Some Inverse Problems Of Nonlinear Time-Fractional Diffusion Equations

In recent years,the inverse problems of mathematical and physical equations have attracted more and more attention of scholars at home and abroad because of their intersection and wide application.Among them,various inverse problems in the linear case and the general bounded domain have been widely studied,while the relevant studies in the nonlinear case and some special domains are relatively few.Therefore,this paper studies the inverse problems of several kinds of nonlinear time fractional diffusion equations,these include the inverse problem of nonlinear time-fractional diffusion equation in semi-infinite domain,the inverse source problem of time-fractional diffusion equation in spherical domain and the inverse problem of nonlinear time-fractional diffusion equation in cylindrical domain.These inverse problems are seriously ill-posed,so this paper proposes several regularization methods to obtain approximate solutions.In chapter 3,we mainly study the backward problem of nonlinear time-fractional diffusion equation in semi-infinite domain.For this problem,we analyze its ill-posed nature,propose variational regularization to obtain stable approximate solution and prove the existence and uniqueness of the solution.At the same time,we also obtain the error estimate between the exact solution and the approximate solution.Finally,numerical examples verify the effectiveness of the method.In chapter 4,we mainly consider two kinds of inverse problems in symmetric domain.One is the inverse source problem of time-fractional diffusion equation in spherical domain,which is to identify f(r)in the source term f(r)q(t)in the form of separated variables.We propose a new iterative variational regularization method to obtain stable solutions for its illposed nature,The error estimates of convergence are given under both a priori and a posteriori rule.Finally,three numerical examples with different smoothing degrees are used to verify the effectiveness of the proposed iterative variational regularization.The second is the inverse problem of nonlinear fractional diffusion equation in cylindrical domain.We prove the existence and uniqueness of the solution of the nonlinear problem,and give the ill-posed analysis of the inverse problem.We use the generalized quasi-boundary value regularization method to obtain the stable approximate solution and obtain the error estimates of the exact solution and the approximate solution,Finally,numerical experiments show that the method is effective for solving nonlinear problems.The inverse problems of time-fractional diffusion equations in the three special regions mentioned above can find practical application background in the field of engineering,and play a certain theoretical support role in the development of scientific theory and technology today.