Frequency domain methods for the solutions of N-order fuzzy differential equations

Abstract

The solutions of the fuzzy differential equations in Refs. [8–10, 12, 14, 16] have so far been obtained by integration or fuzzy integration in the time domain. It is often more convenient, particularly, in the fuzzy random vibration problems [12] and the fuzzy stochastic dynamic systems [14] to obtain the solutions by integration or fuzzy integration in the frequency domain. This is accomplished by the use to generalized fuzzy harmonic analysis [12, 14]. In the second paper of a series of reports on fuzzy differenrial equations, we continue studying the nth-order fuzzy differential equation X ∼ (n)(t) + a n−1(t) X ∼ n−1(t) + ⋯ + a 0(t) X ∼ (t) = F(t) , where X ( n) ( t), X ( n−1) ( t), … X (1)( t) are nth, ( n−1)th…, 1st same-oorder (or reverse-order) derived functions of an unknown fuzzy set-valued function X( t), respectively; X( t) is known fuzzy set-valued function; a i ( t), i = 0,1, …, n − 1, are deterministic functions of parameter t. The solving processes of frequency domain for nth-order fuzzy differential equations are put forward. One example is considered in order to demonstrate the rationality and validity of the methods. The work provides an indispensable mathematical tool for setting up the theories of fuzzy stochastic differential equations [8], fuzzy dynamical systems [2], fuzzy random vibration [12], fuzzy stochastic dynamical systems [14, 18–20] and fuzzy stochastic systems [21–23].