Three Regular Solutions For A Class Of Fractional Differential Equations

The time fractional diffusion equation is a natural extension of the integer-order diffusion equation,which is a popular research field in recent years.It has a wide range of applications,such as in fluid mechanics,physics and so on.To solve the diffusion equation of inverse problem has important significance:because it is generally seen as a surface when observing a substance,it is difficult to speculate on its internal structure and initial morphology,what's more the solution of the inverse problem can provide more information for our observation.The diffusion equation studied in this paper is:Solve the initial value of the known final value,which is also the inverse problem.There are three methods that I used:Tikhonov regularization method,a modified regularization method,Landweber iterative regularization method,and finally get the result that the regular solution converges to the exact solution.Firstly,we study the posterior error estimation of time fractional diffusion equation where the filter is q(?,?)=?2/?+?2.Secondly,this paper studies the prior and posterior estimates of the modified filter q(?,?)=??/(?+?v?)1/v(Modification of regularized filters based on Tikhonov).Then,this paper studies the prior and posterior error estimates of Landweber's iterative regularization method.Finally,summarize this article and proposes further research directions that can be expected.Numerical solutions are also presented in the appendix as a supplement.The innovation of this paper is to apply a Tikhonov modified regularization method mentioned in this paper to the fractional diffusion equation,and to perform a posterior analysis on the Tikhonov regularization method.