Fractional Versions of the Fundamental Theorem of Calculus

The concept of fractional integral in the Riemann-Liouville, Liouville, Weyl and Riesz sense is presented. Some properties involving the particular Riemann-Liouville integral are mentioned. By means of this concept we present the fractional derivatives, specifically, the Riemann-Liouville, Liouville, Caputo, Weyl and Riesz versions are discussed. The so-called fundamental theorem of fractional calculus is presented and discussed in all these different versions.


Introduction
Fractional calculus, a popular name used to denote the calculus of non integer order, is as old as the calculus of integer order as created independently by Newton and Leibniz.In contrast with the calculus of integer order, fractional calculus has been granted a specific area of mathematics only in 1974, after the first international congress dedicated exclusively to it.Before this congress there were only sporadic independent papers, without a consolidated line [1,2].
During the 1980s fractional calculus attracted researchers and explicit applications began to appear in several fields.We mention the doctoral thesis, published as an article [3], which seems to be the first one in the subject and the classical book by Miller and Ross [1], where one can see a timeline from 1645 to 1974.After the decade of 1990, completely consolidated, there appeared some specific journals and several textbooks were published.These facts lent a great visibility to the subject and it gained prestige around the world.An interesting timeline from 1645 to 2010 is presented in references [4][5][6].We recall here that an important advantage of using fractional differential equations in applications is their non-local property.The use of fractional calculus is more realistic and this is one reason why fractional calculus has become more popular.
Nowadays, fractional calculus can be considered a frontier area in mathematics in the sense that there is as much research on its applications as there is on the calculus of integer order.Several applications in all areas of knowledgement are collected, presented and discussed in different books as follow [7][8][9][10][11][12].
The main objective of this paper is to explain what is meant by calculus of non integer order and collect any different versions of the fractional derivatives associated with a particular fractional integral.Specifically, we recover the concepts of fractional integral and fractional derivative in different versions and present a new version of the so-called fundamental theorem of fractional calculus (FTFC), which is interpreted as a generalization of the classical fundamental theorem of calculus.We mention three recent works where FTFC is discussed, Tarasov's book [12], a paper by Tarasov [13] and a paper by Dannon [14] in which a particular case of the parameter associated with the derivative is presented.The paper is written as follows: in section two, we first review the concept of fractional integral in the Riemann-Liouville sense, which can be interpreted as a generalization of the integral of integer order and in the Liouville sense, which is a particular case of the Riemann-Liouville one.We review also the concept of fractional integral in the Weyl sense and in the Riesz sense.Section two present also the concepts of derivative as proposed by Riemann-Liouville, Liouville, Caputo, Weyl and Riesz, showing the real importance and applications.Some properties are also presented, among which one associated with the semigroup property.Our main result appears in section three, in which we present and demonstrate the many faces of the FTFC, in all different versions and which are interpreted as a generalization of the fundamental theorem of calculus.Applications are presented in section four.

Fractional Calculus
The integral and derivatives of non integer order have several applications and are used to solve problems in different fields of knowledge, specifically, involving a fractional differential equation with boundary value conditions and/or initial conditions [7,9,11,12].They can be seen as generalizations of the integral and derivatives of integer order.On the other hand, we mention two papers, by Heymans & Podlubny [15] and Podlubny [16] that provide an interesting geometric interpretation, and discuss applications of fractional calculus, with integral and derivatives of non integer order.Also, we mention a recent paper in which the authors discuss a fractional differential equation with integral boundary value conditions [17].We remember that, there are several ways to introduce the concepts of fractional integral and fractional derivatives, which are not necessarily coincident with each other [18].The so-called Grünwald-Letnikov derivative, which will be not discussed in this paper, is convenient and useful to affront problems involving a numerical treatment [19].
In this section, the concept of fractional integral in the Riemann-Liouville, Liouville, Weyl and Riesz sense is presented.Some properties involving the particular Riemann-Liouville integral are mentioned.By means of this concept we present the fractional derivatives, specifically, the Riemann-Liouville, Liouville, Caputo, Weyl and Riesz versions are discussed.

Fractional Integral of Riemann-Liouville
The fractional integral of Riemann-Liouville is an integral that generalizes the concept of integral in the classical sense, and which can be obtained as a generalization of the Cauchy-Riemann integral.As we have already said, before we define the fractional integral Riemann-Liouville.
The fractional integrals of Riemann-Liouville of order    with , on the left and on the right, are defined respectively by , where I is the identity operator.

Fractional Derivative of Riemann-Liouville
After we introduce the fractional integral in the Riemann-Liouville sense, we define the fractional derivative of Riemann-Liouville, which is the most used by mathematicians, particularly, into the problems in which the initial conditions are not involved.
(Riemann-Liouville derivative) Let    , with , where    denotes the integer part of  , the fractional derivatives in the Riemann-Liouville sense, on the left and on the right, are defined by and respectively.Note that the derivatives in Equations ( 1) and (2) exist

Fractional Integral and Derivative in the Liouville Sense
An interesting particular case of the fractional integral of Riemann-Liouville and the corresponding fractional derivative, is the so-called Liouville fractional integral and f x whose derivatives up to order 1 n  are absolutely continuous on .
the Liouville fractional derivative.This case is obtained by substitution a and in the expressions associated with the fractional integral in the Riemann-Liouville sense.

  b  
(Liouville integral and derivative) The fractional integrals in the Liouville sense on the real axis, on the left and on the right, for and , are defined by and respectively.
The corresponding fractional derivatives in the Liouville sense are given by and  

Fractional Integral and Derivative in the Weyl Sense
The operations involving the fractional integral and the fractional derivative, as above defined by means of the Riemann-Liouville operators are convenient for a function represented by a series of power but not for functions defined, for example, by means of Fourier series, because   f x is a periodic function with period 2 , the  I a   f x   cannot be periodic.For this reason, it is convenient to introduce the fractional integral and the fractional derivatives in the so-called Weyl sense.First, some remarks on the Fourier series are presented.Let

 
f x be a periodic function with period 2 , defined on the real axis, with null average value, i.e., representing the Fourier series of   f x where are the corresponding Fourier coefficients.Note that, by hypotesis, as the function has null average value, we have .
Equations ( 3) and (5 efined in ), respectively.If we define the Fourier series of   , in the same way, the frace Weyl tional integral in th sense on the right and the fractional derivative on the right, defined by al Derivative in the Caputo Sense he

Fraction
The differential operator of non integer order in t Caputo sense is similar to the differential operator of non integer order in the Riemann-Liouville sense.The capital difference is that: in the Caputo sense, the derivative acts first on the function, after we evaluate the integral and in the Riemann-Liouville sense, the derivatives acts on the integral, i.e., we first evaluate the integral and after we calculate the derivative.The derivative in the Caputo sense is more restritive than the Riemann-Liouville one.We also note that, both derivatives are defined by means of the Riemann-Liouvile fractional integral.The importance associated with this derivative is that, the derivative in the Caputo sense can be used, for example, in the case of a fractional differential equation with initial conditions which have a well known interpretation, as in the calculus of integer order [20,21]. and , with Re 0.
The so-called fractional integral of order  in the Riesz sense, denoted by   , which is also known by Riesz potential and is defined by the Fourier convolution product .
Fractional Integral in the Riesz Sense and the clidean First of all, we introduce the fractional integral corresponding fractional derivative in the Eu with

 
Re 0   , and space n  , but for our purpose we discuss specifically the case : is the Riesz kernel and   n   is defined in [9], as follows good" [9], with   , the fractional operator ciently (Riesz integral) As we have already said, we take in articular, , then p :  which is the Riesz fractional integral.We can also write Equation (8) in terms of the Liouville integral ient co s.For this end, we introduce a conven nvolution product, i.e., we rewrite Equation (8) in the following form with    


Applying the Fourier transform in both sides of Equation (9), we get where the functions   f  and   ĝ  are the Fourier transforms of the functions   f x and   g x , respectively.Thus, we can write, Then, rewriting Equation ( 8) in terms of the Liouville integrals, we get Finally, we can write the fractional integral in the Riesz sense, in terms of a sum of two Liouville integrals and    For the best of our knowledged this is a new result.
. ntroduced in problems that can be treat as a Fo volution product.In this section, we introduc tional derivative and express it in terms of the Liouville derivative.An example of a specific convolution product e proved.As definition but we are interested in a par- , is defined for n l  where are defined in [9].The deri s of the nsform is vative in term In the particular case, , we have : with 0 1 .
Thus, considering 0 1    , we Equation ( 12) in terms of the fractional derivative in the sense, as follows can write Liouville with 0 1 In what follow we express the Riesz derivative   f x in terms of a tion product.Using Equation (13) we get Thus, the convenient convolution product is where (14) ng the Fourier transform in both sides of Equa-, we obtain the Fourier transform of the function with 0 1    .Using this result we prove a theorem involving the Fourier convolution of two particular functions.
Theorem 1 The Fourier convolution product of the functions   Thus, the Fourier transform of the convolution product can be written as To recover the convolution product, we apply the corresponding inverse Fourier transform in both sides of the last equation, and we get in Equation ( 16) is complex, because 0 1    .

The Fundamental Theorem of Fractional
tion of different versions of the fractional integral operator and the corresponding fractional derivative it is natural to introduce the co onding FTFC associated with these different versions.Then, we pr he proof, otherwise, we present the proof.As we have already said, in all cases we first write the theorem in general form, consider a particular case and finally, we recover, as a convenient limit, the fundamental theorem in the corresponding classical version.

Calculus
After the presenta rresp esent in this section the so-called FTFC, in the Riemann-Liouville, Caputo, Liouville, Weyl and Riesz versions.The results that are known we mention the reference where one can see t such that and in the case and for     The Equations ( 17) and ( 18) follow from [11].In the pa substitute rticular case 1In the case 7) and (18). 1 We will now show that the Theorem 2 in which we consider the fractional operator in the uto s .
Proof. ( llows fro in [9].follows fro ma 2.22 i then, for x   , we have: Proof.(1) Using Part (1) of the Theorem 2, follows (2) Using Part (2) of the Theorem 2, we have that: if     In the same way, we have   where the so-ca e space of Lizorkin functions, d in [9], and let 0    .
Proof.(1) See Proper [9].(2) For ty 2.35 in 0 1    , , using Eq 9) we have uation ( , and . us, using Equation (1 get with   Considering 0 1    , and Equation 10) and Equation ( 13), we can write using ( Using the Theorem 1, we tain . means of the The rem 4, for 0 1 On and, us the other h ing Theorem 6, for 0 1 . In this case, we can rite, ve or in the following form, , we get in the same way, that

Applications
In this section, using the FTFC, fractional differential equations are solved, one of them associated with the Riemann-Liouville case and the other involving the case.
The next application, we discuss the same problem which has been discussed by Jafari & Momani [22] usin another methodology, the so-called modified homotoy perturbation method.We solve the equation using the variables and the FTFC (Riemann-Liouville).
Example 2 Consider the initial value problem involving the fractional diffusion equation g method of separation of where x x x  for and x , with , Suppose a solution with the form tion ( 22) into the fractional diffusion 21), we get Substituting Equa equation, Equation ( where  is a real constant.
We first consider the fractional equation Substituting Equation (7) into Equation ( 23), we get an equivalent equation on both sides of the last equation we have , e E 3 .
We note that, in the paper by Jafari & Momani [22] its solution is presented with a misprint, ., as can be verified t a s at the & Momani [22] is different from ours because it solution is not a solution of Equation (21).
As a particular case, we consider the problem associat ion, that is i.e his solution is not olution of Equation ( 21).We remark, in passing, th solution presented in the paper by Jafari and derivative) We define the fractional integral and the respective fractional derivatives in the W the Caputo sense, are defined in terms of the f onal integral operator of Riemann-Liouville as racti

2 7 .
Fractional Derivative in the Riesz SenseThe fractional derivative in the Riesz sense has been i urier cone this fracwill bwe have already mentioned, we present the general ticular case involving the parameter.(Riesz derivative) The fractional derivative of   f x in the Riesz sense, with n   x so e problem, i.e., the fractional diffusion equation and the initial condition, After a brief introduction about ger order, popularly known as fractional calculus, we the Liouville sense.We then discussed the formulation of ed by Riemann-Liouville and, interchanging the integral tive, we introduced the formulation proposed by Caputo.fraction any l equations.A natural continuation of this work resides in the fact that we can obtai sociated with fractional differential equations invol al ic is as in Figure1.thecalculus of non-intepresented concept of fractional integral in the Riemannfractional derivatives as introduc with the deriva-We presented also the al integral and fractional derivatives in the Liouville, Weyl and Riesz sense.As our main result, we colleted and showed the m faces of the FTFC, associated wit Ri h the emann-Liouville, Caputo, Liouville, Weyl and Riesz version.As applications, we discussed two examples involving fractional differentia n solutions asving so fractional derivatives as proposed by Riesz and Weyl.A study in this direction is being developed[23].
Figure 1.Graphics of    in the case 0.8  