Research Article Generalization of Fuzzy Laplace Transforms of Fuzzy Fractional Derivatives about the General Fractional Order −1<< Amal Khalaf Haydar and Ruaa Hameed Hassan Department of Mathematics, Faculty of Education for Girls, Kufa University, Najaf, Iraq Correspondence should be addressed to Amal Khalaf Haydar; amalkh.hayder@uokufa.edu.iq Received 17 July 2015; Accepted 13 October 2015 Academic Editor: Zhen-Lai Han Copyright © 2016 A. K. Haydar and R. H. Hassan. Tis 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. Te main aim in this paper is to use all the possible arrangements of objects such that  1 of them are equal to 1 and  2 (the others) of them are equal to 2, in order to generalize the defnitions of Riemann-Liouville and Caputo fractional derivatives (about order 0<<) for a fuzzy-valued function. Also, we fnd fuzzy Laplace transforms for Riemann-Liouville and Caputo fractional derivatives about the general fractional order −1<< under H-diferentiability. Some fuzzy fractional initial value problems (FFIVPs) are solved using the above two generalizations. 1. Introduction Fuzzy Fractional Diferential Equations (FFDEs) can ofer a more comprehensive account of the process or phenomenon. Tis has recently captured much attention in FFDEs. As the derivative of a function is defned in the sense of Riemann- Liouville, Gr¨ unwald-Letnikov, or Caputo in fractional calcu- lus, the used derivative is to be specifed and defned in FFDEs as well [1]. Many researchers have worked on the feld of Fuzzy Fractional Diferential Equations (FFDEs); for example, Salahshour et al. [2] dealt with the solutions of FFDEs under Riemann-Liouville H-diferentiability by fuzzy Laplace trans- forms; Mazandarani and Kamyad [1] presented the solution to FFIVP under Caputo-type fuzzy fractional derivatives by a modifed fractional Euler method; Wu and Baleanu [3] proposed a novel modifcation of the variational iteration method (VIM) by means of the Laplace transform; they extended the method successfully to fractional diferen- tial equations; Ahmadian et al. [4] reveal a computational method based on using a Tau method with Jacobi polyno- mials for the solution of fuzzy linear fractional diferential equations of order 0< V <1, and Allahviranloo et al. [5] introduced the fuzzy Caputo fractional diferential equations under the generalized Hukuhara diferentiability. Tis paper is arranged as follows. Basic concepts are given in Section 2. In Section 3, the general formula of the fuzzy Riemann-Liouville fractional derivatives and the general formula of Laplace transforms of the fuzzy Riemann- Liouville fractional derivatives for a fuzzy-valued function  are found. In Section 4, the general formula of the fuzzy Caputo fractional derivatives and the general formula of Laplace transforms of the fuzzy Caputo fractional derivatives for a fuzzy-valued function  are found. In Section 5, conclu- sions are drawn. 2. Basic Concepts In this section, we give the basic concepts which are needed in the next sections. We denote   [,] as the space of all continuous fuzzy-valued functions on [,]. Also, we denote the space of all Lebesgue integrable fuzzy-valued functions on the bounded interval [,]⊂ R by   [,]. Teorem 1 (see [6]). Let  be a positive integer. Let  −1  be continuous on  = [0,∞);  is the class of piecewise continuous functions on   = (0,∞) and integrable on any fnite subinterval of  = [0, ∞) and let  > 0. Ten, one fnds the following: Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2016, Article ID 6380978, 13 pages http://dx.doi.org/10.1155/2016/6380978