Numerical algorithms for the forward and backward fractional Feynman-Kac equations

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by  Weihua Deng

NonamemanuscriptNo.
(willbeinsertedbytheeditor)
Numerical algorithms for the forward and backward
fractional Feynman-Kac equations
WeihuaDeng ,1,MinghuaChen1,EliBarkai2
∗
4
1
0
2 Received:date/Accepted:date

n
a
J Abstract TheFeynman-Kacequationsareatypeofpartialdifferentialequationsdescribing
thedistributionoffunctionalsofdiffusivemotion.Theprobabilitydensityfunction(PDF)of
3
Brownianfunctionals satisfiestheFeynman-Kacformula, beingaSchro¨dinger equationin

] imaginarytime.Thefunctionalsofno-Brownianmotion,oranomalousdiffusion,followthe
h
fractionalFeynman-Kacequation[J.Stat.Phys.141,1071-1092,2010],wherethefractional
p
substantialderivativeisinvolved.Basedonrecentlydevelopeddiscretizedschemesforfrac-
-
p tionalsubstantialderivatives[arXiv:1310.3086],thispaperfocusesonprovidingalgorithms
m fornumericallysolvingtheforwardandbackwardfractionalFeynman-Kacequations;since
o thefractionalsubstantialderivativeisnon-localtime-spacecoupledoperator,newchallenges
c areintroducedcomparingwiththegeneralfractionalderivative.Twoways(finitedifference
s. and finite element) of discretizing the space derivative are considered. For the backward
c fractionalFeynman-Kacequation,thenumericalstabilityandconvergenceofthealgorithms
i
s withfirstorderaccuracyaretheoreticallydiscussed;andtheoptimalestimatesareobtained.
y Foralltheprovidedschemes,includingthefirstorderandhighorderones,ofbothforward
h andbackwardFeynman-Kac equations,extensivenumerical experiments areperformed to
p
showtheireffectiveness.
[

Keywords fractional Feynman-Kacequation fractionalsubstantialderivative optimal
1 · ·
v convergent order numerical stabilityandconvergence numerical inversionofLaplace
· ·
4 transforms
5
6
0
1 Introduction
.
1
0 Letting x(t) be a trajectory of a Brownian particle andU(x) be a prescribed function, the
4 Brownian functional can be defined as A= tU[x(t )]dt [18], which has many physical
1 0
applications. In 1949, inspiring by Feynman’s path integrals Kac derives a Schro¨dinger-
: R
v likeequationforthedistributionofthefunctionalsofdiffusivemotion[15].Withtherapid
i development on the study of non-Brownian motion, or anomalous diffusion [19,21], the
X
r
Correspondingauthor.E-mail:[email protected]
a ∗
1SchoolofMathematicsandStatistics,LanzhouUniversity,Lanzhou730000,P.R.China
2DepartmentofPhysicsandAdvancedMaterialsandNanotechnologyInstitute,Bar-IlanUniversity,Ramat
Gan52900,Israel.E-mail:[email protected]
2
functionalsofanomalousdiffusionnaturallyattracttheinterestsofphysicists.Inparticular,
Carmi, Turgeman, and Barkai derive the forward and backward fractional Feynman-Kac
equationsfordescribingthedistributionofthefunctionalsofanomalousdiffusion[4,5,25],
whichinvolvesthefractionalsubstantialderivative[12].BeingthesameformofBrownian
functional,thefunctionalofanomalousdiffusioncanalsobedefinedas
t
A= U[x(t )]dt , (1.1)
Z0
wherex(t) isatrajectory of non-Brownian particle; and therearealot of different choice
to prescribeU(x). For example, we can takeU(x)=1 in a given domain and to be zero
otherwise, which characterizes the time spent by a particle in the domain; this functional
canbeusedinkineticstudiesofchemicalreactionsthattakeplaceexclusivelyinthedomain
[2,5]. For inhomogeneous disorder dispersive systems, the motion of the particles is non-
Brownian,andU(x)istakenasxorx2[5].
In recent decades, the numerical methods for fractional partial differential equations
(PDEs)are well developed, including finitedifference methods [8,9,20,23,26], finite ele-
ment[10,11,14],spectralmethod[16,17],etc.However,itseemsthattherearenopublished
worksfornumericallysolvingfractionalPDEswithfractional substantialderivative.Frac-
tional substantial derivative isanon-local time-spacecoupled operator; discretizing itand
numericallysolvingthecorrespondingequationsundoubtedlyintroducesomenewdifficul-
ties comparing with the fractional derivative. We detailedly discuss the properties and ef-
fectivelynumericaldiscretizationsofthefractionalsubstantialderivativesin[6].Thispaper
focuses on numerically solving the forward and backward fractional Feynman-Kac equa-
tionswiththefractionalsubstantialderivativebeingdiscretizedbythewaysgivenin[6]and
theclassicalspatialderivativeistreatedbyfinitedifference andfiniteelementmethod, re-
spectively.ForthebackwardFeynman-Kacequation,wetheoreticallyprovethenumerical
stabilityandconvergenceofitsfirstorderscheme.Foralltheproposedschemes,including
thefirstorderandhighorderones,ofbothforwardandbackwardfractionalFeynman-Kac
equations,theextensivenumericalexperimentsareperformedtoshowtheireffectiveness.
Thedefinitionsoffractionalsubstantialcalculusaregivenasfollows[6].
Definition1 Let n >0, r be a constant, and P(t) be piecewise continuous on (0,¥ ) and
integrableonanyfinitesubinterval[0,¥ ).ThenthefractionalsubstantialintegralofP(t)of
ordern isdefinedas
sItn P(t)= G (1n )Z0t(t−t )n −1e−rU(x)(t−t)P(t )dt , t>0,
whereU(x)isaprescribedfunctionin(1.1).
Definition2 Letm >0,r beaconstant,andP(t)be(m-1)-timescontinuouslydifferentiable
on(a,¥ )anditsm-timesderivativebeintegrableonanyfinitesubintervalof[a,¥ ),where
misthesmallestintegerthatexceedsm .ThenthefractionalsubstantialderivativeofP(t)of
orderm isdefinedas
sDtm P(t)=sDtm[sItm−m P(t)],
where
¶ m
sDm= +r U(x) .
t ¶ t
(cid:18) (cid:19)
3
TheforwardandbackwardfractionalFeynman-Kacequationderivedin[4,5,25]are
¶ ¶ 2
¶ tP(x,r ,t)=k a ¶ x2sDt1−a P(x,r ,t)−r U(x)P(x,r ,t), (1.2)
and
¶ ¶ 2
¶ tP(x,r ,t)=k a sDt1−a ¶ x2P(x,r ,t)−r U(x)P(x,r ,t), (1.3)
whereP(x,r ,t):= ¥ P(x,A,t)e r AdA;for(1.2),P(x,A,t)denotesthejointprobabilityden-
0 −
sityfunction(PDF)offindingtheparticleon(x,A)attimet;whilefor(1.3),P(x,A,t)isthe
R
jointPDFoffindingtheparticleonAattimet withtheinitialpositionoftheparticleatx;
the functional A is defined as (1.1); the diffusion coefficient k a is apositive constant and
a (0,1);whenU(x)=0,both(1.2)and(1.3)reducestothecelebratedfractionalFokker-
∈
Planckequation[3,21].Infact,fromthedefinitionoffractionalsubstantialderivative, Eq.
(1.3)canberewrittenas
¶ 2
sDtP(x,r ,t)=sDt1−a k a ¶ x2P(x,r ,t) ; (1.4)
(cid:20) (cid:21)
thenwecanfurthergetitsequivalentform(seetheAppendix)
scDta P(x,t)=sDta [P(x,t)−e−rU(x)tP(x,0)]
=sDta P(x,t)−t−Ga(e1−rUa(x))tP(x,0)=k a ¶¶x22P(x,t), (1.5)
−
hereandinthefollowingP(x,r ,t)isreplacedbyP(x,t)sincer istakenasafixedconstant.
For(1.2),fromthedefinitionoffractionalsubstantialderivative,wecanalsorecastitas
¶ 2
sDtP(x,t)=k a ¶ x2sDt1−a P(x,t); (1.6)
butitshouldbenotedthatthetwooperators ¶ 2 andD1 a donotcommute.
¶ x2 t−
The outline of this paper is as follows. In Section 2, for (1.2) and (1.3)we derive the
numericalschemeswithfinitedifferencemethodtodiscretizethespacederivative;andthe-
oreticallyprovethatthefirstordertimediscretizationschemeisunconditionallystableand
convergent for (1.3). In Section 3, for (1.3) the time semi-discretized and full discretized
schemes of finite element method are provided; stability and convergence of the schemes
arerigourously established; moreover, theoptimal convergent rateisobtained. Toconfirm
thetheoreticalresultsandshowtheeffectivenessofthefirstorderandhighorderschemes,
theextensivenumericalresultsareprovidedinSection4.Weconcludethepaperwithsome
remarksinthelastsection.
2 FinitedifferenceforfractionalFeynman-Kacequation
In this section wefocuses on deriving the difference schemes for the backward fractional
Feynman-Kacequation (1.4)andtheoretically provethattheprovided firstorder timedis-
cretizationschemeof(1.4)isunconditionallystableandconvergent;thedifferenceschemes
fortheforwardfractionalFeynman-Kacequation(1.2)aregivenasaremark.
4
LettingT >0,W =(0,l),rewriting(1.5)andmakingitsubjecttothegiveninitialand
boundaryconditions,wehave
¶ 2
scDta P(x,t)=sDta [P(x,t)−e−rU(x)tP(x,0)]=k a ¶ x2P(x,t), 0<t≤T, x∈W , (2.1)
withinitialandboundaryconditions
P(x,0)=f (x), x W ,
∈ (2.2)
P(0,t)=y (t), P(l,t)=y (t), 0<t T.
1 2
≤
2.1 Derivationofthedifferencescheme
Letthemeshpointsx =ihfori=0,1,...,M,andt =nt ,n=0,1,...,N,whereh=l/M
i n
andt =T/Naretheuniformspacestepsizeandtimesteplength,respectively.DenotePnas
i
thenumericalapproximationtoP(x,t ).Toapproximate(2.1),weutilizethesecondorder
i n
centraldifferenceformulaforthespatialderivative;thatis
¶ 2P(x,t) P(x ,t ) 2P(x,t )+P(x ,t )
¶ x2 (xi,tn)= i+1 n − h2i n i−1 n +O h2 .
(cid:12)
From(3.8)of[6],wek(cid:12)nowthatthefractionalsubstantialderivativehasq(cid:0)-th(cid:1)orderapproxi-
(cid:12)
mations,i.e.,
n
sDta P(x,t)|(xi,tn)=t −a (cid:229) diq,k,a P(xi,tn−k)+O(t q), q=1,2,3,4,5, (2.3)
k=0
with
diq,k,a =e−rUiktlkq,a , Ui=U(xi), q=1,2,3,4,5,
wherel1,a ,l2,a ,l3,a ,l4,a andl5,a aredefinedby(2.2),(2.4),(2.6),(2.8)and(2.10)in[7],
k k k k k
respectively.Inthefollowing,wedothedetailedtheoreticalanalysisforthefirstordertime
discretizationschemeof(2.1).Forthesimplification,wedenoted1,a byda ;then
i,k i,k
n
sDta P(x,t)|(xi,tn)=t −a (cid:229) dia,kP(xi,tn−k)+O(t );
k=0
(2.4)
n
sDta [e−rU(x)tP(x,0)](xi,tn)=t −a (cid:229) dia,ke−rUi(n−k)t P(xi,0)+O(t ),
k=0
wherethecoefficients
a
dia,k=e−rUikt gk, gk=(−1)k k , (2.5)
(cid:18) (cid:19)
with
a +1
g =1, g = 1 g , k 1.
0 k − k k−1 ≥
(cid:18) (cid:19)
ThenEq.(2.1)canberewrittenas
n n
t −a (cid:229) dia,kP(xi,tn−k)−t −a (cid:229) dia,ke−rUi(n−k)tP(xi,0)
k=0 k=0 (2.6)
P(x ,t ) 2P(x,t )+P(x ,t )
=k a i+1 n − h2i n i−1 n +rin,
5
with
rn C (t +h2), (2.7)
| i|≤ P
whereC isaconstantdependingonlyonP.
P
Multiplying(2.6)byt a ,wehavethefollowingequation
n 1 n 1
(cid:229)− dia,kP(xi,tn−k)−(cid:229)− dia,ke−rUi(n−k)tP(xi,0)
k=0 k=0 (2.8)
=k a t a P(xi+1,tn)−2P(hx2i,tn)+P(xi−1,tn)+Rni,
with
Rn = t a rn C t a (t +h2). (2.9)
| i| | i|≤ P
From(2.5)and(2.8),theresultingdiscretizationof(2.1)canberewrittenas
k t a n 1 n 1
Pin− ah2 Pin+1−2Pin+Pin−1 =k(cid:229)=−0dia,ke−rUi(n−k)tPi0−k(cid:229)=−1dia,kPin−k, n≥1; (2.10)
(cid:0) (cid:1)
or
k t a n 1 n 1
Pin− ah2 Pin+1−2Pin+Pin−1 =k(cid:229)=−0e−rUint gkPi0−k(cid:229)=−1e−rUiktgkPin−k, n≥1, (2.11)
(cid:0) (cid:1)
with i=1,2,...,M 1. It is worthwhile to noting that the second term on the right hand
−
sideof(2.10)or(2.11),respectively,automaticallyvanisheswhenn=1.
Remark2.1 Ifweutilizetheq-th order approximation of(2.3)todiscretizethetimefrac-
tionalsubstantialderivativeof(2.1),theresultingdiscretizationof(2.1)is
k t a n 1 n 1
diq,0,a Pin− ah2 Pin+1−2Pin+Pin−1 =k(cid:229)=−0diq,k,a e−rUi(n−k)tPi0−k(cid:229)=−1diq,k,a Pin−k, (2.12)
(cid:0) (cid:1)
whichgivesalocaltruncationerrorofO t q+h2 ,q=2,3,4,5.
Remark2.2 Using the second order cen(cid:0)tral diff(cid:1)erence formula for the spatial derivative
leadsto
¶ 2
¶ x2sDt1−a P(xi,tn)
= sDt1−a P(xi+1,tn)−2sDt1−a P(xi,tn)+sDt1−a P(xi−1,tn)+O h2 .
h2
(cid:0) (cid:1)
Furtherapplying(2.3)toapproximatethetimefractionalsubstantialderivatives,thenweget
thediscretizationschemesof(1.6):
k t a
diq,0,1Pin− ah2 (diq+,11−,0a Pin+1−2diq,0,1−a Pin+diq−,11−,0a Pin−1)
n k t a n (2.13)
=−k(cid:229)=1diq,k,1Pin−k+ ah2 k(cid:229)=1(diq+,11−,ka Pin+−1k−2diq,k,1−a Pin−k+diq−,11−,ka Pin−−1k),
withthelocaltruncationerrorO t q+h2 ,q=1,2,3,4,5.
(cid:0) (cid:1)
6
2.2 Stabilityandconvergence
Inthissubsection,weprovethatthescheme(2.10)isunconditionallystableandconvergent
indiscreteL2normandL¥ normundertheassumptionthat0 r U h .First,weintroduce
i
≤ ≤
some relevant notations and properties of discretized inner product given in [13]. Denote
un= un 0 i M,n 0 andvn= vn 0 i M,n 0 ,whicharegridfunctions.And
{ i| ≤ ≤ ≥ } { i| ≤ ≤ ≥ }
(un) =(un un)/h, (un) =(un un )/h;
i x i+1− i i x i − i−1
M 1
(un,vn)= (cid:229) − unvnh, un =(un,un)1/2;
i i || ||
i=1
M
(un,vn]=(cid:229) unvnh, un] =(un,un]1/2.
i i || |
i=1
Inparticular,ifun=0andun =0,thereexists
0 M
l2
(un,(vn) )= (un,vn] and un 2 un]2, (2.14)
x x − x x || || ≤ 8|| x |
wherelmeanstheoneappearedinW =(0,l).
Lemma2.1 Thecoefficientsg definedin(2.5)satisfy
k
n 1 ¥
g =1; g <0, (k 1); (cid:229)− g >0; (cid:229) g =0; (2.15)
0 k k k
≥
k=0 k=0
and
1 n(cid:229)−1 (cid:229)¥ 1
< g = g , for n 1. (2.16)
na G (1 a ) k − k≤ na ≥
− k=0 k=n
Proof From [22, p.208], it is easy to get (2.15). Next we prove (2.16). Denoting v =
n
−na (cid:229) ¥k=ngk=na (cid:229) nk=−01gk,n≥1,accordingto[9],thereexists
v <v , i.e., (cid:229) n g < na n(cid:229)−1g , for n 1, (2.17)
n+1 n k (n+1)a k ≥
k=0 k=0
and
1 n(cid:229)−1 (cid:229)¥
< g = g , for n 1.
na G (1 a ) k − k ≥
− k=0 k=n
Nextweprovethefollowinginequalitybymathematicalinduction
n(cid:229)−1g = (cid:229)¥ g 1 , for n 1. (2.18)
k − k≤ na ≥
k=0 k=n
Itisobviousthat(2.18)holdswhenn=1orn=2.Supposingthat
s(cid:229)−1g = (cid:229)¥ g 1 , s=1,2,...,n 1,
i − i≤ sa −
i=0 i=s
andusing(2.17),weobtain
n(cid:229)−1 (n 1)a n(cid:229)−2 (n 1)a 1 1
gk< −na gk≤ −na (n 1)a = na , for n≥2.
k=0 k=0 −
Thenthedesiredinequality(2.16)holds.
7
Theorem2.1 When0 r U h ,thedifferencescheme(2.11)isunconditionallystable.
i
≤ ≤
Proof LetPn betheapproximatesolutionofPn,whichistheexactsolutionofthescheme
i i
(2.11). Taking en =Pn Pn, i=1,2,...,M 1, then from (2.11) we get the following
i i − i −
perturbationeequation
e
eni −t a k a eni+1−2he2ni +eni−1 =n(cid:229)−1e−rUint gke0i −n(cid:229)−1e−rUiktgkein−k, (2.19)
k=0 k=1
withen=en =0.Multiplying(2.19)byhenandsummingupforifrom1toM 1,then
0 M i −
hM(cid:229) −1(eni)2−t a k a hM(cid:229) −1eni+1−2he2ni +eni−1eni
i=1 i=1
(2.20)
M 1n 1 M 1n 1
=h (cid:229) − (cid:229)− e−rUintgke0ieni −h (cid:229) − (cid:229)− e−rUiktgkein−keni.
i=1 k=0 i=1 k=1
Nextweestimate(2.20).Firstly,wehave
M 1
h (cid:229) − (en)2= en 2, (2.21)
i || ||
i=1
andfrom(2.14),itleadsto
−t a k a hM(cid:229) −1eni+1−2he2ni +eni−1eni
i=1
M 1
=−t a k a h (cid:229) − ((eni)x)xeni =−t a k a (en,(enx)x] (2.22)
i=1
=t a k a (enx,exn]=t a k a ||enx]|2≥ 8t la2k a ||en||2≥0.
Sincee−rUint [e−h T,1]andfrom(2.5)and(2.15),weobtain
∈
gk≤dia,k=e−rUikt gk≤e−h Tgk<0, k≥1, (2.23)
and
n 1 n 1 n 1
0<e−rUint (cid:229)− gk= (cid:229)− e−rUint gk (cid:229)− gk, n 1. (2.24)
≤ ≥
k=0 k=0 k=0
Therefore,accordingto(2.24)and(2.23),weobtain
hM(cid:229) −1n(cid:229)−1e−rUint gke0ieni ≤hM(cid:229) −1n(cid:229)−1e−rUint gk(e0i)2+2(eni)2
i=1 k=0 i=1 k=0
(2.25)
1n(cid:229)−1g e0 2+ en 2 ,
≤ 2 k || || || ||
k=0
(cid:0) (cid:1)
and
−hM(cid:229) −1n(cid:229)−1e−rUiktgkeni−keni ≤−hM(cid:229) −1n(cid:229)−1e−rUiktgk(ein−k)22+(eni)2
i=1 k=1 i=1 k=1
(2.26)
≤−21n(cid:229)−1gk ||en−k||2+||en||2 .
k=1
(cid:16) (cid:17)
8
From(2.20-2.26),thereexists
en 2 1n(cid:229)−1gk e0 2+ en 2 1n(cid:229)−1gk en−k 2+ en 2
|| || ≤ 2 || || || || −2 || || || ||
k=0 k=1
(cid:0) (cid:1) (cid:16) (cid:17) (2.27)
= 1+21nk(cid:229)=−11gk!||e0||2−21nk(cid:229)=−11gk||en−k||2.
Nextweprovethat en 2 e0 2 bymathematicalinduction.Forn=1,(2.27)holdsob-
|| || ≤|| ||
viously.Supposing
es 2 e0 2, for s=1,2,...,n 1,
|| || ≤|| || −
andusing(2.27),thenweget
||en||2≤ 1+12nk(cid:229)=−11gk!||e0||2−21nk(cid:229)=−11gk||en−k||2
1+1n(cid:229)−1g e0 2 1n(cid:229)−1g e0 2= e0 2.
≤ 2k=1 k!|| || −2k=1 k|| || || ||
Hence,theproofiscomplete.
Lemma2.2 LetR 0;e k 0,k=0,1,...,Nandsatisfy
≥ ≥
n 1
e n (cid:229)− gke n−k+R, n 1, (2.28)
≤− ≥
k=1
thenwehavethefollowingestimates:
(a)when0<a <1,
1
e n n(cid:229)−1g − R na G (1 a )R; (2.29)
k
≤ k=0 ! ≤ −
(b)whena 1,
→
e n nR. (2.30)
≤
Proof Itisworthtonotingthatthefirsttermontherighthandsideof(2.28)automatically
vanisheswhenn=1.
(1)Case0<a <1:Weprovethefollowingestimatebythemathematicalinduction,
1
e n n(cid:229)−1g − R.
k
≤ k=0 !
Eq.(2.28)holdsobviouslyforn=1.Supposingthat
1
e s s(cid:229)−1g − R, s=1,2,...,n 1,
i
≤ i=0 ! −
9
thenform(2.28)wehave
1 1
e n n(cid:229)−1gke n−k+R n(cid:229)−1gk n−(cid:229) k−1gi − R+R n(cid:229)−1gk n(cid:229)−1gi − R+R
≤−k=1 ≤−k=1 i=0 ! ≤−k=1 i=0 !
1 1
n(cid:229)−1 n(cid:229)−1 − n(cid:229)−1 −
1 g g R+R g R.
k i i
≤ −k=0 ! i=0 ! ≤ i=0 !
Accordingto(2.16)andtheaboveinequality,itleadsto
1
e n n(cid:229)−1g − R na G (1 a )R.
k
≤ k=0 ! ≤ −
(2) Now weconsider the casea 1. SinceG (1 a ) ¥ as a 1 inthe estimate
→ − → →
(2.29). Therefore, we need to look for an estimateof other form. We prove the following
estimatebythemathematicalinduction:
e n nR.
≤
Eq.(2.28)holdsobviouslyforn=1.Supposingthat
e s sR, s=1,2,...,n 1,
≤ −
thus,from(2.28)weget
n 1 n 1 n 1
e n (cid:229)− gke n−k+R (cid:229)− gk(n k)R+R (cid:229)− gk(n 1)R+R (n 1)R+R=nR.
≤− ≤− − ≤− − ≤ −
k=1 k=1 k=1
Theorem2.2 Let Pn be the approximate solution of P(x,t ) computed by the difference
i i n
scheme(2.11)withtheassumption0 r U h .Then
i
≤ ≤
P(x,t ) Pn C G (1 a )l1/2Ta (t +h2), 0<a <1,
|| i n − i ||≤ P −
whereC isdefinedby(2.7)and(x,t ) (0,l) (0,T],i=1,2,...,M 1; n=1,2,...,N.
P i n
∈ × −
Proof Similartotheproofof[9],letP(x,t )betheexactsolutionof(2.1)atthemeshpoint
i n
(x,t ),ande n=P(x,t ) Pn.Subtracting(2.8)from(2.11)andusinge 0=0,weobtain
i n i i n − i i
k t a n 1
ein− ah2 ein+1−2ein+ein−1 =−k(cid:229)=−1e−rUiktgkein−k+Rni, n≥1, (2.31)
(cid:0) (cid:1)
whereRnisdefinedby(2.9).
i
Multiplying(2.31)byhe nandsummingupforifrom1toM 1,thereexists
i −
hM(cid:229) −1(ein)2−t a k a hM(cid:229) −1ein+1−2he2in+ein−1ein
i=1 i=1
(2.32)
M 1n 1 M 1
=−h (cid:229) − (cid:229)− e−rUiktgkein−kein+h (cid:229) − Rniein.
i=1 k=1 i=1
10
ItfollowsfromtheproofofTheorem2.1that
M 1
h (cid:229) − (e n)2= e n 2;
i || ||
i=1
−t a k a hM(cid:229) −1ein+1−2he2in+ein−1ein≥0; (2.33)
i=1
−hM(cid:229) −1n(cid:229)−1e−rUiktgkein−kein≤−12n(cid:229)−1gk ||e n−k||2+||e n||2 .
i=1 k=1 k=1
(cid:16) (cid:17)
Accordingto(2.9),(2.7)and(2.16),weobtain[9]
hM(cid:229) −1 Rne n =t a hM(cid:229) −1 rne n t a hM(cid:229) −1 t a (rn)2+(cid:229) kn=−01gk(e n)2
i=1 | i i| i=1 | i i|≤ i=1 "2(cid:229) kn=−01gk i 2t a i #
= t 2a h M(cid:229) −1(rn)2+1n(cid:229)−1g e n 2
2(cid:229) n 1g i 2 k|| ||
k=−0 k i=1 k=0
≤ t 2a na G2(1−a )h(M−1)CP2(t +h2)2+21n(cid:229)−1gk||e n||2 (2.34)
k=0
≤ Ta G (12−a )lCP2t a (t +h2)2+21n(cid:229)−1gk||e n||2
k=0
=C2t a (t +h2)2+1n(cid:229)−1g e n 2,
k
2 2 || ||
k=0
where(x,t ) (0,l) (0,T],and
i n
∈ ×
C =lC2G (1 a )Ta . (2.35)
2 P −
Accordingto(2.33)and(2.34),thereexists
||e n||2≤−12n(cid:229)−1gk ||e n−k||2+||e n||2 +C22t a (t +h2)2+21n(cid:229)−1gk||e n||2
k=1 k=0
(cid:16) (cid:17) (2.36)
=−12n(cid:229)−1gk||e n−k||2+C22t a (t +h2)2+21||e n||2,
k=1
thatis
n 1
e n 2 (cid:229)− gk e n−k 2+C2t a (t +h2)2. (2.37)
|| || ≤− || ||
k=1
Accordingto(2.35)-(2.37)andLemma2.2,wehave
e n 2 na G (1 a )C t a (t +h2)2=lC2G (1 a )Ta G (1 a )Ta (t +h2)2.
|| || ≤ − 2 P − −
Hence
P(x,t ) Pn = e n C G (1 a )l1/2Ta (t +h2).
|| i n − i || || i||≤ P −