Numerical approach for solution to an uncertain fractional differential equation

Abstract

Uncertain fractional differential equation (UFDE) is of importance tool for the description of uncertain dynamic systems. Generally we may not obtain its analytic solutions in most cases. This paper focuses on proposing a numerical method for solving UFDE involving Caputo derivative. First, the concept of α-path to an UFDE with initial value conditions is introduced, which is a solution of the corresponding fractional differential equation (FDE) involving with the same initial value conditions. Then the relations between its solution and associate α-path are investigated. Besides, a formula is derived for calculating expected value of a monotonic function with respect to solutions of UFDEs. Based on the established relations, numerical algorithms are designed. Finally, some numerical experiments of nonlinear UFDEs are given to demonstrate the effectiveness of the numerical algorithms.