The Existence Of Solutions To Backward Problems For Two Types Of Fractional Diffusion Equations

As a kind of inverse problem,backward problems have been applied to many fields such as biomedical science,physics,information engineering,and so on.Many practical problems need to restore the original state according to the known data information,where the research of backward problems plays a vital role,and the research on the fractional backward problem has also become a hot topic.The paper mainly studies the existence of mild solutions to the backward problem for two types of fractional diffusion equations.In Chapter 1,we introduce the research background and current situation of backward problems of fractional diffusion equations,and then the main work of the paper,related definitions and theorems are given.In Chapter 2,we study the existence of mild solutions for the backward problem of fractional diffusion equations with the hyper-Bessel operator.Under different conditions,different fixed point theorems are used to obtain the existence results.Since the backward problem is ill-posed,the Fourier truncation method is selected to obtain the regular solution of the equation,and then we use Banach fixed point theorem to prove the existence and uniqueness of the regular solution.Finally,we give the error estimate between the regular solution and the mild solution.In Chapter 3,we study the existence of mild solutions for the backward problem of fractional stochastic diffusion equations with two Caputo derivatives.When the right-hand term is linear,we use the properties of the Mittag-Leffler function and the method of eigenvalue expansion to obtain the existence,uniqueness and regularity of the mild solution.When the right-hand term is nonlinear,the Schauder fixed point theorem is used to prove the existence of the mild solution.