Waves damped wave and observation

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1Waves, damped wave and observation? Kim Dang PHUNG Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China. E-mail: kim dang Abstract This talk describes some applications of two kinds of obser- vation estimate for the wave equation and for the damped wave equation in a bounded domain where the geometric control con- dition of C. Bardos, G. Lebeau and J. Rauch may failed. 1 The wave equation and observation We consider the wave equation in the solution u = u(x, t) ? ? ? ∂2t u?∆u = 0 in ?? R , u = 0 on ∂?? R , (u, ∂tu) (·, 0) = (u0, u1) , (1.1) living in a bounded open set ? in Rn, n ≥ 1, either convex or C2 and connected, with boundary ∂?. It is well-known that for any initial data (u0, u1) ? H2(?) ? H10 (?) ? H10 (?), the above problem is well-posed and have a unique strong solution. Linked to exact controllability and strong stabilization for the wave equation (see [Li]), it appears the following observability problem which consists in proving the following estimate ?(u0, u1)?2H10 (?)?L2(?) ≤ C ∫ T 0 ∫ ? |∂tu (x, t)|2 dxdt ?This work is supported by the NSF of China under grants

  • weight function

  • wave equation

  • damped wave

  • french-chinese summer

  • null initial

  • equation implies

  • unique strong

  • ct ?

  • also give theirs


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Signaler un problème

∗Waves,dampedwaveandobservation
KimDangPHUNG
YangtzeCenterofMathematics,SichuanUniversity,
Chengdu610064,China.
E-mail:kimdangphung@yahoo.fr
Abstract
Thistalkdescribessomeapplicationsoftwokindsofobser-
vationestimateforthewaveequationandforthedampedwave
equationinaboundeddomainwherethegeometriccontrolcon-
ditionofC.Bardos,G.LebeauandJ.Rauchmayfailed.

1Thewaveequationandobservation

1

Weconsiderthewaveequationinthesolution
u
=
u
(
x,t
)


t
2
u

Δ
u
=0inΩ
×
R
,

u
=0on

Ω
×
R
,(1.1)
(
u,∂
t
u
)(

,
0)=(
u
0
,u
1
),
livinginaboundedopensetΩin
R
n
,
n

1,eitherconvexor
C
2
and
connected,withboundary

Ω.Itiswell-knownthatforanyinitialdata
(
u
0
,u
1
)

H
2
(Ω)

H
01
(Ω)
×
H
01
(Ω),theaboveproblemiswell-posed
andhaveauniquestrongsolution.
Linkedtoexactcontrollabilityandstrongstabilizationforthewave
equation(see[Li]),itappearsthefollowingobservabilityproblemwhich
consistsinprovingthefollowingestimate
ZZTk
(
u
0
,u
1
)
k
2
H
1
(Ω)
×
L
2
(Ω)

C
|

t
u
(
x,t
)
|
2
dxdt
0ω0∗
ThisworkissupportedbytheNSFofChinaundergrants10525105and10771149.
PartofthistalkwasdonewhentheauthorvisitedFudanUniversitywithafinan-
cialsupportfromthe”French-ChineseSummerInstituteonAppliedMathematics”
(September1-21,2008).

2KimDangPHUNG
forsomeconstant
C>
0independentontheinitialdata.Here,
T>
0
and
ω
isanon-emptyopensubsetinΩ.Duetofinitespeedofpropa-
gation,thetime
T
havetobechosenlargeenough.Dealingwithhigh
frequencywavesi.e.,waveswhichpropagatesaccordingthelawofge-
ometricaloptics,thechoiceof
ω
cannotbearbitrary.Inotherwords,
theexistenceoftrappedrays(e.g,constructedwithgaussianbeams(see
[Ra])impliestherequirementofsomekindofgeometricconditionon
(
ω,T
)(see[BLR])inorderthattheaboveobservabilityestimatemay
.dlohNow,wecanaskwhatkindofestimatewemayhopeinageometry
withtrappedrays.Letusintroducethequantity
k
(
u
0
,u
1
)
k
H
2

H
01
(Ω)
×
H
01
(Ω)
,=Λk
(
u
0
,u
1
)
k
H
01
(Ω)
×
L
2
(Ω)
whichcanbeseenasameasureofthefrequencyofthewave.Inthis
paper,wepresentthetwofollowinginequalities
ZZ2
C
Λ
1

T
2
k
(
u
0
,u
1
)
k
H
01
(Ω)
×
L
2
(Ω)

e
|

t
u
(
x,t
)
|
dxdt
(1.2)
ω0dnaZ
C
Λ
1

Z
22k
(
u
0
,u
1
)
k
H
01
(Ω)
×
L
2
(Ω)

C
|

t
u
(
x,t
)
|
dxdt
(1.3)
ω0where
β

(0
,
1),
γ>
0.Wewillalsogivetheirsapplicationstocontrol
theory.
Thestrategytogetestimate(1.2)isnowwell-known(see[Ro2],[LR])
andasketchoftheproofwillbegiveninAppendixforcompleteness.
Moreprecisely,wehavethefollowingresult.
Theorem1.1.-
Forany
ω
non-emptyopensubsetin
Ω
,forany
β

(0
,
1)
,thereexist
C>
0
and
T>
0
suchthatforanysolution
u
of
(1.1)withnon-identicallyzeroinitialdata
(
u
0
,u
1
)

H
2
(Ω)

H
01
(Ω)
×
H
01
(Ω)
,theinequality(1.2)holds.
Now,wecanaskwhetherisitpossibletogetanotherweightfunction
ofΛthantheexponentialone,andinparticularapolynomialweight
functionwithageometry(Ω

)withtrappedrays.Herewepresentthe
followingresult.
Theorem1.2.-
Thereexistsageometry


)
withtrappedrays
suchthatforanysolution
u
of(1.1)withnon-identicallyzeroinitialdata

Waves,dampedwaveandobservation3
(
u
0
,u
1
)

H
2
(Ω)

H
01
(Ω)
×
H
01
(Ω)
,theinequality(1.3)holdsforsome
C>
0
and
γ>
0
.
TheproofofTheorem1.2isgivenin[Ph1].WiththehelpofTheorem
2.1below,itcanalsobededucedfrom[LiR],[BuH].

2Thedampedwaveequationandourmo-
tivation

Weconsiderthefollowingdampedwaveequationinthesolution
w
=
w
(
x,t
)
½
2∂
t
w

Δ
w
+1
ω

t
w
=0inΩ
×
(0
,
+

),(2.1)
w
=0on

Ω
×
(0
,
+

),
livinginaboundedopensetΩin
R
n
,
n

1,eitherconvexor
C
2
and
connected,withboundary

Ω.Here
ω
isanon-emptyopensubsetin
Ωwithtrappedraysand1
ω
denotesthecharacteristicfunctionon
ω
.
Further,forany(
w,∂
t
w
)(

,
0)

H
2
(Ω)

H
01
(Ω)
×
H
01
(Ω),theabove
problemiswell-posedforany
t

0andhaveauniquestrongsolution.
¢¡¢¡Denoteforany
g

C
[0
,
+

);
H
01
(Ω)

C
1
[0
,
+

);
L
2
(Ω),
Z´³1E
(
g,t
)=
|r
g
(
x,t
)
|
2
+
|

t
g
(
x,t
)
|
2
dx
.
2ΩThenforany0

t
0
<t
1
,thestrongsolution
w
satisfiesthefollowing
formula
ZZt12E
(
w,t
1
)

E
(
w,t
0
)+
|

t
w
(
x,t
)
|
dxdt
=0.(2.2)
ωt0

2.1Thepolynomialdecayrate

Ourmotivationforestablishingestimate(1.3)comesfromthefollowing
result.
Theorem2.1.-
Thefollowingtwoassertionsareequivalent.Let
.0>δ

4KimDangPHUNG
(i)
Thereexists
C>
0
suchthatforanysolution
w
of(2.1)withthe
non-nullinitialdata
(
w,∂
t
w
)(

,
0)=(
w
0
,w
1
)

H
2
(Ω)

H
01
(Ω)
×
H
01
(Ω)
,wehave
´³Z
C
EE
((
∂tw,w
0
,
)0)1

Z
k
(
w
0
,w
1
)
k
2
H
01
(Ω)
×
L
2
(Ω)

C
|

t
w
(
x,t
)
|
2
dxdt
.
ω0(ii)
Thereexists
C>
0
suchthatthesolution
w
of(2.1)withtheinitial
data
(
w,∂
t
w
)(

,
0)=(
w
0
,w
1
)

H
2
(Ω)

H
01
(Ω)
×
H
01
(Ω)
satisfies
C2E
(
w,t
)

t
δ
k
(
w
0
,w
1
)
k
H
2

H
01
(Ω)
×
H
01
(Ω)

t>
0.
Remark.-
Itisnotdifficulttosee(e.g.,[Ph2])byaclassicalde-
compositionmethod,atranslationintimeand(2.2),thattheinequality
(1.3)withtheexponent
γ
forthewaveequationimpliestheinequality
of(
i
)inTheorem2.1withtheexponent
δ
=2
γ/
3forthedampedwave
equation.Andconversely,theinequalityof(
i
)inTheorem2.1withthe
exponent
δ
forthedampedwaveequationimpliestheinequality(1.3)
withtheexponent
γ
=
δ/
2forthewaveequation.
ProofofTheorem2.1.-
(
ii
)

(
i
).Supposethat
C2E
(
w,T
)

T
δ
k
(
w
0
,w
1
)
k
H
2

H
01
(Ω)
×
H
01
(Ω)

T>
0.
Thereforefrom(2.2)
ZZTCE
(
w,
0)

δ
k
(
w
0
,w
1
)
k
2
H
2

H
01
(Ω)
×
H
01
(Ω)
+
|

t
w
(
x,t
)
|
2
dxdt
.
Tω0Bychoosing
Ã
2
!
1

k
(
w
0
,w
1
)
k
H
2

H
01
(Ω)
×
H
01
(Ω)
T
=2
CE
(
w,
0),
wegetthedesiredestimate
δ/1Z

k
(
w
0
,w
1)
k
2
H
2

H
01(Ω)
×
H
01(Ω)

Z
2
C
E
(
w,
0)
E
(
w,
0)

2
|

t
w
(
x,t
)
|
2
dxdt
.
ω0(
i
)

(
ii
).Conversely,supposetheexistenceofaconstant
c>
1such
thatthesolution
w
of(2.1)withthenon-nullinitialdata(
w,∂
t
w
)(

,
0)=
(
w
0
,w
1
)

H
2
(Ω)

H
01
(Ω)
×
H
01
(Ω)satisfies
´³Z
c
E
(
w,
0
E
)(+
wE,
(0

)
tw,
0)1

Z
E
(
w,
0)

c
|

t
w
(
x,t
)
|
2
dxdt
.
ω0

Waves,dampedwaveandobservation5
Weobtainthefollowinginequalitiesbyatranslationonthetimevariable
andbyusing(2.2).

s

0
´³δ/1E
(
w,s
)
R
s
+
c
(
E
(
w,
0
E
)(+
wE,
(
s∂
)
tw,
0))
R
|

t
w
(
x,t
)
|
2
(
E
(
w,
0)+
E
(

t
w,
0))

c

s
µ
ωE
³
(
w,
0)+
E
(

t
w,
0)
´
dxdt


Ew,s
+
c
E
(
w,
0
E
)(+
wE,
(
s∂
)
tw,
0)1


E
(
w,s
)


c
E
(
w,
0)+
E
(

t
w,
0)

E
(
w,
0)+
E
(

t
w,
0)
.

Denoting
G
(
s
)=
E
(
w,
0
E
)(+
wE,
(
s∂
)
t
w,
0)
,wededuceusingthedecreasingof
G
tahtõ
1

1

!"õ
1

1

!#
Gs
+
cG
(
s
)

G
(
s
)

cG
(
s
)

Gs
+
cG
(
s
)

whichgives
õ¶!
δ/1c1Gs
+
cG
(
s
)

1+
cG
(
s
).
´³δ¢¡Let
c
1
=
1
c
+
c
1


1
>
0anddenoting
d
(
s
)=
ccs
1
.Wedistinguish
1twocases.
³´
1

³´
δ
If
c
1
s

c
G
(1
s
)
,then
G
(
s
)

cc
1
s
1
and
G
((1+
c
1
)
s
)

d
(
s
).
³´
1

³´
1

If
c
1
s>c
G
1(
s
)
,then
s
+
c
µ
G
(1
s
)
<
(1

+
c
1
)
s
andthedecreasing
³
1
´
1

of
G
gives
G
((1+
c
1
)
s
)

Gs
+
c
G
(
s
)
andthen
cG
((1+
c
1
)
s
)

1+
cG
(
s
).
Consequently,wehavethat

s>
0,

n

N
,
n

1,
´³hG
((1+
c
1
)
s
)

max
d
(
s
)
,
1+
cc
d
(1+
sc
1
)
,
∙∙∙
,
¸
³´
n
³´³´
n
+1
³´
scsc,
1+
c
d
(1+
c
1
)
n
,
1+
c
G
(1+
c
1
)
n
.

Now,remarkthatwithourchoiceof
c
1
,weget
¶µsc1+
cd
(1+
c
)=
d
(
s
)

s>
0.
1

6KimDangPHUNG
Thus,wededucethat

n

1
µ³´
n
+1
³´¶
G
((1+
c
1
)
s
)

max
d
(
s
)
,
1+
cc
G
(1+
sc
1
)
n
µ³´
n
+1


max
d
(
s
)
,
1+
cc
because
G

1,
andconcludethat

s>
0
E
(
w,s
)
µ
s
¶µ
c
(1+
c
1
)

δ
1
E
(
w,
0)+
E
(

t
w,
0)=
G
(
s
)

d
1+
c
1
=
c
1
s
δ
.
Thiscompletestheproof.

2.2Theapproximatecontrollability

Thegoalofthissectionconsistsingivinganapplicationofestimate
(1.2).
Forany
ω
non-emptyopensubsetinΩ,forany
β

(0
,
1),let
T>
0be
giveninTheorem1.1.
¢¡2Let(
v
0
,v
1
,v
0
d
,v
1
d
)

H
2
(Ω)

H
01
(Ω)
×
H
01
(Ω)and
u
thesolution
of(1.1)withinitialdata(
u,∂
t
u
)(

,
0)=(
v
0
,v
1
).
Foranyinteger
N>
0,letusintroduce
ihNXf
N
(
x,t
)=

1
ω

t
w
(2
`
+1)
(
x,t
)+

t
w
(2
`
)
(
x,T

t
),(2.3)
0=`¢¡where
w
(0)

C
[0
,T
];
H
2
(Ω)

H
01
(Ω)isthesolutionofthedamped
waveequation(2.1)withinitialdata
´³w
(0)
,∂
t
w
(0)
(

,
0)=(
v
0
d
,

v
1
d
)

(
u,


t
u
)(

,T
)inΩ,
¢¡andfor
j

0,
w
(
j
+1)

C
[0
,T
];
H
2
(Ω)

H
01
(Ω)isthesolutionofthe
dampedwaveequation(2.1)withinitialdata
´³´³w
(
j
+1)
,∂
t
w
(
j
+1)
(

,
0)=

w
(
j
)
,∂
t
w
(
j
)
(

,T
)inΩ.
Introduce

°°
(
j
)(
j
)
°°
2
M
=
j
su

0
p
°
w
(

,
0)
,∂
t
w
(

,
0)
°
H
2
(Ω)
×
H
01
(Ω)
.

Waves,dampedwaveandobservation
Ourmainresultisasfollows.

7

Theorem2.2.-
Supposethat
M<
+

.Thenthereexists
C>
0
suchthatforall
N>
0
,thecontrolfunction
f
N
givenby(2.3),
drivesthesystem


t
2
v

Δ
v
=1
ω
×
(0
,T
)
f
N
in
Ω
×
(0
,T
),
v
=0
on

Ω
×
(0
,T
),
(
v,∂
t
v
)(

,
0)=(
v
0
,v
1
)
in
Ω,
tothedesireddata
(
v
0
d
,v
1
d
)
approximatelyattime
T
i.e.,
C2k
v
(

,T
)

v
0
d
,∂
t
v
(

,T
)

v
1
d
k
H
01
(Ω)
×
L
2
(Ω)

2
β
M
,
[ln(1+2
N
)]
andsatisfies
k
f
N
k
L

(0
,T
;
L
2
(Ω))

C
(
N
+1)
k
(
v
0
,v
1
,v
0
d
,v
1
d
)
k
(
H
01
(Ω)
×
L
2
(Ω)
)
2
.

Remark.-
Forany
ε>
0,wecanchoose
N
suchthat
³

CM
´
1

CM
'
ε
2
and(2
N
+1)
'
e
ε
,
[ln(1+2
N
)]
2
β
inorderthat
k
v
(

,T
)

v
0
d
,∂
t
v
(

,T
)

v
1
d
k
H
01
(Ω)
×
L
2
(Ω)

ε
,
dnah
(
C

M
)
1

i
εk
f
k
L

(0
,T
;
L
2
(Ω))

e
k
(
v
0
,v
1
,v
0
d
,v
1
d
)
k
(
H
01
(Ω)
×
L
2
(Ω)
)
2
.
In[Zu],amethodwasproposedtoconstructanapproximatecontrol.
Itconsistsofminimizingafunctionaldependingontheparameter
ε
.
However,noestimateofthecostisgiven.Ontheotherhand,estimate
oftheform(1.2)wasoriginallyestablishedby[Ro2]togivethecost(see
also[Le]).Here,wepresentanewwaytoconstructanapproximate
controlbysuperposingdifferentwaves.Givenacosttonotovercome,
weconstructasolutionwhichwillbeclosedintheabovesensetothe
desiredstate.Ittakesideasfrom[Ru]and[BF]likeaniterativetime
reversalconstruction.

82.2.1Proof

KimDangPHUNG

Considerthesolution
ihNXV
(

,t
)=
w
(2
`
+1)
(

,t
)+
w
(2
`
)
(

,T

t
).
0=`Wededucethatfor
t

(0
,T
)
¤£NP


t
2
V
(

,t
)

Δ
V
(

,t
)=

1
ω

t
w
(2
`
+1)
(

,t
)+

t
w
(2
`
)
(

,T

t
),
0=`
V
=0on

Ω
×
(0
,T
),
(
V,∂
t
V
)(

,
0)=0inΩ
.
¢¡Now,fromthedefinitionof
w
(0)
,thepropertyof
w
(
j
+1)
,∂
t
w
(
j
+1)
(

,
0)
andachangeofvariable,weobtainthat
¢¡¢¡(
V,∂
t
V
)(

,T
)=
w
(0)
,


t
w
(0)
(

,
0)+
w
(2
N
+1)
,∂
t
w
(2
N
+1)
(

,T
)
¢¡=(
v
0
d
,v
1
d
)

(
u,∂
t
u
)(

,T
)+
w
(2
N
+1)
,∂
t
w
(2
N
+1)
(

,T
).
Finally,thesolution
v
=
V
+
u
satisfies


t
2
v

Δ
v
=1
ω
×
(0
,T
)
f
N
inΩ
×
(0
,T
),
v
=0on

Ω
×
(0
,T
),

(
v,∂
t
v
)(

,
0)=(
v
0
,v
1
)in
¡
Ω,
¢

(
v,∂
t
v
)(

,T
)=(
v
0
d
,v
1
d
)+
w
(2
N
+1)
,∂
t
w
(2
N
+1)
(

,T
)inΩ
.
Clearly,
´³k
v
(

,T
)

v
0
d
,∂
t
v
(

,T
)

v
1
d
k
2
H
1
(Ω)
×
L
2
(Ω)
=2
Ew
(2
N
+1)
,T
.
0¢¡Itremainstoestimate
Ew
(2
N
+1)
,T
.Weclaimthat
´³∃
C>
0

N

1
Ew
(2
N
+1)
,T

CM
.
[ln(1+2
N
)]
2
β

Indeed,fromTheorem1.1,wecaneasilyseebyaclassicaldecom-
positionmethodthatthereexist
C>
0and
T>
0suchthatforany
,0≥j°°°
w
(
j
+1)
(

,
0)
,∂
t
w
(
j
+1)
(

,
0)
°
2
12
Ã
H
0
(Ω)
×
L
(Ω)
!
β/1k
w
(
j
+1)
(

,
0)
,∂
t
w
(
j
+1)
(

,
0)
k
H
2(Ω)
×
H
1(Ω)

C
exp
C
k
w
(
j
+1)
(

,
0)
,∂
t
w
(
j
+1)
(

,
0)
k
H
01(Ω)
×
L
2(Ω)
R
T
R¯¯
2
0
ω
¯

t
w
(
j
+1)
(
x,t
)
¯
dxdt
.

Waves,dampedwaveandobservation
Since
³´³´
Ew
(
j
+1)
,
0=
Ew
(
j
)
,T

j

0,

wededucefrom(2.2)thatforany
j

0
¡¢Ã!
1
/
(2
β
)
(
j
+1)
M
Ew,
0

C
exp
C
k
w
(
j
+1)
(

,
0)
,∂
t
w
(
j
+1)
(

,
0)
k
2
H
1(Ω)
×
L
2(Ω)
£
E
¡
w
(
j
)
,T
¢

E
¡
w
(
j
+1)
,T
¢¤
.
0

9

teL´³d
j
=
Ew
(
j
+1)
,T
.
Byusingthedecreasingpropertyofthesequence
d
j
,thatis
d
j

d
j

1
,
weobtainthatforanyinteger0

j

2
N
³
M
´
1
/
(2
β
)
d
j

Ce
C
d
2
N
[
d
j

1

d
j
].

Bysummingover[0
,
2
N
],wededucethat
³
M
´
1
/
(2
β
)
(2
N
+1)
d
2
N

Ce
C
d
2
N
[
d

1

d
2
N
].

Finally,usingthefactthat
d

1

M
,itfollowsthat
Cd
2
N

[ln(1+2
N
)]
2
β
M
.
Thiscompletestheproofofourclaim.

Ontheotherhand,thecomputationoftheboundof
f
N
isimmediate.
Therefore,wecheckthatforsome
C>
0and
T>
0,

k
f
N
k
L

(0
,T
;
L
2
(Ω))

C
(
N
+1)
k
(
v
0
,v
1
,v
0
d
,v
1
d
)
k
(
H
01
(Ω)
×
L
2
(Ω)
)
2
,

C2k
v
(

,T
)

v
0
d
,∂
t
v
(

,T
)

v
1
d
k
H
01
(Ω)
×
L
2
(Ω)

2
β
M
,
[ln(1+2
N
)]
forany
β

(0
,
1)andanyinteger
N>
0.Thiscompletestheproofof
ourTheorem.

10KimDangPHUNG
2.2.2Numericalexperiments

Here,weperformnumericalexperimentstoinvestigatethepracticalap-
plicabilityoftheapproachproposedtoconstructanapproximatecon-
trol.Forsimplicity,weconsiderasquaredomainΩ=(0
,
1)
×
(0
,
1),
ω
=(0
,
1
/
5)
×
(0
,
1).Thetimeofcontrollabilityisgivenby
T
=4.
Forconveniencewerecallsomewell-knownformulas.Denoteby
{
e
j
}
j

1
theHilbertbasisin
L
2
(Ω)formedbytheeigenfunctionsoftheoperator

Δwitheigenvalues
{
λ
j
}
j

1
,suchthat
k
e
j
k
L
2
(Ω)
=1and0

1
<
λ
2

λ
3
≤∙∙∙
,i.e.,
¢¡½λ
j
=
π
2
k
j
2
+
`
j
2
,
k
j
,`
j

N

,
e
j
(
x
1
,x
2
)=2sin(
πk
j
x
1
)sin(
π`
j
x
2
).
Thesolutionof


t
2
v

Δ
v
=
f
inΩ
×
(0
,T
),

v
=0on

Ω
×
(0
,T
),
(
v,∂
t
v
)(

,
0)=(
v
0
,v
1
)inΩ,
where
f
isintheform
Xf
(
x
1
,x
2
)=

1
ω
f
j
(
t
)
e
j
(
x
1
,x
2
),
1≥jisgivenbytheformula
½P
G
0
¡p¢
11
¡p¢
v
(
x
1
,x
2
,t
)=
G
l

i
+
m

a
j
cos

j
+
a
j

λ
j
sin

j
1=j¾¢p¡Rt+

10
sin(
t

s
)
λ
j
R
j
(
s
)
dse
j
(
x
1
,x
2
),
λjerehwP
G
P¯¯
2
v
0
(
x
1
,x
2
)=
G
l

i
+
m

a
j
0
e
j
(
x
1
,x
2
),
λ
j
¯
a
j
0
¯
<
+

,

j
=1
j

1
P
G
P¯¯
2
v
1
(
x
1
,x
2
)=
G
l

i
+
m

a
j
1
e
j
(
x
1
,x
2
),
¯
a
j
1
¯
<
+

,
j
=1
j

1
¢R¡GP

R
j
(
t
)=

G
l

i
+
m

ω
e
i
e
j
dx
1
dx
2
f
i
(
t
).
1=iHere,
G
willbethenumberofGalerkinmode.Thenumericalresultsare
shownbelow.Theapproximatesolutionofthedampedwaveequation
isestablishedviaasystemofODEsolvedbyMATLAB.