Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 31 March 2021 doi:10.20944/preprints202103.0789.v1 Article Fractional Line Integral Gabriel Bengochea 1,∗ , Manuel Ortigueira 2 1 Academia de Matemáticas, Universidad Autónoma de la Ciudad de México, Ciudad de México, México; [email protected] 2 CTS-UNINOVA and DEE, NOVA School of Science and Technology of NOVA University of Lisbon, Quinta da Torre, 2829 – 516 Caparica, Portugal; [email protected] * Correspondence: [email protected] ‡ These authors contributed equally to this work. Version March 30, 2021 submitted to Mathematics 1 Abstract: This paper proposes a definition of fractional line integral, generalising the concept of fractional 2 definite integral. The proposal replicates the properties of the classic definite integral, namely the 3 fundamental theorem of integral calculus. It is based on the concept of fractional anti-derivative used 4 to generalise the Barrow formula. To define the fractional line integrals the Grünwald-Letnikov and 5 Liouville directional derivatives are introduced and their properties described. The integral is defined 6 first for broken line paths and afterwards to any regular curve. 7 Keywords: Fractional Integral; Grünwald-Letnikov Fractional Derivative; Fractional Line Integral; 8 Liouville Fractional Derivative 9 MSC: Primary 26A33; Secondary 26A42 10 1. Introduction 11 It is no use to refer the great evolution that made Fractional Calculus invade many scientifical and 12 technical areas [5,7–9,20]. Advances in various aspects of fractional calculus led to a question: why there 13 are no fractional conterparts of some classic results? In fact and notwithstanding the progresses there are 14 several singular situations. One of them was until recently the non existence of definition for “fractional 15 definite integral”. This gap has been filled by Ortigueira and Machado [14]. Here, we will try to fill in 16 another gap, by introducing a definition of fractional line integral. This generalization was motivated 17 by the results presented in [19] where classic theorems of vectorial calculus were introduced but for 18 integrations over rectangular lines. With the integral introduced here, the Green theorem, for example, 19 can be generalised. To do it, we profit the results stated in [17] to propose a fractional line integral. We 20 use directional derivatives. However, as not all the directional derivatives are suitable, according to the 21 considerations done in [17] we opted for the directional derivatives resulting from the generalization of 22 the Grünwald-Letnikov (GL) and Liouville (L). These are introduced and their main properties listed. It is 23 important to refer the presentation of the Liouville directional regularised derivative. We consider the 24 classic definite Riemann integral and list its properties that serve as guide to defining the fractional definite 25 integral, since we require they have the same properties. The fractional definite integral is expressed in 26 terms of the anti-derivative generalising the Barrow formula. 27 For introducing the fractional line integral (FLI) we start by defining it on a segment of an oriented 28 straight-line. This procedure is enlarged to broken-line paths formed by a sequence of connected Submitted to Mathematics, pages 1 – 12 www.mdpi.com/journal/mathematics © 2021 by the author(s). Distributed under a Creative Commons CC BY license. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 31 March 2021 doi:10.20944/preprints202103.0789.v1 Version March 30, 2021 submitted to Mathematics 2 of 12 29 straight-line segments. Using a standard procedure consiting in approximating a curve by a sequence 30 of broken-lines we introduce the integration over any simple rectifiable line. Its main properties are 31 presented. 32 The paper outlines as follows. In Section2 we introduce the required background. In Section3 we describe 33 the GL and L directional derivatives that we will use in definition of the fractional line integral. The 34 corresponding properties are also presented. In Section4 we introduce the fractional line integral and 35 main properties. 36 2. Background 37 2.1. Functional framework 38 The theory we will develop below is expected to be useful in generalising the classic vectorial 39 theorems suitable for dealing with fractionalisations of important equations of Physics as it is the case 40 of Maxwell equations [12]. Therefore, we will need a framework involving functions f (t), t 2 R, that 41 are of exponential order (to have Laplace transform) or absolutely or square integrable (to have Fourier 42 transform). In particular, we will assume also that they 43 1. are almost everywhere continuous; 44 2. have bounded variation; 3. verify: 1 + j f (x)j < A , g, A 2 R , for x < x¥. jxjg+1 45 In particular, we can have f (x) ≡ 0, x < x¥. 46 Remark 1. If a given function has bounded support, we will extend it to R, with a null extrapolation. This keeps 47 our framework with the maximum generality. 48 2.2. Suitable fractional derivatives In [14] a discussion on the problem of fractional derivative (FD) definition was initiated and formalised through the introduction of two defining criteria. In [16] it was shown that for applications we require derivatives verifying the strict sense criterion that exiges functions defined on R in order to keep valid some classic relations. In this line of thought, we used such derivatives to be the base of the introduction of a definite fractional integral [17].They are the Grünwald-Letnikov (GL) and (regularised) Liouville (rL) derivatives [6,10,16,21]. The forward Grünwald-Letnikov and Liouville derivatives are given by ¥ n (−a)n ∑ (−1) n! f (x − nh) = Da f (x) = lim n 0 , (1) h!0+ ha where (a)k represents the Pochhammer symbol for the raising factorial, and " # 1 Z ¥ N (−1)n f (n)(x) Da f (x) = f (x − t) − u(a) tm t−a−1dt, (2) L (− ) ∑ G a 0 n=0 n! Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 31 March 2021 doi:10.20944/preprints202103.0789.v1 Version March 30, 2021 submitted to Mathematics 3 of 12 where N is the integer part of a, so that a − 1 < N ≤ a, N 2 N0. In spite of our work is based in the derivatives GL and rL we include one version of directional derivative for the usual Liouville derivative (L) [6,21] 1 dm Z x RL a ( ) = ( − )m−a−1 ( ) D+ f x : m x x f x dx, (3) G(m − a) dx −¥ and the Liouville-Caputo derivative (LC) [5] 1 Z x dm LC a ( ) = ( − )m−a−1 ( ) D+ f x : x x m f x dx, (4) G(m − a) −¥ dx + 49 where m − 1 < a ≤ m, m 2 Z . 50 The study of the equivalence of the two was done in [13]. It is a simple task to show that they are 51 really equivalent for functions with LT or FT. Without intending to explore existence problems (see [13]) 52 we can say that f (x) must decrease to zero as x goes to −¥, in agreement with our assumption above 2.1. 53 Remark 2. The concept of “forward” is tied here to the causality in the sense of “going from past to future.” This 54 implies not only an order but also a direction on the real line. These derivatives enjoy relevant characteristics namely the index law (29)[10,13]. This means that given a FD of order a > 0, there is a FD, of negative order, that we will call “anti-derivative” and verifying DaD−a f (x) = D−aDa f (x) = f (x). (5) It can be shown [14] that the GL, rL, or LC, FD of the constant function are identically null. The L derivative of a constant function does not exist, since the integral is divergent. It must be stressed here that, for negative order (anti-derivative), rL, L, and LC are equal: Z ¥ Z ¥ a 1 −a−1 1 −a DL f (x) = f (x − t)t dt = − f (x − t)dt , a < 0. (6) G(−a) 0 G(−a + 1) 0 55 Remark 3. Everything what will be done here can be replicated for the backward derivatives. We will not do it, 56 since it is not so interesting. 57 2.3. Order 1 definite integral 58 Consider a closed interval [a, b] ⊂ R where f (x) is continuous. There are several ways of introducing 59 the definite integral [1,2]. Probably, the simplest is through the Riemann sum. Definition 1. Divide the interval [a, b] into N small intervals with lengths Di, i = 1, 2, ··· , N. Let xi (i = 1, 2, ··· , N) be any point inside each small interval. We call the define integral of f (x) over [a, b] the limit [1] N S = lim f (xi)Di (7) ( )! ∑ max Di 0 i=1 R b 60 The sum S will be represented by a f (x)dx. Using the time scale approach, we can define a nabla derivative and its inverse [15]. This is given by ¥ (−1) f (x) = lim Di f (x − hi) ( )! ∑ sup Di 0 i=0 Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 31 March 2021 doi:10.20944/preprints202103.0789.v1 Version March 30, 2021 submitted to Mathematics 4 of 12 = i−1 where hi ∑k=0 Dk. With this anti-derivative we can rewrite (7) as Z b f (x)dx = f (−1)(b) − f (−1)(a) (8) a b−a By simplicity, we can use equal length intervals Di = N = h, i = 1, 2, ··· , N, h > 0, and xi uniformly spaced, so that we can set, for example, xi = a + (i − 1)h = b − ih, i = 1, 2, ··· , N.