Research On Regularization Method And Algorithm For Source Term Identification Problem Of Three Kinds Of Partial Differential Equations

Biharmonic equations can describe some equations in elastic mechanics.Fractional Rayleigh-Stokes problem is an important problem in physics,which can describe some knowledge of non-Newtonian fluid mechanics.As a new fractional differential operator,Caputo-Fabrizio fractional derivative is widely used in biomedicine,electromagnetics and signal processing.This paper studies the space-time fractional diffusion equation with Caputo-Fabrizio fractional derivative.Therefore,it is practically significant to do further research on these three kinds of physical equations,especially for the inverse problems of these three kinds of equations.This paper studies the source term identification problems of these three kinds of equations,all of which are ill-posed problems and need to be solved by regularization method.The second chapter discusses the source term identification problem of the non-homogeneous biharmonic equation,which is an ill-posed problem.This chapter deals with this problem by using the Landweber iterative regularization method,which does not cause saturation.When the regular solution is obtained,the error estimation between the regular solution and the exact solution is obtained by using the posteriori and priori regularization parameter selection rules.The feasibility and effectiveness of this method are proved by numerical algorithm.The third chapter of this paper mainly studies the source term identification inverse problem of Rayleigh-Stokes,which is an ill-posed problem.This chapter solves this problem by using the fractional Landweber iterative regularization method.When the regular solution is obtained,the corresponding convergence error estimation is obtained by using the regularization parameter selection rules.In the numerical examples,the results of two regularization methods are given successively.In solving this problem,fractional Landweber iterative regularization is more effective than Landweber iterative regularization.The fourth chapter discusses the source term identification problems of the space-time fractional diffusion equation with Caputo-Fabrizio fractional derivative,which is an ill-posed problem.Based on the assumption of a priori bound condition,the optimal error bound of the error estimation formula is obtained.This problem is solved by using the improved quasi-boundary regularization method and Landweber iterative regularization method respectively.Compared with the improved quasi-boundary regularization method,the priori and posteriori convergence error estimation obtained by the Landweber iterative regularization method are both the optimal order.Finally,numerical examples of different properties prove that the two regularization methods are very effective in dealing with this problem.The source term identification problems of the space-time fractional diffusion equation with Caputo-Fabrizio fractional derivative studied in the fourth chapter is a relatively novel inverse problem.The theoretical and numerical results obtained in this paper can fully demonstrate that the used regularization method can solve the given ill-posed problem.