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

Abstract

In the first paper of a series of reports on fuzzy differential equations, we consider the nth-order fuzzy differential equation X ( n) ( t) + a n−1 ( t)X ( n−1) ( t) + … + a o ( t)X( t) = X( t), where X ( n) ( t), X ( n−1) ( t), …, X 1( t) are nth, (n − 1)th,… , 1st same-order (or reverse-order) derived functions of unknown fuzzy set-valued function X( t), respectively; F(t) is a known fuzzy set-valued function; a i ( t), i = 0, 1, … , n - 1 are deterministic functions of time t. And the dime domain methods of the solutions of the n-order, inhomogeneous fuzzy differential equations with variable coefficients and constant coefficients are put forward. Two examples are 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 [3], fuzzy random vibration [12], fuzzy stochastic dynamical systems [15, 18–20] and fuzzy stochastic systems [21–23].