Soft Computing https://doi.org/10.1007/s00500-020-04913-9 FOUNDATIONS A fuzzy generalized power series method under generalized Hukuhara differentiability for solving fuzzy Legendre differential equation Khadijeh Sabzi 1 · Tofigh Allahviranloo 1,2 · Saeid Abbasbandy 3 © Springer-Verlag GmbH Germany, part of Springer Nature 2020 Abstract In this paper, first the fuzzy generalized power series method, in which the coefficients are fuzzy numbers, is introduced, and then, the conditions of the uniqueness of the solution and its convergence for the fuzzy differential equation are investigated. Then, using the fuzzy generalized power series method, the fuzzy Legendre differential equation is considered as a case study, and finally, for further illustration some related examples are solved. Keywords Fuzzy analytic functions · Ordinary point of fuzzy equation · Fuzzy comparison test · Uniqueness fuzzy analytic solution · Fuzzy generalized power series method · Fuzzy Legendre differential equation 1 Introduction Modeling uncertain problem is useful in practical science, and the fuzzy set theory is a powerful tool to this mod- eling. Practically, fuzzy differential equations, FDEs, have strong ability to model a large varieties of natural phenom- ena in different science. The biology and medicine are topics as application of mathematics and FDEs models the behav- ior of these topics. Data and variables in our environment are our assumptions in mathematical modeling, and most of them have vagueness; then, the knowledge about differen- tial equations is incomplete and has ambiguity in real-world problems. The topics of fuzzy differential equations or fuzzy initial value problems have been rapidly growing in recent years and Communicated by A. Di Nola. B Tofigh Allahviranloo allahviranloo@yahoo.com Khadijeh Sabzi khadijehsabzi@yahoo.com Saeid Abbasbandy abbasbandy@yahoo.com 1 Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran 2 Facullty of Engineering and Natural Sciences, Bahcesehir University, Istanbul, Turkey 3 Department of Mathematics, Imam Khomeini International University, Qazvin 34149-16818, Iran studied by several authors (Abbasbandy et al. 2005, 2004; Allahviranloo et al. 2008, 2007; Barkhordari and Kiani 2013; Rodriguez-Lopez 2008; Bede et al. 2007; Chen et al. 2008), and they were first formulated by Kaleva (1987) and Seikala (1987) in time-dependent form. Because of the importance of the concept of fuzzy derivative, it was first introduced by Chang and Zadeh (1972) and followed up by Dubios and Prade (1982) who used the extension principle in their approach. Then, Kaleva formulated fuzzy differential equa- tions, in terms of Hukuhara derivative (Kaleva 1987). For the first time, the Hukuhara difference and derivative of a set-valued function are introduced by Hukuhara in Hukuhara (1967) that was a beginning of the concept of set-valued dif- ferential equations (see e.g. Kaleva 1987; Puri and Ralescu 1983; Bede and Gal 2005). To develop this approach to the fuzzy sets, Bede and Gal introduced the generalized differ- ential of a fuzzy number-valued function. In this case, the FDEs are mapped to a convex cone as two-level wise mod- els of differential equations. But the variety of fuzzy model depends on the type of differentiability, and in two cases of differentiability they have two different solutions. The rela- tionships between two types of generalized differentiability “weakly and strongly” have shown by authors of Stefanini and Bede (2009). In general, it is not easy to derive an analytical solution for the most of the fuzzy differential equations. Therefore, it is vital to develop some reliable and efficient techniques to solve the fuzzy differential equations, and the numerical solution of fractional differential equations has attached considerable 123