Regularization Methods For Inverse Problems Of Two Kinds Of Fractional Diffusion Equations

The fractional order diffusion equation has a wide range of applications in mathematics and physics,which can more accurately describe some anomalous diffusion phenomena,and can be applied to geophysics,thermodynamics,biomedicine,astromechanics,electromagnetism and other fields.Therefore,it is of great significance and practical application value for the study of fractional order diffusion equation.In this paper,we study the application of two kinds of fractional diffusion equations in mathematical models,that is,the problem for the simultaneously inversion of the source term and the initial value in the time-fractional diffusion equation and the fractional Rayleigh-Stokes equation,which are important ill-posed problems,for these two kinds of problems we use two terminal measurement data to invert the source term and the initial value at the same time,and for its ill-posedness,we deal with it through three different regularization methods,namely Landweber iterative regularization method,fractional Tikhonov regularization method and fractional Landweber iteration regularization method.According to expression of the solution,we derive the uniqueness and conditional stability for the problem,and we give the regularization parameter selection method and convergence analysis of the corresponding regularization method.Finally,the effectiveness and feasibility of each regularization method are verified by corresponding numerical examples.