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AdvancesinAppliedMathematics
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,2023,12(6),2965-2978
PublishedOnlineJune2023inHans.https://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2023.126299
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GalerkinSpectralMethodBased
onChebyshevPolynomialsandIts
ApplicationonNumericalSolution
ofPartialDifferentialEquations
JiaWang
1
,RonghuaCheng
2
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,2023,12(6):2965-2978.DOI:10.12677/aam.2023.126299
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1
SchoolofStatisticsandMathematics,YunnanUniversityofFinanceandEconomics,Kunming
Yunnan
2
OfficeofAcademicAffairs,YunnanUniversityofFinanceandEconomics,KunmingYunnan
Received:May28
th
,2023;accepted:Jun.23
rd
,2023;published:Jun.30
th
,2023
Abstract
GalerkinspectralmethodbasedonChebyshevpolynomialshasbeenwidelyusedto
numericallysolvetheboundaryvalueproblemandinitialboundaryvalueproblemof
partialdifferentialequation.Howeverdetailedintroductionofthemethodandits
applicationhavebeenrarelyseeninChineseJournals.Inthispaper,wepresentthe
detailedimplementation procedureofChebyshevGalerkinspectralmethodby means
ofsolvingtheboundaryvalueproblemofHelmholtzequation,initialboundaryvalue
problemoftime-dependentSchrodingerequationandinitialboundaryvalueproblem
ofwaveequation,respectively.Ouralgorithmisbuilton:Firstweassumethatthe
unknownfunctioncanbeapproximatedbytheexpansionofChebyshevpolynomials;
nextwe plugthisexpansioninto thedifferential equation; thenweusetheweak formu-
lation of theequationand make it zero,and obtainthe discretesystemwhich satisfied
thecoefficientsofapproximationexpansionofunknownfunction;finallysolvingthe
discretesystemgivesustheapproximatedvalueofunknownfunction.Galerkinspec-
tralmethodbasedonChebyshevpolynomialshasthemeritofhigh-orderaccuracy,
andsimpleimplementprocedure.Ournumericalalgorithmandnumericalexamples
haveshownallofthesemeritsoftheChebyshevspectralcollocationmethod.
Keywords
ChebyshevPolynomials,GalerkinSpectralMethod,NumericalComputation
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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DOI:10.12677/aam.2023.1262992968
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>
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¯
K
u
t
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Lu
+
f,x
∈
Ω = (
a,b
)
,
(2.5)
B
−
u
= 0
x
=
a,B
+
u
= 0
x
=
b,
(2.6)
u
(
x,
0) =
g
(
x
)
,x
∈
Ω
.
(2.7)
ù
p
·
‚
b
½
L,B
þ
´
,
˜®
•
/
ª
‡
©
Ž
f
§
u
=
u
(
x,t
)
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•
¼
ê
.
Ù
{
¼
ê
þ
´
®
•
¼
ê
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o
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½
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e
µ
1
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&
Ä
¼
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1
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>
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1
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1
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|
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m
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u
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(
t
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n
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x
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u
(
x,t
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x,t
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n
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f
n
(
t
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n
(
x
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g
(
x
)
≈
g
N
(
x
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N
X
n
=0
ˆ
g
n
φ
n
(
x
)
.
ò
þ
ã
Ð
m
ª
“
\
•
§
|
t
u
=
Lu
+
f
†
u
(
x,
0) =
g
(
x
)
Ĵ
§
¦
e
ª
ð
¤
á
µ
∂u
N
∂t
−
(
L
N
u
N
+
f
N
)
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w
= 0
,
∀
v
=
ψ
l
,l
= 0
,
1
,
···
,N,
(
u
N
(
x,
0)
−
g
N
(
x
)
,v
)
w
= 0
,
∀
v
=
ψ
l
,l
= 0
,
1
,
···
,N,.
DOI:10.12677/aam.2023.1262992969
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n
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u
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u
n
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t
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1
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n
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x
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b
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J
a
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ò
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u
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•
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2
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±
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J
¼
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sin(
nx
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½
cos(
nx
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½
e
(
inx
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(
i
2
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−
1)
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•
Ä
¼
ê
¶
X
J
™
•
¼
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u
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½
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a,b
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§
k
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ª
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J
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•
¼
ê
u
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,
+
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±
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(
d
ž
b
½
a
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J
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ª
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•
Ä
¼
ê
;
X
J
™
•
¼
ê
u
´
½
Â
•
[
−∞
,
+
∞
]
þ
š
±
Ï
¼
ê
(
d
ž
b
½
a
=
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=+
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p
I
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−
1
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b
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n
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x
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b
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m
S
N
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V
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±
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(
x
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x
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x
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T
k
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(
x
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j
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(
x
),
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§
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§
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N
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u
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2
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j
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u
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x
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N
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j
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j
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r
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q
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j
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u
j
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j
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j
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x
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k
(
x
))
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f
(
x
)
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k
(
x
))
ω
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= 0
,
1
,
···
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−
2
.
DOI:10.12677/aam.2023.1262992970
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b
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j
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u
j
a
kj
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(
x
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k
(
x
))
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,
1
,
···
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−
2
.
ù
p
-
¼
ê
w
=
w
(
x
) = 1
/
√
1
−
x
2
,
a
kj
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−
(
φ
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j
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x
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φ
k
(
x
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k
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k
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2
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k
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k
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j
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k
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N
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k
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u
1
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k
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N
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,
k
f
k
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f
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x
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k
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x
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M
P
i
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f
(
x
i
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φ
k
(
x
i
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i
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i
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!
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ω
0
=
ω
M
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π/
2
M
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ω
k
=
π/M
é
u
1
≤
i
≤
M
−
1.
X
J
·
‚
2
½
Â
F
= (
f
0
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1
,
···
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N
−
2
)
T
;
b
u
= (
b
u
0
,
b
u
1
,
···
,
b
u
N
−
2
)
T
;
B
= (
b
kj
)
0
≤
k,j
≤
N
−
2
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a
kj
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0
≤
k,j
≤
N
−
2
,
@
o
·
‚
Œ
X
e
l
Ñ
/
ª
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5
•
§
|
(
αB
+
A
)
b
u
=
F.
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L
¦
)
þ
ã
‚
5
•
§
|
§
·
‚
Ò
Ð
m
ª
u
N
=
P
N
−
2
n
=0
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u
n
φ
n
(
x
)
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X
ê
|
¤
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u
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u
0
,
b
u
1
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b
u
N
−
2
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l
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·
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§
>
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¯
K
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u
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fin
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I
2
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|
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.
(3.5)
T
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¼
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u
N
∈
V
2
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,
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α
(
u
N
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)
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−
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u
N
,v
)
w
= (
f,v
)
w
,
∀
v
∈
V
2
N
,
(3.6)
ù
p
-
¼
ê
ω
=
ω
(
x
1
,x
2
) = Π
2
i
=1
(1
−
x
2
i
)
−
1
/
2
.
S
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•
(
u,v
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R
Ω
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x
.
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d
x
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dx
1
dx
2
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b
½
¼
ê
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m
V
2
N
=
span
{
φ
k
(
x
1
)
φ
j
(
x
2
) :
k,j
= 0
,
1
,
···
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−
2
}
.
DOI:10.12677/aam.2023.1262992971
A^
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Z
§
¤
I
u
¿
…
b
½
k
C
q
u
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u
N
=
N
−
2
X
k,j
=0
b
u
kj
φ
k
(
x
1
)
φ
j
(
x
2
)
,
·
‚
ª
f
(3.6)
¥
v
=
φ
l
(
x
1
)
φ
m
(
x
2
)
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,
1
,
···
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−
2,
¿
3
ª
f
(3.6)
m
>
|
^
chebyshev-
Gauss-quadrature
ú
ª
,
k
f
kj
= (
f
(
x
1
,x
2
)
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k
(
x
1
)
φ
j
(
x
2
))
ω
=
M
X
p,q
=0
f
(
x
1
p
,x
2
q
)
φ
l
(
x
1
p
)
φ
m
(
x
2
q
)
ω
p
ω
q
.
Ù
¥
x
1
p
=cos
πp/M,x
2
q
=cos
πq/M,p,q
=0
,...,M
•
chebyshev-Gauss-Lobatto
!
:
,
ω
0
=
ω
0
=
ω
M
=
ω
M
=
π/
2
M
,
ω
k
=
ω
k
=
π/M
é
u
1
≤
i
≤
M
−
1.
@
o
3
‘
œ
/
e
•
§
(3.6)
C
¤
X
e‚
5
•
§
|
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b
UB
+
A
b
UB
+
B
b
UA
T
=
G.
(3.7)
ù
p
Ý
b
U
= (
b
u
kj
)
k,j
=0
,
1
,
···
,N
−
2
,G
= (
f
kj
)
k,j
=0
,
1
,
···
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−
2
.
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)
þ
ã
‚
5
•
§
|
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•
)
b
U
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X
e
µ
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k
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Ñ
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±
9
é
Λ
§
¦
A
−
1
B
=
E
Λ
E
−
1
.
(2)
O
Ž
G
∗
=
E
−
1
A
−
1
G
.
(3)
¦
)
•
§
|
(
αλ
i
+1)
Bv
i
+
λ
i
Av
i
=
g
i
,
l
¥
v
i
.
ù
p
é
Ý
Λ =
diag
(
λ
0
,λ
1
,
···
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N
−
2
).
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i
=(
v
i
0
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i
1
,
···
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iN
−
2
)
T
,
g
i
=(
g
i
0
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i
1
,
···
,
g
iN
−
2
)
T
,
i
=0
,
1
,
···
,N
−
2.
·
‚
5
¿
G
∗
=
(
g
0
,g
1
,
···
,g
N
−
2
)
T
§
V
= (
v
0
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1
,
···
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N
−
2
)
T
.
(4)
•
b
U
=
EV
§
•
Ò
´
Ð
m
ª
¥
X
ê
.
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)
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>
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¯
K
Chebyshev-Galerkin
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3
!
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§
©
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ž
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Å
Ä
•
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>
Š
¯
K
9
¹
ž
9
D
•
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>
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¯
K
Chebyshev-Galerkin
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O
L
§
"
k
•
Ä
X
e
˜
‚
5
Å
Ä
•
§
∂u
∂t
=
∂u
∂x
,
(3.8)
±
9
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A
>
.
^
‡
u
(1
,t
) = 0(3.9)
†
Ð
©
^
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u
(
x,
0) =
f
(
x
)
.
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Ä
u
Chebyshev
õ
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ª
E
Ä
¼
ê
DOI:10.12677/aam.2023.1262992972
A^
ê
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Z
§
¤
I
u
φ
n
(
x
) =
T
n
(
x
)
−
1
,
÷
v
>
.
^
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‚
Ï
é
X
e
C
q
Ð
m
ª
u
N
(
x,t
)
u
(
x,t
)
≈
u
N
(
x,t
) =
N
X
n
=1
a
n
(
t
)
φ
n
(
x
) =
N
X
n
=1
a
n
(
t
)(
T
n
(
x
)
−
1)
.
5
¿
ù
p
¦
Ú
ª
f
´
l
n
= 1
m
©
§
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l
n
= 0
m
©
,
ù
´
Ï
•
φ
0
(
x
) = 0.
¦
{
‘
R
N
(
x,t
) =
∂u
N
∂t
−
∂u
N
∂x
u
¼
ê
φ
k
(
x
)
∈
L
2
w
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DOI:10.12677/aam.2023.1262992973
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DOI:10.12677/aam.2023.1262992974
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1.
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DOI:10.12677/aam.2023.1262992975
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[1]Gottlieb,D.andOrszag,S.A.(1977)NumericalAnalysisofSpectralMethods:Theoryand
Applications.SIAM,Philadelphia,PA.https://doi.org/10.1137/1.9781611970425
[2]Canuto, C., Hussaini, M.Y., Quarteroni, A.,
etal.
(1988) Spectral Methods in FluidDynamics.
Springer,Berlin.https://doi.org/10.1007/978-3-642-84108-81
[3]Funaro,D.(1992)PolynomialApproximationofDifferentialEquations.Springer,Berlin.
https://doi.org/10.1007/978-3-540-46783-0
[4]Funaro,D.(1997)SpectralElementsforTransport-DominatedEquations.Springer,Berlin.
https://doi.org/10.1007/978-3-642-59185-3
[5]Guo,B.Y.(1998)SpectralMethodsandTheirApplications.WorldScientific,Singapore.
[6]Peyret,R.(2002)SpectralMethodsforIncompressibleViscousFlow.Springer-Verlag,New
York.https://doi.org/10.1007/978-1-4757-6557-1
[7]Shen,J.andTang,T.(2006)SpectralandHigh-OrderSpectralswithApplication.Science
Press,Beijing.
[8]Shen,J.,Tang,T.andWang,L.L.(2011)SpectralMethods.Springer,Heidelberg.
https://doi.org/10.1007/978-3-540-71041-7
[9]
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DOI:10.12677/aam.2023.1262992977
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[13]Brif,C.,Chakrabarti,R.andRabitz,H.(2010)ControlofQuantumPhenomena:Past,
Present,andFuture.
NewJournalofPhysics
,
12
,2181-2188.
https://doi.org/10.1088/1367-2630/12/7/075008
DOI:10.12677/aam.2023.1262992978
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