VERSITA [ ractional & Applied Conalysis Calculus An International Journal for Theory and Applications VOLUME 16, NUMBER 1 (2013) RESEARCH PAPER (Print) ISSN 1311-0454 (Electronic) ISSN 1314-2224 GREEN'S THEOREM FOR GENERALIZED FRACTIONAL DERIVATIVES Tatiana Odzijewicz 1, Agnieszka B. Malinowska 2, Delf im F. M. Torres 1 Abstract We study three types of generalized partial fractional order operators. An extension of Green's theorem, by considering partial fractional deriva- tives with more general kernels, is proved. New results are obtained, even in the particular case when the generalized operators are reduced to the standard partial fractional derivatives and fractional integrals in the sense of Riemann-Liouville or Caputo. MSC 2010: Primary 26B20, Secondary 35R11 Key Words and Phrases: fractional calculus, generalized operators, Green's theorem Editorial Note: This work received the "Grünwald-Letnikov Award: Best Student Paper (Theory)", at the 2012 Symposium on Fractional Dif- ferentiation and Its Applications (FDA' 2012), Hohai University, Nanjing. 1. Introduction In 1828, the English mathematician George Green (1793-1841), who up to his forties was working as a baker and a miller, published an essay where he introduced a formula connecting the line integral around a simple closed curve with a double integral. Within years, this result turned out to be useful in many fields of mathematics, physics and engineering [4, 6, 15, 17]. C 2013 Diogenes Co., Sofia pp. 64-75, DOI: 10.2478/s13540-013-0005-z
GREEN'S THEOREM FOR GENERALIZED 65 Generalizations of Green's theorem have chosen different directions, and are known as the Kelvin-Stokes and the Gauss-Ostrogradsky theorems. In this paper, in contrast with previous works, we want to state a Green's theorem for generalized partial fractional derivatives. The notions of generalized fractional derivatives were introduced in [1, 8], and then developed in [11, 12]. A fractional version of the Green theorem has been already shown for the Riemann-Liouville integrals and Caputo derivatives [18], and for fractional operators in the sense of Jumarie [3]. However, generalized fractional operators have never been considered in this aspect. Our result may be useful in the theory of fractional calculus (see, e.g., [7, 9, 14, 16]), in particular for the two-dimensional fractional calculus of variations, where the derivation of Euler-Lagrange equations uses, as a key step in the proof, Green's theorem [3, 5, 10, 13]. The paper is organized as follows. In Section 2 a common review of or- dinary and partial generalized fractional calculus' operators is given. Our results are then formulated and proved in Section 3: we show the two- dimensional integration by parts formula for generalized Riemann-Liouville partial fractional integrals (Theorem 3.1) and Green's theorem for gener- alized partial fractional derivatives (Theorem 3.2). 2. Basic Notions In this section we give definitions of generalized ordinary and partial fractional operators. By the choice of a certain kernel, these operators can be reduced to the standard fractional integrals and derivatives. For more on the subject, we refer the reader to [1, 2, 8, 11, 12]. 2.1. Generalized fractional operators DEFINITION 2.1 (Generalized fractional integral). The operator Ky is given by t b a (Kgf) (t)