On Solutions To Caputo-Hadamard Uncertain Fractional Differential Equations

The theory of fractional differential equations focuses on studying arbitrary real order or complex order differential equations,which is an extension of integral order differential equations.Compared with integer order differential equations,fractional differential equations have memory and hereditary effects,which can describe the evolutionary process and the non-local properties of the dynamical systems more accurately.In real world,most of systems applied in practice are disturbed by not only objective noises but also subjective noises.A system with objective noises can be depicted with the help of probability theory.But when a system is interfered by subjective noises or lack of enough data,we need use the uncertainty theory to model the subjective noises.So the dynamical systems can be described by uncertain differential equations.Furthermore,to model systems with memory effects in uncertain environment,uncertain factors can be incorporated into fractional differential equations,then we study uncertain fractional differential equations.This paper is devoted to investigating Caputo-Hadamard uncertain fractional differential equations,whose form contains logarithmic functions of arbitrary order and is well suited to study logarithmic problems in uncertain environment.First,the definitions of Hadamard uncertain fractional differential equations and Caputo-Hadamard uncertain fractional differential equations are proposed and analytical solutions to some types of Caputo-Hadamard uncertain fractional differential equations are provided.Then,an existence and uniqueness theorem of solution to Caputo-Hadamard uncertain fractional differential equations under the Lipschitz condition and linear growth condition is studied.Next,comparison principles for Caputo-Hadamard fractional differential equations is modified.To solve the solutions of Caputo-Hadamard uncertain fractional differential equations,we will introduce some numerical approach.A concept of α-path is introduced to connect Caputo-Hadamard uncertain fractional differential equations and CaputoHadamard fractional differential equations.Then it is proved that α-path is just an inverse uncertainty distribution of the solution.Meanwhile,numerical algorithms are proposed to calculating the inverse uncertainty distribution and expected value of solutions,and some examples are given to test the validity and accuracy of the algorithm.