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Fractional Versions of the Fundamental Theorem of Calculus

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Fractional Versions of the Fundamental Theorem of Calculus Applied Mathematics, 2013, 4, 23-33 http://dx.doi.org/10.4236/am.2013.47A006 Published Online July 2013 (http://www.scirp.org/journal/am) Fractional Versions of the Fundamental Theorem of Calculus Eliana Contharteze Grigoletto, Edmundo Capelas de Oliveira Department of Applied Mathematics, University of Campinas, Campinas, Brazil Email: [email protected], [email protected] Received April 27, 2013; revised May 27, 2013; accepted June 7, 2013 Copyright © 2013 Eliana Contharteze Grigoletto, Edmundo Capelas de Oliveira. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT The concept of fractional integral in the Riemann-Liouville, Liouville, Weyl and Riesz sense is presented. Some prop- erties involving the particular Riemann-Liouville integral are mentioned. By means of this concept we present the frac- tional 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. Keywords: Fractional Integral; Fractional Derivative; Riemann-Liouville Derivative; Liouville Derivative; Caputo Derivative; Weyl Derivative and Riesz Derivative 1. Introduction much research on its applications as there is on the cal- culus of integer order. Several applications in all areas of Fractional calculus, a popular name used to denote the knowledgement are collected, presented and discussed in calculus of non integer order, is as old as the calculus of different books as follow [7-12]. integer order as created independently by Newton and The main objective of this paper is to explain what is Leibniz. In contrast with the calculus of integer order, meant by calculus of non integer order and collect any fractional calculus has been granted a specific area of different versions of the fractional derivatives associated mathematics only in 1974, after the first international with a particular fractional integral. Specifically, we re- congress dedicated exclusively to it. Before this congress there were only sporadic independent papers, without a cover the concepts of fractional integral and fractional consolidated line [1,2]. derivative in different versions and present a new version During the 1980s fractional calculus attracted re- of the so-called fundamental theorem of fractional cal- searchers and explicit applications began to appear in culus (FTFC), which is interpreted as a generalization of several fields. We mention the doctoral thesis, published the classical fundamental theorem of calculus. We men- as an article [3], which seems to be the first one in the tion three recent works where FTFC is discussed, Tara- subject and the classical book by Miller and Ross [1], sov’s book [12], a paper by Tarasov [13] and a paper by where one can see a timeline from 1645 to 1974. After Dannon [14] in which a particular case of the parameter the decade of 1990, completely consolidated, there ap- associated with the derivative is presented. The paper is peared some specific journals and several textbooks were written as follows: in section two, we first review the published. These facts lent a great visibility to the subject concept of fractional integral in the Riemann-Liouville and it gained prestige around the world. An interesting sense, which can be interpreted as a generalization of the timeline from 1645 to 2010 is presented in references integral of integer order and in the Liouville sense, which [4-6]. We recall here that an important advantage of us- is a particular case of the Riemann-Liouville one. We ing fractional differential equations in applications is review also the concept of fractional integral in the Weyl their non-local property. The use of fractional calculus is sense and in the Riesz sense. Section two present also the more realistic and this is one reason why fractional cal- concepts of derivative as proposed by Riemann-Liouville, culus has become more popular. Liouville, Caputo, Weyl and Riesz, showing the real im- Nowadays, fractional calculus can be considered a portance and applications. Some properties are also pre- frontier area in mathematics in the sense that there is as sented, among which one associated with the semigroup Copyright © 2013 SciRes. AM 24 E. C. GRIGOLETTO, E. C. DE OLIVEIRA property. Our main result appears in section three, in and if Re 0 or 0 , then which we present and demonstrate the many faces of the IIaaIa an d IIbb I b . FTFC, in all different versions and which are interpreted (Riemann-Liouville integrals) Let1 f xaL,1 b , as a generalization of the fundamental theorem of calcu- with ab . The fractional integrals of Rie- lus. Applications are presented in section four. mann-Liouville of order with Re 0 , on the left and on the right, are defined respectively by 2. Fractional Calculus 1 x f The integral and derivatives of non integer order have I:a f xx 1 d , with a , a several applications and are used to solve problems in x different fields of knowledge, specifically, involving a and fractional differential equation with boundary value con- 1 b f I: f xx d , with b . ditions and/or initial conditions [7,9,11,12]. They can be b 1 x x seen as generalizations of the integral and derivatives of 00 integer order. On the other hand, we mention two papers, If 0 , we have IIab :=I, where I is the iden- by Heymans & Podlubny [15] and Podlubny [16] that tity operator. provide an interesting geometric interpretation, and dis- cuss applications of fractional calculus, with integral and 2.2. Fractional Derivative of Riemann-Liouville derivatives of non integer order. Also, we mention a re- After we introduce the fractional integral in the Rie- cent paper in which the authors discuss a fractional dif- mann-Liouville sense, we define the fractional derivative ferential equation with integral boundary value condi- of Riemann-Liouville, which is the most used by mathe- tions [17]. We remember that, there are several ways to maticians, particularly, into the problems in which the introduce the concepts of fractional integral and frac- initial conditions are not involved. tional derivatives, which are not necessarily coincident (Riemann-Liouville derivative) Let , with with each other [18]. The so-called Grünwald-Letnikov Re 0 and n Re 1 , where denotes derivative, which will be not discussed in this paper, is the integer part of , the fractional derivatives in the convenient and useful to affront problems involving a Riemann-Liouville sense, on the left and on the right, are numerical treatment [19]. defined by In this section, the concept of fractional integral in the n Riemann-Liouville, Liouville, Weyl and Riesz sense is d n D:Iaaf xf n x (1) presented. Some properties involving the particular Rie- dx mann-Liouville integral are mentioned. By means of this and concept we present the fractional derivatives, specifically, n n d n the Riemann-Liouville, Liouville, Caputo, Weyl and Ri- D:1Ibbf xf n x, (2) dx esz versions are discussed. respectively. Note that the derivatives in Equations (1) and (2) exist 2.1. Fractional Integral of Riemann-Liouville for2 f xa ACn ,b . The fractional integral of Riemann-Liouville is an inte- If, in particular, n , then gral that generalizes the concept of integral in the classi- n n cal sense, and which can be obtained as a generalization Da f xfx of the Cauchy-Riemann integral. As we have already said, and before we define the fractional integral Riemann-Liou- D=(1)nnf xf ()n()x. ville. b Definition 1 (Spaces IL and IL The ap ap 2.3. Fractional Integral and Derivative in the spaces ILap and ILap are defined, for Liouville Sense Re 0 and p 1, by An interesting particular case of the fractional integral of p IL:apf xfx : I a gxgx , L, ab Riemann-Liouville and the corresponding fractional de- rivative, is the so-called Liouville fractional integral and and 1 p b 1 p p p IL:apf xfx : I a hxhx , L, ab fx Lab ,if fx d x ,for1 p . a respectively. 2Space of complex functions f x whose derivatives up to order Property 2.1 (Semigroup) If Re 0 or 0 n 1 are absolutely continuous on ab, . Copyright © 2013 SciRes. AM E. C. GRIGOLETTO, E. C. DE OLIVEIRA 25 the Liouville fractional derivative. This case is obtained W inx Iefx cinn by substitution a and b in the expres- n sions associated with the fractional integral in the Rie- n0 and mann-Liouville sense. inx (Liouville integral and derivative) The fractional inte- Wefx cinn , n grals in the Liouville sense on the real axis, on the left n0 and on the right, for x and Re 0 , are defined respectively, with and Re 0 . by In the particular case 0 1, if fxL0,21 , 1 x f W Ifx d (3) then IIf xfx and WDx f xf x , 1 x where I and D are defined in Equations (3) and (5), and respectively. If we define the Fourier series of f x in 1 f the form I fx d , (4) x 1 x 1 2 f x ceinx , with c einx fxx d , nn0 n 2 respectively. n0 The corresponding fractional derivatives in the Liou- then we obtain for 01 , in the same way, the frac- ville sense are given by tional integral in the Weyl sense on the right and the n fractional derivative on the right, defined by d n D:If xf n x (5) dx W IIf xfx and n and n d n D:1Ifx n fx, (6) x WDf xfx . dx respectively.
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