A Regularization Method For A Class Of Fractional Order Equations

The problem of identifying unknown non-homogeneous sources is a typical ill-posed problem.They are widely used in real life.For example,environmental pollution problems,medical problems,heat diffusion problems and so on.In this paper,the unknown source identification problem of non-homogeneous time fractional equation as follows:wheref(x)is unknown,Ω is a bounded domain in Rn(n=1,2,3),Dtα is the Caputo fractional derivative of order α∈(0,1),A is a symmetric uniformly elliptic operator,φ(t)and g(x)are two given functions.In this paper,we first prove the ill-posed nature of the problem for φ(t)≡1 andφ(t)≠1,applying the filtering regularization and Tikhonov regularization methods are used respectively to fit the ill-posed problem,then estimate its convergence rate under the condition of selecting appropriate prior and posterior parameters.