A novel second order finite difference discrete scheme for fractal mobile/immobile transport model based on equivalent transformative Caputo formulation

0.21 MB·

by  Zhengguang Liu

A novel second order finite difference discrete scheme for fractal
mobile/immobile transport model based on equivalent transformative Caputo
formulation
ZhengguangLiua,XiaoliLia,∗
aSchoolofMathematics,ShandongUniversity,Jinan,Shandong250100,China.
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1
0 Abstract
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In this article, we present a new second order finite difference discrete scheme for fractal mobile/immobile

n transport model based on equivalent transformative Caputo formulation. The new transformative formulation
a
takesthesingularkernelawaytomaketheintegralcalculationmoreefficient. Furthermore,thisdefinitionisalso
J
effective where α is a positive integer. Besides, the T-Caputo derivative also helps to increase the convergence
5
rate of the discretization of α-order(0 < α < 1) Caputo derivative from O(τ2−α) to O(τ3−α), where τ is the

] timestep. Fornumerical analysis, aCrank-Nicholson finitedifferenceschemeto solve fractalmobile/immobile
P
transport model is introduced and analyzed. The unconditional stability and a priori estimates of the scheme
A
are given rigorously. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical
h. experimentstosupportourtheoreticalanalysis.
t
a Keywords:
m
Transformativeformulation, Singularkernel,mobile/immobiletransportmodel,Unconditionalstability,
[ Estimates

2010MSC: 65M06,65M12,65M15,26A33
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v
3
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1. Introductions
1
0
In recent years, many problems in physical science, electromagnetism, electrochemistry, diffusion and gen-
.
1 eral transport theory can be solved by the fractional calculus approach, which gives attractive applications as
0
a new modeling tool in a variety of scientific and engineering fields. Roughly speaking, the fractional models
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can be classified into two principal kinds: space-fractional differential equation and time-fractional one. Nu-
1
: mericalmethodsandtheoryofsolutions oftheproblemsforfractionaldifferentialequations havebeenstudied
v
extensively bymany researchers which mainly coverfiniteelement methods[1–4],mixedfiniteelement meth-
i
X ods [5–8], finite difference methods [9–12], finite volume (element) methods [13, 14], (local) discontinuous
r Galerkin(L)DGmethods[15],spectralmethods[16,17]andsoon.
a
The singular kernel of Caputo fractional derivative causes a lot of difficult problems both in integral cal-
culation and discretization. To take singular kernel away, Caputo and Fabrizio [18] suggest a new definition
of fractional derivative by changing the kernel (t−s)−α with the function exp(−αt−s) and 1 with M(α).
1−α Γ(1−α) 1−α
The Caputo-Fabrizo derivative can portray substance heterogeneities and configurations with different scales,
which noticeably cannot be managing with the renowned local theories. And some related articles have been
consideredbymanyauthors. Atangana[19]introducestheapplicationtononlinearFishera˛´rsreaction-diffusion
equation based on the new fractional derivative. He [20] also analyzes the extension of the resistance, induc-
tance, capacitance electrical circuit to this fractional derivative without singular kernel. A numerical solution
∗Correspondingauthor.
Emailaddresses:[email protected](ZhengguangLiu),[email protected](XiaoliLi)
for the model of resistance, inductance, capacitance(RLC) circuit via the fractional derivative without singular
kernel is considered by Atangana [21]. However, we observe that there are many different actions between
Caputo-FabrizioderivativeandCaputoderivative. Thetwodefinitionsarenotequivalent andcannottransform
intoeachotherinanycases.
Inthispaper,wesuggestanewtransformativeformulationoffractionalderivativenamedT-Caputoforluma,
whichisequivalent withCaputofractionalderivativeinsomecases. Furthermore,thetwodefinitionscantrans-
form into each other. More importantly, the T-Caputo formula also helps to increase the convergence rate of
thediscretization of α-order(0<α<1)Caputo derivative fromO(τ2−α) toO(τ3−α),where τ is thetimestep.
Fornumericalanalysis,wepresentaCrank-Nicholsonfinitedifferenceschemetosolvefractalmobile/immobile
transport model. The unconditional stability and a priori estimates of the scheme are given rigorously. More-
over, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our
theoreticalanalysis.
A fractal mobile/immobile transport model is a type of second order partial differential equations (PDEs),
describing a wide family of problems including heat diffusion and ocean acoustic propagation, in physical or
mathematical systems with a time variable, which behave essentially like heat diffusing through a solid [22].
Significantprogresshasalreadybeenmadeintheapproximationofthetimefractionalorderdispersionequation,
see [23]. Schumer [24] firstly developes the fractional-order, mobile/immobile (MIM) model. The time drift
term∂u/∂tisaddedtodescribethemotiontimeandthushelpstodistinguishthestatusofparticlesconveniently.
This equation is the limiting equation that governs continuous time random walks with heavy tailed random
waitingtimes. Inmostcases,itisdifficult,orinfeasible,tofindtheanalyticalsolutionorgoodnumericalsolution
of the problems. Numerical solutions or approximate analytical solutions become necessary. Liu et al. [25]
givearadialbasisfunctions(RBFs)meshlessapproachformodelingafractalmobile/immobiletransportmodel.
Numericalsimulationofthefractionalordermobile/immobileadvection-dispersionmodelisconsinderedbyLiu
etal. [26]. Furthermore,ZhangandLiu[27]presentanovelnumericalmethodforthetimevariablefractional
order mobile–immobile advection–dispersion model. Thefinitedifference schemesare usedby Ashyralyev and
Cakir [28] for solving one-dimensional fractional parabolic partial differential equations. They [29] also give
theFDMforfractionalparabolicequationswiththeNeumanncondition.
The paper is organized as follows. In Sect.2, we give the definitions and some notations. We introduce
a Crank-Nicholson finite difference scheme for a fractal mobile/immobile transport model in Sect.3. Then in
Sect.4,wegivetheanalysisofstabilityanderrorestimatesforthepresentedmethod. InSect.5,somenumerical
experimentsforthesecondorderfinitedifferencediscretizationarecarriedout.
2. Somenotationsanddefinitions
Firstly,wegivesomedefinitionswhichareusedinthefollowinganalysis.
LetusrecalltheusualCaputofractionaltimederivativeoforderα,givenby
t
1
CDαu(t)= u′(s)(t−s)−αds, 0<α<1.
0 t Γ(1−α)
Z0
Here,wegivethefollowingnewtransformativeformulationoffractionalderivative.
Definition1. Letu(t)∈C2(0,T),α∈(0,1),thenthenewtransformativeformulaoffractionalorderisdefinedas:
t
1
TCDαu(t)= u′′(s)(t−s)1−αds, 0<α<1.
0 t Γ(2−α)
Z0
From the above definition of fractional order transformative formula, we know that the singular kernel
(t−τ)−αinCaputoderivativeisreplacedwith(t−τ)1−α innewonewhichdoesnothavesingularityfor t=τ.
Lemma2. Supposeu(t)∈C2(0,T),α∈(0,1),thenwehave
u′(0)t1−α
TCDαu(t)=CDαu(t)− .
0 t 0 t Γ(2−α)
2
Inparticular,ifthefunctionissuchthatu′(0)=0,thenwehave
TCDαu(t)=CDαu(t).
0 t 0 t
Proof: Notingthat
∂[u′(s)(t−s)1−α]
=u′′(s)(t−s)1−α−(1−α)u′(s)(t−s)−α.
∂s
Thenitiseasytoget
1 ∂[u′(s)(t−s)1−α]
u′(s)(t−s)−α= u′′(s)(t−s)1−α− .
1−α ∂s
– ™
ThustheCaputoderivativecanberewrittenas
t
1
CDαu(t)= u′(s)(t−s)−αds
0 t Γ(1−α)
Z0
1 t ∂[u′(s)(t−s)1−α]
= u′′(s)(t−s)1−α− ds
Γ(2−α) ∂s
Z0 – ™
=TCDαu(t)−u′(s)(t−s)1−α t
0 t 0
u′(0)t1−α (cid:12)
=TCDαu(t)+ . (cid:12)
0 t Γ(2−α)
Thiscompletestheproof.
Definition3. Suppose u(t)∈ Cn+1(0,T), if n>1, and α∈(n−1,n), the fractional transformative formulation
TCDαu(t)isdefinedby
0 t
t
1
TCDαu(t)= u(n+1)(s)(t−s)n−αds, n−1<α<n.
0 t Γ(n+1−α)
Z0
Lemma4. Supposeu(t)∈Cn+1(0,T),α∈(n−1,n),thenwehave
u(n)(0)tn−α
TCDαu(t)=CDαu(t)− .
0 t 0 t Γ(n+1−α)
Inparticular,ifthefunctionissuchthatu(n)(0)=0,thenwehave
TCDαu(t)=CDαu(t), n−1<α<n.
0 t 0 t
Proof: Similarly analysisintheproofofLemma1,wehave
∂[u(n)(s)(t−s)n−α]
=u(n+1)(s)(t−s)n−α−(n−α)u(n)(s)(t−s)n−1−α.
∂s
Thenitiseasytoget
1 ∂[u(n)(s)(t−s)n−α]
u(n)(s)(t−s)n−1−α= u(n+1)(s)(t−s)n−α− .
n−α ∂s
– ™
3
Thustheα-orderCaputoderivativecanberewrittenas
t
1
CDαu(t)= u(n)(s)(t−s)n−1−αds
0 t Γ(n−α)
Z0
1 t ∂[u(n)(s)(t−s)n−α]
= u(n+1)(s)(t−s)n−α− ds
Γ(n+1−α) ∂s
Z0 – ™
=TCDαu(t)−u(n)(s)(t−s)n−α t
0 t 0
u(n)(0)tn−α (cid:12)
=TCDαu(t)+ . (cid:12)
0 t Γ(n+1−α)
Thiscompletestheproof.
Lemma5. Forthenewfractionalordertransformativeformulation,α∈(0,1)wehave
D(n)(TCDαu(t))=CDα(D(n)u(t)).
t 0 t 0 t t
Inparticular,ifthefunctionissuchthatu′(0)=0,thenwehave
D(n)(TCDαu(t))=TCDα(D(n)u(t)).
t 0 t 0 t t
Proof: Webeginconsidering n=1,thenfromdefinition(1)ofTCDαu(t),weobtain
0 t
t
d 1
D(1)(TCDαu(t))= u′′(s)(t−s)1−αds
t 0 t dt Γ(2−α)
‚ Z0 Œ
t
1
= u′′(s)(t−s)1−α + (1−α)u′′(s)(t−s)−αds
Γ(2−α) s=t
– Z0 ™
t (cid:12)
1 (cid:12)
= u′′(s)(t−s)−αds
Γ(1−α)
Z0
=CDα(D(1)u(t)).
0 t t
Particularly,FromLemma2,weknow TCDαu(t)=CDαu(t)ifu′(0)=0. Thuswehave
0 t 0 t
D(1)(TCDαu(t))=TCDα(D(1)u(t)).
t 0 t 0 t t
Itiseasytogeneralizetheproofforany n>1.
Lemma6. Forthenewfractionalordertransformativeformulation,ifα=n,wehave
TCDnu(t)=u(n)(t)−u(n)(0).
0 t
Proof: FromDefinition3,weobtain
t
1
TCDαu(t)= u(n+1)(s)(t−s)n−αds
0 t Γ(n+1−α)
Z0
.=u(n)(s) t
s=0
=u(n)(t)(cid:12)−u(n)(0).
(cid:12)
FromtheLemma6,weobtain
t
1
TCDαu(t)= u(n+1)(s)(t−s)n−αds
0 t Γ(n+1−α)
Z0
u(n)(0)tn−α
CDαu(t)− , n−1<α<n,
= 0 t Γ(n+1−α)
u(n)(t)−u(n)(0) α=n.

 4
Letusconsider, thetransformative formulationofaparticular function,asu(t)=cos(t)fordifferentα(0<
α<1). Itiseasytogetthatu′(0)=sin(0)=0. FromFigure1,weobservetherearenodifferentactionsbetween
transformativeformulationandCaputoderivative. Wealsoconsideranotherfunctionasu(t)=sin(t)whichhas
u′′(0)=0fordifferentα(1<α<2). FromFigure2,transformativeformulationandCaputoderivativehavethe
exactsamesetofstates.
1 1
γ=0.1 γ=0.1
γ=0.3 γ=0.3
γ=0.6 γ=0.6
0.5 γ=0.9 0.5 γ=0.9
0 0
αT-Caputo: Du(t)-0.5 αCaputo: Du(t)-0.5
-1 -1
-1.5 -1.5
-2 -2
0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20
t t
Figure1:Simulationoftransformativeformulation(left)andCaputoderivative,withα=0.1,0.3,0.6,0.9inthetimeinterval[0,20].
1 1
γ=1.1 γ=1.1
γ=1.3 γ=1.3
γ=1.6 γ=1.6
0.5 γ=1.9 0.5 γ=1.9
0 0
αT-Caputo: Du(t)-0.5 αCaputo: Du(t)-0.5
-1 -1
-1.5 -1.5
-2 -2
0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20
t t
Figure2:Simulationoftransformativeformulation(left)andCaputoderivative,withα=1.1,1.3,1.6,1.9inthetimeinterval[0,20].
3. Finitedifferenceschemeforfractalmobile/immobiletransportmodel
Inthissection,weintroducethebasicideasforthenumericalsolutionofthefractalmobile/immobiletrans-
portmodelbythesecondorderfinitedifferencescheme.
Weconsiderthefollowingfractalmobile/immobiletransportmodel:
∂u(x,t) ∂2u(x,t)
+CDαu(x,t)= + f(x,t), (1)
∂t 0 t ∂x2
where(x,t)∈Ω=[0,L]×[0,T],0<α<1, f ∈C[0,T],withtheinitialconditions
u(x,0)=φ(x), 0≤x ≤L, (2)
5
andboundaryconditions
u(0,t)=u(L,t)=0, t>0. (3)
Letting t =0intheequation(1),weget
u′(0)=ψ(x)=φ (x)+ f(x,0).
xx
UsingLemma2,theabovemodelcanbetransformedintothefollowingformulation:
∂u(x,t) ∂2u(x,t) ψ(x)t1−α
+TCDαu(x,t)= + f(x,t)− , (x,t)∈Ω,
∂t 0 t ∂x2 Γ(2−α)

u(x,0)=φ(x), 0≤x ≤L,



 u(0,t)=u(L,t)=0, t>0,

 ψ(x)=φxx(x)+f(x,0), 0≤ x≤ L.




In order to do discretizations, we define Ω ={x |x = ih, h= L/M, 0≤ i ≤ M} to be a uniform mesh of
h i i
interval[0,L]. Similarly,defineΩ ={t , t =iτ, τ=T/N, 0≤i≤N}tobeauniformmeshofinterval[0,T].
τ n n
Thevaluesofthefunctionuatthegridpointsaredenoteduk =u(x ,t ). Uk istheapproximatesolutionatthe
j j k j
point(x ,t ). Incase, wesuppose V ={V, 0≤i ≤ M,V =V =0} andW ={W, 0≤i ≤N,W =W =0}
j k i 0 M i 0 M
aretwogridfunctionsonΩ . g={gn, 0≤n≤N}isgridfunctionsonΩ .
h τ
Forfunctions g,V andW,wegivesomenotations,define L2 discreteinnerproductsandnorms. Define[12]
gn−gn−1 M−1
δ gn= , (V,W)= hVW, kVk2=(V,V).
t τ i i
i=1
X
3.1. TheCrank-Nicholsonfinitedifferencescheme
Fromnowon,letC standforapositivenumberindependentofτandh,butpossiblywithdifferentvaluesat
differentplaces. Wegivesomelemmaswhichusedinstabilityanalysisanderrorestimates.
TheobjectiveofthissectionistoconsidertheCrank-Nicholsonfinitedifferencemethodforequations(1). A
discreteapproximation tothenew transformative formulation TCDαu(x,t) at (x ,t ) canbeobtained bythe
0 t i k+1
2
followingapproximation
TCDαu(x ,t )= 1 tk+21 u′′(x ,s)(t −s)1−αds
0 t i k+12 Γ(2−α) i k+21
Z0
= 1 k tj+21 u′(xi,tj+21)−u′(xi,tj−12) +(s−t )u(3)(x ,c ) (t −s)1−αds
Γ(2−α)  τ j t i j  k+12
Xj=1Ztj−21  
1 t1  
+ 2 u′′(x,s)(t −s)1−αds
Γ(2−α) k+21
Z0
= 1 k tj+21 uij+1−2uij+uij−1 +rj+(s−t )u(3)(x ,c ) (t −s)1−αds (4)
Γ(2−α)Xj=1Ztj−21  τ2 j t i j  k+12
1 t1 u1−2u0+u−1 
+ 2 i i i +r0+(s−t )u(3)(x ,c ) (t −s)1−αds
Γ(2−α)Z−t1 – τ2 0 t i 0 ™ k+12
2
0
1
− u′′(x,s)(t −s)1−αds,
Γ(2−α) k+12
Z−t1
2
6
wherec ∈(x ,x )andforξ ∈(t ,t ),ξ ∈(t ,t ),ξ ∈(t ,t ),ξ ∈(t ,t ),η∈(t ,t )
j j−1 j+1 1 j+1 j+1 2 j j+1 3 j−1 j 4 j−1 j−1 j−1 j+1
2 2 2 2 2 2 2 2
andu(t)∈C4[0,t ],wehave
k+1
2
1
rj= τ u(3)(x ,t )−u(3)(x ,t )
24 t i j+12 t i j−12
1h i
+ τ2 u(4)(x ,ξ )+u(4)(x ,ξ )−u(4)(x ,ξ )−u(4)(x ,ξ )
256 t i 1 t i 2 t i 3 t i 4 (5)
1 h i
= τ2u(4)(x ,η)+O(τ2)
48 t i
=O(τ2).
Inparticular,for j=0,denoteu−1=u0−τu′(x,0)=φ−τψ.Usingthesimplelinearinterpolantofuat(−t ,0),
1
sofors∈(−t ,0),wehaveu′′(x,s)=0. Itisasuitablemethodtosatisfytheconditionu−1=u0−τu′(x,0).
1
2
Combiningtheequation(4)with(5),weobtain
TCDαu(x ,t )= τ2−α k uij+1−2uij+uij−1 M + τ2−α u1i −u0i − ψ M +Rk+21
0 t i k+21 Γ(3−α) j=1 τ2 ! k−j Γ(3−α)‚ τ2 τŒ k i
X (6)
= τ1−α M δ uk+1− k M −M δ uj−M ψ +Rk+21,
Γ(3−α) 0 t i k−j k−j+1 t i k i i
j=1
X€ Š
 
where
M =(j+1)2−α− j2−α, (7)
j
and
Rk+21 = 1 k tj+21 rj+(s−t )u(3)(x ,c ) (t −s)1−αds
i Γ(2−α)Xj=0Ztj−21 (cid:16) j t i j (cid:17) k+12 (8)
=O(τ3−α).
WegivesomeLemmasabout M thatwillbeusedinthefollowinganalysis.
j
Lemma7. Forthedefinition M ,(j=0,1,2,...,N−1),wehave M >0and M ≥M ,∀j≤k.
j j j+1 j
Proof: Observing that x2−α is a monotone increasing functionfor 0<α<1, thenwe have M =(j+1)2−α−
j
j2−α>0. Next,let f(x)=(x+1)2−α−x2−α,wehave
f′(x)=(2−α)[(x+1)1−α−x1−α]≥0, ∀x ≤0.
Thusweobtain
M = f(j+1)≥ f(j)=M .
j+1 j
Thiscompletestheproof.
Lemma8. Forthedefinition M =(j+1)2−α− j2−α,wedenote G =M −M ,(j=0,1,2,...,N−1). Then
j j+1 j+1 j
itholdsthat
G ≥G ≥···≥G ≥0.
1 2 N
Proof:Firstly,usingLemma(7),itiseasytogetG ≥0. Next,forfixed0<α<1,wegivethefollowingfunction
j
f(x)=(x+2)2−α−2(x+1)2−α+x2−α,
thenwehave
f′(x)=(2−α)[(x+2)1−α−2(x+1)1−α+x1−α].
7
UsingTaylor’sexpansion,wehave
1 1
(x+2)1−α=(x+1)1−α+(1−α)(x+1)−α− (1−α)α(x+1)−(α+1)+ (1−α)α(α+1)ξ−(α+2),
2! 3! 1
1 1
x1−α=(x+1)1−α−(1−α)(x+1)−α− (1−α)α(x+1)−(α+1)− (1−α)α(α+1)ξ−(α+2),
2! 3! 2
whereξ ∈(x+1,x+2)andξ ∈(x,x+1).
1 2
Thus,wehave
1
f′(x)=(2−α) −(1−α)α(x+1)−(α+1)+ (1−α)α(α+1) ξ−(α+2)−ξ−(α+2)
3! 1 2
(cid:20) (cid:16) (cid:17)(cid:21)
≤−(2−α)(1−α)α(x+1)−(α+1)
≤0, ∀x ≥0, 0<α<1.
ItmeansthatG >G ,∀j≥1. Thiscompletestheproof.
j j+1
Thediscretizationoffirstordertimederivativeisstatedas:
∂u(xi,tk+21) = uki+1−uki +O(τ2), (9)
∂t τ
andthesecondorderspatialderivativeisstatedas:
∂2u(xi,tk+12) = 1 uki++11−2uki+1+uki−+11 + uki+1−2uki +uki−1 +O(h2), (10)
∂x2 2 h2 h2 
 
Combining theequation(6)withequations(9)∼(10),wecanobtainthefollowingfinitedifferencescheme,
∀k=0,1,···N−1,
τ1−α k
δ Uk+1+ M δ Uk+1− M −M δ Uj−M ψ
t i Γ(3−α) 0 t i k−j k−j+1 t i k i
j=1
X€ Š (11)
= 1 Uik++11−2Uik+1+Uik−+11 + Uik+1−2Uik+Uik−1 + fk+21 − ψi[(k+ 21)τ]1−α.
2 h2 h2  i Γ(2−α)
NotethatG =M −M ,(j=0,1,2,...,N−1),thenwehave
j+1 j+1 j
k 1 k−1
− M −M δ Uj= G Uk| + G −G Uj−G U0| . (12)
k−j k−j+1 t i τ 1 i k≥1 k−j+1 k−j i k i k≥1
j=1 j=1
X€ Š X€ Š
 
Letβ = τ1−α ,thenabovescheme(11)canberewrittenas
Γ(3−α)
τ τ τ
− Uk+1+ +1+β Uk+1− Uk+1
2h2 i+1 h2 i 2h2 i−1
• (cid:129) ‹ ˜
τ τ τ k−1
= Uk + − +1+β−G | Uk+ Uk +β G −G Uj+G U0|
2h2 i+1 h2 1 k≥1 i 2h2 i−1 k−j k−j+1 i k i k≥1 (13)
(cid:129) ‹ Xj=1€ Š
+τβM ψ +τfk+21 − τψi[(k+12)τ]1−α. 
k i i Γ(2−α)
8
4. Stabilityanalysisandoptimalerrorestimates
4.1. Stabilityanalysis
We analyze thestability of thedifferenceschemebya Fourier analysis. Let Uk betheapproximate solution
i
of(13),anddefine
ρk=Uk−Uk, 1≤i≤M, 0≤k≤N. e
i i i
Thenwehave
e
τ τ τ
− ρk+1+ +1+β ρk+1− ρk+1
2h2 i+1 h2 i 2h2 i−1
• τ (cid:129) τ ‹ ˜ τ k−1 (14)
= ρk + − +1+β−βG | ρk+ ρk +β G −G ρj+βG ρ0| .
2h2 i+1 h2 1 k≥1 i 2h2 i−1 k−j k−j+1 i k i k≥1
(cid:129) ‹ Xj=1€ Š
 
Asthesamedefinitionin[30],wedefinethegridfunction
0, 0≤ x≤ x ,
1
2
ρk(x)= ρik, xi−1 ≤ x≤ xi+1, 1≤i≤M−1,
2 2
0, x ≤ x ≤x .
 M−1 M
2

Wecanexpandρk(x)inaFourierseries
∞
ρk(x)= dk(l)ei2πLlx, k=1,2,...,N,
l=−∞
X
wherediscreteFouriercoefficientsd (l)are
k
L
1
dk(l)= L ρk(ξ)e−i2Lπlξdξ. (15)
Z0
ThenwehavetheParsevalequalityforthediscreteFouriertransform
L ∞
|ρk(x)|2dx= |d (l)|2.
k
Z0 l=−∞
X
Introducethefollowingnorm
M−1 1/2 L 1/2
kρkk = h|ρk|2 = |ρk|2dx .
2 i i
i=1 ! Z0 !
X
Thenweobtain
∞
kρkk2= |d (l)|2.
2 k
l=−∞
X
Based on the above analysis, we can suppose the solution of equation (14) has the following form ρk =
m
d eimhγ where L=1andγ=2πl.
k
Lemma9. Supposethatd (l)(k=1,2,...,N)aredefinedby(15),thenfor0<α<1,wehave
k
|d |≤|d |, k=1,2,...,N.
k 0
9
Proof: Substitutingρk =d eimhγ intoequation(14),wehave
m k
τ τ τ
− d ei(m+1)hγ+ +1+β d eimhγ− d ei(m−1)hγ
2h2 k+1 h2 k+1 2h2 k+1
• τ (cid:129) τ ‹ τ ˜
= d ei(m+1)hγ+ − +1+β−βG | d eimhγ+ d ei(m−1)hγ
2h2 k h2 1 k≥1 k 2h2 k (16)
• (cid:129) ‹
k−1
+β G −G d eimhγ+βG d eimhγ| .
k−j k−j+1 j k 0 k≥1

j=1
X€ Š

Bysimplycalculation,wecanget
τ τ τ τ
− (eihγ+e−ihγ)+ +1+β d = (eihγ+e−ihγ)+ − +1+β−βG | d
2h2 h2 k+1 2h2 h2 1 k≥1 k
• (cid:129) ‹˜ • k−1 (cid:129) ‹˜ (17)
+β G −G d +βG d | .
k−j k−j+1 j k 0 k≥1
j=1
X€ Š
Notingthateihγ+e−ihγ=2cos(hγ),thusequation(17)canberewrittenasthefollowingformulation:
τ τ τ τ
− cos(hγ)+ +1+β d = cos(hγ)+ − +1+β−βG | d
h2 h2 k+1 h2 h2 1 k≥1 k
• (cid:129) ‹˜ • k−1 (cid:129) ‹˜ (18)
+β G −G d +βG d | .
k−j k−j+1 j k 0 k≥1
j=1
X€ Š
Firstly,lettingk=0inequation(18)toobtain
τ cos(hγ)− τ +1+β −1−cos(hγ)τ+1+β
|d |= h2 h2 |d |= h2 |d |≤|d |. (19)
1 (cid:12)−τ cos(hγ)+ τ +1+β(cid:12) 0 (cid:12) 1−cos(hγ)τ+1+β (cid:12) 0 0
(cid:12) h2 h2 (cid:12) (cid:12)(cid:12) h2 (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Nowsupposethatwehave(cid:12)provedthat|d |≤|d |,n(cid:12)=1,2,.(cid:12)..,k,thenusingthe(cid:12)equation(18),weobtain
n 0 (cid:12) (cid:12)
|d |≤ −1−chos2(hγ)τ+1+β−βG1 |d |+ 1 βk−1 G −G |d |+βG |d | .
k+1 (cid:12)(cid:12)(cid:12) 1−cohs2(hγ)τ+1+β (cid:12)(cid:12)(cid:12) k |1−chos2(hγ)τ+1+β| Xj=1€ k−j k−j+1Š j k 0 
(cid:12) (cid:12)  (20)
(cid:12) (cid:12) 
Observin(cid:12)gthatGj≥0andGj−Gj+1(cid:12)≥0inLemma8,thenweobtain
k−1 k−1
β G −G |d |+βG |d |≤β G −G +G |d |=βG |d | (21)
k−j k−j+1 j k 0 k−j k−j+1 k 0 1 0
 
j=1 j=1
X€ Š X€ Š
 
Combiningtheequation(20)withequation(21),wecanobtain
|−1−cos(hγ)τ+1+β−βG |+βG
|d |≤ h2 1 1 |d |. (22)
k+1  |1−cos(hγ)τ+1+β|  0
h2
 
If−1−cos(hγ)τ+1+β−βG >0,thenwehave
h2 1
−1−cos(hγ)τ+1+β
|d |≤ h2 |d |≤|d |. (23)
k+1 |1−cos(hγ)τ+1+β| 0 0
h2
 
10