Uncorrected Author Proof Journal of Intelligent & Fuzzy Systems xx (20xx) x–xx DOI:10.3233/JIFS-151940 IOS Press 1 Fractional relaxation-oscillation differential equations via fuzzy variational iteration method 1 2 3 A. Armand a,∗ , T. Allahviranloo b , S. Abbasbandy b and Z. Gouyandeh c 4 a Young Researchers and Elite Club, Yadegar-e-Imam Khomeini (RAH)- Shahre-Rey Branch, Islamic Azad University, Tehran, Iran 5 6 b Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran 7 c Department of Mathematics, Najafabad Branch, Islamic Azad University, Najafabad, Isfahan, Iran 8 Abstract. The fuzzy variation iteration method is investigated to solve a linear fractional differential equation under Caputo generalized Hukuhara differentiability. This method is based on the use of Lagrange multipliers for identification of optimal value in the correction functionals by using fuzzy integration by parts. In this scheme, the correction, functional can make without converting fuzzy fractional differential equation to two crisp equations. To this, derivative of the product of two functions and integration by parts is obtained for fuzzy valued functions. The effectiveness of the proposed method is verified by solving two of the important applications of these equations are fractional relaxation and oscillation differential equations. 9 10 11 12 13 14 Keywords: Caputo generalized Hukuhara derivative, fuzzy integration by parts, fuzzy variation iteration method, fractional relaxation-oscillation differential equation, basset equation, Bagley-Torvik equation 15 16 1. Introduction 17 The application of fractional calculus has attracted 18 more attention in the past few years because of sig- 19 nificant advantage of the fractional order models 20 in comparison with integer order models. Also, the 21 fuzzy set theory is a powerful tool for modeling 22 uncertain problems. These vagueness in fractional 23 order models may be appearing in each part of the 24 problem like initial condition, boundary condition 25 or etc. Therefore, solving fractional order problem 26 in the sense of real conditions leads to use inter- 27 val or fuzzy calculations (see e.g [18, 21, 23, 26]). 28 The notion of fuzzy set was introduced by Zadeh ∗ Corresponding author. A. Armand, Young Researchers and Elite Club, Yadegar-e-Imam Khomeini (RAH)- Shahre- Rey Branch, Islamic Azad University, Tehran, Iran. E-mail: atefeh.armand@ymail.com. [27] as an extension of the classical notion of set. 29 Using the concepts of integral and Hukuhara deriva- 30 tive for set-valued functions, introduced by Hukuhara 31 [16], similar concepts of fuzzy functions were intro- 32 duced and studied by several authors in [10, 12, 25]. 33 Hukuhara derivative was the starting point of the 34 topic of set differential equations and later also for 35 fuzzy fractional differential equations. By the con- 36 cept of Hukuhara differentiability, Agarwal et al. 37 [3] introduced the concept of Riemann- Liouville 38 fractional derivative to solve uncertain fractional dif- 39 ferential equations and it is considered in e.g [9, 17]. 40 The Caputo fractional derivatives defined based on 41 Hukuhara difference in [13] and the Caputo frac- 42 tional differential equations are investigated under 43 strongly generalized Hukuhara differentiability. In 44 order to overcome some shortcomings of Hukuhara 45 differentiability approach like as its solution has 46 1064-1246/16/$35.00 © 2016 – IOS Press and the authors. All rights reserved