Soft Comput (2014) 18:2461–2469 DOI 10.1007/s00500-014-1224-x METHODOLOGIES AND APPLICATION Laplace transform formula on fuzzy nth-order derivative and its application in fuzzy ordinary differential equations M. Barkhordari Ahmadi · N. A. Kiani · N. Mikaeilvand Published online: 12 February 2014 © Springer-Verlag Berlin Heidelberg 2014 Abstract In this paper, the Laplace transform formula on the fuzzy nth-order derivative by using the strongly general- ized differentiability concept is investigated. Also, the related theorems and properties are proved in detail and, it is used in an analytic method for fuzzy two order differential equation. The method is illustrated by solving some examples. Keywords Fuzzy number · Fuzzy valued function · Generalized differentiability · Fuzzy differential equation · Laplace transform 1 Introduction The topic of fuzzy differential equations (FDEs) has been rapidly growing in recent years. The concept of the fuzzy derivative was first introduced by Chang and Zadeh (1972); it was followed up by Dubios and Prade (1982), who used the extension principle in their approach. Other methods have been discussed by Puri and Ralescu (1983) and Goetschel and Voxman (1986). Kandel (1980), Kandel and Byatt (1978) Communicated by T. Allahviranloo. M. B. Ahmadi (B ) Department of Mathematics, Bandar Abbas Branch, Islamic Azad University, Bandar Abbas, Iran e-mail: mahnazbarkhordarii@gmail.com N. A. Kiani Viroquant Research Group Modeling, Bioquant, Heidelberg University, Im Neuenheimer Feld 267, 69120 Heidelberg, Germany N. Mikaeilvand Department of Mathematics, Ardabil Branch, Islamic Azad University, Ardabil, Iran applied the concept of FDEs to the analysis of fuzzy dynami- cal problems. The FDE and the initial value problem (Cauchy problem) were rigorously treated by Kaleva (1987, 1990), Seikkala (1987), Ouyang and Wu (1989), Kloeden (1991) and Menda (1988), and by other researchers (see Bede et al. 2006a; Buckly and Feuring 2003, 1999; Buckly 2006; Wu and Shiji 1998; Ding 1997; Jowers et al. 2007). The numer- ical methods for solving fuzzy differential equations are introduced in Abbasbandy and Allahviranloo (2002, 2004), Allahviranloo (2002), Ghanbari (2009). A thorough theoretical research of fuzzy Cauchy prob- lems was given by Kaleva (1987), Seikkala (1987), Ouyang and Wu (1989) and Kloeden (1991) and Wu (2000). Kaleva (1987) discussed the properties of differentiable fuzzy set- valued functions by means of the concept of H-differentia- bility due to Puri and Ralescu (1983), gave the existence and uniqueness theorem for a solution of the fuzzy differential equation y ′ = f (t ; y ); y (t 0 ) = y 0 when f satisfies the Lipschitz condition. Further, Song and Wu (2000) investi- gate fuzzy differential equations, and generalize the main results of Kaleva (1987). Seikkala (1987), defined the fuzzy derivative which is the generalization of Hukuhara deriv- ative, and showed that fuzzy initial value problem y ′ = f (t ; y ); y (t 0 ) = y 0 has a unique solution, for the fuzzy process of a real variable whose values are in the fuzzy num- ber space (E, D), where f satisfies the generalized Lipschitz condition. Strongly generalized differentiability was introduced in Bede and Gal (2005) and studied in Bede et al. (2006a). The existence and uniqueness theorem of solution of Nth-order fuzzy differential equations under Nth-order generalized dif- ferentiability was studied by Salahshour (2011). The strongly generalized derivative is defined for a larger class of fuzzy- valued function than the H-derivative, and fuzzy differential equations can have solutions which have a decreasing length 123