多项时间分数阶抛物型方程反源问题的拟逆方法 A Quasi-Inverse Method for Inverse Problems of Parabolic Equationsof the Time Fractional Order

doi:10.12677/AAM.2023.126288

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AdvancesinAppliedMathematicsA^êÆ?Ð,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

,2023;accepted:Jun.19

,2023;published:Jun.27

,2023

©ÙÚ^:…!.õ‘žm©êÔ.•§‡¯K[_•{[J].A^êÆ?Ð,2023,12(6):2861-2875.

DOI:10.12677/aam.2023.126288

…!

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

2023byauthor(s)andHansPublishersInc.

This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).

http://creativecommons.org/licenses/by/4.0/

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DOI:10.12677/aam.2023.1262882864A^êÆ?Ð

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DOI:10.12677/aam.2023.1262882865A^êÆ?Ð

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DOI:10.12677/aam.2023.1262882866A^êÆ?Ð

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−1

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−1

(n)

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

h(0)λ

(n)

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(T)+

(s)λ

(T−s)

(n)

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(T−s)



≥|h(0)|λ

(n)

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(T−s)

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DOI:10.12677/aam.2023.1262882867A^êÆ?Ð

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≤E

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hg,φ



(T−s)

−1

(n)

,α

(T−s)h(s)ds









p+β

(14)

dÚn3.2,•3˜‡~êC

>0¦

kfk

≤C

p+β

kgk

p+β

.(15)

Ïd,

kfk≤C

p+β

.(16)

y..

52‡¯K(1)´¾.(1))XJ•3,ŒUØëY•6u ªàêâ.duh(t)3[0,T]

þ˜‡ëY¼ê,•3˜‡~êC

>0¦C

=sup

t∈[0,T]

|h(t)|<+∞.·‚k



(T−s)

−1

(n)

,α

(T−s)h(s)ds



≤C



(T−s)

−1

(n)

,α

(T−s)ds



= C

(n)

,1+α

(T)

≤

1+λ

≤

(17)

Ïd



(T−s)

−1

(n)

,α

(T−s)h(s)ds



−1

≥

.(18)

dλ

→∞,fØëY•6uêâ,‡¯K(1)´¾.

DOI:10.12677/aam.2023.1262882868A^êÆ?Ð

…!

4.këêÀ5KÚÂñ„Ý

3!¥,·‚JÑl¯K(4)%C¯K(1)ØO.·‚ lKzëêkÀ5K¥

Holder.ØO.nØ(J3e¡½n4.1¥•ã.

½n4.1h(t)÷vbH1.éb≥β,¯K(4)´·½.d,XJ¯Kf(1))÷v

kfk

≤E,p>0,E>ε,(19)

Úf

´¯K(4)),Ke•ã¤á:

(i)XJ0 <p<b,Kα=





p+β

,•3˜‡~êC

¦

−fk≤C

p+β

.(20)

(ii)XJp≥b,Kα=





p+β

,•3˜‡~êC

¦

−fk≤C

p+β

.(21)

Äk,·‚JÑ˜(J5y²½n4.1.

Ún4.2éb≥β,¯K(4)´·½.d,XJf

∈D(A

b−β

),v(t) ∈D(A

),t∈[0,T),K•3˜

‡~êC

¦

k≤C

−

y†(6)aq,¯K(4))•3,)LˆªXe¤«:

v(t) =

∞

n=1



(t−s)

−1

(n)

,α

(t−s)h(s)dshf

,φ

ids



†Ún2.7aq,·‚k

,φ

(T−s)

−1

(n)

,α

(T−s)



(I+αA

,φ



h(s)ds

= (I+αλ

)hf

,φ

(T−s)

−1

(n)

,α

(T−s)h(s)ds.

(22)

Ïd,3¯K(4)¥f

LˆªXe:

∞

n=1

,φ

iφ

(I+αλ

)

(T−s)

−1

(n)

,α

(T−s)h(s)ds

,(23)

v(t) =

∞

n=1

(t−s)

−1

(n)

,α

(t−s)h(s)dshg

,φ

iφ

(I+αλ

)

(T−s)

−1

(n)

,α

(T−s)h(s)ds

.(24)

DOI:10.12677/aam.2023.1262882869A^êÆ?Ð

…!

d(13),·‚k

(T−s)

−1

(n)

,α

(T−s)h(s)ds

(T−s)

−1

(n)

,α

(T−s)h(s)ds

≤

(25)

l(18)Ú(25),·‚

≤

(T−s)

−1

(n)

,α

(T−s)h(s)ds

≤

(26)

?˜Ú,d(26)·‚k

b−β

∞

n=1

2(b−β)

,φ



(1+αλ

)

(T−s)

−1

(n)

,α

(T−s)h(s)ds



≤

∞

n=1

2(b−β)

2β

,φ

(1+αλ

)

≤

<+∞.

(27)

ùÒy²f

∈D(A

b−β

).Ó,·‚k±eO

kv(t)k

≤

∞

n=1

2β



(t−s)

−1

(n)

,α

(t−s)h(s)ds



,φ

(1+αλ

)

≤

∞

n=1

2β



(t−s)

−1

(n)

,α

(t−s)h(s)ds



,φ

≤

∞

n=1



−t

(n)

,1+α

(t)



2β

,φ

≤



−C

1+λ



2β

,φ

≤

<+∞.

(28)

Ïdv(t) ∈D(A

).,˜•¡

∞

n=1

,φ



(1+αλ

)

(T−s)

−1

(n)

,α

(T−s)h(s)ds



.(29)

DOI:10.12677/aam.2023.1262882870A^êÆ?Ð

…!

éb>β,^YoungØª,·‚k

1+αλ

≥

b−β

·1

b−β





≥α

,(30)

1+αλ

≥α

.(31)

é¤kb≥β.

ddŒ„,

≤

∞

n=1

−

2β

,φ



(T−s)

−1

(n)

,α

(T−s)h(s)ds



.(32)

dÚn3.3Ú(32),•3˜‡~êC

¦

k≤C

−

k.(33)

y..

3e©¥,·‚^f

1α

5L«e¡¯K)











j=1

∂

ω(x,t)+A

ω(x,t) = (I+αA

1α

h(t),(x,t) ∈Ω

:= Ω×I,

ω(x,0) = 0,x∈Ω,

ω(x,t) = 0,(x,t) ∈∂Ω×I,

(34)

t= Tž,kω(x,T) = g.(34))LˆªXe:

1α

∞

n=1

hg,φ

(1+αλ

)

(T−s)

−1

(n)

,α

(T−s)h(s)ds

.(35)

Ún4.3XJf

1α

´¯K(34))Úf

´¯K(4)),K

−f

1α

k≤C

−

ε.

y·‚wf

−f

1α

´¯K(4)),•Ò´^g

−g“Og

.^Ún4.2,·‚k

−f

1α

k≤C

−

−gk≤C

−

ε.

y..

Ún4.4XJé˜~êp,E>0,kfk

≤E,K•3˜‡~êC

>0¦

kf−f

1α

k≤

(

,p<b,

αE,p≥b.

DOI:10.12677/aam.2023.1262882871A^êÆ?Ð

…!

y·‚k

kf−f

1α

∞

n=1

hf−f

1α

,φ

∞

n=1







hg,φ

(T−s)

−1

(n)

,α

(T−s)h(s)ds

−

hg,φ

(1+αλ

)

(T−s)

−1

(n)

,α

(T−s)h(s)ds







∞

n=1

hg,φ



(1+αλ

)

(T−s)

−1

(n)

,α

(T−s)h(s)ds



∞

n=1



αλ

b−p

1+αλ



hf,φ

(36)

XJp<b,^YoungØª,·‚k

1+αλ

≥

·1

b−p



b−p



b−p

≥α

b−p

.(37)

Ïd,

−f

1α

≤

∞

n=1

hf,φ

≤α

.(38)

XJp≥b,K

−f

1α

≤

∞

n=1

2(b−p)

hf,φ

≤λ

2(b−p)

.(39)

y..

yXJp<b,dÚn4.3ÚÚn4.4,·‚k

kf−f

k≤kf−f

1α

k+kf

−f

1α

≤α

E+C

−

ε.

(40)

Àα=





p+β

,•3˜‡~êC

>0¦

kf−f

k≤C

p+β

.(41)

y..

yXJp≥b,·‚k

kf−f

k≤kf−f

1α

k+kf

−f

1α

≤C

αE+C

−

ε.

(42)

DOI:10.12677/aam.2023.1262882872A^êÆ?Ð

…!

Àα=





b+β

,•3˜‡~êC

>0¦

kf−f

k≤C

b+β

.(43)

5.o(†Ð"

§·½.ÄuMittag-Leﬄer¼ê5Ÿ,3kKzëêÀJ5Ke,·‚Kz)Ø

ü‘[_‡¥-‡(Jí2õ‘[_‡¯K¥,¿…‡¯K^‡-½5

ž‡üÐŠÚ‘¯K„™Œ•,I‡?˜ÚïÄ.

ë•©z

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