A type of uncertain differential equations with analytic solution

Introduction Uncertainty theory, as a branch of axiomatic mathematics dealing with human’s belief degree, was founded by Liu [1] in 2007 and refined by Liu [2] in 2010. During the past 6 years, many researchers have contributed in this area. For example, Peng and Iwamura [3] gave a sufficient condition for the uncertainty distribution of an uncertain variable. Liu and Ha [4] gave a formula to calculate the expected value of a function of multiple uncertain variables. Chen and Dai [5] showed that a normal uncertain variable has the maximum entropy given the expected value and variance. Especially, Liu [6] proposed uncertain programming as a type of mathematical programming involving uncertain variables. In order to model the evolution of an uncertain phenomenon, Liu [7] proposed a concept of uncertain process. Meanwhile, Liu [7] gave an uncertain renewal process as an example. After that, Liu [2] proposed an uncertain renewal reward process, and Yao and Li [8] proposed an uncertain alternating renewal process. In addition, Zhang et al. [9] proposed an uncertain delayed renewal process. In 2009, Liu [10] mathematically defined a type of uncertain process, named canonical Liu process, which has independent and stationary uncertain normal increments and of which almost all the sample paths are Lipschitz continuous. In addition, Liu [11] proved the extreme value theorems for an independent increment uncertain process. In 2009, Liu [10] founded an uncertain calculus to deal with the integral and differential of an uncertain process with respect to Liu process, which are called Liu integral and Liu differential afterwards. Then Liu and Yao [12] studied an uncertain integral with respect to multiple Liu processes. After that, Chen and Ralescu [13] proposed an uncertain integral with respect to the general Liu process. Inspired by the Liu integral, Yao [14] proposed an uncertain calculus with respect to an uncertain renewal process.