# On Riemann-Liouville and Caputo Derivatives

Authors: Changpin Li; Deliang Qian; YangQuan Chen
Journal: Discrete Dynamics in Nature and Society
 (2011)

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## Abstract

Recently, many models are formulated in terms of fractional derivatives, such as in control processing, viscoelasticity, signal processing, and anomalous diffusion. In the present paper, we further study the important properties of the Riemann-Liouville (RL) derivative, one of mostly used fractional
derivatives. Some important properties of the Caputo derivative which have not been discussed elsewhere are simultaneously mentioned. The partial fractional derivatives are also introduced. These discussions are beneficial in understanding fractional calculus and modeling fractional equations in science and engineering.


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## Body

## 1. Introduction

Fractional calculus is not a new topic;  in reality it has almost the same history as that of the classical calculus [1]. Since the occurrence of fractional (or fractional-order) derivative, the theories of fractional calculus (fractional derivative plus fractional integral) has undergone a significant and even heated development, which has been primarily contributed by pure but not applied mathematicians; the reader can refer to an encyclopedic book [2] and many references cited therein. In the last few decades, however, applied scientists and engineers realized that differential equations with fractional derivative provided a natural framework for the discussion of various kinds of real problems modeled by the aid of fractional derivative, such as viscoelastic systems, signal processing, diffusion processes, control processing, fractional stochastic systems, allometry in biology and ecology ([3–17] and huge cited references therein).Different from classical (or integer-order) derivative, there are several kinds of definitions for fractional derivatives. These definitions are generally not equivalent with each other. In the following, we introduce several definitions [7, 14].Definition 1.1.
Yα, the convolution kernel of orderα∈R+for fractional integrals, is defined by
(1.1)Yα(t)=t+α-1Γ(α)∈Lloc⁡1(R+),
where Γ is the well-known Euler Gamma function and
(1.2)t+α-1={tα-1,t>0,0,t≤0.Definition 1.2.
The fractional integral (or the Riemann-Liouville integral)D0,t-αwith fractional orderα∈R+of functionx(t)is defined as
(1.3)D0,t-αx(t)=Yα*x(t)=1Γ(α)∫0t(t-τ)α-1x(τ)dτ.Yα has an important convolution property (or semigroup property), that is, Yα*Yβ=Yα+β for arbitrary α>0 and β>0. This implies that D0,t-α·D0,t-β=D0,t-α-β.Definition 1.3.
The Grünwald-Letnikov fractional derivative with fractional orderαis defined by,ifx(t)∈Cm[0,t],(1.4)GLD0,tαx(t)=∑k=0m-1x(k)(0)t-α+kΓ(-α+k+1)+1Γ(m-α)∫0t(t-τ)m-α-1x(m)(τ)dτ,
where m-1≤α<m∈Z+.This is not the original definition. The initial definition is given by a limit, that is,(1.5)GLD0,tαx(t)=lim⁡h→0,nh=th-α∑k=0n(-1)k(pk)x(t-kh).
The limit expression is not convenient for analysis but often used for numerical approximation.Definition 1.4.
The Riemann-Liouville derivative of fractional orderαof functionx(t)is given as
(1.6)RLD0,tαx(t)=dmdtmD0,t-(m-α)x(t)=1Γ(m-α)dmdtm∫0t(t-τ)m-α-1x(τ)dτ,
where m-1≤α<m∈Z+.From Definitions1.3 and 1.4,  one can see that RLD0,tαx(t)=D0,tαGLx(t) if x(t)∈Cm[0,t] which can be verified via integration by parts. This fact and the original definition of GLD0,tα provide a numerical method for fractional differential equation with Riemann-Liouville derivative [18].Definition 1.5.
The Riesz fractional derivative of fractional orderαof functionx(t)is given as
(1.7)RDαx(t)=-12cos⁡⁡(πα/2)Γ(m-α)⋅dmdtm(∫-∞t(t-τ)m-α-1x(τ)dτ+(-1)m∫t∞(t-τ)m-α-1x(τ)dτ),
in which m-1≤α<m∈Z+.This derivative was induced by the Riemann-Liouville derivative and is useful in physics.Definition 1.6.
The Caputo derivative of fractional orderαof functionx(t)is defined as
(1.8)D0,tαCx(t)=D0,t-(m-α)dmdtmx(t)=1Γ(m-α)∫0t(t-τ)m-α-1x(m)(τ)dτ,
in which m-1<α<m∈Z+.From this definition, one can see thatCD0,tαx(n)(t)=DC0,tα+nx(t).Comparing this definition with the Riemann-Liouville one, functions which are derivable in the Caputo sense are much “fewer" than those which are derivable in the Riemann-Liouville sense.The following definition is also used in mathematical analysis.Definition 1.7.
Y-α(α∈R+) is the generalized function in the sense of Schwartz, as the unique convolution inverse ofY+αin the convolution algebraD+′(R):with the use of the Dirac distribution, which is the neutral element of convolution; this readsY+α*Y-α=δ. With the notation, the generalized fractional derivative of order α of a casual function or distribution isGD0,tαx(t)=Y-α*x(t).From this definition and the semigroup property ofYα, one has GD0,tα·GD0,tβ=DG0,tα+β, where α>0,β>0. These definitions for fractional derivatives are not equivalent. There are some discussions available, say, in [9, 14].In the realm of the fractional differential equations, Caputo derivative and Riemann-Liouville ones are mostly used. It seems that the former is more welcome since the initial value of fractional differential equation with Caputo derivative is the same as that of integer differential equation; for example, the initial value condition of fractional differential equationCD0,tαx(t)=f(t,x) with α∈(0,1), t>0 is posed as x(0)=x0. But for the fractional differential equation RLD0,tαx(t)=f(t,x) with α∈(0,1),  t>0, its initial value condition involves fractional integral (and/or derivative), its initial value condition is given as [DRL0,tα-1x(t)]t=0=x0 (if α∈(1,2), then its initial value conditions are given as [DRL0,tα-2x(t)]t=0=x0, [DRL0,tα-1x(t)]t=0=x0(1)). Most people think that these fractional-order initial values are not easy to measure. This makes an illusion; that is, RL derivative seems to be used in less situations. But in reality, this is not the case. Physical and geometric interpretations for RL derivative can be found in [19]. It makes it possible to observe and/or measure values of RL integral and derivative(s).On the other hand, besides the smooth requirement, Caputo derivative does not coincide with the classical derivative [9], say, for α∈(m-1,m), m∈Z+,(1.9)lim⁡α→(m-1)+D0,tαCx(t)=x(m-1)(t)-x(m-1)(0),lim⁡α→m+D0,tαCx(t)=xm(t),
while RL derivative is in-line with the classical derivative, this can be seen from the following equations for α∈(m-1,m), m∈Z+, for α∈(m-1,m), m∈Z+: (1.10)lim⁡α→(m-1)+DRL0,tαx(t)=x(m-1)(t),lim⁡α→m+DRL0,tαx(t)=x(m)(t).
Furthermore, fractional-order initial value condition(s) for RL-type differential equation can be given as usual. For example, the initial value condition [DRL0,tα-1x(t)]t=0=x0 for equation DRL0,tαx(t)=f(t,x) with α∈(0,1), t>0 can be replaced by [t1-αx(t)]t=0=x0/Γ(α) [20]. Of course, for 1<α∈(m-1,m), m∈Z+, we can use the formula D0,tαCx(t)=DRL0,tα(x(t)-∑k=0m-1tkx(k)(0)/k!) [9] to change corresponding fractional-order initial values into integer-order initial values. It has been found that RL derivative is very useful to characterize anomalous diffusion, Lévy flights and traps [21, 22], and so forth.Here, we have no intention of mentioning which derivative is more widely utilized, but we must stress that every derivative has its own serviceable range. Since there are much more studies on properties of Caputo derivative [9, 10, 14], in this paper we focus on further studying the properties of RL derivative, which is helpful in understanding RL derivative and modeling fractional equations by the aid of RL derivative. And some extra properties of Caputo derivative are also introduced. The outline of the rest paper is organized as follows. In Section 2, we further study the important properties of RL derivative which have not appeared elsewhere. In the following section, we generalize the RL derivative to the RL partial derivative. The last section includes conclusions.


## 2. Further Properties of RL and Caputo Derivatives

We first list the known properties [9, 10, 14] just for reference.Property 1.
(1)
Forα∈(0,1),  x(t)∈Lloc1(R+),(2.1)RLD0,tαx(t)=GD0,tαx(t)-δ(t)[DRL0,tα-1x(t)]t=0=ddt∫0tY1-α(t-τ)x(τ)dτ.(2)
Forn-1<α<n∈Z+,  x(t)∈Lloc1(R+),(2.2)DRL0,tαx(t)=D0,tαGx(t)-δ(t)[DRL0,tα-1x(t)]t=0-∑k=1n-1Y-k[DRL0,tα-k-1x(t)]t=0=dndtn∫0tYn-α(t-τ)x(τ)dτ.(3)
Assumeα>0, then  (dn/dtn)DRL0,tαx(t)=DRL0,tn+αx(t);  (dn/dtn)DRL0,t-αx(t)=DRL0,tn-αx(t) if  n-α>0, and  (dn/dtn)DRL0,t-αx(t)=D0,tn-αx(t) if  n-α<0, hold for any  n∈Z+.(4)
DRL0,tα·D0,t-αx(t)=x(t)  for all  α>0.  More generally,  DRL0,tα·D0,t-βx(t)=DRL0,tα-βx(t)  for all  β>0. If  α<β, then  DRL0,tα-βx(t)=D0,tα-βx(t).(5)
D0,t-α·DRL0,tαx(t)=x(t)-∑k=1n(tα-k[DRL0,tα-kx(t)]t=0/Γ(α-k+1)), where  n-1<α<n∈Z+.(6)
D0,t-n·DRL0,tnx(t)=x(t)-∑k=0n-1(tk/k!)x(k)(0).(7)
DRL0,tαc=(t-α/Γ(1-α))c,  where  α>0  and   c  is an arbitrary constant.(8)
ℒ[DRL0,tαx(t)]=sαX(s)-∑k=0n-1sk[DRL0,tα-k-1x(t)]t=0,  in which  ℒ  is the Laplace transform, and  ℒ[x(t)]=X(s).(9)
D0,tαCx(t)=DRL0,tα(x(t)-∑k=0n-1(tk/k!)x(k)(0)),  where  n-1<α<n∈Z+.For Caputo derivative,D0,tαC·D0,t-β=D0,tα-βC generally does not hold for all α, β>0.From (4) in Property1, one has very interesting conclusions as follows.Conclusion.
Ifx(t) is defined in the interval  [a,b] and  (1/Γ(α))∫at(t-τ)α-1x(τ)dτ=0 for  α>0 and for all  t∈[a,b], then  x(t)≡0.Proof.
The condition implies thatD0,t-αx(t)=0. Taking the RL derivative operator in both sides and applying Property 1(4) yields x(t)≡0 in [a,b].Conclusion.
The following equation(2.3)CD0,tαx(t)=f(x),α∈(0,1),x∈R,x(0)=x0,
does not have a periodic solution if x0 does not solve f(x)=0, where f(x) is continuous.Proof.
The above equation is equivalent to the following Volterra integral equation [23]:(2.4)x(t)=x0+D0,t-αf(x(t)).
If x(t) has a periodic solution with period T>0, then setting t=T in the above formula and using Conclusion 1 lead to f(x(T))≡0; that is, x0 solves f(x)=0 due to x(T)≡x0, which is contradictory to the assumption.  So the result holds.But the above conclusion is not suitable for the nonautonomous fractional system with the Caputo derivative. The counterexample is constructed as follows:(2.5)DC0,tαx(t)=x(t)+∑k=0+∞(-1)t2k+1k[(2k+1)!!Γ(2k+2-α)t-α+1(2k+1)!!],α∈(0,1),x(0)=0,
and has a periodic solution x(t)=sint.Some discussions on the periodic solution of the Caputo-type fractional differential equation can be referred to [24].For the RL derivative case, the corresponding equation does not have the integer-order initial value condition(s). Its Cauchy problem is often posed as follows [2, 20]:(2.6)DRL0,tαx(t)=f(x),α∈(0,1),x∈R,t>0,DRL0,tα-1x(t)|t=0=x0(ort1-αx(t)|t=0=x0Γ(α)).Conclusion.
Assume thatf(x) is continuous,  x is a function of  t>0 and that  limt→0+ is not bounded, but  limt→0+f(x(t)) exists. Then (2.6) does not have a periodic solution.Proof.
Equation (2.6) is equivalent to the following integral equation [2, 20]:
(2.7)x(t)=x0Γ(α)tα-1+D0,t-αf(x(t)).
Iflimt→0+ is bounded, then the case is trivial so it is omitted here. We only show interests in the case that limt→0+ is not bounded. Suppose that (2.6) has a periodic solution with period T>0, then, for arbitrary small δ>0, one has x(δ)=x(δ+T). From (2.7), |x(δ+T)| has a bound independent of δ for arbitrary δ>0 due to the assumption of f(x), but |x(t)| approaches to +∞ as δ→0+. This completes Conclusion 3.The previous conclusion can be very smoothly generalized to the higher-dimensional case. In the following, we further study the important nature of RL derivative.Property 2.
(1)
Composition with the integral operator: forα>0,  β>0, then  DRL0,tα-β=DRL0,tα·D0,t-β≠D0,t-β·DRL0,tα.(2)
Composition with the integer derivative operator: forα∈(n-1,n),  n∈Z+,  m∈Z+, then  (dm/dtm)·DRL0,tα=DRL0,tα+m≠DRL0,tα·(dm/dtm).(3)
Composition with Caputo operator: forα∈(n-1,n),  n∈Z+,  (α≠)β∈(m-1,m),  m∈Z+, then  DRL0,tα+β-m(dm/dtm)=DRL0,tα·D0,tβC≠D0,tβC·DRL0,tα=D0,t-(m-β)·DRL0,tα+m.(4)
Composition with the generalized fractional derivative operator:  forα∈(n-1,n),  n∈Z+,  β>0, then  (dn/dtn)[Yn-α-β*]=DRL0,tα·D0,tβG≠D0,tβG·DRL0,tα=Y-β*DRL0,tα.Proof.
(1) Can be regarded as the direction conclusion of Property1(4) and (5).
(2) Can be derived by the direct computation.
(3) Means that the RL derivative operators cannot commute with each other unless the involved initial value conditions are homogeneous [14].
(4) Can be proved by Property1(2) and corresponding definitions.Although the Riemann-Liouville integral operatorD0,t-α(α∈R+) has the semigroup property, that is, D0,t-α·D0,t-β=D0,t-α-β (α>0, β>0), RL derivative operator DRL0,tα does not have this character, that is, DRL0,tα·DRL0,tβ≠DRL0,tα+β and DRL0,tβ·DRL0,tα≠DRL0,tα+β [14]. However, we have following interesting result.Property 3.
Ifx(t)∈C1[0,T],  αi∈(0,1)(i=1,2) (the trivial case  αi=0  or   1 is simple and removed here), and  α1+α2∈(0,1], then  DRL0,tα1·DRL0,tα2x(t)=DRL0,tα1+α2x(t).Proof.
According to Property1(3), one gets(2.8)D0,tα2C⋅D0,tα1Cx(t)=D0,tα2C{DRL0,tα1[x(t)-x(0)]}=DRL0,tα2{DRL0,tα1[x(t)-x(0)]-D0,tα1Cx(t)|t=0}=DRL0,tα2{DRL0,tα1[x(t)-x(0)]}=DRL0,tα2⋅DRL0,tα1x(t)-t-α1-α2Γ(1-α1-α2)x(0).
Similarly,
(2.9)D0,tα2C⋅D0,tα1Cx(t)=DRL0,tα2⋅DRL0,tα1x(t)-t-α1-α2x(0)Γ(1-α1-α2).
On the other hand,(2.10)D0,tα1+α2Cx(t)=DRL0,tα1+α2x(t)-t-α1-α2Γ(1-α1-α2)x(0).
Ifα1+α2=1, then (t-α1-α2/Γ(1-α1-α2))x(0) is automatically equal to zero because Γ(0)=∞. By using Theorem 3.3 of [9], one obtains that DRL0,tα2·DRL0,tα1x(t)=DRL0,tα1·DRL0,tα2x(t)=DRL0,tα1+α2x(t).
The following result is for comparison nature of fractional derivatives.Property 4.
(1) Ifα∈(n-1,n),  n∈Z+, and  x(k)(0)≥0(k=0,1,…,n-1), then  CD0,tα≥DRL0,tα.
(2) Ifα∈(n-1,n),  n∈Z+,  DRL0,tαx(t)≥DRL0,tαy(t), and  [DRL0,tα-k-1x(t)]t=0≥[DRL0,tα-k-1x(t)]t=0  (k=0,1,…,n-1), then  x(t)≥y(t). Parallelly, if  α∈(n-1,n), n∈Z+,  CD0,tαx(t)≥CD0,tαy(t), and  x(k)(0)≥y(k)(0)(k=0,1,…,n-1), then  x(t)≥y(t).Proof.
(1) It is just the direction conclusion of Property1(3).
(2) We only show the first part. The proof of the second part (the general case of Lemma 10 [25]) can be similarly given.
SettingDRL0,tαx(t)=ξ(t)+DRL0,tαy(t) and taking the Laplace transform in both sides, one has(2.11)sαX(s)=sαY(s)+L[ξ(t)]+∑k=0n-1sk{[DRL0,tα-k-1x(t)]t=0-DRL0,tα-k-1y(t)]t=0}.
It immediately follows from dividing by sα and taking the inverse Laplace transform in both sides that
(2.12)x(t)=y(t)+D0,t-αξ(t)+∑k=0n-1{[DRL0,tα-k-1x(t)]t=0-DRL0,tα-k-1y(t)]t=0}Yα-k.
The last two addends in the right side of the above equality are nonnegative. This completes the proof.Property 5.
Let𝒜={x(t)∈R,  t≥0,  x(t) is analytical for any  t≥0}.  If  α∈(0,1), then RL derivative operator  DRL0,tα defined in  𝒜 can be expressed as(2.13)DRL0,tα=∑k=0∞[dk⋅/dtk]t=0Γ(k-α+1)tk-α.
More generally, if n-1<α<n∈Z+, then DRL0,tα defined in 𝒜 has also the following form:
(2.14)DRL0,tα=∑k=0∞[dk⋅/dtk]t=0Γ(k-α+1)tk-α.The proof is easy so it is left out here.Remark 2.1.
(1) For an arbitrary functionx(t)∈𝒜, according to the expressions of Caputo differential operator [10] and RL differential operator, one can also easy get Property 1(3).
(2) Even ifx(t)∈𝒜 (it implies that x(0) exists), [DRL0,tαx(t)]t=0(α>0) may not exist unless the initial value x(0)=0.The following example shows that a function is not derivable at one point in the classical sense but is derivable at the same point in RL sense.Example 2.2.
Consider(2.15)x(t)={1-t,0<t≤1,t-1,1<t<1+ε,ε>0.
x(t) exists (right) derivative in the classical sense at t=0 but does not exist derivative in the same sense at t=1.  By simple calculation, one has(2.16)DRL0,tαx(t)={t-αΓ(1-α)-t1-αΓ(2-α),0<t≤1,t-αΓ(1-α)-t1-α-2(t-1)1-αΓ(2-α),t∈(1,1+ε),
where α∈(0,1).From the above example,x′(0+) exists, but x′(1) does not exist; it is quite the reverse for the RL derivative, that is, DRL0,tαx(t) does not exist at t=0 but exists at t=1. So we cannot in general terms say that RL derivative is more general than the classical derivative unless the initial time (or the origin) is excluded. From the above example, we also see that DRL0,tαx(t)>0 if t<1-α, but x(t) is not monotonously increasing for t∈(0,1-α). The RL derivative DRL0,tαx(t)>0 only means that D0,t-(1-α)x(t) is monotonously increasing with respect to t but does not imply that x(t) is monotonously increasing. Geometrically speaking, the value DRL0,tαx(t) at point t relates to an “area." On the other hand, its Caputo derivative exists in the whole interval (0,1+ɛ), although its classical derivative does not at t=1. So we cannot regard RL and Caputo derivatives as the generalization of the typical derivative in rigorous mathematical meaning.Definition1.4 is sometimes called the “left RL fractional derivative." Correspondingly, the right RL fractional derivative with α order (α∈(m-1,m),m∈Z+) is defined as (2.17)DRLt,bαx(t)=(-1)mΓ(m-α)dmdtm∫tb(τ-t)m-α-1x(τ)dτ,
in which t∈(0,b).The Riesz fractional derivative (Definition1.5) is actually induced by left and right RL derivatives. The properties of right RL derivative can be similarly given.


## 3. Partial RL Derivative

Present studies on the anomalous diffusion are often restricted in one space dimension, say [22, 26–28] and references cited therein, where the involved RL derivative is defined with order α∈(0,1) in one spatial dimension. If the anomalous diffusion phenomenon appears in R2 or in higher spatial dimensions, how do we model it? In another words, how do we define the partial RL derivative? In this section, we first introduce the partial RL derivatives which were mentioned in [2], and then we define the partial Caputo derivatives in a similar manner.Supposeαi∈(0,1), i=1,2, α=α1+α2. If we define (3.1)∂RLx1α1x2α2α1+α2u(x1,x2)=∂α2∂x2α2(∂α1∂x1α1u(x1,x2))=∂α2∂x2α2(1Γ(1-α1)∂∂x1∫0x1(x1-ξ)-α1u(ξ,x2)dξ)=1Γ(1-α2)∂∂x2∫0x2(x2-τ)-α2Γ(1-α1)∂∂x1∫0x1(x1-ξ)-α1u(ξ,τ)dξdτ,
then (3.2)∂RLx1α1x2α2α1+α2u(x1,x2)=1Γ(1-α1)⋅Γ(1-α2)∂2∂x1∂x2∫0x2∫0x1(x2-τ)-α2(x1-ξ)-α1u(ξ,τ)dξdτ.According to the classical calculus, if(3.3)∂2∂x1∂x2∫0x2∫0x1(x2-τ)-α2(x1-ξ)-α1u(ξ,τ)dξdτ,∂2∂x2∂x1∫0x1∫0x2(x2-τ)-α2(x1-ξ)-α1u(ξ,τ)dτdξ
exist in a neighborhood of (x1,x2) and are continuous at this point (x1,x2), then (3.4)RL∂x1α1x2α2α1+α2u(x1,x2)=RL∂x2α2x1α1α1+α2u(x1,x2).
If α=α1+α2∈(0,1), then the above partial RL derivative can characterize subdiffusion in R2.The case withα1=0 or α2=0 was simply mentioned in [7],(3.5)∂RLx2α2α2u(x1,x2)=∂α2∂x2α2u(x1,x2)=1Γ(1-α2)∂∂x2∫0x2(x2-τ)-α2u(x1,τ)dτ,∂RLx1α1α1u(x1,x2)=∂α1∂x1α1u(x1,x2)=1Γ(1-α1)∂∂x1∫0x1(x1-ξ)-α1u(ξ,x2)dξ.
Now we give the definition of the partial RL derivative as follows.Definition 3.1.
The partial RL derivative with orderα1+α2 (α1th order in x1-direction andα2th order inx2 direction) is defined as follows:
(3.6)∂RLx1α1x2α2α1+α2u(x1,x2)=1Γ(m-α1)⋅Γ(n-α2)∂m+n∂x1m∂x2n⋅∫0x2∫0x1(x2-τ)n-α2-1(x1-ξ)m-α1-1u(ξ,τ)dξdτ,
where α1∈(m-1,m),α2∈(n-1,n), m, n∈Z+.In the right side of the above equality, if the derivative value of the integral has no relation to partial differential sequence, then the value of the left side of the above equation does not either.The definition in more higher-dimensional space is given in the following.Definition 3.2.
The partial RL derivative with order∑i=1ℓαi (αith order in xi-direction,  i=1,…,ℓ) is defined as follows:
(3.7)∂RLx1α1⋯xlαlα1+⋯+αlu(x1,…,xl)=(∂m1+⋯+ml)/(∂x1m1⋯∂xlml)∏i=1lΓ(mi-αi)∫0xl⋯∫0x1(xl-ξl)ml-αl-1⋯(x1-ξ1)m1-α1-1udξ1⋯dξl,
where αi∈(mi-1,mi), mi∈Z+, i=1,…,ℓ.It is easy to show that (may refer to [9]) (3.8)lim⁡αi→mi-∂RLx1α1⋯xlαlα1+⋯+αlu(x1,…,xl)=∂RLx1α1⋯xi-1αi-1xi+1αi+1⋯xlαlα1+⋯+αi-1+αi+1+⋯+αl∂mi∂ximiu(x1,…,xl),lim⁡αi→(mi-1)+∂RLx1α1⋯xlαlα1+⋯+αlu(x1,…,xl)=∂RLx1α1⋯xi-1αi-1xi+1αi+1⋯xlαlα1+⋯+αi-1+αi+1+⋯+αl∂mi-1u(x1,…,xl)∂ximi-1.We can similarly define the partial Caputo derivative.Definition 3.3.
A two-dimensional case: the partial Caputo derivative with orderα1+α2 (α1th order in x1-direction andα2th order in x2-direction) is defined as follows:
(3.9)∂Cx1α1x2α2α1+α2u(x1,x2)=1Γ(m-α1)⋅Γ(n-α2)⋅∫0x2∫0x1(x2-τ)n-α2-1(x1-ξ)m-α1-1∂m+n∂ξm∂τnu(ξ,τ)dξdτ,
where α1∈(m-1,m),α2∈(n-1,n), m, n∈Z+.Definition 3.4.
A higher-dimensional case: the partial Caputo derivative with order∑i=1ℓαi (αith order in xi-direction,i=1,…,ℓ) is defined as follows:
(3.10)∂RLx1α1⋯xlαlα1+⋯+αlu(x1,…,xl)=1∏i=1lΓ(mi-αi)⋅∫0xl⋯∫0x1(xl-ξl)ml-αl-1⋯(x1-ξ1)m1-α1-1∂m1+⋯+ml∂ξ1m1⋯∂ξlmlu(ξ1,…,ξl)dξ1⋯dξl,
where αi∈(mi-1,mi),mi∈Z+, i=1,…,ℓ.In the right sides of the above equalities of Definitions3.2–3.4, if the derivative values of the integrals do not relate to partial differential sequences, then the values of the left sides of the above equations do not either.One can also get(3.11)lim⁡αi→mi-C∂x1α1⋯xlαlα1+⋯+αlu(x1,…,xl)=C∂x1α1⋯xi-1αi-1xi+1αi+1⋯xlαlα1+⋯+αi-1+αi+1+⋯+αl∂mi∂ximiu(x1,…,xl),lim⁡αi→(mi-1)+C∂x1α1⋯xlαlα1+⋯+αlu(x1,…,xl)=C∂x1α1⋯xi-1αi-1xi+1αi+1⋯xlαlα1+⋯+αi-1+αi+1+⋯+αl∂mi-1u(x1,…,xl)∂ximi-1-C∂x1α1⋯xi-1αi-1xi+1αi+1⋯xlαlα1+⋯+αi-1+αi+1+⋯+αl∂mi-1u(x1,…,xi-1,0,xi+1,…,xl)∂ximi-1.By the way, the partial (RL) fractional is also defined here.Definition 3.5.
The partial (RL) integral with order∑i=1ℓαi (αi th order in xi-direction,  i=1,…,ℓ) is defined as follows:
(3.12)∂RLx1α1⋯xlαl-(α1+⋯+αl)u(x1,…,xl)=1∏i=1lΓ(αi)⋅∫0xl⋯∫0x1(xl-ξl)αl-1⋯(x1-ξ1)α1-1udξ1⋯dξl,
where αi∈R+,i=1,…,ℓ.Example 3.6.
Letu=u(x1,…,xℓ)=x1⋯xℓ.
(1) By simple calculation, one hasRL∂x1α1⋯xℓαℓα1+⋯+αℓu=∏i=1ℓxi1-αi/Γ(2-αi), in which αi>0, i=1,2,…,ℓ. If there exists an (2≤)αi∈Z+, then RL∂x1α1⋯xℓαℓα1+⋯+αℓu=0 because Γ(-k)=∞, k=0,1,2,…. This coincides with the property of classical derivative.
(2) By almost the same calculation, one hasC∂x1α1⋯xℓαℓα1+⋯+αℓu=∏i=1ℓ(xi1-αi/Γ(2-αi)), in which 1≥αi>0,i=1,2,…,ℓ. For same αi's values, C∂x1α1⋯xℓαℓα1+⋯+αℓu=∂RLx1α1⋯xℓαℓα1+⋯+αℓu due to the zero initial value condition. If there exists an i such that αi>1, then C∂x1α1⋯xℓαℓα1+⋯+αℓu=0.
(3) One also has∂x1α1⋯xℓαℓ-(α1+⋯+αℓ)u=∏i=1ℓ(xi1+αi/Γ(2+αi)), in which αi>0,i=1,2,…,ℓ.


## 4. Conclusions

In this paper, we further studied the important properties of the RL derivatives. We also discussed some properties of the Caputo derivative which have not been studied elsewhere. And we generalized the fractional derivative defined in the real line to the partial fractional derivatives in higher space dimensions. How to generalize the fractional derivatives in the real plane to those in the complex plane is our future work.



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*Source: 562494-2011-03-24.xml*