Reduction And Well-posedness Of Fractional Differential Equations

It has been found that fractional calculus is very suitable for describing the phenomena with features such as "memory","long distance interaction" and "heredity",which is the main advantage of fractional calculus compared to classical calculus.It is well known that the ordinary differential equations which can obtain the analytical solutions are relatively few,and it is also impossible to obtain the exact solution for the high-dimensional differ-ential systems.Whether for finite dimensional differential systems or infinite dimensional differential systems,it becomes particularly critical to obtain lower-dimensional systems by proper reduction methods,and then analyze their dynamic properties.In the theory of ordinary differential equations,there are at least two reduction methods,namely,center manifold reduction and Lyapunov-Schmidt reduction.Motivated by these ideas,in this dissertation,we mainly study the reduction methods for high-dimensional fractional dif-ferential equations and discuss the problem of the well-posedness of fractional differential equations,besides,we study the properties of Hadamard-type fractional derivative and give the definition of the finite part integrals and analyze the dependence of solution on param-eters for Hadamard-type fractional differential equations.This dissertation aims at the study on the reduction methods for fractional differential equations and discuss the problem of the well-posedness and the dependence of solution on parameters of fractional differential equations.The main four contents of this dissertation are given as follows:(1)In view of suitable contraction map,we prove the existence of the fractional center manifold of Caputo fractional differential system as 0<a<1 and get the corresponding error and convergence order of the approximation of fractional center manifold.When the derivative order is approaching to 1,the result can be consistent with the classical case.(2)The Lyapunov-Schmidt reduction for Caputo fractional differential equations with different fractional order(0<?<1,1<?<2)is studied respectively.The Fredholm principle for some fractional differential operators is proposed,and then the Caputo frac-tional differential system is reduced by Lyapunov-Schmidt reduction.Finally,an example is presented to illustrate our derived method,and the corresponding equivalent system is obtained by bifurcation calculation.(3)The well-posedness of solutions for four types of fractional differential equations are studied,respectively.Firstly,we summarize the definite conditions for the solution of left Riemann-Liouville fractional differential equation as 0<?<1;Secondly,we fur-ther propose the well-posed conditions for right Riemann-Liouville fractional differential equation as 0<?<1;Then we conclude the proper initial conditions of Riesz fractional differential equations as 1<?<2;Finally,we study the well-posed problem of Hadamard fractional differential equation as 0<?<1.(4)First,we study some properties of Hadamard-type fractional operators.Then,we define the finite part integral,and we prove that the equivalence of finite part integral and Hadamard type fractional derivative in an appropriate function space;In the sequel,the definite conditions of certain class of Hadamard-type fractional differential equations are proposed.Finally,we prove a novel Gronwall inequality with weak singularity and analyze the dependence of solutions of Hadamard-type fractional differential equations on the derivative order along with the proposed initial value conditions and small perturbation terms.