Nonparametric estimation for uncertain fractional differential equations

After uncertainty theory was established, it has become a new branch of mathematics and been applied to describing the indeterministic phenomena as an uncertain dynamic system. As an important part of uncertainty theory, uncertain fractional differential equation is a good tool to model some complex dynamic systems with uncertainties. Due to the lack of corresponding information, models of parametric uncertain fractional differential equation are not always available. Therefore, the requirement for nonparametric estimation of uncertain fractional differential equation is urgent. Utilizing the thought of Legendre polynomial approximation, we propose a method to estimate the nonparametric uncertain fractional differential equations. Then, the error analysis of this method is given. After that, the numerical simulation is applied to illustrating the validity of error analysis and show the errors between the approximating functions and actual functions. The stability of this method has also been verified without highly dense observations. Finally, by means of uncertain hypothesis test and Kolmogorov–Smirnov test, we prove the superiority of uncertain fractional differential equations for some examples.