Regularization Methods And Algorithms For Inverse Problems Of Several Classes Of Partial Differential Equations

Diffusion equation inversion is always one of the hot problems in mathematical physics equation inverse problem.From the classical parabolic equation inversion problem to the anomalous diffusion equation inversion problem(fractional diffusion equation inversion problem),it has a wide range of application background in practical science and engineering.In this paper,we mainly study the inversion problems of several kinds of fractional diffusion equations,which are ill-posed and require regularization methods to solve the ill-poseness.In Chapter 2,we study the inversion of the initial value of the time fractional pseudo-parabolic equation.Firstly,it is proved that the inversion initial value problem is ill-posed,and then the conditional stability result is given under the prior bound condition.The improved quasi-boundary regularization method and fractional Landweber iterative regularization method are applied to solve this ill-posed problem.Secondly,based on the prior bound and Morozov inconsistency principle,the prior error estimation formula and the posterior error estimation formula between the exact solution and the regular solution are derived respectively.Finally,the rationality of the two regularization methods is verified by numerical examples.In Chapter 3,the inversion of the source term and the initial value of the time fractional diffusion equation are studied.Firstly,it is proved that the source term and the initial value problem are ill-posed,then the conditional stability results are given under the prior bound conditions,and the modified Tikhonov regularization method is applied to solve the ill-posed problems.Secondly,based on the priori bound and Morozov inconsistency principle,the prior and posterior error estimators between the exact and regular solutions are derived respectively.Finally,the regularization method is used to verified by numerical examples.In Chapter 4,the inversion of the source term and the initial value of the spacetime fractional diffusion equation are studied.It is first proved that the inversion source term and initial value problem are ill-posed,and then conditional stability results are given under a priori bound conditions,and the ill-posedness of this problem is solved by applying the regularization method combining the modified quasi-reversibility and the modified quasi-boundary.Secondly,based on the prior bound and Morozov inconsistency principle,the prior and posterior error estimation formulas between the exact solution and the regular solution are derived respectively.Finally,the regularization method is used to verified by numerical examples.The source term and initial value problems studied in Chapter 3 and Chapter 4 of this paper are the in-depth study of the fractional diffusion equation inversion problem in Chapter 2,which is a relatively novel inverse problem topic.The theoretical results obtained and the regularization method adopted in this paper can solve this kind of ill-posed problems well.