FREQUENCY-DOMAIN METHODS FOR THE SOLUTIONS OF N-ORDER FUZZY DIFFERENTIAL-EQUATIONS

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 integrati on in the frequency domain. This is accomplished by the use to general ized fuzzy harmonic analysis [12, 14]. In the second paper of a series of reports on fuzzy differential equations, we continue studying the nth-order fuzzy differential equation (X) under tilde((n))(t) + a(n-1) (t)(X) under tilde((n-1))(t) + ... + a(0)(t)(X) under tilde(t) = (F) u nder tilde(t), where (X) under tilde((n))(t), (X) under tilde((n-1))(t ), ..., (X) under tilde((1))(t) are nth, (n - 1)th, ..., 1st same-orde r (or reverse-order) derived functions of an unknown fuzzy set-valued function (X) under tilde(t), respectively; (F) under tilde(t) is a kno wn fuzzy set-valued function; a(i)(t), i = 0, 1, ..., n - 1, are deter ministic 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 val idity of the methods. The work provides an indispensable mathematical tool for setting up the theories of fuzzy stochastic differential equa tions [8], fuzzy dynamical systems [2], fuzzy random vibration [12], f uzzy stochastic dynamical systems [14, 18-20] and fuzzy stochastic sys tems [21-23]. (C) 1998 Elsevier Science B.V.