Well-posedness and numerical algorithm for the tempered fractional ordinary differential equations

0.24 MB·English

by  Can Li

Well-posedness and numerical algorithm for the
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tempered fractional ordinary di erential equations
Can Lia
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1 aBeijingComputational ScienceResearchCenter,Beijing10084,P.R.China.
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DepartmentofAppliedMathematics, SchoolofSciences, Xi’anUniversityofTechnology,
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Xi’an,Shaanxi710054, P.R.China.
n
a
J Weihua Dengb

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b SchoolofMathematicsandStatistics,
]
A GansuKeyLaboratoryofAppliedMathematics andComplexSystems,Lanzhou
C University, Lanzhou730000, P.R.China.
.
h
Lijing Zhaoc
t
a
m
c SchoolofMathematicsandStatistics,
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GansuKeyLaboratoryofAppliedMathematics andComplexSystems,Lanzhou

1 University, Lanzhou730000, P.R.China.
v
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0 Abstract
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1. Trapped dynamics widely appears in nature, e.g., the motion of particles in viscous cyto-
0 plasm. Thefamous continuous timerandom walk(CTRW)model withpower lawwaiting
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time distribution (having diverging first moment) describes this phenomenon. Because of
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: thefinitelifetimeofbiologicalparticles,sometimesitisnecessarytotemperthepowerlaw
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measure such that the waiting time measure has convergent first moment. Then the time
i
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operator ofthe Fokker-Planck equation corresponding totheCTRWmodel withtempered
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waiting time measure is the so-called tempered fractional derivative. This paper focus on
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discussing theproperties ofthetimetempered fractional derivative, andstudying thewell-
posednessandtheJacobi-predictor-correctoralgorithmforthetemperedfractionalordinary
differential equation. By adjusting the parameter of the proposed algorithm, any desired
convergence ordercanbeobtainedandthecomputationalcostlinearlyincreaseswithtime.
Andtheeffectiveness ofthealgorithm isnumerically confirmed.
Keywords: temperedfractional operators, well-posedness, Jacobi-predictor-corrector
algorithm, convergence.
Emailaddresses: [email protected](CanLi),[email protected](Weihua
Deng),[email protected](LijingZhao).
PreprintsubmittedtoElsevier
1 Introduction
The fractional calculus has a long history. The origin of fractional calculus can be
traced back to the letter between Leibniz and L’Hoˆpital in 1695. In the past three
centuries,thedevelopmentofthetheoriesoffractionalcalculusiswellcontributed
by manymathematiciansand physicists.And from thelastcentury, thebookscov-
ering fractional calculus began to emerge, such as Oldham and Spanier (1974),
Samko, Kilbas and Marichev (1993), Podlubny (1999), and so on. In recent years,
moretheoriesandexperimentsshowthatabroadrangeofnon-classicalphenomena
appearedintheappliedsciencesandengineeringcanbedescribedbyfractionalcal-
culus [33,26,35]. Because of its good mathematical features, nowadays fractional
calculus has become a powerful tool in depicting the anomalous kinetics which
arises in physics, chemistry, biology, finance, and other complex dynamics [26].
In practical applications, several different kinds of fractional derivatives, such as
Riemann-Liouvillefractionalderivative,Caputofractionalderivative[33,35],Riesz
fractional derivative[35], and Hilferfractionalderivative[19,42]are introduced.
One of the typical applications for fractional calculus is the description of anoma-
lousdiffusionbehavioroflivingparticles;and thetempered fractionalcalculusde-
scribes the transitionbetween normal and anomalous diffusions(or the anomalous
diffusionin finite time or bounded physical space). In the continuoustimerandom
walk(CTRW)model,foraLe´vyflightparticle,thescalinglimitoftheCTRWwith
a jump distribution function φ(x) ∼ x−(1+α)(1 < α < 2) exhibits superdiffusive dy-
namics. The corresponding stable Le´vy distribution for particle displacement con-
tains arbitrarily large jumps and has divergent spatial moments. However, the infi-
nitespatialmomentsmaynotbefeasibleforsomephysicalprocesses[8].Oneway
toovercomethedivergenceofthemomentsofLe´vydistributionsintransportmod-
elsistoexponentiallytempertheLe´vymeasure.Thenthespacefractionaloperator
will be replaced by the spatially tempered fractional operator in the corresponding
models [7,8,36]. This paper concentrates on the time tempered fractional deriva-
tive,whicharisesintheFokker-PlanckequationcorrespondingtotheCTRWmodel
with tempered power law waiting time distribution [34,17]. Tempering the power
law waiting time measure makes its first moment finite and the trapped dynamics
more physical. Sometimes it is necessary/reasonable to make the first moment of
thewaitingtimemeasurefinite,e.g., thebiologicalparticlesmovinginviscouscy-
toplasm and displaying trapped dynamical behavior just have finite lifetime. The
time tempered diffusion dynamics describes the coexistence/transition of subdif-
fusionand normaldiffusionphenomenon(orthe subdiffusionin finite time)which
wasempiricallyconfirmedinanumberofsystems[8,27].Moreapplicationsforthe
tempered fractional derivatives and tempered differential equations can be found,
for instance, in poroelasticity [18], finance [7], ground water hydrology [27,28],
and geophysicalflows[29].
Tempered fractional calculus can be recognized as the generalization of fractional
2
calculus.Tothebestofourknowledge,thedefinitionsoffractionalintegrationwith
weak singular and exponential kernels were firstly reported in Buschman’s earlier
work [4]. For the other different definitions of the tempered fractional integration,
see the books [39,35,28] and references therein. This work continues previous ef-
forts [25] to explore the time tempered fractional derivative. The well-posedness,
including existence, uniqueness, and stability, of the tempered fractional ordinary
differential equation (ODE) is discussed, and the properties of the time tempered
fractional derivative are analyzed. Then the Jacobi-predictor-corrector algorithm
forthetemperedfractionalODEisprovided,whichhasthestrikingbenefits:1.any
desired convergence order can be obtained by simply adjusting the parameter (the
number of interpolationpoints); the computationalcost increases linearly with the
time t instead of t2 usually taken place for nonlocal time dependent problem. And
extensivenumericalexperimentsare performed toconfirm theseadvantages.
In Section 2, we introduce the definitions and show the properties of the tempered
fractionalcalculus,includingthegeneralizationsofthetemperedfractional deriva-
tivesintheRiemann-LiouvilleandCaputosense,andthecompositeproperty.More
basic properties are listed and proved in Appendix A; the expression and proper-
ties of the tempered fractional calculus in Laplace space are proposed and proved
in Appendix B. In Section 3, we discuss the initial value problem of the tempered
fractional ODE: first derive the Volterra integral formulation of the tempered frac-
tionalODE;thenprovethewell-posednessoftheconsideredproblem.TheJacobi-
Predictor-CorrectoralgorithmforthetemperedfractionalODEisdesignedanddis-
cussed in Section 4, and two numerical examples are solved by the algorithm to
showitspowerfulness.
2 Preliminaries
In this section, we first give the definitions and some properties of the tempered
fractionalcalculus.Let[a,b]beafiniteintervalonthereal lineR. Denote L([a,b])
as the integrable space which includes the Lebesgue measurable functions on the
finiteinterval[a,b],i.e.,
b
L([a,b]) = u : kuk = |u(t)|dt < ∞ .
L([a,b])
(cid:26) Za (cid:27)
And let AC[a,b] be the space of real-values functions u(t) which are absolutely
continuous on [a,b]. For n ∈ N+, we denote ACn[a,b] as the space of real-values
functions u(t) which have continuous derivatives up to order n − 1 on [a,b] such
that dn−1u(t) ∈ AC[a,b], i.e.,
dxn−1
dn−1
ACn[a,b] = u : [a,b] → R, u(t) ∈ AC[a,b] .
dxn−1
(cid:26) (cid:27)
3
And denote byCn[a,b] the space of functions u(t) which are n times continuously
differentiableon [a,b].
Definition 1(Riemann-Liouvilletemperedfractional integral[4,8]) Supposethatthe
real function u(t) is piecewise continuous on (a,b) and u(t) ∈ L([a,b]), σ > 0,λ ≥
0. TheRiemann-Liouvilletemperedfractionalintegraloforder σis definedto be
1 t
Iσ,λu(t) = e−λt Iσ eλtu(t) = e−λ(t−s)(t− s)σ−1u(s)ds, (1)
a t a t Γ(σ) Z
a
(cid:0) (cid:1)
where Iσ denotestheRiemann-Liouvillefractionalintegral
a t
1 t
Iσu(t) = (t− s)σ−1u(s)ds. (2)
a t Γ(σ) Z
a
Obviously, the tempered fractional integral (1) reduces to the Riemann-Liouville
fractional integralifλ = 0. In practical applications,sometimesthefractional inte-
gral (1)isrepresented as D−σ,λu(t).
a t
Definition 2(Riemann-Liouvilletemperedfractional derivative[3,8]) Forn−1 <
α < n,n ∈ N+,λ ≥ 0. The Riemann-Liouville tempered fractional derivative is
defined by
e−λt dn t eλsu(s)
Dα,λu(t) = e−λt Dα eλtu(t) = ds, (3)
a t a t Γ(n−α)dtn Z (t− s)α−n+1
a
(cid:0) (cid:1)
where Dα(eλtu(t))denotestheRiemann-Liouvillefractionalderivative[33]
a t
dn 1 dn t (eλsu(s))
Dα(eλtu(t)) = In−α(eλtu(t)) = ds. (4)
a t dtn a t Γ(n−α)dtn Z (t− s)α−n+1
a
(cid:0) (cid:1)
Remark 1([3]) ThevariantsoftheRiemann-Liouvilletemperedfractionalderiva-
tives aredefinedas
Dα,λu(t)−λαu(t), 0 < α < 1,
a t
aDαt,λu(t) = aDαt,λu(t)−αλα−1dud(tt) −λαu(t), 1 < α < 2. (5)

Definition 3(fractional substantialderivative[16,40,6]) Forn−1 < α < n,n ∈ N+,
andλ(x)beinganygivenfunctiondefinedinspacedomain.TheRiemann-Liouville
fractionalsubstantialderivativeisdefined by
d n d n t e−λ(x)·(t−s)u(s)
Dα,λ(x)u(t) = +λ(x) In−α,λ(x)u(t) = +λ(x) ds, (6)
s (cid:18)dt (cid:19) a t (cid:18)dt (cid:19) Za (t− s)α−n+1
4
where In−α,λ(x) denotestheRiemann-Liouvillefractionalintegraland
a t
d n d d
+λ(x) = +λ(x) ··· +λ(x) . (7)
(cid:18)dt (cid:19) (cid:18)d t (cid:19) (cid:18) d t (cid:19)
ntimes
| {z }
Remark 2 The fractional substantial derivative (6) is equivalent to the Riemann-
Liouville tempered fractionalderivative (3) if λ(x) is a nonnegative constant func-
tion.In fact,usingintegrationbypartsleadsto
d n t e−λ(x)·(t−s)u(s)
+λ(x) ds
(cid:18)dt (cid:19) (cid:20)Za (t− s)α−n+1 (cid:21)
d n−1 d t e−λ(x)·(t−s)u(s)
= +λ(x) +λ(x) ds
(cid:18)dt (cid:19) (cid:20)(cid:18)dt (cid:19)Za (t− s)α−n+1 (cid:21)
d n−1 d t eλ(x)su(s)
= +λ(x) e−λ(x)t ds
(cid:18)dt (cid:19) (cid:20) dt Za (t− s)α−n+1 (cid:21)
d n−2 d2 t eλ(x)su(s)
= +λ(x) e−λ(x)t ds
(cid:18)dt (cid:19) (cid:20) dt2 Za (t− s)α−n+1 (cid:21)
= ···
= Dα,λ(x)u(t).
a t
n
The tempered n-th order derivative of u(t) equals to d +λ u(t), which can be
dt
simply/resonablydenoted as Dn,λu(t). (cid:16) (cid:17)
Definition 4(Caputo tempered fractional derivative[35,41]) Forn−1 < α < n,n ∈
N+,λ ≥ 0. TheCaputo tempered fractionalderivativeis definedas
e−λt t 1 dn(eλsu(s))
CDα,λu(t) = e−λt CDα eλtu(t) = ds, (8)
a t a t Γ(n−α) Z (t− s)α−n+1 dsn
a
(cid:0) (cid:1)
whereCDα,λ(eλtu(t))denotestheCaputofractionalderivative[33]
a t
1 t 1 dn(eλsu(s))
CDα(eλtu(t)) = ds. (9)
a t Γ(n−α) Z (t− s)α−n+1 dsn
a
Remark 3 TheequivalentformsofRiemann-Liouvilletemperedfractionalderiva-
tive(3)andCaputotemperedfractionalderivative(8)are Dα,λu(t) = Dn,λ In−α,λu(t)
a t a t
andCDα,λu(t) = In−α,λDn,λu(t),respectively.
a t a t
Notethatwhenλ = 0,theRiemann-Liouville(Caputo)temperedfractional deriva-
tivereduces totheRiemann-Liouville(Caputo)fractional derivative.
Proposition1 Let u(t) ∈ ACn[a,b]andn−1 < α < n. Then forallt ≥ a, holds
n−1 e−λt(t−a)k−α dk
CDα,λ u(t) = Dα,λ u(t) − eλtu(t) . (10)
a t (cid:0) (cid:1) a t (cid:0) (cid:1) Xk=0 Γ(k−α+1)(cid:20)dtk(cid:0) (cid:1)(cid:12)(cid:12)(cid:12)t=a(cid:21)
(cid:12)
(cid:12)
5
Proof. Take v(t) = eλtu(t) in the equation for the Riemann-Liouville and Caputo
fractional derivatives[33,23,35]
n−1 (t−a)k dkv(t)
CDαv(t) = Dα v(t)− ,
a t a t k! dtk t=a
(cid:18) Xk=0 (cid:12) (cid:19)
(cid:12)
(cid:12)
yielding
n−1 (t−a)k dk
CDα eλtu(t) = Dα eλtu(t)− eλtu(t) .
a t a t k! dtk t=a
(cid:0) (cid:1) (cid:18) Xk=0 (cid:0) (cid:1)(cid:12)(cid:12) (cid:19)
(cid:12)
Multiplyingbothsidesoftheaboveequationbye−λt,we obtain
n−1 (t−a)k dk
e−λt CDα eλtu(t) = e−λt Dα eλtu(t)− eλtu(t) .
a t a t k! dtk t=a
(cid:0) (cid:1) (cid:18) Xk=0 (cid:0) (cid:1)(cid:12)(cid:12) (cid:19)
(cid:12)
Furthermore,usingthedefinitionsofRiemann-LiouvilleandCaputotemperedfrac-
tionalderivatives,weget that
n−1 (t−a)k dk
CDα,λ u(t) = Dα,λ u(t) − e−λt Dα eλtu(t) . (11)
a t (cid:0) (cid:1) a t (cid:0) (cid:1) Xk=0 a t (cid:18) k! (cid:19)(cid:20)dtk(cid:0) (cid:1)(cid:12)(cid:12)(cid:12)t=a(cid:21)
(cid:12)
(cid:12)
Using the linearity properties presented in Proposition 4 and the formula of power
function
(t−a)k 1 1 Γ(k+1)(t−a)k−α
Dα = Dα (t−a)k = ,
a t k! k!a t k! Γ(k−α+1)
(cid:18) (cid:19) (cid:18) (cid:19)
wededucethedesiredresult from(11). (cid:3)
Proposition2 (Compositeproperties)
(1) Let u(x) ∈ L([a,b]) and In−α,λu(t) ∈ ACn[a,b]. Then the Riemann-Liouville
tempered fractionalderivativeandintegralhavethecompositeproperties
n−1 e−λt(t−a)α−k−1
Iα,λ[ Dα,λu(t)] = u(t)− Dα−k−1(eλtu(t)) , (12)
a t a t Γ(α−k) a t t=a
Xk=0 (cid:2) (cid:12)(cid:12) (cid:3)
(cid:12)
and
Dα,λ[ Iα,λu(t)] = u(t). (13)
a t a t
(2) Let u(t) ∈ ACn[a,b] and n − 1 < α < n. Then the Caputo tempered fractional
derivative and the Riemann-Liouville tempered fractional integral have the com-
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positeproperties
n−1 (t−a)k dk(eλtu(t))
Iα,λ[CDα,λu(t)] = u(t)− e−λt , (14)
a t a t Xk=0 k! (cid:20) dtk (cid:12)(cid:12)t=a(cid:21)
(cid:12)
(cid:12)
(cid:12)
and
CDα,λ[ Iα,λu(t)] = u(t) if α ∈ (0,1). (15)
a t a t
Proof. From the definitions of Riemann-Liouville tempered fractional integral and
derivative,wehave
Iα,λ[ Dα,λu(t)] = e−λt Iα eλt( Dα,λu(t))
a t a t a t a t
= e−λt Iα(cid:2)eλt e−λt Dα(eλ(cid:3)tu(t))
a t a t (16)
= e−λt Iα(cid:2) D(cid:0)α(eλtu(t)) . (cid:1)(cid:3)
a t a t
(cid:2) (cid:3)
(I)
| {z }
Thanksto thecompositionformula[33,23,35]
n−1 (t−a)α−k−1
Iα[ Dαv(t)] = v(t)− Dα−k−1(v(t)) ,
a t a t Γ(α−k) a t t=a
Xk=0 (cid:2) (cid:12)(cid:12) (cid:3)
(cid:12)
weget
n−1 (t−a)α−k−1
(I) = Iα[ Dαeλtu(t)] = eλtu(t)− Dα−k−1(eλtu(t)) .
a t a t Γ(α−k) a t t=a
Xk=0 (cid:2) (cid:12)(cid:12) (cid:3)
(cid:12)
Insertingtheaboveformulainto(16)leadsto (12).
Again from the definitions of Riemann-Liouville tempered fractional integral and
derivative,thereexists
Dα,λ[ Iα,λu(t)] = e−λt Dα eλt( Iα,λu(t))
a t a t a t a t
= e−λt Dα(cid:2)eλt e−λt Iα(eλ(cid:3)tu(t))
a t a t
= e−λt Dα(cid:2) Iα(cid:0)(eλtu(t)) . (cid:1)(cid:3)
a t a t
(cid:2) (cid:3)
Furthermore, using the compositeproperties of Riemann-Liouvillefractional inte-
gral and derivative[33,23,35]
Dα Iα(v(t)) = v(t), (17)
a t a t
(cid:2) (cid:3)
weget
Dα,λ[ Iα,λu(t)] = e−λt Dα Iα(eλtu(t)) = u(t),
a t a t a t a t
by takingv(t) = eλtu(t)in Eq.(17). (cid:2) (cid:3)
7
Similarly,usingthecompositepropertiesofCaputofractionalderivative[33,23,35]
n−1 (t−a)k dkv(t)
Iα[CDαv(t)] = v(t)− ,
a t a t Xk=0 k! (cid:20) dtk (cid:21)(cid:12)(cid:12)t=a
(cid:12)
(cid:12)
(cid:12)
and
CDα Iα(v(t)) = v(t), if α ∈ (0,1), (18)
a t a t
wecan get (14)and (15). (cid:2) (cid:3)
(cid:3)
Remark 4 Fora constantC,
Dα,λC = Ce−λt Dαeλt, CDα,λC = Ce−λtCDαeλt. (19)
a t a t a t a t
Obviously, Dα,λ(C) , CDα,λ(C). And CDα,λ(C) is no longer equal to zero, being
a t a t a t
differentfromCDα(C) ≡ 0.
a t
3 Well-posedness ofthe tempered fractional ordinary differential equations
In this section, we consider the ODEs with Riemann-Liouville and Caputo tem-
pered fractional derivatives,respectively,i.e.,
Dα,λu(t) = f(t,u(t)), n−1 < α < n,λ ≥ 0,
a t (20)
 Dα−k−1 eλtu(t) = g , k = 0,1,2,··· ,n−1,
 a t t=a k
and (cid:2) (cid:0) (cid:1)(cid:3)(cid:12)(cid:12)(cid:12)
CDα,λu(t) = f(t,u(t)), n−1 < α < n,λ ≥ 0,
a t
 dk (21)
The Cauchy proble(cid:20)mdtsk((e2λ0tu)(ta)n)(cid:21)d(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)t=(a21=)ccka,nkb=e 0c,o1n,v2e,r·te·d· ,tno−th1e.equivalent Volterra
integralequationsofthesecond kindundersomeproperconditions.
Lemma 1 Ifthefunction f(t,u(t))andu(t)belongto L([a,b]),thenu(t)issolution
of the initial value problem (20) if and only if u(t) is the solution of the Volterra
integralequationofthesecondkind
n−1 e−λt(t−a)α−k−1 1 t
u(t) = g + e−λ(t−s)(t− s)α−1f(s,u(s))ds. (22)
k Γ(α−k) Γ(α) Z
Xk=0 a
In particular, if 0 < α < 1, then u(t) satisfies the Cauchy problem (20) if and only
8
ifu(t)satisfiesthefollowingintegralequation
e−λt(t−a)α−1 1 t
u(t) = g + e−λ(t−s)(t− s)α−1f(s,u(s))ds. (23)
0 Γ(α) Γ(α) Z
a
Proof. For the linear Cauchy problems (20) and (21), the conclusion is directly
reached by the Laplace transform given in Appendix B. Now we prove the more
general case.
Necessity. Performing theintegraloperator Iα,λ on bothsides ofthefirst equation
a t
of (20), wehave
n−1 e−λt(t−a)α−k−1 1 t
u(t) = g + e−λ(t−s)(t− s)α−1f(s,u(s))ds,
k Γ(α−k) Γ(α) Z
Xk=0 a
where we use the composite property (1) given in Proposition 2. Then Eq. (22) is
obtained.
Suf ficiency.Applyingtheoperator Dα,λ tobothsidesofEq. (22)results in
a t
n−1 Dα,λ(e−λt(t−a)α−k−1)
Dα,λu(t) = g a t + Dα,λ Iα,λf(t,u(t)) = f(t,u(t)), (24)
a t k Γ(α−k) a t a t
Xk=0
whereweusethefact
Dα,λ(e−λt(t−a)α−k−1) e−λt(t−a)α−k−1−α e−λt(t−a)−k−1
a t = = = 0, k = 0,1,2,··· ,n−1,
Γ(α−k) Γ(−k) ∞
andthecompositeproperty(13).Nowweshowthatthesolutionof (22)satisfiesthe
initialconditiongiveninEq.(20).Multiplyingeλt andthenperformingtheoperator
Dα−j−1 on bothsidesofEq. (22), for 0 ≤ j < n−2 < n−1 < α < n, wehave
a t
n−2 (t−a)j−k
Dα−j−1(eλtu(t)) = g + Dα−j−1 Iα eλtf(t,u(t)) ,
a t kΓ(j−k+1) a t a t
Xk=0 (cid:0) (cid:1) (25)
n−2 (t−a)j−k
= g + D−j−1 eλtf(t,u(t)) ,
kΓ(j−k+1) a t
Xk=0 (cid:0) (cid:1)
wheretheformula
Γ(α−k)
Dα−j−1 (t−a)α−k−1 = Dα−j−1 (t−a)α−k−1 = (t−a)j−k, 0 ≤ k < n−2,
a t a t Γ(j−k+1)
(cid:0) (cid:1) (cid:0) (cid:1)
isutilized.
9
Takingalimitt → a in theaboveequation,weobtain
n−2 (t−a)j−k
lim Dα−j−1(eλtu(t)) = lim g +lim D−j−1 eλtf(t,u(t)) , (26)
t→aa t t→a kΓ(j−k+1) t→a a t
Xk=0 (cid:0) (cid:1)
withthesecondtermintherighthandsidebeingequaltozero;andforitsfirstterm,
wehave
n−2 (t−a)j−k j−1 (t−a)j−k n−2 (t−a)j−k
lim g =lim g +g +lim g
t→a kΓ(j−k+1) t→0 kΓ(j−k+1) j t→a kΓ(j−k+1)
Xk=0 Xk=0 kX=j+1
j−1 g n−2 g (t−a)j−k
= k ·0+g +lim k
Γ(j−k+1) j t→a ∞
Xk=0 kX=j+1
=g .
j
(27)
(cid:3)
By the similartechnique in proving Lemma1, we obtain the followingconclusion
fortheCauchy problem(21).
Lemma 2 Ifthefunction f(t,u)iscontinuous,thenu(t)isthesolutionoftheinitial
valueproblem(21)ifandonlyifu(t)isthesolutionoftheVolterraintegralequation
ofthesecond kind
n−1 e−λt(t−a)k 1 t
u(t) = c + e−λ(t−s)(t− s)α−1f(s,u(s))ds. (28)
k Γ(k+1) Γ(α) Z
Xk=0 a
Inparticular,if 0 < α < 1,thenu(t)satisfiestheCauchyproblemifandonlyifu(t)
satisfiesthefollowingintegralequation
1 t
u(t) = u(a)e−λ(t−a) + e−λ(t−s)(t− s)α−1f(s,u(s))ds. (29)
Γ(α) Z
a
3.1 Existenceanduniqueness
Manyauthors haveconsidered the existenceand uniquenessofthe solutionsto the
nonlinear ODEs with fractional derivatives; see, e.g., [32,1,15,14,13,20,21,42,45].
Fortheglobalexistenceanduniformasymptoticstabilityresultsoffractionalfunc-
tionaldifferentialequationscorrespondingto(23),onecansee[24,2].Inthefollow-
ing,we discusstheexistenceand uniquenessofthesolutionsofthenonlineartem-
pered fractional differential equations based on the equivalent Volterra equations
presented in Lemmas 1 and 2. We shall employ the Banach fixed point theorem to
proveit.Let f : [a,b]×B → Rbeacontinuousfunctionsuchthat f(t,u) ∈ L([a,b])
10