| The research on the inverse problem of diffusion equation has been of great practical significance in engineering fields such as geological exploration and groundwater pollution prevention,and it is also one of the hotspots in inverse problem research.In this paper,we consider the problem of inversion of the source terms and initial values of one-dimensional time fractional and integer order diffusion equations through local observation data.And,we propose a modified quasi-boundary value method for the spatial correlation source terms of time fractional order equations based on the terminal global observation data.The first chapter mainly introduces the research significance of the inverse problem of diffusion equation and the research status of domestic and foreign scholars in this area.In the second chapter,we quote some of the well-known results used in this paper.The third chapter mainly concerns on inverse problems for the one-dimensional diffusion equation based on the local observation data,including the fractional and the integer order diffusion equation.In this chapter,we consider to reconstruct the initial distribution of the diffusion equation through the left endpoint observation data.Firstly,we give the ill-posedness analysis for the backward problem.Then the uniqueness of the backward problem is proven by Laplace transformation technique and analytic continuation method.Next,the inverse problem is transformed into Tikhonov type optimization problems,and the conjugate gradient method is adopted to solve the optimization problem with the help of the variational adjoint technique.Several numerical examples are tested to show the efficiency and stability of the proposed method.In chapter 4,we research the inversion source problem for one-dimensional time fractional order and integer order diffusion equation.We consider inverting the spatially correlated source terms of the diffusion equation with the same observation data as used in Chapter 3.First,we analyze the ill-posedness of the problem,and then use the Laplace transform and analytical continuation techniques to prove the uniqueness of the inverse source problem.Then,the inverse source problem is transformed into a variational optimization problem by Tikhonov regularization method.The gradient of the functional is derived based on the idea of variational adjoint method,and then the conjugate gradient method is used to solve the problem.Finally,we give several numerical examples to show the effectiveness of the proposed method.Chapter 5 mainly studies inverse source problem of time fractional diffusion equation.Base on the classic Quasi-boundary value method,we propose a modified quasi-boundary value method to solve an inverse source problem for a time fractional diffusion equation.Under some boundedness assumption,the corresponding convergence rate estimates are derived by using an a priori and an a posteriori regularization parameter choice rules,respectively.Based on the superposition principle,we propose a direct inversion algorithm in a parallel manner.Finally,several numerical examples are given to illustrate the effectiveness of the algorithm.Some conclusions and future works are contained in the sixth chapter. |
