Regularization Methods And Algorithms For The Inverse Problems Of Fractional Rayleigh-stokes Equation And Potential-free Field Schr(?)dinger Equation

Fractional Rayleigh-Stokes problem is an important problem in physics.It plays an important role in describing the behavior of some non-Newtonian fluids.Schr(?)dinger equation is a basic physical equation describing the behavior of nonrelativistic quantum mechanics.As a simple form of Schr(?)dinger equation,the potential-free field Schr(?)dinger equation has important applications in calculating the energy levels of hydrogen atoms and harmonic oscillators,the solution of Airy wave packets.Therefore,it is of practical significance to study these two kinds of physical equations,especially the inverse problems.In this paper,we study the initial value problem of fractional Rayleigh-Stokes equation and the inverse problem of the potential-free field Schr(?)dinger equation,which are ill-posed problems and need to be solved by regularization method.In the second chapter of this paper,we mainly consider the initial value identification problem of the homogeneous Rayleigh-Stokes equation for a generalized second-grade fluid with the Riemann-Liouville fractional derivative model.The problem is ill-posed,i.e.,the solution(if it exists)does not depend continuously on the data.In this chapter,we use Landweber iterative regularization method to solve the problem.Based on a conditional stability result,the convergent error estimates between the exact solution and the regularization solution by using an a prior regularization parameter choice rule and an a posterior regularization parameter choice rule are obtained.The effectiveness and stability of the method are verified by numerical examples.In the third chapter,we mainly study the inverse Schr(?)dinger problem of potential-free field,which is ill-posed.Based on the assumption of a priori bound condition,we give the optimal error bound of this problem.Furthermore,we use two different regularization methods to solve this problem.Compared with the Landweber iterative normalization method,the convergent error estimate under an a priori regularization choice rule obtained by a modified kernel method is optimal,the convergent error estimate under an a posteriori regularization choice rule is order-optimal.Finally,the effectiveness and stability of these methods are proved by numerical examples.In the fourth chapter,based on the third chapter,we continue to study an inverse time-fractional Schr(?)dinger problem of potential-free field.This is an ill-posed problem.Based on the assumption of a priori bound condition,we give the result of the optimal error bound.Furthermore,we introduce an modified kernel method to obtain the optimal convergence error estimate under an a priori regularization choice rule and the order-optimal convergence error estimate under an a posteriori regularization choice rule.Finally,some numerical examples are given to illustrate the effectiveness and stability of this method.For the initial value identification of the fractional Rayleigh-Stokes equation,some scholars have studied it.Under an a prior bound condition,the error estimate between the exact solution and the regular solution based on the prior regularization parameter choice rule is given.As we all know,the prior regularization parameter is based on the smoothness condition of the exact solution,which is difficult to give in advance in fact.The posterior regularization parameter choice rule based on the data error level information and the measurement data itself is more practical.In the second chapter,we use the Landweber iterative regularization method to give not only the convergent error estimate between the exact solution and the regularization solution under the prior regularization parameter choice rule,but also the convergent error estimate under the posterior regularization parameter choice rule,and verify the method by some numerical examples.The inverse Schr(?)dinger problem of potential-free field studied in chapter three and chapter four is a relatively new inverse problem,which is the first time studied by the author.Some theoretical results and regularization methods in this paper can solve this kind of ill-posed problems well.