JID:FSS AID:7644 /FLA [m3SC+; v1.298; Prn:12/04/2019; 10:40] P.1(1-21) Available online at www.sciencedirect.com ScienceDirect Fuzzy Sets and Systems ••• (••••) •••–••• www.elsevier.com/locate/fss 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 Differential calculus of fuzzy multi-variable functions and its applications to fuzzy partial differential equations R. Alikhani ∗ , F. Bahrami, S. Parvizi Department of Mathematics, University of Tabriz, Tabriz, Iran Received 27 May 2018; received in revised form 7 January 2019; accepted 5 April 2019 Abstract In this study, we deal with the concept of directional derivatives for fuzzy multi-variable functions and give explicit expressions for their directional derivatives. We give some applications of this concept as Green’s identity for fuzzy multi-variable functions and Neumann boundary value problems. The fuzziness variations of a fuzzy multi-variable function according to the changing of the directions at a given point are examined. Moreover, we provide solutions to wave equations with fuzzy initial values. Some examples are presented to illustrate the results.  2019 Published by Elsevier B.V. Keywords: Fuzzy multi-variable function; Strongly generalized directional derivative; Fuzzy partial differentiable equation; Fuzzy wave equation 1. Introduction Physical models often have some uncertainty which can be considered as coming from different sources. Fuzzy sets appear as a suitable instrument to model the uncertainties raised by impreciseness and vagueness. Fuzzy sets theory, is a natural way to model these systems subject to uncertainties. It has been explored in various fields due to its great applicability and functionality. As soon as the idea of a function with fuzzy values was born, it raised the idea of some kind of fuzzy derivatives and fuzzy differential equations as well. Over the past few years, the calculus of fuzzy functions and ordinary differential equations with fuzzy data have been studied extensively both theoretically and numerically [3,4,6,8–11,16,24,25]. The modeling of some applied problems with uncertain data has given rise to fuzzy partial differential (FPDEs). However, much less attention has been paid to multi-variable functions and consequently partial differential equations with fuzzy data [1,2,5,12–15,17,18,20–23,26]. For instance, in [1], the authors proposed difference method for solving FPDEs. Later the authors in [15] have ap- plied fuzzy logic and appropriate rule-based systems to construct the solutions for fuzzy partial differential equations which model the reoccupation of ants in a region of attraction. The authors in [12] also have employed the Zadeh’s * Corresponding author. E-mail address: alikhani@tabrizu.ac.ir (R. Alikhani). https://doi.org/10.1016/j.fss.2019.04.011 0165-0114/ 2019 Published by Elsevier B.V.