On Riemann-Liouville and Caputo Derivatives

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.

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. 𝑌_𝛼, the convolution kernel of order𝛼∈𝑅⁺for fractional integrals, is defined by 𝑌_𝛼(𝑡)=𝑡₊^𝛼−1Γ(𝛼)∈𝐿¹_loc𝑅⁺, (1.1) where Γ is the well-known Euler Gamma function and 𝑡₊^𝛼−1=𝑡^𝛼−1,𝑡>0,0,𝑡≤0. (1.2)

Definition 1.2. The fractional integral (or the Riemann-Liouville integral)𝐷^−𝛼_0,𝑡with fractional order𝛼∈𝑅⁺of function𝑥(𝑡)is defined as 𝐷^−𝛼_0,𝑡𝑥(𝑡)=𝑌_𝛼(𝑡)=1∗𝑥Γ(𝛼)^𝑡₀(𝑡−𝜏)^𝛼−1𝑥(𝜏)𝑑𝜏. (1.3)

𝑌_𝛼 has an important convolution property (or semigroup property), that is, 𝑌_𝛼∗𝑌_𝛽=𝑌_𝛼+𝛽 for arbitrary 𝛼>0 and 𝛽>0. This implies that 𝐷^−𝛼_0,𝑡⋅𝐷^−𝛽_0,𝑡=𝐷^{−𝛼−𝛽}_0,𝑡.

Definition 1.3. The Grünwald-Letnikov fractional derivative with fractional order𝛼is defined by, if𝑥(𝑡)∈𝐶^𝑚[0,𝑡], _GL𝐷^𝛼_0,𝑡𝑥(𝑡)=^𝑚−1_𝑘=0𝑥^(𝑘)(0)𝑡^−𝛼+𝑘Γ()+1−𝛼+𝑘+1Γ(𝑚−𝛼)^𝑡₀(𝑡−𝜏)^{𝑚−𝛼−1}𝑥^(𝑚)(𝜏)𝑑𝜏, (1.4) where 𝑚−1≤𝛼<𝑚∈𝑍⁺.

This is not the original definition. The initial definition is given by a limit, that is, _GL𝐷^𝛼_0,𝑡𝑥(𝑡)=lim_{ℎ→0,𝑛ℎ=𝑡}ℎ^𝑛−𝛼_𝑘=0(−1)^𝑘𝑝𝑘𝑥(𝑡−𝑘ℎ). (1.5) 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 _RL𝐷^𝛼_0,𝑡𝑥(𝑡)=𝑑^𝑚𝑑𝑡^𝑚𝐷^{−(𝑚−𝛼)}_0,𝑡𝑥(𝑡)=1Γ()𝑑𝑚−𝛼𝑑𝑡^𝑚𝑚𝑡₀(𝑡−𝜏)^{𝑚−𝛼−1}𝑥(𝜏)𝑑𝜏, (1.6) where 𝑚−1≤𝛼<𝑚∈𝑍⁺.

From Definitions 1.3 and 1.4, one can see that _RL𝐷^𝛼_0,𝑡𝑥(𝑡)=_GL𝐷^𝛼_0,𝑡𝑥(𝑡) if 𝑥(𝑡)∈𝐶^𝑚[0,𝑡] which can be verified via integration by parts. This fact and the original definition of _GL𝐷^𝛼_0,𝑡 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 _𝑅𝐷^𝛼𝑥(𝑡)1=−()Γ()⋅𝑑2cos𝜋𝛼/2𝑚−𝛼_𝑚𝑑𝑡^𝑚𝑡_−∞(𝑡−𝜏)^{𝑚−𝛼−1}𝑥(𝜏)𝑑𝜏+(−1)^𝑚∞_𝑡(𝑡−𝜏)^{𝑚−𝛼−1}𝑥(𝜏)𝑑𝜏, (1.7) 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 _𝐶𝐷^𝛼_0,𝑡𝑥(𝑡)=𝐷^{−(𝑚−𝛼)}_0,𝑡𝑑^𝑚𝑑𝑡^𝑚𝑥(𝑡)=1Γ(𝑚−𝛼)^𝑡₀(𝑡−𝜏)^{𝑚−𝛼−1}𝑥^(𝑚)(𝜏)𝑑𝜏, (1.8) in which 𝑚−1<𝛼<𝑚∈𝑍⁺.

From this definition, one can see that _𝐶𝐷^𝛼_0,𝑡𝑥^(𝑛)(𝑡)=_𝐶𝐷^𝛼+𝑛_0,𝑡𝑥(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 is_𝐺𝐷^𝛼_0,𝑡𝑥(𝑡)=𝑌_−𝛼∗𝑥(𝑡).

From this definition and the semigroup property of 𝑌_𝛼, one has _𝐺𝐷^𝛼_0,𝑡⋅_𝐺𝐷^𝛽_0,𝑡=_𝐺𝐷^𝛼+𝛽_0,𝑡, 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 equation _𝐶𝐷^𝛼_0,𝑡𝑥(𝑡)=𝑓(𝑡,𝑥) with 𝛼∈(0,1), 𝑡>0 is posed as 𝑥(0)=𝑥₀. But for the fractional differential equation _RL𝐷^𝛼_0,𝑡𝑥(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 [_RL𝐷^𝛼−1_0,𝑡𝑥(𝑡)]_𝑡=0=𝑥₀ (if 𝛼∈(1,2), then its initial value conditions are given as [_RL𝐷^𝛼−2_0,𝑡𝑥(𝑡)]_𝑡=0=𝑥₀, [_RL𝐷^𝛼−1_0,𝑡𝑥(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⁺,lim_{()𝛼→𝑚−1+𝐶}𝐷^𝛼_0,𝑡𝑥(𝑡)=𝑥^(𝑚−1)(𝑡)−𝑥^(𝑚−1)(0),lim_{𝛼→𝑚+𝐶}𝐷^𝛼_0,𝑡𝑥(𝑡)=𝑥^𝑚(𝑡), (1.9) 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⁺: lim_{()𝛼→𝑚−1+RL}𝐷^𝛼_0,𝑡𝑥(𝑡)=𝑥^(𝑚−1)(𝑡),lim_{𝛼→𝑚+RL}𝐷^𝛼_0,𝑡𝑥(𝑡)=𝑥^(𝑚)(𝑡). (1.10) Furthermore, fractional-order initial value condition(s) for RL-type differential equation can be given as usual. For example, the initial value condition [_RL𝐷^𝛼−1_0,𝑡𝑥(𝑡)]_𝑡=0=𝑥₀ for equation _RL𝐷^𝛼_0,𝑡𝑥(𝑡)=𝑓(𝑡,𝑥) with 𝛼∈(0,1), 𝑡>0 can be replaced by [𝑡^1−𝛼𝑥(𝑡)]_𝑡=0=𝑥₀/Γ(𝛼) [20]. Of course, for 1<𝛼∈(𝑚−1,𝑚), 𝑚∈𝑍⁺, we can use the formula _𝐶𝐷^𝛼_0,𝑡𝑥(𝑡)=_RL𝐷^𝛼_0,𝑡(𝑥(𝑡)−∑^𝑚−1_𝑘=0𝑡^𝑘𝑥^(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)For 𝛼∈(0,1), 𝑥(𝑡)∈𝐿¹_loc(𝑅⁺),_RL𝐷^𝛼_0,𝑡𝑥(𝑡)=_𝐺𝐷^𝛼_0,𝑡𝑥(𝑡)−𝛿(𝑡)_RL𝐷^𝛼−1_0,𝑡𝑥(𝑡)_𝑡=0=𝑑𝑑𝑡^𝑡₀𝑌_1−𝛼(𝑡−𝜏)𝑥(𝜏)𝑑𝜏. (2.1)(2)For 𝑛−1<𝛼<𝑛∈𝑍⁺, 𝑥(𝑡)∈𝐿¹_loc(𝑅⁺), _RL𝐷^𝛼_0,𝑡𝑥(𝑡)=_𝐺𝐷^𝛼_0,𝑡𝑥(𝑡)−𝛿(𝑡)_RL𝐷^𝛼−1_0,𝑡𝑥(𝑡)_𝑡=0−^𝑛−1_𝑘=1𝑌_−𝑘RL𝐷^{𝛼−𝑘−1}_0,𝑡𝑥(𝑡)_𝑡=0=𝑑^𝑛𝑑𝑡^𝑛𝑡₀𝑌_𝑛−𝛼(𝑡−𝜏)𝑥(𝜏)𝑑𝜏. (2.2)(3)Assume 𝛼>0, then (𝑑^𝑛/𝑑𝑡^𝑛)_RL𝐷^𝛼_0,𝑡𝑥(𝑡)=_RL𝐷^𝑛+𝛼_0,𝑡𝑥(𝑡); (𝑑^𝑛/𝑑𝑡^𝑛)_RL𝐷^−𝛼_0,𝑡𝑥(𝑡)=_RL𝐷^𝑛−𝛼_0,𝑡𝑥(𝑡) if 𝑛−𝛼>0, and (𝑑^𝑛/𝑑𝑡^𝑛)_RL𝐷^−𝛼_0,𝑡𝑥(𝑡)=𝐷^𝑛−𝛼_0,𝑡𝑥(𝑡) if 𝑛−𝛼<0, hold for any 𝑛∈𝑍⁺.(4)_RL𝐷^𝛼_0,𝑡⋅𝐷^−𝛼_0,𝑡𝑥(𝑡)=𝑥(𝑡) for all 𝛼>0. More generally, _RL𝐷^𝛼_0,𝑡⋅𝐷^−𝛽_0,𝑡𝑥(𝑡)=_RL𝐷^𝛼−𝛽_0,𝑡𝑥(𝑡) for all 𝛽>0. If 𝛼<𝛽, then _RL𝐷^𝛼−𝛽_0,𝑡𝑥(𝑡)=𝐷^𝛼−𝛽_0,𝑡𝑥(𝑡).(5)𝐷^−𝛼_0,𝑡⋅_RL𝐷^𝛼_0,𝑡𝑥(𝑡)=𝑥(𝑡)−∑^𝑛_𝑘=1(𝑡^𝛼−𝑘[_RL𝐷^𝛼−𝑘_0,𝑡𝑥(𝑡)]_𝑡=0/Γ(𝛼−𝑘+1)), where 𝑛−1<𝛼<𝑛∈𝑍⁺.(6)𝐷^−𝑛_0,𝑡⋅_RL𝐷^𝑛_0,𝑡𝑥(𝑡)=𝑥(𝑡)−∑^𝑛−1_𝑘=0(𝑡^𝑘/𝑘!)𝑥^(𝑘)(0).(7)_RL𝐷^𝛼_0,𝑡𝑐=(𝑡^−𝛼/Γ(1−𝛼))𝑐, where 𝛼>0 and 𝑐 is an arbitrary constant.(8)ℒ[_RL𝐷^𝛼_0,𝑡𝑥(𝑡)]=𝑠^𝛼𝑋(𝑠)−∑^𝑛−1_𝑘=0𝑠^𝑘[_RL𝐷^{𝛼−𝑘−1}_0,𝑡𝑥(𝑡)]_𝑡=0, in which ℒ is the Laplace transform, and ℒ[𝑥(𝑡)]=𝑋(𝑠).(9)_𝐶𝐷^𝛼_0,𝑡𝑥(𝑡)=_RL𝐷^𝛼_0,𝑡(𝑥(𝑡)−∑^𝑛−1_𝑘=0(𝑡^𝑘/𝑘!)𝑥^(𝑘)(0)), where 𝑛−1<𝛼<𝑛∈𝑍⁺.

For Caputo derivative, _𝐶𝐷^𝛼_0,𝑡⋅𝐷^−𝛽_0,𝑡=_𝐶𝐷^𝛼−𝛽_0,𝑡 generally does not hold for all 𝛼, 𝛽>0.

From (4) in Property 1, one has very interesting conclusions as follows.

Conclusion. If 𝑥(𝑡) is defined in the interval [𝑎,𝑏] and (1/Γ(𝛼))∫^𝑡_𝑎(𝑡−𝜏)^𝛼−1𝑥(𝜏)𝑑𝜏=0 for 𝛼>0 and for all 𝑡∈[𝑎,𝑏], then 𝑥(𝑡)≡0.

Proof. The condition implies that 𝐷^−𝛼_0,𝑡𝑥(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_𝐶𝐷^𝛼_0,𝑡𝑥(𝑡)(𝑥)()𝑥(0)=𝑓,𝛼∈0,1,𝑥∈𝑅,=𝑥₀, (2.3) 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]:𝑥(𝑡)=𝑥₀+𝐷^−𝛼_0,𝑡𝑓(𝑥(𝑡)). (2.4) 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: _𝐶𝐷^𝛼_0,𝑡𝑥(𝑡)=𝑥(𝑡)+^+∞_𝑘=0(−1)^𝑘𝑡^2𝑘+1(2𝑘+1)!!Γ()𝑡2𝑘+2−𝛼^−𝛼+1()(),𝑥(0)2𝑘+1!!,𝛼∈0,1=0, (2.5) 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]:_RL𝐷^𝛼_0,𝑡𝑥(𝑡)=𝑓(𝑥),𝛼∈(0,1),𝑥∈𝑅,𝑡>0,_RL𝐷^𝛼−1_0,𝑡𝑥(𝑡)_𝑡=0=𝑥₀or𝑡^1−𝛼𝑥(𝑡)_𝑡=0=𝑥₀Γ(𝛼). (2.6)

Conclusion. Assume that 𝑓(𝑥) is continuous, 𝑥 is a function of 𝑡>0 and that lim_𝑡→0⁺ is not bounded, but lim_𝑡→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]: 𝑥(𝑡)=𝑥₀Γ(𝛼)𝑡^𝛼−1+𝐷^−𝛼_0,𝑡𝑓(𝑥(𝑡)). (2.7)
If lim_𝑡→0⁺ is bounded, then the case is trivial so it is omitted here. We only show interests in the case that lim_𝑡→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: for 𝛼>0, 𝛽>0, then _RL𝐷^𝛼−𝛽_0,𝑡=_RL𝐷^𝛼_0,𝑡⋅𝐷^−𝛽_0,𝑡≠𝐷^−𝛽_0,𝑡⋅_RL𝐷^𝛼_0,𝑡.(2)Composition with the integer derivative operator: for 𝛼∈(𝑛−1,𝑛), 𝑛∈𝑍⁺, 𝑚∈𝑍⁺, then (𝑑^𝑚/𝑑𝑡^𝑚)⋅_RL𝐷^𝛼_0,𝑡=_RL𝐷^𝛼+𝑚_0,𝑡≠_RL𝐷^𝛼_0,𝑡⋅(𝑑^𝑚/𝑑𝑡^𝑚).(3)Composition with Caputo operator: for 𝛼∈(𝑛−1,𝑛), 𝑛∈𝑍⁺, (𝛼≠)𝛽∈(𝑚−1,𝑚), 𝑚∈𝑍⁺, then _RL𝐷^{𝛼+𝛽−𝑚}_0,𝑡(𝑑^𝑚/𝑑𝑡^𝑚)=_RL𝐷^𝛼_0,𝑡⋅_𝐶𝐷^𝛽_0,𝑡≠_𝐶𝐷^𝛽_0,𝑡⋅_RL𝐷^𝛼_0,𝑡=𝐷^{−(𝑚−𝛽)}_0,𝑡⋅_RL𝐷^𝛼+𝑚_0,𝑡.(4)Composition with the generalized fractional derivative operator: for 𝛼∈(𝑛−1,𝑛), 𝑛∈𝑍⁺, 𝛽>0, then (𝑑^𝑛/𝑑𝑡^𝑛)[𝑌_{𝑛−𝛼−𝛽}∗]=_RL𝐷^𝛼_0,𝑡⋅_𝐺𝐷^𝛽_0,𝑡≠_𝐺𝐷^𝛽_0,𝑡⋅_RL𝐷^𝛼_0,𝑡=𝑌_−𝛽∗_RL𝐷^𝛼_0,𝑡.

Proof. (1) Can be regarded as the direction conclusion of Property 1(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 Property 1(2) and corresponding definitions.

Although the Riemann-Liouville integral operator 𝐷^−𝛼_0,𝑡(𝛼∈𝑅⁺) has the semigroup property, that is, 𝐷^−𝛼_0,𝑡⋅𝐷^−𝛽_0,𝑡=𝐷^{−𝛼−𝛽}_0,𝑡 (𝛼>0, 𝛽>0), RL derivative operator _RL𝐷^𝛼_0,𝑡 does not have this character, that is, _RL𝐷^𝛼_0,𝑡⋅_RL𝐷^𝛽_0,𝑡≠_RL𝐷^𝛼+𝛽_0,𝑡 and _RL𝐷^𝛽_0,𝑡⋅_RL𝐷^𝛼_0,𝑡≠_RL𝐷^𝛼+𝛽_0,𝑡 [14]. However, we have following interesting result.

Property 3. If 𝑥(𝑡)∈𝐶¹[0,𝑇], 𝛼_𝑖∈(0,1)(𝑖=1,2) (the trivial case 𝛼_𝑖=0  or   1 is simple and removed here), and 𝛼₁+𝛼₂∈(0,1], then _RL𝐷^𝛼1_0,𝑡⋅_RL𝐷^𝛼2_0,𝑡𝑥(𝑡)=_RL𝐷^𝛼1+𝛼2_0,𝑡𝑥(𝑡).

Proof. According to Property 1(3), one gets_𝐶𝐷^𝛼2_0,𝑡⋅_𝐶𝐷^𝛼1_0,𝑡𝑥(𝑡)=_𝐶𝐷^𝛼2_0,𝑡RL𝐷^𝛼1_0,𝑡[𝑥(𝑡)(0=−𝑥)]_RL𝐷^𝛼2_0,𝑡RL𝐷^𝛼1_0,𝑡[𝑥(𝑡)−𝑥(0)]−_𝐶𝐷^𝛼1_0,𝑡𝑥(𝑡)_𝑡=0=_RL𝐷^𝛼2_0,𝑡RL𝐷^𝛼1_0,𝑡[𝑥(𝑡)(0=−𝑥)]_RL𝐷^𝛼2_0,𝑡⋅_RL𝐷^𝛼1_0,𝑡𝑥(𝑡)−𝑡^{−𝛼1−𝛼2}Γ1−𝛼₁−𝛼₂𝑥(0). (2.8) Similarly, _𝐶𝐷^𝛼2_0,𝑡⋅_𝐶𝐷^𝛼1_0,𝑡𝑥(𝑡)=_RL𝐷^𝛼2_0,𝑡⋅_RL𝐷^𝛼1_0,𝑡𝑥(𝑡)−𝑡^{−𝛼1−𝛼2}𝑥(0)Γ1−𝛼₁−𝛼₂. (2.9)
On the other hand,_𝐶𝐷^𝛼1+𝛼2_0,𝑡𝑥(𝑡)=_RL𝐷^𝛼1+𝛼2_0,𝑡𝑥(𝑡)−𝑡^{−𝛼1−𝛼2}Γ1−𝛼₁−𝛼₂𝑥(0). (2.10)
If 𝛼₁+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 _RL𝐷^𝛼2_0,𝑡⋅_RL𝐷^𝛼1_0,𝑡𝑥(𝑡)=_RL𝐷^𝛼1_0,𝑡⋅_RL𝐷^𝛼2_0,𝑡𝑥(𝑡)=_RL𝐷^𝛼1+𝛼2_0,𝑡𝑥(𝑡).
The following result is for comparison nature of fractional derivatives.

Property 4. (1) If 𝛼∈(𝑛−1,𝑛), 𝑛∈𝑍⁺, and 𝑥^(𝑘)(0)≥0(𝑘=0,1,…,𝑛−1), then _𝐶𝐷^𝛼_0,𝑡≥_RL𝐷^𝛼_0,𝑡.
(2) If 𝛼∈(𝑛−1,𝑛), 𝑛∈𝑍⁺, _RL𝐷^𝛼_0,𝑡𝑥(𝑡)≥_RL𝐷^𝛼_0,𝑡𝑦(𝑡), and [_RL𝐷^{𝛼−𝑘−1}_0,𝑡𝑥(𝑡)]_𝑡=0≥[_RL𝐷^{𝛼−𝑘−1}_0,𝑡𝑥(𝑡)]_𝑡=0  (𝑘=0,1,…,𝑛−1), then 𝑥(𝑡)≥𝑦(𝑡). Parallelly, if 𝛼∈(𝑛−1,𝑛), 𝑛∈𝑍⁺, _𝐶𝐷^𝛼_0,𝑡𝑥(𝑡)≥_𝐶𝐷^𝛼_0,𝑡𝑦(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 Property 1(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.
Setting _RL𝐷^𝛼_0,𝑡𝑥(𝑡)=𝜉(𝑡)+_RL𝐷^𝛼_0,𝑡𝑦(𝑡) and taking the Laplace transform in both sides, one has𝑠^𝛼𝑋(𝑠)=𝑠^𝛼𝑌(𝑠)𝜉(𝑡++ℒ)_{𝑛−1𝑘=0}𝑠^𝑘_RL𝐷^{𝛼−𝑘−1}_0,𝑡𝑥(𝑡)_𝑡=0−_RL𝐷^{𝛼−𝑘−1}_0,𝑡𝑦(𝑡)_𝑡=0. (2.11) It immediately follows from dividing by 𝑠^𝛼 and taking the inverse Laplace transform in both sides that 𝑥(𝑡)=𝑦(𝑡)+𝐷^−𝛼_0,𝑡𝜉(𝑡)+^𝑛−1_𝑘=0RL𝐷^{𝛼−𝑘−1}_0,𝑡𝑥(𝑡)_𝑡=0−_RL𝐷^{𝛼−𝑘−1}_0,𝑡𝑦(𝑡)_𝑡=0𝑌_𝛼−𝑘. (2.12) The last two addends in the right side of the above equality are nonnegative. This completes the proof.

Property 5. Let 𝒜={𝑥(𝑡)∈𝑅, 𝑡≥0, 𝑥(𝑡) is analytical for any 𝑡≥0}. If 𝛼∈(0,1), then RL derivative operator _RL𝐷^𝛼_0,𝑡 defined in 𝒜 can be expressed as_RL𝐷^𝛼_0,𝑡=^∞_𝑘=0𝑑^𝑘⋅/𝑑𝑡^𝑘_𝑡=0Γ()𝑡𝑘−𝛼+1^𝑘−𝛼. (2.13) More generally, if 𝑛−1<𝛼<𝑛∈𝑍⁺, then _𝑅𝐿𝐷^𝛼_0,𝑡 defined in 𝒜 has also the following form: _RL𝐷^𝛼_0,𝑡=^∞_𝑘=0𝑑^𝑘⋅/𝑑𝑡^𝑘_𝑡=0Γ()𝑡𝑘−𝛼+1^𝑘−𝛼. (2.14)

The proof is easy so it is left out here.

Remark 2.1. (1) For an arbitrary function 𝑥(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 if 𝑥(𝑡)∈𝒜 (it implies that 𝑥(0) exists), [_RL𝐷^𝛼_0,𝑡𝑥(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. Consider𝑥(𝑡)=1−𝑡,0<𝑡≤1,𝑡−1,1<𝑡<1+𝜀,𝜀>0. (2.15)
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_RL𝐷^𝛼_0,𝑡𝑥(𝑡)=𝑡^−𝛼Γ()−𝑡1−𝛼Γ(^1−𝛼)𝑡2−𝛼,0<𝑡≤1,_−𝛼Γ()−𝑡1−𝛼^1−𝛼−2(𝑡−1)^1−𝛼Γ()(),2−𝛼,𝑡∈1,1+𝜀 (2.16) 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, _RL𝐷^𝛼_0,𝑡𝑥(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 _RL𝐷^𝛼_0,𝑡𝑥(𝑡)>0 if 𝑡<1−𝛼, but 𝑥(𝑡) is not monotonously increasing for 𝑡∈(0,1−𝛼). The RL derivative _RL𝐷^𝛼_0,𝑡𝑥(𝑡)>0 only means that 𝐷^{−(1−𝛼)}_0,𝑡𝑥(𝑡) is monotonously increasing with respect to 𝑡 but does not imply that 𝑥(𝑡) is monotonously increasing. Geometrically speaking, the value _𝑅𝐿𝐷^𝛼_0,𝑡𝑥(𝑡) at point 𝑡 relates to an “area." On the other hand, its Caputo derivative exists in the whole interval (0,1+𝜀), 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.

Definition 1.4 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 _RL𝐷^𝛼_𝑡,𝑏𝑥(𝑡)=(−1)^𝑚Γ()𝑑𝑚−𝛼𝑑𝑡^𝑚𝑚𝑏_𝑡(𝜏−𝑡)^{𝑚−𝛼−1}𝑥(𝜏)𝑑𝜏, (2.17) in which 𝑡∈(0,𝑏).

The Riesz fractional derivative (Definition 1.5) 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.

Suppose 𝛼_𝑖∈(0,1), 𝑖=1,2, 𝛼=𝛼₁+𝛼₂. If we define _RL𝜕^𝛼1+𝛼2_{𝑥𝛼11𝑥𝛼22}𝑢𝑥₁,𝑥₂=𝜕^𝛼2𝜕𝑥^𝛼2₂𝜕^𝛼1𝜕𝑥^𝛼1₁𝑢𝑥₁,𝑥₂=𝜕^𝛼2𝜕𝑥^𝛼2₂1Γ1−𝛼₁𝜕𝜕𝑥₁^𝑥1₀𝑥₁−𝜉^−𝛼1𝑢𝜉,𝑥₂=1𝑑𝜉Γ1−𝛼₂𝜕𝜕𝑥₂^𝑥2₀(𝑥₂−𝜏)^−𝛼2Γ1−𝛼₁𝜕𝜕𝑥₁^𝑥1₀𝑥₁−𝜉^−𝛼1𝑢(𝜉,𝜏)𝑑𝜉𝑑𝜏, (3.1) then _RL𝜕^𝛼1+𝛼2_{𝑥𝛼11𝑥𝛼22}𝑢𝑥₁,𝑥₂=1Γ1−𝛼₁⋅Γ1−𝛼₂𝜕²𝜕𝑥₁𝜕𝑥₂^𝑥2₀^𝑥1₀𝑥₂−𝜏^−𝛼2𝑥₁−𝜉^−𝛼1𝑢(𝜉,𝜏)𝑑𝜉𝑑𝜏. (3.2)

According to the classical calculus, if 𝜕²𝜕𝑥₁𝜕𝑥₂^𝑥2₀^𝑥1₀𝑥₂−𝜏^−𝛼2𝑥₁−𝜉^−𝛼1𝑢()𝜕𝜉,𝜏𝑑𝜉𝑑𝜏,₂𝜕𝑥₂𝜕𝑥₁^𝑥1₀^𝑥2₀𝑥₂−𝜏^−𝛼2𝑥₁−𝜉^−𝛼1𝑢(𝜉,𝜏)𝑑𝜏𝑑𝜉 (3.3) exist in a neighborhood of (𝑥₁,𝑥₂) and are continuous at this point (𝑥₁,𝑥₂), then _RL𝜕^𝛼1+𝛼2_{𝑥𝛼11𝑥𝛼22}𝑢𝑥₁,𝑥₂=_RL𝜕^𝛼1+𝛼2_{𝑥𝛼22𝑥𝛼11}𝑢𝑥₁,𝑥₂. (3.4) If 𝛼=𝛼₁+𝛼₂∈(0,1), then the above partial RL derivative can characterize subdiffusion in 𝑅².

The case with 𝛼₁=0 or 𝛼₂=0 was simply mentioned in [7],_RL𝜕^𝛼2_𝑥𝛼22𝑢𝑥₁,𝑥₂=𝜕^𝛼2𝜕𝑥^𝛼2₂𝑢𝑥₁,𝑥₂=1Γ1−𝛼₂𝜕𝜕𝑥₂^𝑥2₀𝑥₂−𝜏^−𝛼2𝑢𝑥₁,𝜏𝑑𝜏,_RL𝜕^𝛼1_𝑥𝛼11𝑢𝑥₁,𝑥₂=𝜕^𝛼1𝜕𝑥^𝛼1₁𝑢𝑥₁,𝑥₂=1Γ1−𝛼₁𝜕𝜕𝑥₁^𝑥1₀𝑥₁−𝜉^−𝛼1𝑢𝜉,𝑥₂𝑑𝜉. (3.5) Now we give the definition of the partial RL derivative as follows.

Definition 3.1. The partial RL derivative with order 𝛼₁+𝛼₂ (𝛼₁th order in 𝑥₁-direction and𝛼₂th order in𝑥₂ direction) is defined as follows: _RL𝜕^𝛼1+𝛼2_{𝑥𝛼11𝑥𝛼22}𝑢𝑥₁,𝑥₂=1Γ𝑚−𝛼₁⋅Γ𝑛−𝛼₂𝜕^𝑚+𝑛𝜕𝑥^𝑚₁𝜕𝑥^𝑛₂⋅^𝑥2₀^𝑥1₀𝑥₂−𝜏^{𝑛−𝛼2−1}𝑥₁−𝜉^{𝑚−𝛼1−1}𝑢(𝜉,𝜏)𝑑𝜉𝑑𝜏, (3.6) 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∑^ℓ_𝑖=1𝛼_𝑖 (𝛼_𝑖th order in 𝑥_𝑖-direction,  𝑖=1,…,ℓ) is defined as follows: _RL𝜕^{𝛼1+⋯+𝛼ℓ}_{𝑥𝛼11⋯𝑥𝛼ℓℓ}𝑢𝑥₁,…,𝑥_ℓ=𝜕^𝑚1+⋯+𝑚_ℓ/𝜕𝑥^𝑚1₁⋯𝜕𝑥^𝑚ℓ_ℓ∏^ℓ_𝑖=1Γ𝑚_𝑖−𝛼_𝑖^𝑥ℓ0⋯^𝑥1₀(𝑥_ℓ−𝜉_ℓ)^{𝑚ℓ−𝛼ℓ−1}⋯𝑥₁−𝜉₁^{𝑚1−𝛼1−1}𝑢𝑑𝜉₁⋯𝑑𝜉_ℓ, (3.7) where 𝛼_𝑖∈(𝑚_𝑖−1,𝑚_𝑖), 𝑚_𝑖∈𝑍⁺, 𝑖=1,…,ℓ.

It is easy to show that (may refer to [9]) lim_{𝛼𝑖→𝑚−𝑖RL}𝜕^{𝛼1+⋯+𝛼ℓ}_{𝑥𝛼11⋯𝑥𝛼ℓℓ}𝑢𝑥₁,…,𝑥_ℓ=_RL𝜕^{𝛼1+⋯+𝛼𝑖−1+𝛼𝑖+1+⋯+𝛼ℓ}_{𝑥𝛼11⋯𝑥𝛼𝑖−1𝑖−1𝑥𝛼𝑖+1𝑖+1⋯𝑥𝛼ℓℓ}𝜕^𝑚𝑖𝜕𝑥^𝑚𝑖_𝑖𝑢𝑥₁,…,𝑥_ℓ,lim_{𝛼𝑖→𝑚𝑖−1+RL}𝜕^{𝛼1+⋯+𝛼ℓ}_{𝑥𝛼11⋯𝑥𝛼ℓℓ}𝑢𝑥₁,…,𝑥_ℓ=_RL𝜕^{𝛼1+⋯+𝛼𝑖−1+𝛼𝑖+1+⋯+𝛼ℓ}_{𝑥𝛼11⋯𝑥𝛼𝑖−1𝑖−1𝑥𝛼𝑖+1𝑖+1⋯𝑥𝛼ℓℓ}𝜕^𝑚𝑖−1𝑢𝑥₁,…,𝑥_ℓ𝜕𝑥^𝑚𝑖_𝑖⁻¹. (3.8)

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: _𝐶𝜕^𝛼1+𝛼2_{𝑥𝛼11𝑥𝛼22}𝑢𝑥₁,𝑥₂=1Γ𝑚−𝛼₁⋅Γ𝑛−𝛼₂⋅^𝑥2₀^𝑥1₀𝑥₂−𝜏^{𝑛−𝛼2−1}𝑥₁−𝜉^{𝑚−𝛼1−1}𝜕^𝑚+𝑛𝜕𝜉^𝑚𝜕𝜏^𝑛𝑢(𝜉,𝜏)𝑑𝜉𝑑𝜏, (3.9) 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∑^ℓ_𝑖=1𝛼_𝑖 (𝛼_𝑖th order in 𝑥_𝑖-direction,𝑖=1,…,ℓ) is defined as follows: _RL𝜕^{𝛼1+⋯+𝛼ℓ}_{𝑥𝛼11⋯𝑥𝛼ℓℓ}𝑢𝑥₁,…,𝑥_ℓ=1∏^ℓ_𝑖=1Γ𝑚_𝑖−𝛼_𝑖⋅^𝑥ℓ₀⋯^𝑥1₀𝑥_ℓ−𝜉_ℓ^{𝑚ℓ−𝛼ℓ−1}⋯(𝑥₁−𝜉₁)^{𝑚1−𝛼1−1}𝜕^𝑚1+⋯+𝑚_ℓ𝜕𝜉^𝑚1₁⋯𝜕𝜉^𝑚ℓ_ℓ𝑢𝜉₁,…,𝜉_ℓ𝑑𝜉₁⋯𝑑𝜉_ℓ, (3.10) where 𝛼_𝑖∈(𝑚_𝑖−1,𝑚_𝑖),𝑚_𝑖∈𝑍⁺, 𝑖=1,…,ℓ.

In the right sides of the above equalities of Definitions 3.2–3.4, 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 lim_{𝛼𝑖→𝑚−𝑖𝐶}𝜕^{𝛼1+⋯+𝛼ℓ}_{𝑥𝛼11⋯𝑥𝛼ℓℓ}𝑢𝑥₁,…,𝑥_ℓ=_𝐶𝜕^{𝛼1+⋯+𝛼𝑖−1+𝛼𝑖+1+⋯+𝛼ℓ}_{𝑥𝛼11⋯𝑥𝛼𝑖−1𝑖−1𝑥𝛼𝑖+1𝑖+1⋯𝑥𝛼ℓℓ}𝜕^𝑚𝑖𝜕𝑥^𝑚𝑖_𝑖𝑢𝑥₁,…,𝑥_ℓ,lim_{𝛼𝑖→𝑚𝑖−1+𝐶}𝜕^{𝛼1+⋯+𝛼ℓ}_{𝑥𝛼11⋯𝑥𝛼ℓℓ}𝑢𝑥₁,…,𝑥_ℓ=_𝐶𝜕^{𝛼1+⋯+𝛼𝑖−1+𝛼𝑖+1+⋯+𝛼ℓ}_{𝑥𝛼11⋯𝑥𝛼𝑖−1𝑖−1𝑥𝛼𝑖+1𝑖+1⋯𝑥𝛼ℓℓ}𝜕^𝑚𝑖−1𝑢𝑥₁,…,𝑥_ℓ𝜕𝑥^𝑚𝑖_𝑖⁻¹−_𝐶𝜕^{𝛼1+⋯+𝛼𝑖−1+𝛼𝑖+1+⋯+𝛼ℓ}_{𝑥𝛼11⋯𝑥𝛼𝑖−1𝑖−1𝑥𝛼𝑖+1𝑖+1⋯𝑥𝛼ℓℓ}𝜕^𝑚𝑖−1𝑢𝑥₁,…,𝑥_𝑖−1,0,𝑥_𝑖+1,…,𝑥_ℓ𝜕𝑥^𝑚𝑖_𝑖⁻¹. (3.11)

By the way, the partial (RL) fractional is also defined here.

Definition 3.5. The partial (RL) integral with order∑^ℓ_𝑖=1𝛼_𝑖 (𝛼_𝑖 th order in 𝑥_𝑖-direction,  𝑖=1,…,ℓ) is defined as follows: _RL𝜕^{−(𝛼1+⋯+𝛼ℓ)}_{𝑥𝛼11⋯𝑥𝛼ℓℓ}𝑢𝑥₁,…,𝑥_ℓ=1∏^ℓ_𝑖=1Γ𝛼_𝑖⋅^𝑥ℓ₀⋯^𝑥1₀𝑥_ℓ−𝜉_ℓ^𝛼ℓ−1⋯𝑥₁−𝜉₁^𝛼1−1𝑢𝑑𝜉₁⋯𝑑𝜉_ℓ, (3.12) where 𝛼_𝑖∈𝑅⁺,𝑖=1,…,ℓ.

Example 3.6. Let 𝑢=𝑢(𝑥₁,…,𝑥_ℓ)=𝑥₁⋯𝑥_ℓ.
(1) By simple calculation, one has _RL𝜕^{𝛼1+⋯+𝛼ℓ}_{𝑥𝛼11⋯𝑥𝛼ℓℓ}𝑢=∏^ℓ_𝑖=1𝑥^1−𝛼𝑖_𝑖/Γ(2−𝛼_𝑖), in which 𝛼_𝑖>0, 𝑖=1,2,…,ℓ. If there exists an (2≤)𝛼_𝑖∈𝑍⁺, then _RL𝜕^{𝛼1+⋯+𝛼ℓ}_{𝑥𝛼11⋯𝑥𝛼ℓℓ}𝑢=0 because Γ(−𝑘)=∞, 𝑘=0,1,2,…. This coincides with the property of classical derivative.
(2) By almost the same calculation, one has _𝐶𝜕^{𝛼1+⋯+𝛼ℓ}_{𝑥𝛼11⋯𝑥𝛼ℓℓ}𝑢=∏^ℓ_𝑖=1(𝑥^1−𝛼𝑖_𝑖/Γ(2−𝛼_𝑖)), in which 1≥𝛼_𝑖>0,𝑖=1,2,…,ℓ. For same 𝛼_𝑖's values, _𝐶𝜕^{𝛼1+⋯+𝛼ℓ}_{𝑥𝛼11⋯𝑥𝛼ℓℓ}𝑢=_RL𝜕^{𝛼1+⋯+𝛼ℓ}_{𝑥𝛼11⋯𝑥𝛼ℓℓ}𝑢 due to the zero initial value condition. If there exists an 𝑖 such that 𝛼_𝑖>1, then _𝐶𝜕^{𝛼1+⋯+𝛼ℓ}_{𝑥𝛼11⋯𝑥𝛼ℓℓ}𝑢=0.
(3) One also has 𝜕^{−(𝛼1+⋯+𝛼ℓ)}_{𝑥𝛼11⋯𝑥𝛼ℓℓ}𝑢=∏^ℓ_𝑖=1(𝑥^1+𝛼𝑖_𝑖/Γ(2+𝛼_𝑖)), in which 𝛼_𝑖>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.

Acknowledgments

The present work was supported in part by the National Natural Science Foundation of China under Grant no. 10872119 and the Key Disciplines of Shanghai Municipality under Grant no. S30104.