
C
4
.
(18)
d
λ
β
n
→∞
,
f
Ø
ë
Y
•
6
u
ê
â
,
‡
¯
K
(1)
´
¾
.
DOI:10.12677/aam.2023.1262882868
A^
ê
Æ
?
Ð
…
!
4.
k
ë
ê
À
5
K
Ú
Â
ñ
„
Ý
3
!
¥
,
·
‚
J
Ñ
l
¯
K
(4)
%
C
¯
K
(1)
Ø
O
.
·
‚
l
K
z
ë
ê
k
À
5
K
¥
Holder
.
Ø
O
.
n
Ø
(
J
3
e
¡
½
n
4.1
¥
•
ã
.
½
n
4.1
h
(
t
)
÷
v
b
H1
.
é
b
≥
β
,
¯
K
(4)
´·
½
.
d
,
X
J
¯
K
f
(1)
)
÷
v
k
f
k
p
≤
E,p>
0
,E>ε,
(19)
Ú
f
α
´
¯
K
(4)
)
,
K
e
•
ã
¤
á
:
(i)
X
J
0
<p<b
,
K
α
=
ε
E
b
p
+
β
,
•
3
˜
‡
~
ê
C
2
¦
k
f
α
−
f
k≤
C
2
ε
p
p
+
β
E
β
p
+
β
.
(20)
(ii)
X
J
p
≥
b
,
K
α
=
ε
E
b
p
+
β
,
•
3
˜
‡
~
ê
C
3
¦
k
f
α
−
f
k≤
C
3
ε
b
p
+
β
E
β
p
+
β
.
(21)
Ä
k
,
·
‚
J
Ñ
˜
(
J
5
y
²
½
n
4.1.
Ú
n
4.2
é
b
≥
β
,
¯
K
(4)
´·
½
.
d
,
X
J
f
α
∈
D
(
A
b
−
β
),
v
(
t
)
∈
D
(
A
b
),
t
∈
[0
,T
),
K
•
3
˜
‡
~
ê
C
5
¦
k
f
α
k≤
C
5
α
−
β
b
k
g
ε
k
.
y
†
(6)
a
q
,
¯
K
(4)
)
•
3
,
)
L
ˆ
ª
X
e
¤
«
:
v
(
t
) =
∞
X
n
=1
Z
t
0
(
t
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
t
−
s
)
h
(
s
)
ds
h
f
α
,φ
n
i
ds
φ
n
.
†
Ú
n
2.7
a
q
,
·
‚
k
h
g
ε
,φ
n
i
=
Z
T
0
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
(
I
+
αA
b
)
f
α
,φ
n
h
(
s
)
ds
= (
I
+
αλ
b
n
)
h
f
α
,φ
n
i
Z
T
0
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
h
(
s
)
ds.
(22)
Ï
d
,
3
¯
K
(4)
¥
f
α
L
ˆ
ª
X
e
:
f
α
=
∞
X
n
=1
h
g
ε
,φ
n
i
φ
n
(
I
+
αλ
β
n
)
R
T
0
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
h
(
s
)
ds
,
(23)
Ú
v
(
t
) =
∞
X
n
=1
R
t
0
(
t
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
t
−
s
)
h
(
s
)
ds
h
g
ε
,φ
n
i
φ
n
(
I
+
αλ
b
n
)
R
T
0
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
h
(
s
)
ds
.
(24)
DOI:10.12677/aam.2023.1262882869
A^
ê
Æ
?
Ð
…
!
d
(13),
·
‚
k
1
R
T
0
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
h
(
s
)
ds
=
λ
β
n
λ
β
n
R
T
0
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
h
(
s
)
ds
≤
λ
β
n
C
3
(25)
l
(18)
Ú
(25),
·
‚
λ
β
n
C
4
≤
1
R
T
0
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
h
(
s
)
ds
≤
λ
β
n
C
2
(26)
?
˜
Ú
,
d
(26)
·
‚
k
k
f
α
k
2
b
−
β
=
∞
X
n
=1
λ
2(
b
−
β
)
n
h
g
ε
,φ
n
i
2
(1+
αλ
b
n
)
R
T
0
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
h
(
s
)
ds
2
≤
∞
X
n
=1
λ
2(
b
−
β
)
n
λ
2
β
n
h
g
ε
,φ
n
i
2
C
2
2
(1+
αλ
b
n
)
2
≤
k
g
ε
k
2
C
2
2
α
2
<
+
∞
.
(27)
ù
Ò
y
²
f
α
∈
D
(
A
b
−
β
).
Ó
,
·
‚
k
±
e
O
k
v
(
t
)
k
2
b
≤
∞
X
n
=1
λ
2
b
n
λ
2
β
n
R
t
0
(
t
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
t
−
s
)
h
(
s
)
ds
2
h
g
ε
,φ
n
i
2
C
2
2
(1+
αλ
b
n
)
2
≤
∞
X
n
=1
λ
2
β
n
R
t
0
(
t
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
t
−
s
)
h
(
s
)
ds
2
h
g
ε
,φ
n
i
2
C
2
2
α
2
≤
∞
X
n
=1
−
t
α
1
E
(
n
)
α
0
,
1+
α
1
(
t
)
2
λ
2
β
n
h
g
ε
,φ
n
i
2
C
2
2
α
2
≤
−
C
1
t
α
1
1+
λ
β
n
t
α
1
2
λ
2
β
n
h
g
ε
,φ
n
i
2
C
2
2
α
2
≤
C
1
2
k
g
ε
k
2
b
C
2
2
α
2
<
+
∞
.
(28)
Ï
d
v
(
t
)
∈
D
(
A
b
).
,
˜
•
¡
k
f
α
k
2
=
∞
X
n
=1
h
g
ε
,φ
n
i
2
(1+
αλ
b
n
)
R
T
0
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
h
(
s
)
ds
2
.
(29)
DOI:10.12677/aam.2023.1262882870
A^
ê
Æ
?
Ð
…
!
é
b>β
,
^
Young
Ø
ª
,
·
‚
k
1+
αλ
b
n
≥
b
−
β
b
·
1
b
b
−
β
+
β
b
α
β
b
λ
β
n
b
β
≥
α
β
b
λ
β
n
,
(30)
½
1+
αλ
b
n
≥
α
β
b
λ
β
n
.
(31)
é
¤
k
b
≥
β
.
d
d
Œ
„
,
k
f
α
k
2
≤
∞
X
n
=1
α
−
2
β
b
h
g
ε
,φ
n
i
2
λ
β
n
R
T
0
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
h
(
s
)
ds
2
.
(32)
d
Ú
n
3.3
Ú
(32),
•
3
˜
‡
~
ê
C
5
¦
k
f
α
k≤
C
5
α
−
β
b
k
g
ε
k
.
(33)
y
.
.
3
e
©
¥
,
·
‚
^
f
1
α
5
L
«
e
¡
¯
K
)
m
P
j
=1
q
j
∂
α
j
0
+
ω
(
x,t
)+
A
β
ω
(
x,t
) = (
I
+
αA
b
)
f
1
α
h
(
t
)
,
(
x,t
)
∈
Ω
T
:= Ω
×
I,
ω
(
x,
0) = 0
,x
∈
Ω
,
ω
(
x,t
) = 0
,
(
x,t
)
∈
∂
Ω
×
I,
(34)
t
=
T
ž
,
k
ω
(
x,T
) =
g
.(34)
)
L
ˆ
ª
X
e
:
k
f
1
α
k
2
=
∞
X
n
=1
h
g,φ
n
i
2
φ
n
(1+
αλ
b
n
)
R
T
0
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
h
(
s
)
ds
.
(35)
Ú
n
4.3
X
J
f
1
α
´
¯
K
(34)
)
Ú
f
α
´
¯
K
(4)
)
,
K
k
f
α
−
f
1
α
k≤
C
5
α
−
β
b
ε.
y
·
‚
w
f
α
−
f
1
α
´
¯
K
(4)
)
,
•
Ò
´
^
g
ε
−
g
“
O
g
ε
.
^
Ú
n
4.2,
·
‚
k
k
f
α
−
f
1
α
k≤
C
5
α
−
β
b
k
g
ε
−
g
k≤
C
5
α
−
β
b
ε.
y
.
.
Ú
n
4.4
X
J
é
˜
~
ê
p
,
E>
0,
k
f
k
p
≤
E
,
K
•
3
˜
‡
~
ê
C
8
>
0
¦
k
f
−
f
1
α
k≤
(
α
β
b
,p<b,
C
6
αE,p
≥
b.
DOI:10.12677/aam.2023.1262882871
A^
ê
Æ
?
Ð
…
!
y
·
‚
k
k
f
−
f
1
α
k
2
=
∞
X
n
=1
h
f
−
f
1
α
,φ
n
i
2
=
∞
X
n
=1
h
g,φ
n
i
R
T
0
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
h
(
s
)
ds
−
h
g,φ
n
i
(1+
αλ
b
n
)
R
T
0
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
h
(
s
)
ds
2
=
∞
X
n
=1
α
2
λ
2
b
n
h
g,φ
n
i
2
(1+
αλ
b
n
)
R
T
0
(
T
−
s
)
α
1
−
1
E
(
n
)
α
0
,α
1
(
T
−
s
)
h
(
s
)
ds
2
=
∞
X
n
=1
αλ
b
−
p
n
1+
αλ
b
n
2
λ
2
p
n
h
f,φ
n
i
2
.
(36)
X
J
p<b
,
^
Young
Ø
ª
,
·
‚
k
1+
αλ
b
n
≥
p
b
·
1
b
p
+
b
−
p
b
α
b
−
p
b
λ
b
−
p
n
b
b
−
p
≥
α
b
−
p
b
λ
b
−
p
n
.
(37)
Ï
d
,
k
f
α
−
f
1
α
k
2
≤
∞
X
n
=1
α
2
p
b
λ
2
p
n
h
f,φ
n
i
2
≤
α
2
p
b
E
2
.
(38)
X
J
p
≥
b
,
K
k
f
α
−
f
1
α
k
2
≤
∞
X
n
=1
α
2
λ
2(
b
−
p
)
1
λ
2
p
n
h
f,φ
n
i
2
≤
λ
2(
b
−
p
)
1
α
2
E
2
.
(39)
y
.
.
y
3
·
‚
5
y
²
½
n
4.1.
·
‚
Ä
k
‰
Ñ
½
n
4.1
1
˜
Ü
©
y
²
L
§
.
y
X
J
p<b
,
d
Ú
n
4.3
Ú
Ú
n
4.4,
·
‚
k
k
f
−
f
α
k≤k
f
−
f
1
α
k
+
k
f
α
−
f
1
α
k
≤
α
p
b
E
+
C
5
α
−
β
b
ε.
(40)
À
α
=
ε
E
b
p
+
β
,
•
3
˜
‡
~
ê
C
2
>
0
¦
k
f
−
f
α
k≤
C
2
ε
p
p
+
β
E
β
p
+
β
.
(41)
y
.
.
X
·
‚
y
²
½
n
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