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AdvancesinAppliedMathematicsA^êÆ?Ð,2023,12(6),2965-2978
PublishedOnlineJune2023inHans.https://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2023.126299
ÄuChebyshevõ‘ªGalerkinÌ•{3
¦) ‡©•§¯K¥A^
ZZZ
1
§§§¤¤¤IIIuuu
2∗
1
H㲌ÆÖ?,H&²
2
H㲌ÆÚO†êÆÆ,H&²
ÂvFϵ2023c528F¶¹^Fϵ2023c623F¶uÙFϵ2023c630F
Á‡
ÄuChebyshevõ‘ªGalerkinÌ•{2•A^u ‡©•§>Š¯K†Ð>Š¯KOŽ
¥§•[0T•{äNA^L§©Ù"©ÏL¦)äN~f(Helmholtz•§
>Š¯K!¹ž˜ÅÄ•§Ð>Š¯K±9¹ž‚59D•§Ð>Š¯K)5•
[0ÄuChebyshevõ‘ªGalerkinÌ•{¢yL§"kb½•§™•¼êU^Ä
uChebyshevõ‘ªÐmª5%C§,òT™•¼ê%CÐmª“\‡©•§ƒ¥§2•
§f/ª¿¦Ù•"§?™•¼êÐmª¥Xê¤÷v•§|§•ªÏL¦)T•
§|™•¼êCq&E.ÄuChebyshevõ‘ªGalerkinÌ•{äk°Ýp!¢yL§
{ ü`:§© ÏLŽ{¢yL§9êŠ~f0ÄuChebyshevõ‘ªGalerkinÌ•
{ù`:"
'…c
Chebyshevõ‘ª§GalerkinÌ•{§êŠOŽ
GalerkinSpectralMethodBased
onChebyshevPolynomialsandIts
ApplicationonNumericalSolution
ofPartialDifferentialEquations
JiaWang
1
,RonghuaCheng
2∗
∗ÏÕŠö"
©ÙÚ^:Z,¤Iu.ÄuChebyshevõ‘ªGalerkinÌ•{3¦) ‡©•§¯K¥A^[J].A^êÆ?
Ð,2023,12(6):2965-2978.DOI:10.12677/aam.2023.126299
Z§¤Iu
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/
1.Úó
Ôn!zÆ!)Ô!ó§+•Nõy– Œ±^‡©•§5£ã,‡©•§(~‡©•§
DOI:10.12677/aam.2023.1262992966A^êÆ?Ð
Z§¤Iu
½ ‡©•§))Û)Ø´¦),ISÆö&?ïć©•§êŠOŽ•{,~Xk•©
{!k•{!Ì•{.Ì•{ÏÙ°ÝpA:É2•'5,'uÌ•{ƒ'=©©z
õ:GottliebŠö(1977)•@ØãÌ•{n†A^[1];Canuto<(1988)3Ù;ÍØãÌ
•{36Nå ÆA^[2]¶Funaro(1992,1997)3¦;Í¥©OØã˜Ì•{ÄnØ!
˜Ì•{3¦)é6‘ÓÌ•§¦)¥A^[3,4];HÕ(1998)3ÙØÍ¥•[ØãÌ
•{Än†nØ©ÛL§[5];Peyret (2002)3Ö¥Øã¦)ØŒØ Ê56NÌ•{[6];
•CShenÚTang(2006),Shen,TangÚWang(2011)<3Ù=©;Í¥•JøŒþÄÌ•{
Úp°ÝŽ{9ƒA•{Âñ5©Û9Ø©ÛnØ[7,8].
3ISïÄ©z¥§•[0GalerkinÌ•{¦) ‡©•§Ž{L§©zé,¦Ö
öØN´)T•{.Ò·‚¤•,3·IkXeïÄö&?GalerkinÌ•{:š7(2013)‰
Ño·Üšàg>Š¯KPetrov-GalerkinÌ•{[9];Çu(2020,2021)©O3©Ù¥æ
^ChebyshevGalerkinÌ•{¦)‘š‚5‡A*Ñ•§Ún ‡©•§[10,11].]˜
(2021)‰Ñ˜aÌGalerkin.|¢òÿ{OŽŒ‚5ý¯KÂñ5©Û[12].
©Äuc<ïÄ,•[0XÛÄuChebyshevõ‘ªEGalerkinÌ•{5©Oꊦ
)Helmholtz•§>Š¯K!¹ž‚5Ž™•§Ð>Š¯K.T•{Œ^u?n˜E,
‡©•§§§U3¢y{üz•§Óž,ŒˆpÌ°Ý,ù椕T•{˜ŒA^d
Š.3äN¢yL§¥§kb½•§¥™•¼êU^ÄuChebyshevõ‘ªÐmª5%C,,
òT™•¼ê%CÐmª“\•§ƒ¥,2•§f/ª¿¦•§f/ª•",?
™•¼êÐmª¥Xê¤÷v•§|,•ªÏL¦)T•§|™•¼êCq&E.
©(SüXeµ12!ÏL‡©•§>Š¯KÚÐ>Š¯K§0GalerkinÌ•{A
^n.313!¥§©O•[ÛHelmholtz•§>Š¯K!¹ž‚5Ž™•§Ð>Š¯
KÄuChebyshevõ‘ªGalerkinÌ•{Ž{L§.314!¥§ÏLA‡äNêŠ~f§y
¢313!¥0•{Œ¢y¦)˜‘!‘Helmholtz•§>Š¯K!˜‘!‘¹ž‚5
Ž™•§Ð>Š¯K,êŠ(Jy¢¤JÑꊕ{p°Ý†p5.315!ƒ¥§·
‚o(¤JÑꊕ{`":.
2.GalerkinÌ•{¦)‡©•§ÄL§
!̇0‡©•§>Š¯KGalerkinÌ•{¦)ÄL§†‡©•§Ð>Š¯K
GalerkinÌ•{¦)ÄL§"
2.1.GalerkinÌ•{¦)>Š¯K
•Ä¦)‡©•§>Š¯K
Lu= f,x∈Ω = (a,b),(2.1)
B
−
u= 0x= a,B
+
u= 0x= b,(2.2)
DOI:10.12677/aam.2023.1262992967A^êÆ?Ð
Z§¤Iu
ùpu= u(x)´˜™•¼ê§f=f(x)´˜®•¼ê"·‚„b½L,B
−
,B
+
þ´,˜®•Žf(Œ
U¹k‡©/ª)"
@oGalerkinÌ•{¦)T>Š¯K˜„OÚ½Xeµ
1˜Ú.À½Ü·Á&ļêφ
n
= φ
n
(x),n= 0,1,2,···,N§§‚þ÷v>.^‡(2.2)§•
Ò´µ
B
−
φ
n
= 0x= a,B
+
φ
n
= 0x= b,n= 0,1,···,N.
1Ú.|^Á&ļêφ
n
,n= 0,1,2,···,NÐmª§Cqµ
u(x) ≈u
N
(x) =
N
X
n=0
ˆu
n
φ
n
(x) ≈u(x),
f(x) ≈f
N
(x) =
N
X
n=0
ˆ
f
n
φ
n
(x) ≈f(x).
òþãÐmª“\•§Lu
N
= f
N
ƒ¥§2©OSȧµ
(Lu
N
,ψ
l
)
w
= (f
N
,ψ
l
)
w
l= 0,1,···,N,
ùp¼êSȽ•(u,v)
w
=
R
Ω
u(x)v(x)w(x)dx,w=w(x)´,˜-¼ê§ψ
l
= ψ
l
(x),l=
0,1,···,N¡•ÿÁ¼ê"
1nÚ.Šâþª§'uXêˆu
n
,n= 0,1,···,N“ꕧ|"
1oÚ.|^ꊕ{¦)†Xêˆu
n
,n= 0,1,···,Nƒ'“ꕧ|"
1ÊÚ.ò¦Xêˆu
n
,n= 0,1,···,N“ \Ðmªu
N
(x)ƒ¥,·‚Ò™•¼ê˜‡
ÌCq"
Remark2.1.Ï~§Á&ļêφ
n
=φ
n
(x),n=0,1,2,···,N†ÿÁ¼êψ
l
=ψ
l
(x),l=
0,1,···,N/ªƒÓ"¦‚/ªØ˜ž§·‚¡þ¡Ì•{•Petro-GalerkinÌ•{"
Remark2.2.Á&ļêφ
n
=φ
n
(x),n=0,1,2,···,NÏ~l,˜¼ê{P
n
(x)}
∞
n=0
¥
§T¼ê'u-¼êw= w(x)µ
(P
i
,P
j
)
w
=
Z
Ω
P
i
(x)P
j
(x)w(x)dx= δ
ij
=



1i= j,
0i6= j.
Remark2.3.XJ؇¦Á&ļêφ
n
(x),n=0,1,2,···,N÷v >.^‡(2.2)§ @oòe¡Ð
mª
u
N
(x) =
N
X
n=0
ˆu
n
φ
n
(x).
DOI:10.12677/aam.2023.1262992968A^êÆ?Ð
Z§¤Iu
“\>Š¯K(2.1)-(2.2)ƒ¥§džXêˆu
0
,ˆu
1
,···,ˆu
N
÷veª
N
X
n=0
ˆu
n
(Lφ
n
,φ
i
)
w
= (f,φ
i
),i= 0,1,2,···,N−2(2.3)
B
−
u
N
(a) = 0,B
+
u
N
(b) = 0.(2.4)
ÏL¦)þ㕧|(2.3)-(2.4)ŒXêˆu
0
,ˆu
1
,···,ˆu
N
.dž§*ªf(2.3)§·‚uyÙ¥
i6= N−1,Nùü‘"ù«•{q¡•TauÌ•{§§´dLanczosu1938cJÑ"TauÌ•{
nØØ3Gottlieb!Orszag(1977)„kCanuto(1988)Ö¥kù«Ø0"3¢SA^
ƒ¥§<‚kž•^§5)‡©•§>Š¯K"
2.2.GalerkinÌ•{¦)Ð>Š¯K
XJ·‚•Ä‡©•§Ð>Š¯K
u
t
= Lu+f,x∈Ω = (a,b),(2.5)
B
−
u= 0x= a,B
+
u= 0x= b,(2.6)
u(x,0) = g(x),x∈Ω.(2.7)
ùp·‚b½L,Bþ´,˜®•/ª‡©Žf§u=u(x,t)•™•¼ê.Ù{¼êþ´®•¼
ê"
@oÌ•{˜„OÚ½Xeµ
1˜Ú.ÀÁ&ļêφ
n
,n= 0,1,2,···,N,¦§‚÷v>.^‡µ
B
−
φ
n
= 0x= a,B
+
φ
n
= 0x= b,n= 0,1,···,N.
1Ú.|^Á&ļêφ
n
= φ
n
(x),n= 0,1,2,···,NÐmª§µ
u(x,t) ≈u
N
(x,t) =
N
X
n=0
ˆu
n
(t)φ
n
(x),
u(x,t) ≈f
N
(x,t) =
N
X
n=0
ˆ
f
n
(t)φ
n
(x),
g(x) ≈g
N
(x) =
N
X
n=0
ˆg
n
φ
n
(x).
òþãÐmª“\•§|
t
u= Lu+f†u(x,0) = g(x)ƒ¥§¦eªð¤áµ

∂u
N
∂t
−(L
N
u
N
+f
N
),v

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.2023.1262992969A^êÆ?Ð
Z§¤Iu
1nÚ.Šâþª§'uXêˆu
n
(t)(n= 0,1,···,N)~‡©•§|Њ¯K"
1oÚ.|^ꊕ{£~Xk•©{¤¦)†Xêˆu
n
(t)(n= 0,1,···,N)ƒ'~‡©•
§|Њ¯K"
1ÊÚ.ò¦Xêˆu
n
(t)(n=0,1,···,N)“\Ðmªu
N
(x,t)ƒ¥,·‚Ò˜‡ÌC
q"
3þ¡Á&ļêφ
n
(x)ÀL§ƒ¥§XJ™•¼êu´½Â•[a,b]þ±Ï¼ê(ù
pb½a=0,b=2π¶XJa6=0,b6=2π,Œ‰˜‚5C†§ò™•¼êu½Â•d[a,b]C
•[0,2π])§·‚Œ±ÀJ¼êsin(nx)½cos(nx)½e
(inx)
(i
2
=−1)Š•Ä¼ê¶XJ™•¼êu´
½Â•[a,b]þš±Ï¼ê§k‰C†§ò™•¼êu½Â•d[a,b]C•[−1,1]),ŒÀJõ‘
ª§~XChebyshevõ‘ª,Legendreõ‘ªŠ•Ä¼ê¶ XJ™•¼êu´½Â•[0,+∞]þ
š±Ï¼ê(džb½a=0,b=+∞),ŒÀJHermitõ‘ªŠ•Ä¼ê;XJ™•¼êu´½Â
•[−∞,+∞]þš±Ï¼ê(džb½a=−∞,b=+∞),ŒÀJLagureeõ‘ªŠ•Ä¼ê"3
e˜!¥§·‚ò0ÄuChebyshevõ‘ªGalerkinÌ•{OL§"
3.ÄuChebyshevõ‘ªGalerkinÌ•{
3.1.)Helmholtz•§Chebyshev-Galerkin•{
!¥§©O0˜‘9‘Helmholtz>Š¯KChebyshev-Galerkin•{
k•Ä˜‘Helmholtz•§>Š¯K¦)
αu−u
00
= finΩ = I,u|
Ω
= 0,(3.1)
ùpI= (−1,1)"
b½T
n
(x)•ngChebyshevõ‘ª§¿…b½¼ê˜mS
N
,V
N
©O•
S
N
= span{T
0
(x),T
1
(x),...,T
N
(x)},V
N
= {v∈S
N
: v(±1) = 0}.
é²w§V
N
= span{φ
0
(x),φ
1
(x),...,φ
N−2
(x)},ùpφ
k
(x) = T
k
(x)−T
k+2
(x).
|^Chebyshev-Galerkin•{§b½¼êu≈u
N
=
N−2
P
j=0
bu
j
φ
j
(x),òƒ“\˜‘Helmholtz•
§•§αu
N
−u
00
N
= f,=α
N−2
P
j=0
bu
j
φ
j
(x)−
N−2
P
j=0
bu
j
φ
00
j
(x) = f(x).ù†‰§‡¦r"
CqOŽL§¥§·‚‡¦eãf/ª¤á
(α
N−2
X
j=0
bu
j
φ
j
(x)−
N−2
X
j=0
bu
j
φ
00
j
(x),φ
k
(x))
ω
= (f(x),φ
k
(x))
ω
,k= 0,1,···,N−2.
DOI:10.12677/aam.2023.1262992970A^êÆ?Ð
Z§¤Iu
u´
α
N−2
X
j=0
bu
j
b
kj
+
N−2
X
j=0
bu
j
a
kj
= (f(x),φ
k
(x))
ω
,k= 0,1,···,N−2.
ùp-¼êw= w(x) = 1/
√
1−x
2
,a
kj
= −(φ
00
j
(x),φ
k
(x))
ω
±9b
kj
= (φ
j
(x),φ
k
(x))
ω
.ÏLOŽ
Υ
b
kj
= b
jk
=





(c
k
+1)
2
π,j= k
−
π
2
,j= k−2 ½j= k+2;
0,Ù{œ/
(3.2)
±9
a
kj
=







2π(k+1)(k+2),j= k
4π(k+1),j= k+2,k+4,k+6,···.
0j>k½j+kÛ
(3.3)
Ù¥Xêc
0
= c
N
= 2,c
k
= 1éu1 ≤k≤N−1.
|^chebyshev-Gauss-quadratureúª,kf
k
=(f(x),φ
k
(x))
w
=
M
P
i=0
f(x
i
)φ
k
(x
i
)ω
i
,Ù¥x
i
=
cosπi/M,i= 0,...,M•chebyshev-Gauss-Lobatto!:,ω
0
=ω
M
=π/2M,ω
k
=π/Méu1 ≤i≤
M−1.XJ·‚2½Â
F= (f
0
,f
1
,···,f
N−2
)
T
;
bu= (bu
0
,bu
1
,···,bu
N−2
)
T
;
B= (b
kj
)
0≤k,j≤N−2
,A= (a
kj
)
0≤k,j≤N−2
,
@o·‚ŒXelÑ/ª‚5•§|
(αB+A)bu= F.(3.4)
ÏL¦)þã‚5•§|§·‚ÒÐmªu
N
=
P
N−2
n=0
bu
n
φ
n
(x)ƒ¥Xê|¤•
þbu= (bu
0
,bu
1
,···,bu
N−2
)
T
.l§·‚Ò¯K˜«Chebyshev-GalerkinÌCq"
2•Ä‘Helmholtz•§>Š¯K¦)
αu−∆u= finΩ = I
2
,u|
Ω
= 0.(3.5)
T•{Ïé¼êu
N
∈V
2
N
,¦
α(u
N
,v)
w
−(∆u
N
,v)
w
= (f,v)
w
,∀v∈V
2
N
,(3.6)
ùp-¼êω= ω(x
1
,x
2
) = Π
2
i=1
(1−x
2
i
)
−1/2
.SÈ•(u,v)
ω
=
R
Ω
uvωdx.¿…§dx= dx
1
dx
2
.b
½¼ê˜m
V
2
N
= span{φ
k
(x
1
)φ
j
(x
2
) : k,j= 0,1,···,N−2}.
DOI:10.12677/aam.2023.1262992971A^êÆ?Ð
Z§¤Iu
¿…b½kCq
u≈u
N
=
N−2
X
k,j=0
bu
kj
φ
k
(x
1
)φ
j
(x
2
),
·‚ªf(3.6)¥v= φ
l
(x
1
)φ
m
(x
2
),l,m= 0,1,···,N−2, ¿3ªf(3.6)m>|^chebyshev-
Gauss-quadratureúª,k
f
kj
= (f(x
1
,x
2
),φ
k
(x
1
)φ
j
(x
2
))
ω
=
M
X
p,q=0
f(x
1p
,x
2q
)φ
l
(x
1p
)φ
m
(x
2q
)ω
p
ω
q
.
Ù¥x
1p
=cosπp/M,x
2q
=cosπq/M,p,q=0,...,M•chebyshev-Gauss-Lobatto!:,ω
0
=ω
0
=
ω
M
= ω
M
= π/2M,ω
k
= ω
k
= π/Méu1 ≤i≤M−1.
@o3‘œ/e•§(3.6)C¤Xe‚5•§|
αB
b
UB+A
b
UB+B
b
UA
T
= G.(3.7)
ùpÝ
b
U= (bu
kj
)
k,j=0,1,···,N−2
,G= (f
kj
)
k,j=0,1,···,N−2
.
¦)þã‚5•§|™•)
b
U•{Xeµ
(1)kéÑŒ_E±9éΛ§¦A
−1
B= EΛE
−1
.
(2)OŽG
∗
= E
−1
A
−1
G.
(3)¦)•§|(αλ
i
+1)Bv
i
+λ
i
Av
i
= g
i
,l¥v
i
.ùpéÝΛ = diag(λ
0
,λ
1
,···,λ
N−2
).
v
i
=(v
i0
,v
i1
,···,v
iN−2
)
T
,g
i
=(g
i0
,g
i1
,···,g
iN−2
)
T
,i=0,1,···,N−2.·‚5¿G
∗
=
(g
0
,g
1
,···,g
N−2
)
T
§V= (v
0
,v
1
,···,v
N−2
)
T
.
(4)•
b
U= EV§•Ò´Ðmª¥Xê.
3.2.)Ð>Š¯KChebyshev-Galerkin•{
3!ƒ¥§©O0¹ž˜ÅÄ•§Ð>Š¯K9¹ž9D•§Ð>Š¯K
Chebyshev-Galerkin•{OL§"
k•ÄXe˜‚5ÅÄ•§
∂u
∂t
=
∂u
∂x
,(3.8)
±9ƒA>.^‡
u(1,t) = 0(3.9)
†Ð©^‡
u(x,0) = f(x).(3.10)
ÄuChebyshevõ‘ªEļê
DOI:10.12677/aam.2023.1262992972A^êÆ?Ð
Z§¤Iu
φ
n
(x) = T
n
(x)−1,
÷v>.^‡(3.9).·‚ÏéXeCqÐmªu
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¿ùp¦Úªf´ln= 1m©§Ø´ln= 0m©,ù´Ï•φ
0
(x) = 0.
¦{‘
R
N
(x,t) =
∂u
N
∂t
−
∂u
N
∂x
u¼êφ
k
(x) ∈L
2
w
[−1,1],•Ò´
2
π
Z
1
−1
R
N
(x,t)φ
k
(x)
1
√
1−x
2
dx= 0,k= 1,...,N.(3.11)
ùp·‚®²•,ÀJ-¼êω(x) =
1
√
1−x
2
.lªf(3.11)§·‚
N
X
n=1
M
kn
da
n
dt
=
N
X
n=1
S
kn
a
n
(t),k= 1,...,N,
Ù¥
M
kn
=
2
π
Z
1
−1
(T
k
(x)−1)(T
n
(x)−1)
1
√
1−x
2
dx= 2+δ
kn
.
k= nž,M
kn
= 1;k6= nž,M
kn
= 0.

S
kn
=
2
π
Z
1
−1
(T
k
(x)−1)
dT
n
(x)
dx
1
√
1−x
2
dx,k,n= 1,...,N.
dungChebyshevõ‘ªT
n
(x)÷ve¡ðª
dT
n
(x)
dx
= 2n
n−1
X
p=0,p+n•Û
T
p
(x)
c
p
,
ùÝS
kn
ŒU•
S
kn
=
2
π
Z
1
−1
(T
k
(x)−1)2n
n−1
X
p=0,p+n•Û
T
p
(x)
c
p
1
√
1−x
2
dx
= 2n
n−1
X
p=0,p+n•Û
(δ
kp
−δ
0p
).
DOI:10.12677/aam.2023.1262992973A^êÆ?Ð
Z§¤Iu
XJ2½ÂM=(M
kn
)
k,n=1,2,...,N
;S=(S
kn
)
k,n=1,2,...,N
;a=a(t)=(a
1
(t),a
2
(t),...,a
N
(t))
T
.ù
·‚Ò'uXêa
1
(t),a
2
(t),···,a
N
(t)~‡©•§|µ
da
dt
= M
−1
Sa,(3.12)
t= 0ž^‡Œl^‡(3.10)"²LOŽ§Œ•a
n
(0)÷veª
a
n
(0) =
2
π
Z
1
−1
f(x)(T
n
(x)−1)
1
√
1−x
2
dx,n= 1,...,N.(3.13)
3žm••ÏLCrank-Nicolson•{¦)'uXêa
1
(t),a
2
(t),···,a
N
(t)~‡©•§|Њ¯
K(3.12)-(3.13),Xê3ØÓž•CqŠ"
Ó2•ÄXe‚59D•§
∂u
∂t
=
∂
2
u
∂x
2
,(3.14)
±9>.^‡
u(−1,t) = u(1,t) = 0,
†Ð©^‡u(x,0) = f(x).
·‚Œ±ļê•
φ
n
(x) = T
n
(x)−T
n+2
(x),n≥0.
ùpļêÑ÷v>.^‡§•Ò´φ
n
(±1) = 0.
·‚ÏéCqXeÐmª
u
N
(x,t) =
N−2
X
n=0
a
n
(t)φ
n
(x),
¿…‡¦{‘
R
N
(x,t) =
∂u
N
∂t
−
∂
2
u
N
∂x
2
uz‡Ä¼êφ
k
(x)§•Ò´§÷veª
Z
1
−1
R
N
(x,t)φ
k
(x)
1
√
1−x
2
dx= 0,∀k= 0,...,N−2.
ù·‚ÒChebyshev-Galerkin•{
N−2
X
n=0
b
kn
da
n
dt
=
N−2
X
n=0
a
kn
a
n
(t),∀k= 0,1,...,N−2,(3.15)
ùpÝb
kn
½Â„(3.2)±9a
kn
½Â„(3.3).
DOI:10.12677/aam.2023.1262992974A^êÆ?Ð
Z§¤Iu
þãL§ÒXe~‡©•§|
da
dt
= −B
−1
Aa,(3.16)
ùpa= a(t),¿…kЩ^‡
a
n
(0) =
Z
1
−1
f(x)φ
n
(x)
1
√
1−x
2
dx,∀n= 0,1,...,N−2..(3.17)
Ó§3žm••ÏLCrank-Nicolson•{¦)'uXêa
0
(t),a
1
(t),···,a
N−2
(t)~‡©•§|
Њ¯K(3.16)-(3.17),Xê3ØÓž•CqŠ"
4.OŽ(J
~1.·‚k•Ä˜‘‚5>Š¯K
(
−u
xx
+σu= f(x)−1 <x<1
u(−1) = 0,u(1) = 0
Ù¥u= u(x)´™•¼ê.f(x) = σ(1−x
2
)+2,σ= 1.d•§k˜O()u(x) = 1−x
2
.L1Ы
¦)˜‘Helmhotz>Š¯KØ©Û.
Table1.ErrorofChebyshev-GalerkinmethodfortheboundaryvalueproblemofHelmhotzequationin1D
L1.¦)˜‘Helmhotz>Š¯KChebyshev-Galerkin•{Ø©Û
N8163264
error3.331e-163.331e-165.551e-163.331e-16
~2.e¡?Ø‘>Š¯K¦).•Ä
−u
xx
−u
yy
+σu= f−1 <x<1,−1 <y<1,
u= 0,x= −1 ½x= 1 ½y= −1 ½y= 1,
¦).ù´u= u(x,y)´™•¼ê§f(x,y) = 2(−x
2
+1)+2(−y
2
+1)+σ(1−x
2
)(1−y
2
),σ= 1.
d•§k˜O()u(x,y) = (1−x
2
)(1−y
2
).L2Ы¦)‘Helmhotz>Š¯KØ©Û.
Table2.ErrorofChebyshev-GalerkinmethodfortheboundaryvalueproblemofHelmhotzequationin2D
L2.¦)‘Helmhotz>Š¯KChebyshev-Galerkin•{Ø©Û
N
x
= N
y
8163264
error2.220e-167.771e-166.661e-163.331e-16
~3.k•ÄXe˜‚5ÅÄ•§
DOI:10.12677/aam.2023.1262992975A^êÆ?Ð
Z§¤Iu
∂u
∂t
=
∂u
∂x
,
±9ƒA>.^‡
u(1,t) = 0
†Ð©^‡
u(x,0) = e
−50x
2
.
d ¯KkXeO()u(x,t)=e
−50(x+t)
2
,¿…x→1,u(x,t)→0.L3OŽChebyshev-Galerkin
•{¼êψ(x,t)3t= 1ž!:Cq)†3t= 1ž!:O()ƒ.lL3ƒ(JŒ±w
Ѧ^Chebyshev-Galerkin•{äkÌ°Ý.
Table3.ErrorofChebyshev-Galerkinmethodforthefirst-orderlinearwaveequation
L3.)˜‚5ÅÄ•§Chebyshev-Galerkin•{Ø©Û,ùpžmÚ•0.01
N
x
3264128256
error7.341e-47.304e-47.435e-43.331e-16
~4.Ó•ÄXe‚59D•§)
∂u
∂t
=
∂
2
u
∂x
2
,
±9>.^‡
u(−1,t) = u(1,t) = 0,
†Ð©^‡u(x,0)=cos
πx
2
.d¯KkXeO()u(x,t)=e
−
π
2
t
4
cos
πx
2
.L4OŽ˜Ì•{
¼êψ(x,t)3t=1ž!:Cq)†3t=1ž!:O()ƒ.lL4ƒ(JŒ±wѦ^
Chebyshev-Galerkin•{•äkÌ°Ý.
Table4.ErrorofChebyshev-Galerkinmethodforthesecond-orderlinearheatconductionequation
L4.)‚59D•§Chebyshev-Galerkin•{Ø©Û,ùpžmÚ•0.01
N
x
= N
y
3264128256
error1.062e-51.062e-51.062e-51.062e-5
5.(Ø
©ÏL©O¦)˜‘!‘Helmholtz•§>Š¯K!˜‘!‘¹ž‚5Ž™•§Ð
>Š¯K5•[0ÄuChebyshevõ‘ªGalerkinÌ•{¢yL§.ÄuChebyshevõ‘ª
DOI:10.12677/aam.2023.1262992976A^êÆ?Ð
Z§¤Iu
GalerkinÌ•{äk°Ýp!Ž{¢yL§{ü`:.©‰ÑGalerkinÌ•{Ž{•[L
§.¤kêŠOŽ(JÑÏL¤>Matlab§S(Œ†Ï&ŠöéX¢‡ƒ'§S“è).
ÏLOŽuyT•{•kOŽþŒù˜":.ØL‘XOŽÅ5UJp§d¯KAT Œ‘
XOŽÅM‡Jp)û.,,ÄuÙ¦aõ‘ª(~XLegendreõ‘ª!Jacobianõ
‘ª)GalerkinÌ•{í•´aq§©¤0•{Œ±^uOÙ§aõ‘ª
GalerkinÌ•{§¿ŒòƒA^‡©•§¯Kꊦ)ƒ¥"3ò5,ò?ØXÛEÄ
uChebyshevõ‘ªGalerkinÌ{§±BA^••E,‡©•§>Š¯K9‡©•§Ð>
Š¯Kƒ¥;•ò?ØXÛEÄuChebyshevõ‘ªGalerkinÌ{§±BA^••E,
þf››¯KêŠOŽƒ¥[13].
Ä7‘8
H˜e‰ï‘8(2023J0650)]Ï.
ë•©z
[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]š7,´|.o·Üšàg>Š¯KPetrov-GalerkinÌ•{[J].êÆ¢‚†@£,2013,
43(11):255-260.
[10]°ù,Çu. ‘š‚5‡A*Ñ•§ÛÜmäGalerkinÌ{[J].êŠOŽ†OŽÅA^,
2020,41(1):1-18.
DOI:10.12677/aam.2023.1262992977A^êÆ?Ð
Z§¤Iu
[11]Å™,Çu.n ‡©•§ž˜mäGalerkinÌ•{[J].êŠOŽ†OŽÅA^,2021,
42(3):247-262.
[12]4•,]˜,[.OŽŒ‚5ý¯Kõ)˜aÌGalerkin.|¢òÿ{Âñ5©
Û[J].¥I‰Æ:êÆ,2021,51(9):1407-1431.
[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.2023.1262992978A^êÆ?Ð

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